Diffusive Wave Approximation to the Shallow Water Equations: Computational Approach
Collier, Nathan; Radwan, Hany; Dalcin, Lisandro; Calo, Victor M.
2011-01-01
We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity
Diffusive Wave Approximation to the Shallow Water Equations: Computational Approach
Collier, Nathan
2011-05-14
We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity. This robust adaptive time discretization corrects the initial time step size to achieve a user specified bound on the discretization error and allows time step size variations of several orders of magnitude. In particular, in the one dimensional results presented in this work feature a change of four orders of magnitudes for the time step over the entire simulation.
Inan, Ibrahim E.; Kaya, Dogan
2006-01-01
In this Letter by considering an improved tanh function method, we found some exact solutions of the potential Kadomtsev-Petviashvili equation. Some exact solutions of the system of the shallow water wave equation were also found
Integrability of an extended (2+1)-dimensional shallow water wave equation with Bell polynomials
Wang Yun-Hu; Chen Yong
2013-01-01
We investigate the extended (2+1)-dimensional shallow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Bäcklund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method. (general)
Lebedev, D.R.
1979-01-01
Benney's equations of motion of incompressible nonviscous fluid with free surface in the approximation of long waves are analyzed. The connection between the Lie algebra of Hamilton plane vector fields and the Benney's momentum equations is shown
Modification of 2-D Time-Domain Shallow Water Wave Equation using Asymptotic Expansion Method
Khairuman, Teuku; Nasruddin, MN; Tulus; Ramli, Marwan
2018-01-01
Generally, research on the tsunami wave propagation model can be conducted by using a linear model of shallow water theory, where a non-linear side on high order is ignored. In line with research on the investigation of the tsunami waves, the Boussinesq equation model underwent a change aimed to obtain an improved quality of the dispersion relation and non-linearity by increasing the order to be higher. To solve non-linear sides at high order is used a asymptotic expansion method. This method can be used to solve non linear partial differential equations. In the present work, we found that this method needs much computational time and memory with the increase of the number of elements.
Time adaptivity in the diffusive wave approximation to the shallow water equations
Collier, Nathan; Radwan, Hany; Dalcí n, Lisandro D.; Calo, Victor M.
2013-01-01
We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity. This robust adaptive time discretization corrects the initial time step size to achieve a user specified bound on the discretization error and allows time step size variations of several orders of magnitude. In particular, the one dimensional results presented in this work feature a change of four orders of magnitudes for the time step over the entire simulation. © 2011 Elsevier B.V.
Time adaptivity in the diffusive wave approximation to the shallow water equations
Collier, Nathan
2013-05-01
We discuss the use of time adaptivity applied to the one dimensional diffusive wave approximation to the shallow water equations. A simple and computationally economical error estimator is discussed which enables time-step size adaptivity. This robust adaptive time discretization corrects the initial time step size to achieve a user specified bound on the discretization error and allows time step size variations of several orders of magnitude. In particular, the one dimensional results presented in this work feature a change of four orders of magnitudes for the time step over the entire simulation. © 2011 Elsevier B.V.
Energy-preserving H1-Galerkin schemes for shallow water wave equations with peakon solutions
Miyatake, Yuto; Matsuo, Takayasu
2012-01-01
New energy-preserving Galerkin schemes for the Camassa–Holm and the Degasperis–Procesi equations which model shallow water waves are presented. The schemes can be implemented only with cheap H 1 elements, which is expected to be sufficient to catch the characteristic peakon solutions. The keys of the derivation are the Hamiltonian structures of the equations and an L 2 -projection technique newly employed in the present Letter to mimic the Hamiltonian structures in a discrete setting, so that the desired energy-preserving property rightly follows. Numerical examples confirm the effectiveness of the schemes. -- Highlights: ► Numerical integration of the Camassa–Holm and Degasperis–Procesi equation. ► New energy-preserving Galerkin schemes for these equations are proposed. ► They can be implemented only with P1 elements. ► They well capture the characteristic peakon solutions over long time. ► The keys are the Hamiltonian structures and L 2 -projection technique.
Li, Tatsien
2017-01-01
This book focuses on nonlinear wave equations, which are of considerable significance from both physical and theoretical perspectives. It also presents complete results on the lower bound estimates of lifespan (including the global existence), which are established for classical solutions to the Cauchy problem of nonlinear wave equations with small initial data in all possible space dimensions and with all possible integer powers of nonlinear terms. Further, the book proposes the global iteration method, which offers a unified and straightforward approach for treating these kinds of problems. Purely based on the properties of solut ions to the corresponding linear problems, the method simply applies the contraction mapping principle.
A comparative study of diffraction of shallow-water waves by high-level IGN and GN equations
Zhao, B.B. [College of Shipbuilding Engineering, Harbin Engineering University, 150001 Harbin (China); Ertekin, R.C. [Department of Ocean and Resources Engineering, University of Hawai' i, Honolulu, HI 96822 (United States); College of Shipbuilding Engineering, Harbin Engineering University, 150001 Harbin (China); Duan, W.Y., E-mail: duanwenyangheu@hotmail.com [College of Shipbuilding Engineering, Harbin Engineering University, 150001 Harbin (China)
2015-02-15
This work is on the nonlinear diffraction analysis of shallow-water waves, impinging on submerged obstacles, by two related theories, namely the classical Green–Naghdi (GN) equations and the Irrotational Green–Naghdi (IGN) equations, both sets of equations being at high levels and derived for incompressible and inviscid flows. Recently, the high-level Green–Naghdi equations have been applied to some wave transformation problems. The high-level IGN equations have also been used in the last decade to study certain wave propagation problems. However, past works on these theories used different numerical methods to solve these nonlinear and unsteady sets of differential equations and at different levels. Moreover, different physical problems have been solved in the past. Therefore, it has not been possible to understand the differences produced by these two sets of theories and their range of applicability so far. We are thus motivated to make a direct comparison of the results produced by these theories by use of the same numerical method to solve physically the same wave diffraction problems. We focus on comparing these two theories by using similar codes; only the equations used are different but other parts of the codes, such as the wave-maker, damping zone, discretion method, matrix solver, etc., are exactly the same. This way, we eliminate many potential sources of differences that could be produced by the solution of different equations. The physical problems include the presence of various submerged obstacles that can be used for example as breakwaters or to represent the continental shelf. A numerical wave tank is created by placing a wavemaker on one end and a wave absorbing beach on the other. The nonlinear and unsteady sets of differential equations are solved by the finite-difference method. The results are compared with different equations as well as with the available experimental data.
Linear superposition solutions to nonlinear wave equations
Liu Yu
2012-01-01
The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic, triangle, and exponential functions, and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics. The linear superposition solutions to the generalized KdV equation K(2,2,1), the Oliver water wave equation, and the k(n, n) equation are given. The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed, and the reason why the solutions with the forms of hyperbolic, triangle, and exponential functions can form the linear superposition solutions is also discussed
Initial-Boundary Value Problem Solution of the Nonlinear Shallow-water Wave Equations
Kanoglu, U.; Aydin, B.
2014-12-01
The hodograph transformation solutions of the one-dimensional nonlinear shallow-water wave (NSW) equations are usually obtained through integral transform techniques such as Fourier-Bessel transforms. However, the original formulation of Carrier and Greenspan (1958 J Fluid Mech) and its variant Carrier et al. (2003 J Fluid Mech) involve evaluation integrals. Since elliptic integrals are highly singular as discussed in Carrier et al. (2003), this solution methodology requires either approximation of the associated integrands by smooth functions or selection of regular initial/boundary data. It should be noted that Kanoglu (2004 J Fluid Mech) partly resolves this issue by simplifying the resulting integrals in closed form. Here, the hodograph transform approach is coupled with the classical eigenfunction expansion method rather than integral transform techniques and a new analytical model for nonlinear long wave propagation over a plane beach is derived. This approach is based on the solution methodology used in Aydın & Kanoglu (2007 CMES-Comp Model Eng) for wind set-down relaxation problem. In contrast to classical initial- or boundary-value problem solutions, here, the NSW equations are formulated to yield an initial-boundary value problem (IBVP) solution. In general, initial wave profile with nonzero initial velocity distribution is assumed and the flow variables are given in the form of Fourier-Bessel series. The results reveal that the developed method allows accurate estimation of the spatial and temporal variation of the flow quantities, i.e., free-surface height and depth-averaged velocity, with much less computational effort compared to the integral transform techniques such as Carrier et al. (2003), Kanoglu (2004), Tinti & Tonini (2005 J Fluid Mech), and Kanoglu & Synolakis (2006 Phys Rev Lett). Acknowledgments: This work is funded by project ASTARTE- Assessment, STrategy And Risk Reduction for Tsunamis in Europe. Grant 603839, 7th FP (ENV.2013.6.4-3 ENV
Chabchoub, A., E-mail: achabchoub@swin.edu.au [Centre for Ocean Engineering Science and Technology, Swinburne University of Technology, Hawthorn, Victoria 3122 (Australia); Kibler, B.; Finot, C.; Millot, G. [Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 6303 CNRS, Université de Bourgogne, 21078 Dijon (France); Onorato, M. [Dipartimento di Fisica, Università degli Studi di Torino, Torino 10125 (Italy); Istituto Nazionale di Fisica Nucleare, INFN, Sezione di Torino, Torino 10125 (Italy); Dudley, J.M. [Institut FEMTO-ST, UMR 6174 CNRS- Université de Franche-Comté, 25030 Besançon (France); Babanin, A.V. [Centre for Ocean Engineering Science and Technology, Swinburne University of Technology, Hawthorn, Victoria 3122 (Australia)
2015-10-15
The dynamics of waves in weakly nonlinear dispersive media can be described by the nonlinear Schrödinger equation (NLSE). An important feature of the equation is that it can be derived in a number of different physical contexts; therefore, analogies between different fields, such as for example fiber optics, water waves, plasma waves and Bose–Einstein condensates, can be established. Here, we investigate the similarities between wave propagation in optical Kerr media and water waves. In particular, we discuss the modulation instability (MI) in both media. In analogy to the water wave problem, we derive for Kerr-media the Benjamin–Feir index, i.e. a nondimensional parameter related to the probability of formation of rogue waves in incoherent wave trains.
Wave-equation dispersion inversion
Li, Jing
2016-12-08
We present the theory for wave-equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. The dispersion curves are obtained from Rayleigh waves recorded by vertical-component geophones. Similar to wave-equation traveltime tomography, the complicated surface wave arrivals in traces are skeletonized as simpler data, namely the picked dispersion curves in the phase-velocity and frequency domains. Solutions to the elastic wave equation and an iterative optimization method are then used to invert these curves for 2-D or 3-D S-wave velocity models. This procedure, denoted as wave-equation dispersion inversion (WD), does not require the assumption of a layered model and is significantly less prone to the cycle-skipping problems of full waveform inversion. The synthetic and field data examples demonstrate that WD can approximately reconstruct the S-wave velocity distributions in laterally heterogeneous media if the dispersion curves can be identified and picked. The WD method is easily extended to anisotropic data and the inversion of dispersion curves associated with Love waves.
Wave-equation dispersion inversion
Li, Jing; Feng, Zongcai; Schuster, Gerard T.
2016-01-01
We present the theory for wave-equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. The dispersion curves are obtained
The relativistic electron wave equation
Dirac, P.A.M.
1977-08-01
The paper was presented at the European Conference on Particle Physics held in Budapest between the 4th and 9th July of 1977. A short review is given on the birth of the relativistic electron wave equation. After Schroedinger has shown the equivalence of his wave mechanics and the matrix mechanics of Heisenberg, a general transformation theory was developed by the author. This theory required a relativistic wave equation linear in delta/delta t. As the Klein--Gordon equation available at this time did not satisfy this condition the development of a new equation became necessary. The equation which was found gave the value of the electron spin and magnetic moment automatically. (D.P.)
Kuznetsov, N.; Maz'ya, V.; Vainberg, B.
2002-08-01
This book gives a self-contained and up-to-date account of mathematical results in the linear theory of water waves. The study of waves has many applications, including the prediction of behavior of floating bodies (ships, submarines, tension-leg platforms etc.), the calculation of wave-making resistance in naval architecture, and the description of wave patterns over bottom topography in geophysical hydrodynamics. The first section deals with time-harmonic waves. Three linear boundary value problems serve as the approximate mathematical models for these types of water waves. The next section uses a plethora of mathematical techniques in the investigation of these three problems. The techniques used in the book include integral equations based on Green's functions, various inequalities between the kinetic and potential energy and integral identities which are indispensable for proving the uniqueness theorems. The so-called inverse procedure is applied to constructing examples of non-uniqueness, usually referred to as 'trapped nodes.'
Wave Partial Differential Equation
Szöllös, Alexandr
2009-01-01
Práce se zabývá diferenciálními rovnicemi, jejich využitím při analýze vedení, experimenty s vedením a možnou akcelerací výpočtu v GPU s využitím prostředí nVidia CUDA. This work deals with diffrential equations, with the possibility of using them for analysis of the line and the possibility of accelerating the computations in GPU using nVidia CUDA. C
Wave equation of hydrogen atom
Suwito.
1977-01-01
The calculation of the energy levels of the hydrogen atom using Bohr, Schroedinger and Dirac theories is reviewed. The result is compared with that obtained from infinite component wave equations theory which developed recently. The conclusion can be stated that the latter theory is better to describe the composit system than the former. (author)
High-order Finite Difference Solution of Euler Equations for Nonlinear Water Waves
Christiansen, Torben Robert Bilgrav; Bingham, Harry B.; Engsig-Karup, Allan Peter
2012-01-01
is discretized using arbitrary-order finite difference schemes on a staggered grid with one optional stretching in each coordinate direction. The momentum equations and kinematic free surface condition are integrated in time using the classic fourth-order Runge-Kutta scheme. Mass conservation is satisfied...
Wave equations for pulse propagation
Shore, B.W.
1987-01-01
Theoretical discussions of the propagation of pulses of laser radiation through atomic or molecular vapor rely on a number of traditional approximations for idealizing the radiation and the molecules, and for quantifying their mutual interaction by various equations of propagation (for the radiation) and excitation (for the molecules). In treating short-pulse phenomena it is essential to consider coherent excitation phenomena of the sort that is manifest in Rabi oscillations of atomic or molecular populations. Such processes are not adequately treated by rate equations for excitation nor by rate equations for radiation. As part of a more comprehensive treatment of the coupled equations that describe propagation of short pulses, this memo presents background discussion of the equations that describe the field. This memo discusses the origin, in Maxwell's equations, of the wave equation used in the description of pulse propagation. It notes the separation into lamellar and solenoidal (or longitudinal and transverse) and positive and negative frequency parts. It mentions the possibility of separating the polarization field into linear and nonlinear parts, in order to define a susceptibility or index of refraction and, from these, a phase and group velocity. The memo discusses various ways of characterizing the polarization characteristics of plane waves, that is, of parameterizing a transverse unit vector, such as the Jones vector, the Stokes vector, and the Poincare sphere. It discusses the connection between macroscopically defined quantities, such as the intensity or, more generally, the Stokes parameters, and microscopic field amplitudes. The material presented here is a portion of a more extensive treatment of propagation to be presented separately. The equations presented here have been described in various books and articles. They are collected here as a summary and review of theory needed when treating pulse propagation
Wave Equation Inversion of Skeletonized SurfaceWaves
Zhang, Zhendong; Liu, Yike; Schuster, Gerard T.
2015-01-01
We present a surface-wave inversion method that inverts for the S-wave velocity from the Rayleigh dispersion curve for the fundamental-mode. We call this wave equation inversion of skeletonized surface waves because the dispersion curve
Analysis of wave equation in electromagnetic field by Proca equation
Pamungkas, Oky Rio; Soeparmi; Cari
2017-01-01
This research is aimed to analyze wave equation for the electric and magnetic field, vector and scalar potential, and continuity equation using Proca equation. Then, also analyze comparison of the solution on Maxwell and Proca equation for scalar potential and electric field, both as a function of distance and constant wave number. (paper)
Wave equations in higher dimensions
Dong, Shi-Hai
2011-01-01
Higher dimensional theories have attracted much attention because they make it possible to reduce much of physics in a concise, elegant fashion that unifies the two great theories of the 20th century: Quantum Theory and Relativity. This book provides an elementary description of quantum wave equations in higher dimensions at an advanced level so as to put all current mathematical and physical concepts and techniques at the reader’s disposal. A comprehensive description of quantum wave equations in higher dimensions and their broad range of applications in quantum mechanics is provided, which complements the traditional coverage found in the existing quantum mechanics textbooks and gives scientists a fresh outlook on quantum systems in all branches of physics. In Parts I and II the basic properties of the SO(n) group are reviewed and basic theories and techniques related to wave equations in higher dimensions are introduced. Parts III and IV cover important quantum systems in the framework of non-relativisti...
Nonlinear effects in water waves
Janssen, P.A.E.M.
1989-05-01
This set of lecture notes on nonlinear effects in water waves was written on the occasion of the first ICTP course on Ocean Waves and Tides held from 26 September until 28 October 1988 in Trieste, Italy. It presents a summary and unification of my knowledge on nonlinear effects of gravity waves on an incompressible fluid without vorticity. The starting point of the theory is the Hamiltonian for water waves. The evolution equations of both weakly nonlinear, shallow water and deep water gravity waves are derived by suitable approximation of the energy of the waves, resulting in the Korteweg-de Vries equation and the Zakharov equation, respectively. Next, interesting properties of the KdV equation (solitons) and the Zakharov equation (instability of a finite amplitude wave train) are discussed in some detail. Finally, the evolution of a homogeneous, random wave field due to resonant four wave processes is considered and the importance of this process for ocean wave prediction is pointed out. 38 refs, 21 figs
Skeletonized wave equation of surface wave dispersion inversion
Li, Jing; Schuster, Gerard T.
2016-01-01
We present the theory for wave equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. Similar to wave-equation travel
Dutta, Gaurav
2016-10-12
Strong subsurface attenuation leads to distortion of amplitudes and phases of seismic waves propagating inside the earth. The amplitude and the dispersion losses from attenuation are often compensated for during prestack depth migration. However, most attenuation compensation or Qcompensation migration algorithms require an estimate of the background Q model. We have developed a wave-equation gradient optimization method that inverts for the subsurface Q distribution by minimizing a skeletonized misfit function ∈, where ∈ is the sum of the squared differences between the observed and the predicted peak/centroid-frequency shifts of the early arrivals. The gradient is computed by migrating the observed traces weighted by the frequency shift residuals. The background Q model is perturbed until the predicted and the observed traces have the same peak frequencies or the same centroid frequencies. Numerical tests determined that an improved accuracy of the Q model by wave-equation Q tomography leads to a noticeable improvement in migration image quality. © 2016 Society of Exploration Geophysicists.
Dutta, Gaurav; Schuster, Gerard T.
2016-01-01
Strong subsurface attenuation leads to distortion of amplitudes and phases of seismic waves propagating inside the earth. The amplitude and the dispersion losses from attenuation are often compensated for during prestack depth migration. However, most attenuation compensation or Qcompensation migration algorithms require an estimate of the background Q model. We have developed a wave-equation gradient optimization method that inverts for the subsurface Q distribution by minimizing a skeletonized misfit function ∈, where ∈ is the sum of the squared differences between the observed and the predicted peak/centroid-frequency shifts of the early arrivals. The gradient is computed by migrating the observed traces weighted by the frequency shift residuals. The background Q model is perturbed until the predicted and the observed traces have the same peak frequencies or the same centroid frequencies. Numerical tests determined that an improved accuracy of the Q model by wave-equation Q tomography leads to a noticeable improvement in migration image quality. © 2016 Society of Exploration Geophysicists.
Exact solitary waves of the Fisher equation
Kudryashov, Nikolai A.
2005-01-01
New method is presented to search exact solutions of nonlinear differential equations. This approach is used to look for exact solutions of the Fisher equation. New exact solitary waves of the Fisher equation are given
Pelinovsky, Efim; Chaikovskaia, Natalya; Rodin, Artem
2015-04-01
The paper presents the analysis of the formation and evolution of shock wave in shallow water with no restrictions on its amplitude in the framework of the nonlinear shallow water equations. It is shown that in the case of large-amplitude waves appears a new nonlinear effect of reflection from the shock front of incident wave. These results are important for the assessment of coastal flooding by tsunami waves and storm surges. Very often the largest number of victims was observed on the coastline where the wave moved breaking. Many people, instead of running away, were just looking at the movement of the "raging wall" and lost time. This fact highlights the importance of researching the problem of security and optimal behavior of people in situations with increased risk. Usually there is uncertainty about the exact time, when rogue waves will impact. This fact limits the ability of people to adjust their behavior psychologically to the stressful situations. It concerns specialists, who are busy both in the field of flying activity and marine service as well as adults, young people and children, who live on the coastal zone. The rogue wave research is very important and it demands cooperation of different scientists - mathematicians and physicists, as well as sociologists and psychologists, because the final goal of efforts of all scientists is minimization of the harm, brought by rogue waves to humanity.
EXACT TRAVELLING WAVE SOLUTIONS TO BBM EQUATION
无
2009-01-01
Abundant new travelling wave solutions to the BBM (Benjamin-Bona-Mahoni) equation are obtained by the generalized Jacobian elliptic function method. This method can be applied to other nonlinear evolution equations.
Capillary-gravity waves and the Navier-Stokes equation
Behroozi, F.; Podolefsky, N.
2001-01-01
Water waves are a source of great fascination for undergraduates and thus provide an excellent context for introducing some important topics in fluid dynamics. In this paper we introduce the potential theory for incompressible and inviscid flow and derive the differential equation that governs the behaviour of the velocity potential. Next we obtain the harmonic solutions of the velocity potential by a very general argument. These solutions in turn yield the equations for the velocity and displacement of a water element under the action of a harmonic wave. Finally we obtain the dispersion relation for surface waves by requiring that the harmonic solutions satisfy the Navier-Stokes equation. (author)
Karczewska, Anna; Rozmej, Piotr; Infeld, Eryk
2015-11-01
It is well known that the Korteweg-de Vries (KdV) equation has an infinite set of conserved quantities. The first three are often considered to represent mass, momentum, and energy. Here we try to answer the question of how this comes about and also how these KdV quantities relate to those of the Euler shallow-water equations. Here Luke's Lagrangian is helpful. We also consider higher-order extensions of KdV. Though in general not integrable, in some sense they are almost so within the accuracy of the expansion.
Wave Equation Inversion of Skeletonized SurfaceWaves
Zhang, Zhendong
2015-08-19
We present a surface-wave inversion method that inverts for the S-wave velocity from the Rayleigh dispersion curve for the fundamental-mode. We call this wave equation inversion of skeletonized surface waves because the dispersion curve for the fundamental-mode Rayleigh wave is inverted using finite-difference solutions to the wave equation. The best match between the predicted and observed dispersion curves provides the optimal S-wave velocity model. Results with synthetic and field data illustrate the benefits and limitations of this method.
Spatial evolution equation of wind wave growth
WANG; Wei; (王; 伟); SUN; Fu; (孙; 孚); DAI; Dejun; (戴德君)
2003-01-01
Based on the dynamic essence of air-sea interactions, a feedback type of spatial evolution equation is suggested to match reasonably the growing process of wind waves. This simple equation involving the dominant factors of wind wave growth is able to explain the transfer of energy from high to low frequencies without introducing the concept of nonlinear wave-wave interactions, and the results agree well with observations. The rate of wave height growth derived in this dissertation is applicable to both laboratory and open sea, which solidifies the physical basis of using laboratory experiments to investigate the generation of wind waves. Thus the proposed spatial evolution equation provides a new approach for the research on dynamic mechanism of air-sea interactions and wind wave prediction.
Skeletonized wave equation of surface wave dispersion inversion
Li, Jing
2016-09-06
We present the theory for wave equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. Similar to wave-equation travel-time inversion, the complicated surface-wave arrivals in traces are skeletonized as simpler data, namely the picked dispersion curves in the (kx,ω) domain. Solutions to the elastic wave equation and an iterative optimization method are then used to invert these curves for 2D or 3D velocity models. This procedure, denoted as wave equation dispersion inversion (WD), does not require the assumption of a layered model and is less prone to the cycle skipping problems of full waveform inversion (FWI). The synthetic and field data examples demonstrate that WD can accurately reconstruct the S-wave velocity distribution in laterally heterogeneous media.
Electronic representation of wave equation
Veigend, Petr; Kunovský, Jiří, E-mail: kunovsky@fit.vutbr.cz; Kocina, Filip; Nečasová, Gabriela; Valenta, Václav [University of Technology, Faculty of Information Technology, Božetěchova 2, 612 66 Brno (Czech Republic); Šátek, Václav [IT4Innovations, VŠB Technical University of Ostrava, 17. listopadu 15/2172, 708 33 Ostrava-Poruba (Czech Republic); University of Technology, Faculty of Information Technology, Božetěchova 2, 612 66 Brno (Czech Republic)
2016-06-08
The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.
Water Waves The Mathematical Theory with Applications
Stoker, J J
2011-01-01
Offers an integrated account of the mathematical hypothesis of wave motion in liquids with a free surface, subjected to gravitational and other forces. Uses both potential and linear wave equation theories, together with applications such as the Laplace and Fourier transform methods, conformal mapping and complex variable techniques in general or integral equations, methods employing a Green's function. Coverage includes fundamental hydrodynamics, waves on sloping beaches, problems involving waves in shallow water, the motion of ships and much more.
Nonlinear Electrostatic Wave Equations for Magnetized Plasmas
Dysthe, K.B.; Mjølhus, E.; Pécseli, Hans
1984-01-01
The lowest order kinetic effects are included in the equations for nonlinear electrostatic electron waves in a magnetized plasma. The modifications of the authors' previous analysis based on a fluid model are discussed.......The lowest order kinetic effects are included in the equations for nonlinear electrostatic electron waves in a magnetized plasma. The modifications of the authors' previous analysis based on a fluid model are discussed....
Nonlinear wave equation with intrinsic wave particle dualism
Klein, J.J.
1976-01-01
A nonlinear wave equation derived from the sine-Gordon equation is shown to possess a variety of solutions, the most interesting of which is a solution that describes a wave packet travelling with velocity usub(e) modulating a carrier wave travelling with velocity usub(c). The envelop and carrier wave speeds agree precisely with the group and phase velocities found by de Broglie for matter waves. No spreading is exhibited by the soliton, so that it behaves exactly like a particle in classical mechanics. Moreover, the classically computed energy E of the disturbance turns out to be exactly equal to the frequency ω of the carrier wave, so that the Planck relation is automatically satisfied without postulating a particle-wave dualism. (author)
Diffusion phenomenon for linear dissipative wave equations
Said-Houari, Belkacem
2012-01-01
In this paper we prove the diffusion phenomenon for the linear wave equation. To derive the diffusion phenomenon, a new method is used. In fact, for initial data in some weighted spaces, we prove that for {equation presented} decays with the rate {equation presented} [0,1] faster than that of either u or v, where u is the solution of the linear wave equation with initial data {equation presented} [0,1], and v is the solution of the related heat equation with initial data v 0 = u 0 + u 1. This result improves the result in H. Yang and A. Milani [Bull. Sci. Math. 124 (2000), 415-433] in the sense that, under the above restriction on the initial data, the decay rate given in that paper can be improved by t -γ/2. © European Mathematical Society.
New exact travelling wave solutions of bidirectional wave equations
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea. ∗ ... exact travelling wave solutions of system (1) using the modified tanh–coth function method ... The ordinary differential equation is then integrated.
Transport equation and shock waves
Besnard, D.
1981-04-01
A multi-group method is derived from a one dimensional transport equation for the slowing down and spatial transport of energetic positive ions in a plasma. This method is used to calculate the behaviour of energetic charged particles in non homogeneous and non stationary plasma, and the effect of energy deposition of the particles on the heating of the plasma. In that purpose, an equation for the density of fast ions is obtained from the Fokker-Planck equation, and a closure condition for the second moment of this equation is deduced from phenomenological considerations. This method leads to a numerical method, simple and very efficient, which doesn't require much computer storage. Two types of numerical results are obtained. First, results on the slowing down of 3.5 MeV alpha particles in a 50 keV plasma plublished by Corman and al and Moses are compared with the results obtained with both our method and a Monte Carlo type method. Good agreement was obtained, even for energy deposition on the ions of the plasma. Secondly, we have calculated propagation of alpha particles heating a cold plasma. These results are in very good agreement with those given by an accurate Monte Carlo method, for both the thermal velocity, and the energy deposition in the plasma
Topological horseshoes in travelling waves of discretized nonlinear wave equations
Chen, Yi-Chiuan; Chen, Shyan-Shiou; Yuan, Juan-Ming
2014-01-01
Applying the concept of anti-integrable limit to coupled map lattices originated from space-time discretized nonlinear wave equations, we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions. In particular, the coupled map lattices display spatio-temporal chaos on the horseshoes
Topological horseshoes in travelling waves of discretized nonlinear wave equations
Chen, Yi-Chiuan, E-mail: YCChen@math.sinica.edu.tw [Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan (China); Chen, Shyan-Shiou, E-mail: sschen@ntnu.edu.tw [Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan (China); Yuan, Juan-Ming, E-mail: jmyuan@pu.edu.tw [Department of Financial and Computational Mathematics, Providence University, Shalu, Taichung 43301, Taiwan (China)
2014-04-15
Applying the concept of anti-integrable limit to coupled map lattices originated from space-time discretized nonlinear wave equations, we show that there exist topological horseshoes in the phase space formed by the initial states of travelling wave solutions. In particular, the coupled map lattices display spatio-temporal chaos on the horseshoes.
Wave-equation Qs Inversion of Skeletonized Surface Waves
Li, Jing
2017-02-08
We present a skeletonized inversion method that inverts surface-wave data for the Qs quality factor. Similar to the inversion of dispersion curves for the S-wave velocity model, the complicated surface-wave arrivals are skeletonized as simpler data, namely the amplitude spectra of the windowed Rayleigh-wave arrivals. The optimal Qs model is the one that minimizes the difference in the peak frequencies of the predicted and observed Rayleigh wave arrivals using a gradient-based wave-equation optimization method. Solutions to the viscoelastic wave-equation are used to compute the predicted Rayleigh-wave arrivals and the misfit gradient at every iteration. This procedure, denoted as wave-equation Qs inversion (WQs), does not require the assumption of a layered model and tends to have fast and robust convergence compared to full waveform inversion (FWI). Numerical examples with synthetic and field data demonstrate that the WQs method can accurately invert for a smoothed approximation to the subsurface Qs distribution as long as the Vs model is known with sufficient accuracy.
Skeletonized wave-equation Qs tomography using surface waves
Li, Jing
2017-08-17
We present a skeletonized inversion method that inverts surface-wave data for the Qs quality factor. Similar to the inversion of dispersion curves for the S-wave velocity model, the complicated surface-wave arrivals are skeletonized as simpler data, namely the amplitude spectra of the windowed Rayleigh-wave arrivals. The optimal Qs model is then found that minimizes the difference in the peak frequencies of the predicted and observed Rayleigh wave arrivals using a gradient-based wave-equation optimization method. Solutions to the viscoelastic wave-equation are used to compute the predicted Rayleigh-wave arrivals and the misfit gradient at every iteration. This procedure, denoted as wave-equation Qs tomography (WQs), does not require the assumption of a layered model and tends to have fast and robust convergence compared to Q full waveform inversion (Q-FWI). Numerical examples with synthetic and field data demonstrate that the WQs method can accurately invert for a smoothed approximation to the subsur-face Qs distribution as long as the Vs model is known with sufficient accuracy.
Wave-equation Qs Inversion of Skeletonized Surface Waves
Li, Jing; Dutta, Gaurav; Schuster, Gerard T.
2017-01-01
We present a skeletonized inversion method that inverts surface-wave data for the Qs quality factor. Similar to the inversion of dispersion curves for the S-wave velocity model, the complicated surface-wave arrivals are skeletonized as simpler data, namely the amplitude spectra of the windowed Rayleigh-wave arrivals. The optimal Qs model is the one that minimizes the difference in the peak frequencies of the predicted and observed Rayleigh wave arrivals using a gradient-based wave-equation optimization method. Solutions to the viscoelastic wave-equation are used to compute the predicted Rayleigh-wave arrivals and the misfit gradient at every iteration. This procedure, denoted as wave-equation Qs inversion (WQs), does not require the assumption of a layered model and tends to have fast and robust convergence compared to full waveform inversion (FWI). Numerical examples with synthetic and field data demonstrate that the WQs method can accurately invert for a smoothed approximation to the subsurface Qs distribution as long as the Vs model is known with sufficient accuracy.
Radio wave propagation and parabolic equation modeling
Apaydin, Gokhan
2018-01-01
A thorough understanding of electromagnetic wave propagation is fundamental to the development of sophisticated communication and detection technologies. The powerful numerical methods described in this book represent a major step forward in our ability to accurately model electromagnetic wave propagation in order to establish and maintain reliable communication links, to detect targets in radar systems, and to maintain robust mobile phone and broadcasting networks. The first new book on guided wave propagation modeling and simulation to appear in nearly two decades, Radio Wave Propagation and Parabolic Equation Modeling addresses the fundamentals of electromagnetic wave propagation generally, with a specific focus on radio wave propagation through various media. The authors explore an array of new applications, and detail various v rtual electromagnetic tools for solving several frequent electromagnetic propagation problems. All of the methods described are presented within the context of real-world scenari...
Gabor Wave Packet Method to Solve Plasma Wave Equations
Pletzer, A.; Phillips, C.K.; Smithe, D.N.
2003-01-01
A numerical method for solving plasma wave equations arising in the context of mode conversion between the fast magnetosonic and the slow (e.g ion Bernstein) wave is presented. The numerical algorithm relies on the expansion of the solution in Gaussian wave packets known as Gabor functions, which have good resolution properties in both real and Fourier space. The wave packets are ideally suited to capture both the large and small wavelength features that characterize mode conversion problems. The accuracy of the scheme is compared with a standard finite element approach
Relativistic wave equations and compton scattering
Sutanto, S.H.; Robson, B.A.
1998-01-01
Full text: Recently an eight-component relativistic wave equation for spin-1/2 particles was proposed.This equation was obtained from a four-component spin-1/2 wave equation (the KG1/2 equation), which contains second-order derivatives in both space and time, by a procedure involving a linearisation of the time derivative analogous to that introduced by Feshbach and Villars for the Klein-Gordon equation. This new eight-component equation gives the same bound-state energy eigenvalue spectra for hydrogenic atoms as the Dirac equation but has been shown to predict different radiative transition probabilities for the fine structure of both the Balmer and Lyman a-lines. Since it has been shown that the new theory does not always give the same results as the Dirac theory, it is important to consider the validity of the new equation in the case of other physical problems. One of the early crucial tests of the Dirac theory was its application to the scattering of a photon by a free electron: the so-called Compton scattering problem. In this paper we apply the new theory to the calculation of Compton scattering to order e 2 . It will be shown that in spite of the considerable difference in the structure of the new theory and that of Dirac the cross section is given by the Klein-Nishina formula
Skeletonized Least Squares Wave Equation Migration
Zhan, Ge; Schuster, Gerard T.
2010-01-01
of the wave equation. Only the early‐arrivals of these Green's functions are saved and skeletonized to form the migration Green's function (MGF) by convolution. Then the migration image is obtained by a dot product between the recorded shot gathers and the MGF
Some isometrical identities in the wave equation
Saburou Saitoh
1984-01-01
Full Text Available We consider the usual wave equation utt(x,t=c2uxx(x,t on the real line with some typical initial and boundary conditions. In each case, we establish a natural isometrical identity and inverse formula between the sourse function and the response function.
On the so called rogue waves in nonlinear Schrodinger equations
Y. Charles Li
2016-04-01
Full Text Available The mechanism of a rogue water wave is still unknown. One popular conjecture is that the Peregrine wave solution of the nonlinear Schrodinger equation (NLS provides a mechanism. A Peregrine wave solution can be obtained by taking the infinite spatial period limit to the homoclinic solutions. In this article, from the perspective of the phase space structure of these homoclinic orbits in the infinite dimensional phase space where the NLS defines a dynamical system, we examine the observability of these homoclinic orbits (and their approximations. Our conclusion is that these approximate homoclinic orbits are the most observable solutions, and they should correspond to the most common deep ocean waves rather than the rare rogue waves. We also discuss other possibilities for the mechanism of a rogue wave: rough dependence on initial data or finite time blow up.
Partial Differential Equations and Solitary Waves Theory
Wazwaz, Abdul-Majid
2009-01-01
"Partial Differential Equations and Solitary Waves Theory" is a self-contained book divided into two parts: Part I is a coherent survey bringing together newly developed methods for solving PDEs. While some traditional techniques are presented, this part does not require thorough understanding of abstract theories or compact concepts. Well-selected worked examples and exercises shall guide the reader through the text. Part II provides an extensive exposition of the solitary waves theory. This part handles nonlinear evolution equations by methods such as Hirota’s bilinear method or the tanh-coth method. A self-contained treatment is presented to discuss complete integrability of a wide class of nonlinear equations. This part presents in an accessible manner a systematic presentation of solitons, multi-soliton solutions, kinks, peakons, cuspons, and compactons. While the whole book can be used as a text for advanced undergraduate and graduate students in applied mathematics, physics and engineering, Part II w...
Ultra Deep Wave Equation Imaging and Illumination
Alexander M. Popovici; Sergey Fomel; Paul Sava; Sean Crawley; Yining Li; Cristian Lupascu
2006-09-30
In this project we developed and tested a novel technology, designed to enhance seismic resolution and imaging of ultra-deep complex geologic structures by using state-of-the-art wave-equation depth migration and wave-equation velocity model building technology for deeper data penetration and recovery, steeper dip and ultra-deep structure imaging, accurate velocity estimation for imaging and pore pressure prediction and accurate illumination and amplitude processing for extending the AVO prediction window. Ultra-deep wave-equation imaging provides greater resolution and accuracy under complex geologic structures where energy multipathing occurs, than what can be accomplished today with standard imaging technology. The objective of the research effort was to examine the feasibility of imaging ultra-deep structures onshore and offshore, by using (1) wave-equation migration, (2) angle-gathers velocity model building, and (3) wave-equation illumination and amplitude compensation. The effort consisted of answering critical technical questions that determine the feasibility of the proposed methodology, testing the theory on synthetic data, and finally applying the technology for imaging ultra-deep real data. Some of the questions answered by this research addressed: (1) the handling of true amplitudes in the downward continuation and imaging algorithm and the preservation of the amplitude with offset or amplitude with angle information required for AVO studies, (2) the effect of several imaging conditions on amplitudes, (3) non-elastic attenuation and approaches for recovering the amplitude and frequency, (4) the effect of aperture and illumination on imaging steep dips and on discriminating the velocities in the ultra-deep structures. All these effects were incorporated in the final imaging step of a real data set acquired specifically to address ultra-deep imaging issues, with large offsets (12,500 m) and long recording time (20 s).
Mandal, Birendra Nath
2015-01-01
The theory of water waves is most varied and is a fascinating topic. It includes a wide range of natural phenomena in oceans, rivers, and lakes. It is mostly concerned with elucidation of some general aspects of wave motion including the prediction of behaviour of waves in the presence of obstacles of some special configurations that are of interest to ocean engineers. Unfortunately, even the apparently simple problems appear to be difficult to tackle mathematically unless some simplified assumptions are made. Fortunately, one can assume water to be an incompressible, in viscid and homogeneous
Khusnutdinova, K. R.; Stepanyants, Y. A.; Tranter, M. R.
2018-02-01
We study solitary wave solutions of the fifth-order Korteweg-de Vries equation which contains, besides the traditional quadratic nonlinearity and third-order dispersion, additional terms including cubic nonlinearity and fifth order linear dispersion, as well as two nonlinear dispersive terms. An exact solitary wave solution to this equation is derived, and the dependence of its amplitude, width, and speed on the parameters of the governing equation is studied. It is shown that the derived solution can represent either an embedded or regular soliton depending on the equation parameters. The nonlinear dispersive terms can drastically influence the existence of solitary waves, their nature (regular or embedded), profile, polarity, and stability with respect to small perturbations. We show, in particular, that in some cases embedded solitons can be stable even with respect to interactions with regular solitons. The results obtained are applicable to surface and internal waves in fluids, as well as to waves in other media (plasma, solid waveguides, elastic media with microstructure, etc.).
True amplitude wave equation migration arising from true amplitude one-way wave equations
Zhang, Yu; Zhang, Guanquan; Bleistein, Norman
2003-10-01
One-way wave operators are powerful tools for use in forward modelling and inversion. Their implementation, however, involves introduction of the square root of an operator as a pseudo-differential operator. Furthermore, a simple factoring of the wave operator produces one-way wave equations that yield the same travel times as the full wave equation, but do not yield accurate amplitudes except for homogeneous media and for almost all points in heterogeneous media. Here, we present augmented one-way wave equations. We show that these equations yield solutions for which the leading order asymptotic amplitude as well as the travel time satisfy the same differential equations as the corresponding functions for the full wave equation. Exact representations of the square-root operator appearing in these differential equations are elusive, except in cases in which the heterogeneity of the medium is independent of the transverse spatial variables. Here, we address the fully heterogeneous case. Singling out depth as the preferred direction of propagation, we introduce a representation of the square-root operator as an integral in which a rational function of the transverse Laplacian appears in the integrand. This allows us to carry out explicit asymptotic analysis of the resulting one-way wave equations. To do this, we introduce an auxiliary function that satisfies a lower dimensional wave equation in transverse spatial variables only. We prove that ray theory for these one-way wave equations leads to one-way eikonal equations and the correct leading order transport equation for the full wave equation. We then introduce appropriate boundary conditions at z = 0 to generate waves at depth whose quotient leads to a reflector map and an estimate of the ray theoretical reflection coefficient on the reflector. Thus, these true amplitude one-way wave equations lead to a 'true amplitude wave equation migration' (WEM) method. In fact, we prove that applying the WEM imaging condition
Skeletonized wave-equation inversion for Q
Dutta, Gaurav
2016-09-06
A wave-equation gradient optimization method is presented that inverts for the subsurface Q distribution by minimizing a skeletonized misfit function ε. Here, ε is the sum of the squared differences between the observed and the predicted peak/centroid frequency shifts of the early-arrivals. The gradient is computed by migrating the observed traces weighted by the frequency-shift residuals. The background Q model is perturbed until the predicted and the observed traces have the same peak frequencies or the same centroid frequencies. Numerical tests show that an improved accuracy of the inverted Q model by wave-equation Q tomography (WQ) leads to a noticeable improvement in the migration image quality.
Skeletonized wave-equation inversion for Q
Dutta, Gaurav; Schuster, Gerard T.
2016-01-01
A wave-equation gradient optimization method is presented that inverts for the subsurface Q distribution by minimizing a skeletonized misfit function ε. Here, ε is the sum of the squared differences between the observed and the predicted peak/centroid frequency shifts of the early-arrivals. The gradient is computed by migrating the observed traces weighted by the frequency-shift residuals. The background Q model is perturbed until the predicted and the observed traces have the same peak frequencies or the same centroid frequencies. Numerical tests show that an improved accuracy of the inverted Q model by wave-equation Q tomography (WQ) leads to a noticeable improvement in the migration image quality.
The damped wave equation with unbounded damping
Freitas, P.; Siegl, Petr; Tretter, C.
2018-01-01
Roč. 264, č. 12 (2018), s. 7023-7054 ISSN 0022-0396 Institutional support: RVO:61389005 Keywords : damped wave equation * unbounded damping * essential spectrum * quadratic operator funciton with unbounded coefficients * Schrodinger operators with complex potentials Subject RIV: BE - Theoretical Physics OBOR OECD: Atomic, molecular and chemical physics (physics of atoms and molecules including collision, interaction with radiation, magnetic resonances, Mössbauer effect) Impact factor: 1.988, year: 2016
Skeletonized Least Squares Wave Equation Migration
Zhan, Ge
2010-10-17
The theory for skeletonized least squares wave equation migration (LSM) is presented. The key idea is, for an assumed velocity model, the source‐side Green\\'s function and the geophone‐side Green\\'s function are computed by a numerical solution of the wave equation. Only the early‐arrivals of these Green\\'s functions are saved and skeletonized to form the migration Green\\'s function (MGF) by convolution. Then the migration image is obtained by a dot product between the recorded shot gathers and the MGF for every trial image point. The key to an efficient implementation of iterative LSM is that at each conjugate gradient iteration, the MGF is reused and no new finitedifference (FD) simulations are needed to get the updated migration image. It is believed that this procedure combined with phase‐encoded multi‐source technology will allow for the efficient computation of wave equation LSM images in less time than that of conventional reverse time migration (RTM).
2016-01-01
This volume brings together four lecture courses on modern aspects of water waves. The intention, through the lectures, is to present quite a range of mathematical ideas, primarily to show what is possible and what, currently, is of particular interest. Water waves of large amplitude can only be fully understood in terms of nonlinear effects, linear theory being not adequate for their description. Taking advantage of insights from physical observation, experimental evidence and numerical simulations, classical and modern mathematical approaches can be used to gain insight into their dynamics. The book presents several avenues and offers a wide range of material of current interest. Due to the interdisciplinary nature of the subject, the book should be of interest to mathematicians (pure and applied), physicists and engineers. The lectures provide a useful source for those who want to begin to investigate how mathematics can be used to improve our understanding of water wave phenomena. In addition, some of the...
Wave-equation reflection traveltime inversion
Zhang, Sanzong
2011-01-01
The main difficulty with iterative waveform inversion using a gradient optimization method is that it tends to get stuck in local minima associated within the waveform misfit function. This is because the waveform misfit function is highly nonlinear with respect to changes in the velocity model. To reduce this nonlinearity, we present a reflection traveltime tomography method based on the wave equation which enjoys a more quasi-linear relationship between the model and the data. A local crosscorrelation of the windowed downgoing direct wave and the upgoing reflection wave at the image point yields the lag time that maximizes the correlation. This lag time represents the reflection traveltime residual that is back-projected into the earth model to update the velocity in the same way as wave-equation transmission traveltime inversion. No travel-time picking is needed and no high-frequency approximation is assumed. The mathematical derivation and the numerical examples are presented to partly demonstrate its efficiency and robustness. © 2011 Society of Exploration Geophysicists.
Blowing-up Semilinear Wave Equation with Exponential ...
Blowing-up Semilinear Wave Equation with Exponential Nonlinearity in Two Space ... We investigate the initial value problem for some semi-linear wave equation in two space dimensions with exponential nonlinearity growth. ... Current Issue
Exact traveling wave solutions of the Boussinesq equation
Ding Shuangshuang; Zhao Xiqiang
2006-01-01
The repeated homogeneous balance method is used to construct exact traveling wave solutions of the Boussinesq equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions of the Boussinesq equation are successfully obtained
Travelling Waves in Hyperbolic Chemotaxis Equations
Xue, Chuan; Hwang, Hyung Ju; Painter, Kevin J.; Erban, Radek
2010-01-01
Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235-248, 1971). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically. © 2010 Society for Mathematical Biology.
Travelling Waves in Hyperbolic Chemotaxis Equations
Xue, Chuan
2010-10-16
Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235-248, 1971). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically. © 2010 Society for Mathematical Biology.
Cnoidal waves governed by the Kudryashov–Sinelshchikov equation
Randrüüt, Merle; Braun, Manfred
2013-01-01
The evolution equation for waves propagating in a mixture of liquid and gas bubbles as proposed by Kudryashov and Sinelshchikov allows, in a special case, the propagation of solitary waves of the sech 2 type. It is shown that these waves represent the solitary limit separating two families of periodic waves. One of them consists of the same cnoidal waves that are solutions of the Korteweg–de Vries equation, while the other one does not have a corresponding counterpart. It is pointed out how the ordinary differential equations governing traveling-wave solutions of the Kudryashov–Sinelshchikov and the Korteweg–de Vries equations are related to each other.
Cnoidal waves governed by the Kudryashov–Sinelshchikov equation
Randrüüt, Merle, E-mail: merler@cens.ioc.ee [Tallinn University of Technology, Faculty of Mechanical Engineering, Department of Mechatronics, Ehitajate tee 5, 19086 Tallinn (Estonia); Braun, Manfred [University of Duisburg–Essen, Chair of Mechanics and Robotics, Lotharstraße 1, 47057 Duisburg (Germany)
2013-10-30
The evolution equation for waves propagating in a mixture of liquid and gas bubbles as proposed by Kudryashov and Sinelshchikov allows, in a special case, the propagation of solitary waves of the sech{sup 2} type. It is shown that these waves represent the solitary limit separating two families of periodic waves. One of them consists of the same cnoidal waves that are solutions of the Korteweg–de Vries equation, while the other one does not have a corresponding counterpart. It is pointed out how the ordinary differential equations governing traveling-wave solutions of the Kudryashov–Sinelshchikov and the Korteweg–de Vries equations are related to each other.
Bifurcation of the spin-wave equations
Cascon, A.; Koiller, J.; Rezende, S.M.
1990-01-01
We study the bifurcations of the spin-wave equations that describe the parametric pumping of collective modes in magnetic media. Mechanisms describing the following dynamical phenomena are proposed: (i) sequential excitation of modes via zero eigenvalue bifurcations; (ii) Hopf bifurcations followed (or not) by Feingenbaum cascades of period doubling; (iii) local and global homoclinic phenomena. Two new organizing center for routes to chaos are identified; in the classification given by Guckenheimer and Holmes [GH], one is a codimension-two local bifurcation, with one pair of imaginary eigenvalues and a zero eigenvalue, to which many dynamical consequences are known; secondly, global homoclinic bifurcations associated to splitting of separatrices, in the limit where the system can be considered a Hamiltonian subjected to weak dissipation and forcing. We outline what further numerical and algebraic work is necessary for the detailed study following this program. (author)
The damped wave equation with unbounded damping
Freitas, Pedro; Siegl, Petr; Tretter, Christiane
2018-06-01
We analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity. We prove the generation of a contraction semigroup, study the relation between the spectra of the semigroup generator and the associated quadratic operator function, the convergence of non-real eigenvalues in the asymptotic regime of diverging damping on a subdomain, and we investigate the appearance of essential spectrum on the negative real axis. We further show that the presence of the latter prevents exponential estimates for the semigroup and turns out to be a robust effect that cannot be easily canceled by adding a positive potential. These analytic results are illustrated by examples.
Rogue periodic waves of the modified KdV equation
Chen, Jinbing; Pelinovsky, Dmitry E.
2018-05-01
Rogue periodic waves stand for rogue waves on a periodic background. Two families of travelling periodic waves of the modified Korteweg–de Vries (mKdV) equation in the focusing case are expressed by the Jacobian elliptic functions dn and cn. By using one-fold and two-fold Darboux transformations of the travelling periodic waves, we construct new explicit solutions for the mKdV equation. Since the dn-periodic wave is modulationally stable with respect to long-wave perturbations, the new solution constructed from the dn-periodic wave is a nonlinear superposition of an algebraically decaying soliton and the dn-periodic wave. On the other hand, since the cn-periodic wave is modulationally unstable with respect to long-wave perturbations, the new solution constructed from the cn-periodic wave is a rogue wave on the cn-periodic background, which generalizes the classical rogue wave (the so-called Peregrine’s breather) of the nonlinear Schrödinger equation. We compute the magnification factor for the rogue cn-periodic wave of the mKdV equation and show that it remains constant for all amplitudes. As a by-product of our work, we find explicit expressions for the periodic eigenfunctions of the spectral problem associated with the dn and cn periodic waves of the mKdV equation.
Separate P‐ and SV‐wave equations for VTI media
Pestana, Reynam C.; Ursin, Bjø rn; Stoffa, Paul L.
2011-01-01
In isotropic media we use the scalar acoustic wave equation to perform reverse time migration RTM of the recorded pressure wavefleld data. In anisotropic media P- and SV-waves are coupled and the elastic wave equation should be used for RTM. However, an acoustic anisotropic wave equation is often used instead. This results in significant shear wave energy in both modeling and RTM. To avoid this undesired SV-wave energy, we propose a different approach to separate P- and SV-wave components for vertical transversely isotropic VTI media. We derive independent pseudo-differential wave equations for each mode. The derived equations for P- and SV-waves are stable and reduce to the isotropic case. The equations presented here can be effectively used to model and migrate seismic data in VTI media where ε - δ is small. The SV-wave equation we develop is now well-posed and triplications in the SV wavefront are removed resulting in stable wave propagation. We show modeling and RTM results using the derived pure P-wave mode in complex VTI media and use the rapid expansion method REM to propagate the waveflelds in time. © 2011 Society of Exploration Geophysicists.
M. Arshad
Full Text Available In this manuscript, we constructed different form of new exact solutions of generalized coupled Zakharov–Kuznetsov and dispersive long wave equations by utilizing the modified extended direct algebraic method. New exact traveling wave solutions for both equations are obtained in the form of soliton, periodic, bright, and dark solitary wave solutions. There are many applications of the present traveling wave solutions in physics and furthermore, a wide class of coupled nonlinear evolution equations can be solved by this method. Keywords: Traveling wave solutions, Elliptic solutions, Generalized coupled Zakharov–Kuznetsov equation, Dispersive long wave equation, Modified extended direct algebraic method
Traveling wave behavior for a generalized fisher equation
Feng Zhaosheng
2008-01-01
There is the widespread existence of wave phenomena in physics, chemistry and biology. This clearly necessitates a study of traveling waves in depth and of the modeling and analysis involved. In the present paper, we study a nonlinear reaction-diffusion equation, which can be regarded as a generalized Fisher equation. Applying the Cole-Hopf transformation and the first integral method, we obtain a class of traveling solitary wave solutions for this generalized Fisher equation
Islam, Md Hamidul; Khan, Kamruzzaman; Akbar, M Ali; Salam, Md Abdus
2014-01-01
Mathematical modeling of many physical systems leads to nonlinear evolution equations because most physical systems are inherently nonlinear in nature. The investigation of traveling wave solutions of nonlinear partial differential equations (NPDEs) plays a significant role in the study of nonlinear physical phenomena. In this article, we construct the traveling wave solutions of modified KDV-ZK equation and viscous Burgers equation by using an enhanced (G '/G) -expansion method. A number of traveling wave solutions in terms of unknown parameters are obtained. Derived traveling wave solutions exhibit solitary waves when special values are given to its unknown parameters. 35C07; 35C08; 35P99.
Approximate equations at breaking for nearshore wave transformation coefficients
Chandramohan, P.; Nayak, B.U.; SanilKumar, V.
Based on small amplitude wave theory approximate equations are evaluated for determining the coefficients of shoaling, refraction, bottom friction, bottom percolation and viscous dissipation at breaking. The results obtainEd. by these equations...
Temperature waves and the Boltzmann kinetic equation for phonons
Urushev, D.; Borisov, M.; Vavrek, A.
1988-01-01
The ordinary parabolic equation for thermal conduction based on the Fourier empiric law as well as the generalized thermal conduction equation based on the Maxwell law have been derived from the Boltzmann equation for the phonons within the relaxation time approximation. The temperature waves of the so-called second sound in crystals at low temperatures are transformed into Fourier waves at low frequencies with respect to the characteristic frequency of the U-processes. These waves are transformed into temperature waves similar to the second sound waves in He II at frequences higher than the U-processes characteristic. 1 fig., 19 refs
Iyer, Ramakrishnan; Johnson, Clifford V; Pennington, Jeffrey S
2011-01-01
We uncover a remarkable role that an infinite hierarchy of nonlinear differential equations plays in organizing and connecting certain c-hat <1 string theories non-perturbatively. We are able to embed the type 0A and 0B (A, A) minimal string theories into this single framework. The string theories arise as special limits of a rich system of equations underpinned by an integrable system known as the dispersive water wave hierarchy. We observe that there are several other string-like limits of the system, and conjecture that some of them are type IIA and IIB (A, D) minimal string backgrounds. We explain how these and several string-like special points arise and are connected. In some cases, the framework endows the theories with a non-perturbative definition for the first time. Notably, we discover that the Painleve IV equation plays a key role in organizing the string theory physics, joining its siblings, Painleve I and II, whose roles have previously been identified in this minimal string context.
A new auxiliary equation and exact travelling wave solutions of nonlinear equations
Sirendaoreji
2006-01-01
A new auxiliary ordinary differential equation and its solutions are used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the auxiliary equation which has more new exact solutions. More new exact travelling wave solutions are obtained for the quadratic nonlinear Klein-Gordon equation, the combined KdV and mKdV equation, the sine-Gordon equation and the Whitham-Broer-Kaup equations
Soomere, T.
2010-07-01
Most of the processes resulting in the formation of unexpectedly high surface waves in deep water (such as dispersive and geometrical focusing, interactions with currents and internal waves, reflection from caustic areas, etc.) are active also in shallow areas. Only the mechanism of modulational instability is not active in finite depth conditions. Instead, wave amplification along certain coastal profiles and the drastic dependence of the run-up height on the incident wave shape may substantially contribute to the formation of rogue waves in the nearshore. A unique source of long-living rogue waves (that has no analogues in the deep ocean) is the nonlinear interaction of obliquely propagating solitary shallow-water waves and an equivalent mechanism of Mach reflection of waves from the coast. The characteristic features of these processes are (i) extreme amplification of the steepness of the wave fronts, (ii) change in the orientation of the largest wave crests compared with that of the counterparts and (iii) rapid displacement of the location of the extreme wave humps along the crests of the interacting waves. The presence of coasts raises a number of related questions such as the possibility of conversion of rogue waves into sneaker waves with extremely high run-up. Also, the reaction of bottom sediments and the entire coastal zone to the rogue waves may be drastic.
Wave equations on anti self dual (ASD) manifolds
Bashingwa, Jean-Juste; Kara, A. H.
2017-11-01
In this paper, we study and perform analyses of the wave equation on some manifolds with non diagonal metric g_{ij} which are of neutral signatures. These include the invariance properties, variational symmetries and conservation laws. In the recent past, wave equations on the standard (space time) Lorentzian manifolds have been performed but not on the manifolds from metrics of neutral signatures.
The propagation of travelling waves for stochastic generalized KPP equations
Elworthy, K.D.; Zhao, H.Z.
1993-09-01
We study the existence and propagation of approximate travelling waves of generalized KPP equations with seasonal multiplicative white noise perturbations of Ito type. Three regimes of perturbation are considered: weak, milk, and strong. We show that weak perturbations have little effect on the wave like solutions of the unperturbed equations while strong perturbations essentially destroy the wave and force the solutions to die down. For mild perturbations we show that there is a residual wave form but propagating at a different speed to that of the unperturbed equation. In the appendix J.G. Gaines illustrates these different regimes by computer simulations. (author). 27 refs, 13 figs
An approach to rogue waves through the cnoidal equation
Lechuga, Antonio
2014-05-01
Lately it has been realized the importance of rogue waves in some events happening in open seas. Extreme waves and extreme weather could explain some accidents, but not all of them. Every now and then inflicted damages on ships only can be reported to be caused by anomalous and elusive waves, such as rogue waves. That's one of the reason why they continue attracting considerable interest among researchers. In the frame of the Nonlinear Schrödinger equation(NLS), Witham(1974) and Dingemans and Otta (2001)gave asymptotic solutions in moving coordinates that transformed the NLS equation in a ordinary differential equation that is the Duffing or cnoidal wave equation. Applying the Zakharov equation, Stiassnie and Shemer(2004) and Shemer(2010)got also a similar equation. It's well known that this ordinary equation can be solved in elliptic functions. The main aim of this presentation is to sort out the domains of the solutions of this equation, that, of course, are linked to the corresponding solutions of the partial differential equations(PDEs). That being, Lechuga(2007),a simple way to look for anomalous waves as it's the case with some "chaotic" solutions of the Duffing equation.
Local energy decay for linear wave equations with variable coefficients
Ikehata, Ryo
2005-06-01
A uniform local energy decay result is derived to the linear wave equation with spatial variable coefficients. We deal with this equation in an exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data, and its proof is quite simple. This generalizes a previous famous result due to Morawetz [The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961) 561-568]. In order to prove local energy decay, we mainly apply two types of ideas due to Ikehata-Matsuyama [L2-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002) 33-42] and Todorova-Yordanov [Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].
Bifurcations of traveling wave solutions for an integrable equation
Li Jibin; Qiao Zhijun
2010-01-01
This paper deals with the following equation m t =(1/2)(1/m k ) xxx -(1/2)(1/m k ) x , which is proposed by Z. J. Qiao [J. Math. Phys. 48, 082701 (2007)] and Qiao and Liu [Chaos, Solitons Fractals 41, 587 (2009)]. By adopting the phase analysis method of planar dynamical systems and the theory of the singular traveling wave systems to the traveling wave solutions of the equation, it is shown that for different k, the equation may have infinitely many solitary wave solutions, periodic wave solutions, kink/antikink wave solutions, cusped solitary wave solutions, and breaking loop solutions. We discuss in a detail the cases of k=-2,-(1/2),(1/2),2, and parametric representations of all possible bounded traveling wave solutions are given in the different (c,g)-parameter regions.
Exact solitary waves of the Korteveg - de Vries - Burgers equation
Kudryashov, N. A.
2004-01-01
New approach is presented to search exact solutions of nonlinear differential equations. This method is used to look for exact solutions of the Korteveg -- de Vries -- Burgers equation. New exact solitary waves of the Korteveg -- de Vries -- Burgers equation are found.
On a functional equation related to the intermediate long wave equation
Hone, A N W; Novikov, V S
2004-01-01
We resolve an open problem stated by Ablowitz et al (1982 J. Phys. A: Math. Gen. 15 781) concerning the integral operator appearing in the intermediate long wave equation. We explain how this is resolved using the perturbative symmetry approach introduced by one of us with Mikhailov. By solving a certain functional equation, we prove that the intermediate long wave equation and the Benjamin-Ono equation are the unique integrable cases within a particular class of integro-differential equations. Furthermore, we explain how the perturbative symmetry approach is naturally extended to treat equations on a periodic domain. (letter to the editor)
Nonlocal Reformulations of Water and Internal Waves and Asymptotic Reductions
Ablowitz, Mark J.
2009-09-01
Nonlocal reformulations of the classical equations of water waves and two ideal fluids separated by a free interface, bounded above by either a rigid lid or a free surface, are obtained. The kinematic equations may be written in terms of integral equations with a free parameter. By expressing the pressure, or Bernoulli, equation in terms of the surface/interface variables, a closed system is obtained. An advantage of this formulation, referred to as the nonlocal spectral (NSP) formulation, is that the vertical component is eliminated, thus reducing the dimensionality and fixing the domain in which the equations are posed. The NSP equations and the Dirichlet-Neumann operators associated with the water wave or two-fluid equations can be related to each other and the Dirichlet-Neumann series can be obtained from the NSP equations. Important asymptotic reductions obtained from the two-fluid nonlocal system include the generalizations of the Benney-Luke and Kadomtsev-Petviashvili (KP) equations, referred to as intermediate-long wave (ILW) generalizations. These 2+1 dimensional equations possess lump type solutions. In the water wave problem high-order asymptotic series are obtained for two and three dimensional gravity-capillary solitary waves. In two dimensions, the first term in the asymptotic series is the well-known hyperbolic secant squared solution of the KdV equation; in three dimensions, the first term is the rational lump solution of the KP equation.
Lecture Notes for the Course in Water Wave Mechanics
Andersen, Thomas Lykke; Frigaard, Peter
knowledge. The course is at the same time an introduction to the course in coastal hydraulics on the 8th semester. The notes cover the following five lectures: 1. Definitions. Governing equations and boundary conditions. Derivation of velocity potential for linear waves. Dispersion relationship. 2. Particle...... paths, velocities, accelerations, pressure variation, deep and shallow water waves, wave energy and group velocity. 3. Shoaling, refraction, diffraction and wave breaking. 4. Irregular waves. Time domain analysis of waves. 5. Wave spectra. Frequency domain analysis of waves. The present notes are based...
Runge-Kutta Integration of the Equal Width Wave Equation Using the Method of Lines
M. A. Banaja
2015-01-01
Full Text Available The equal width (EW equation governs nonlinear wave phenomena like waves in shallow water. Numerical solution of the (EW equation is obtained by using the method of lines (MOL based on Runge-Kutta integration. Using von Neumann stability analysis, the scheme is found to be unconditionally stable. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Accuracy of the proposed method is discussed by computing the L2 and L∞ error norms. The results are found in good agreement with exact solution.
Traveling waves of the regularized short pulse equation
Shen, Y; Horikis, T P; Kevrekidis, P G; Frantzeskakis, D J
2014-01-01
The properties of the so-called regularized short pulse equation (RSPE) are explored with a particular focus on the traveling wave solutions of this model. We theoretically analyze and numerically evolve two sets of such solutions. First, using a fixed point iteration scheme, we numerically integrate the equation to find solitary waves. It is found that these solutions are well approximated by a finite sum of hyperbolic secants powers. The dependence of the soliton's parameters (height, width, etc) to the parameters of the equation is also investigated. Second, by developing a multiple scale reduction of the RSPE to the nonlinear Schrödinger equation, we are able to construct (both standing and traveling) envelope wave breather type solutions of the former, based on the solitary wave structures of the latter. Both the regular and the breathing traveling wave solutions identified are found to be robust and should thus be amenable to observations in the form of few optical cycle pulses. (paper)
Line Rogue Waves in the Mel'nikov Equation
Shi, Yongkang
2017-07-01
General line rogue waves in the Mel'nikov equation are derived via the Hirota bilinear method, which are given in terms of determinants whose matrix elements have plain algebraic expressions. It is shown that fundamental rogue waves are line rogue waves, which arise from the constant background with a line profile and then disappear into the constant background again. By means of the regulation of free parameters, two subclass of nonfundamental rogue waves are generated, which are called as multirogue waves and higher-order rogue waves. The multirogue waves consist of several fundamental line rogue waves, which arise from the constant background and then decay back to the constant background. The higher-order rogue waves start from a localised lump and retreat back to it. The dynamical behaviours of these line rogue waves are demonstrated by the density and the three-dimensional figures.
Skeletonized wave-equation Qs tomography using surface waves
Li, Jing; Dutta, Gaurav; Schuster, Gerard T.
2017-01-01
We present a skeletonized inversion method that inverts surface-wave data for the Qs quality factor. Similar to the inversion of dispersion curves for the S-wave velocity model, the complicated surface-wave arrivals are skeletonized as simpler data
Li, Jing; Schuster, Gerard T.; Zeng, Zhaofa
2017-01-01
A robust imaging technology is reviewed that provide subsurface information in challenging environments: wave-equation dispersion inversion (WD) of surface waves for the shear velocity model. We demonstrate the benefits and liabilities of the method
Reduction of the equation for lower hybrid waves in a plasma to a nonlinear Schroedinger equation
Karney, C. F. F.
1977-01-01
Equations describing the nonlinear propagation of waves in an anisotropic plasma are rarely exactly soluble. However it is often possible to make approximations that reduce the exact equations into a simpler equation. The use of MACSYMA to make such approximations, and so reduce the equation describing lower hybrid waves into the nonlinear Schrodinger equation which is soluble by the inverse scattering method is demonstrated. MACSYMA is used at several stages in the calculation only because there is a natural division between calculations that are easiest done by hand, and those that are easiest done by machine.
Relativistic covariant wave equations and acausality in external fields
Pijlgroms, R.B.J.
1980-01-01
The author considers linear, finite dimensional, first order relativistic wave equations: (βsup(μ)ideltasub(μ)-β)PSI(x) = 0 with βsup(μ) and β constant matrices. Firstly , the question of the relativistic covariance conditions on these equations is considered. Then the theory of these equations with β non-singular is summarized. Theories with βsup(μ), β square matrices and β singular are also discussed. Non-square systems of covariant relativistic wave equations for arbitrary spin > 1 are then considered. Finally, the interaction with external fields and the acausality problem are discussed. (G.T.H.)
Orbital stability of solitary waves for Kundu equation
Zhang, Weiguo; Qin, Yinghao; Zhao, Yan; Guo, Boling
In this paper, we consider the Kundu equation which is not a standard Hamiltonian system. The abstract orbital stability theory proposed by Grillakis et al. (1987, 1990) cannot be applied directly to study orbital stability of solitary waves for this equation. Motivated by the idea of Guo and Wu (1995), we construct three invariants of motion and use detailed spectral analysis to obtain orbital stability of solitary waves for Kundu equation. Since Kundu equation is more complex than the derivative Schrödinger equation, we utilize some techniques to overcome some difficulties in this paper. It should be pointed out that the results obtained in this paper are more general than those obtained by Guo and Wu (1995). We present a sufficient condition under which solitary waves are orbitally stable for 2c+sυ1995) only considered the case 2c+sυ>0. We obtain the results on orbital stability of solitary waves for the derivative Schrödinger equation given by Colin and Ohta (2006) as a corollary in this paper. Furthermore, we obtain orbital stability of solitary waves for Chen-Lee-Lin equation and Gerdjikov-Ivanov equation, respectively.
Yang Pei; Li Zhibin; Chen Yong
2010-01-01
In this paper, the short-wave model equations are investigated, which are associated with the Camassa-Holm (CH) and Degasperis-Procesi (DP) shallow-water wave equations. Firstly, by means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. Secondly, the equation is solved by homotopy analysis method. Lastly, by the transformations back to the original independent variables, the solution of the original partial differential equation is obtained. The two types of solutions of the short-wave models are obtained in parametric form, one is one-cusp soliton for the CH equation while the other one is one-loop soliton for the DP equation. The approximate analytic solutions expressed by a series of exponential functions agree well with the exact solutions. It demonstrates the validity and great potential of homotopy analysis method for complicated nonlinear solitary wave problems. (general)
Paraxial WKB solution of a scalar wave equation
Pereverzev, G.V.
1993-04-01
An asymptotic method of solving a scalar wave equation in inhomogeneous media is developed. This method is an extension of the WKB method to the multidimensional case. It reduces a general wave equation to a set of ordinary differential equations similar to that of the eikonal approach and includes the latter as a particular case. However, the WKB method makes use of another kind of asymptotic expansion and, unlike the eikonal approach, describes the wave properties, i.e. diffraction and interference. At the same time, the three-dimensional WKB method is more simple for numerical treatment because the number of equations is less than in the eikonal approach. The method developed may be used for a calculation of wave fields in problems of RF heating, current drive and plasma diagnostics with microwave beams. (orig.)
Closed form solutions of two time fractional nonlinear wave equations
M. Ali Akbar
2018-06-01
Full Text Available In this article, we investigate the exact traveling wave solutions of two nonlinear time fractional wave equations. The fractional derivatives are described in the sense of conformable fractional derivatives. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters. To investigate such types of solutions, we implement the new generalized (G′/G-expansion method. The extracted solutions are reliable, useful and suitable to comprehend the optimal control problems, chaotic vibrations, global and local bifurcations and resonances, furthermore, fission and fusion phenomena occur in solitons, the relativistic energy-momentum relation, scalar electrodynamics, quantum relativistic one-particle theory, electromagnetic interactions etc. The results reveal that the method is very fruitful and convenient for exploring nonlinear differential equations of fractional order treated in theoretical physics. Keywords: Traveling wave solution, Soliton, Generalized (G′/G-expansion method, Time fractional Duffing equation, Time fractional Riccati equation
Relativistic transport equation for a discontinuity wave of multiplicity one
Giambo, S; Palumbo, A [Istituto di Matematica, Universita degli Studi, Messina (Italy)
1980-04-14
In the framework of the theory of the singular hypersurfaces, the transport equation for the amplitude of a discontinuity wave, corresponding to a simple characteristic of a quasi-linear hyperbolic system, is established in the context of special relativity.
Anisotropic wave-equation traveltime and waveform inversion
Feng, Shihang; Schuster, Gerard T.
2016-01-01
The wave-equation traveltime and waveform inversion (WTW) methodology is developed to invert for anisotropic parameters in a vertical transverse isotropic (VTI) meidum. The simultaneous inversion of anisotropic parameters v0, ε and δ is initially
Exponential decay for solutions to semilinear damped wave equation
Gerbi, Sté phane; Said-Houari, Belkacem
2011-01-01
This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Intro- ducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data
Anisotropic wave-equation traveltime and waveform inversion
Feng, Shihang
2016-09-06
The wave-equation traveltime and waveform inversion (WTW) methodology is developed to invert for anisotropic parameters in a vertical transverse isotropic (VTI) meidum. The simultaneous inversion of anisotropic parameters v0, ε and δ is initially performed using the wave-equation traveltime inversion (WT) method. The WT tomograms are then used as starting background models for VTI full waveform inversion. Preliminary numerical tests on synthetic data demonstrate the feasibility of this method for multi-parameter inversion.
Continuity relations and quantum wave equations
Goedecke, G.H.; Davis, B.T.
2010-01-01
We investigate the mathematical synthesis of the Schroedinger, Klein-Gordon, Pauli-Schroedinger, and Dirac equations starting from probability continuity relations. We utilize methods similar to those employed by R. E. Collins (Lett. Nuovo Cimento, 18 (1977) 581) in his construction of the Schroedinger equation from the position probability continuity relation for a single particle. Our new results include the mathematical construction of the Pauli-Schroedinger and Dirac equations from the position probability continuity relations for a particle that can transition between two states or among four states, respectively.
A wave equation interpolating between classical and quantum mechanics
Schleich, W. P.; Greenberger, D. M.; Kobe, D. H.; Scully, M. O.
2015-10-01
We derive a ‘master’ wave equation for a family of complex-valued waves {{Φ }}\\equiv R{exp}[{{{i}}S}({cl)}/{{\\hbar }}] whose phase dynamics is dictated by the Hamilton-Jacobi equation for the classical action {S}({cl)}. For a special choice of the dynamics of the amplitude R which eliminates all remnants of classical mechanics associated with {S}({cl)} our wave equation reduces to the Schrödinger equation. In this case the amplitude satisfies a Schrödinger equation analogous to that of a charged particle in an electromagnetic field where the roles of the scalar and the vector potentials are played by the classical energy and the momentum, respectively. In general this amplitude is complex and thereby creates in addition to the classical phase {S}({cl)}/{{\\hbar }} a quantum phase. Classical statistical mechanics, as described by a classical matter wave, follows from our wave equation when we choose the dynamics of the amplitude such that it remains real for all times. Our analysis shows that classical and quantum matter waves are distinguished by two different choices of the dynamics of their amplitudes rather than two values of Planck’s constant. We dedicate this paper to the memory of Richard Lewis Arnowitt—a pioneer of many-body theory, a path finder at the interface of gravity and quantum mechanics, and a true leader in non-relativistic and relativistic quantum field theory.
New exact wave solutions for Hirota equation
2Department of Engineering Sciences, Faculty of Technology and Engineering,. University ... of nonlinear partial differential equations (NPDEs) in mathematical physics. Keywords. ... This method has been successfully applied to obtain exact.
Drift of Spiral Waves in Complex Ginzburg-Landau Equation
Yang Junzhong; Zhang Mei
2006-01-01
The spontaneous drift of the spiral wave in a finite domain in the complex Ginzburg-Landau equation is investigated numerically. By using the interactions between the spiral wave and its images, we propose a phenomenological theory to explain the observations.
Scattering for wave equations with dissipative terms in layered media
Mitsuteru Kadowaki
2011-05-01
Full Text Available In this article, we show the existence of scattering solutions to wave equations with dissipative terms in layered media. To analyze the wave propagation in layered media, it is necessary to handle singular points called thresholds in the spectrum. Our main tools are Kato's smooth perturbation theory and some approximate operators.
Equations of state for light water
Rubin, G.A.; Granziera, M.R.
1983-01-01
The equations of state for light water were developed, based on the tables of Keenan and Keyes. Equations are presented, describing the specific volume, internal energy, enthalpy and entropy of saturated steam, superheated vapor and subcooled liquid as a function of pressure and temperature. For each property, several equations are shown, with different precisions and different degress of complexity. (Author) [pt
Integral Equation Methods for Electromagnetic and Elastic Waves
Chew, Weng; Hu, Bin
2008-01-01
Integral Equation Methods for Electromagnetic and Elastic Waves is an outgrowth of several years of work. There have been no recent books on integral equation methods. There are books written on integral equations, but either they have been around for a while, or they were written by mathematicians. Much of the knowledge in integral equation methods still resides in journal papers. With this book, important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral eq
Dynamic equations for gauge-invariant wave functions
Kapshaj, V.N.; Skachkov, N.B.; Solovtsov, I.L.
1984-01-01
The Bethe-Salpeter and quasipotential dynamic equations for wave functions of relative quark motion, have been derived. Wave functions are determined by the gauge invariant method. The V.A. Fock gauge condition is used in the construction. Despite the transl tional noninvariance of the gauge condition the standard separation of variables has been obtained and wave function doesn't contain gauge exponents
Quadratic algebras in the noncommutative integration method of wave equation
Varaksin, O.L.
1995-01-01
The paper deals with the investigation of applications of the method of noncommutative integration of linear differential equations by partial derivatives. Nontrivial example was taken for integration of three-dimensions wave equation with the use of non-Abelian quadratic algebras
Traveling waves and conservation laws for highly nonlinear wave equations modeling Hertz chains
Przedborski, Michelle; Anco, Stephen C.
2017-09-01
A highly nonlinear, fourth-order wave equation that models the continuum theory of long wavelength pulses in weakly compressed, homogeneous, discrete chains with a general power-law contact interaction is studied. For this wave equation, all solitary wave solutions and all nonlinear periodic wave solutions, along with all conservation laws, are derived. The solutions are explicitly parameterized in terms of the asymptotic value of the wave amplitude in the case of solitary waves and the peak of the wave amplitude in the case of nonlinear periodic waves. All cases in which the solution expressions can be stated in an explicit analytic form using elementary functions are worked out. In these cases, explicit expressions for the total energy and total momentum for all solutions are obtained as well. The derivation of the solutions uses the conservation laws combined with an energy analysis argument to reduce the wave equation directly to a separable first-order differential equation that determines the wave amplitude in terms of the traveling wave variable. This method can be applied more generally to other highly nonlinear wave equations.
Wave Functions for Time-Dependent Dirac Equation under GUP
Zhang, Meng-Yao; Long, Chao-Yun; Long, Zheng-Wen
2018-04-01
In this work, the time-dependent Dirac equation is investigated under generalized uncertainty principle (GUP) framework. It is possible to construct the exact solutions of Dirac equation when the time-dependent potentials satisfied the proper conditions. In (1+1) dimensions, the analytical wave functions of the Dirac equation under GUP have been obtained for the two kinds time-dependent potentials. Supported by the National Natural Science Foundation of China under Grant No. 11565009
On the Stochastic Wave Equation with Nonlinear Damping
Kim, Jong Uhn
2008-01-01
We discuss an initial boundary value problem for the stochastic wave equation with nonlinear damping. We establish the existence and uniqueness of a solution. Our method for the existence of pathwise solutions consists of regularization of the equation and data, the Galerkin approximation and an elementary measure-theoretic argument. We also prove the existence of an invariant measure when the equation has pure nonlinear damping
Invariant measures for stochastic nonlinear beam and wave equations
Brzezniak, Z.; Ondreját, Martin; Seidler, Jan
2016-01-01
Roč. 260, č. 5 (2016), s. 4157-4179 ISSN 0022-0396 R&D Projects: GA ČR GAP201/10/0752 Institutional support: RVO:67985556 Keywords : stochastic partial differential equation * stochastic beam equation * stochastic wave equation * invariant measure Subject RIV: BA - General Mathematics Impact factor: 1.988, year: 2016 http://library.utia.cas.cz/separaty/2016/SI/ondrejat-0453412.pdf
Unified formulation of radiation conditions for the wave equation
Krenk, Steen
2002-01-01
A family of radiation conditions for the wave equation is derived by truncating a rational function approxiamtion of the corresponding plane wave representation, and it is demonstrated how these boundary conditions can be formulated in terms of fictitious surface densities, governed by second......-order wave equations on the radiating surface. Several well-established radiation boundary conditions appear as special cases, corresponding to different choice of the coefficients in the rational approximation. The relation between these choices is established, and an explicit formulation in terms...
A nonlinear wave equation in nonadiabatic flame propagation
Booty, M.R.; Matalon, M.; Matkowsky, B.J.
1988-01-01
The authors derive a nonlinear wave equation from the diffusional thermal model of gaseous combustion to describe the evolution of a flame front. The equation arises as a long wave theory, for values of the volumeric heat loss in a neighborhood of the extinction point (beyond which planar uniformly propagating flames cease to exist), and for Lewis numbers near the critical value beyond which uniformly propagating planar flames lose stability via a degenerate Hopf bifurcation. Analysis of the equation suggests the possibility of a singularity developing in finite time
Zhang Huiqun
2009-01-01
By using a new coupled Riccati equations, a direct algebraic method, which was applied to obtain exact travelling wave solutions of some complex nonlinear equations, is improved. And the exact travelling wave solutions of the complex KdV equation, Boussinesq equation and Klein-Gordon equation are investigated using the improved method. The method presented in this paper can also be applied to construct exact travelling wave solutions for other nonlinear complex equations.
Travelling wave solutions to the Kuramoto-Sivashinsky equation
Nickel, J.
2007-01-01
Combining the approaches given by Baldwin [Baldwin D et al. Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs. J Symbol Comput 2004;37:669-705], Peng [Peng YZ. A polynomial expansion method and new general solitary wave solutions to KS equation. Comm Theor Phys 2003;39:641-2] and by Schuermann [Schuermann HW, Serov VS. Weierstrass' solutions to certain nonlinear wave and evolution equations. Proc progress electromagnetics research symposium, 28-31 March 2004, Pisa. p. 651-4; Schuermann HW. Traveling-wave solutions to the cubic-quintic nonlinear Schroedinger equation. Phys Rev E 1996;54:4312-20] leads to a method for finding exact travelling wave solutions of nonlinear wave and evolution equations (NLWEE). The first idea is to generalize ansaetze given by Baldwin and Peng to find elliptic solutions of NLWEEs. Secondly, conditions used by Schuermann to find physical (real and bounded) solutions and to discriminate between periodic and solitary wave solutions are used. The method is shown in detail by evaluating new solutions of the Kuramoto-Sivashinsky equation
Separation of variables for the nonlinear wave equation in polar coordinates
Shermenev, Alexander
2004-01-01
Some classical types of nonlinear wave motion in polar coordinates are studied within quadratic approximation. When the nonlinear quadratic terms in the wave equation are arbitrary, the usual perturbation techniques used in polar coordinates leads to overdetermined systems of linear algebraic equations for the unknown coefficients. However, we show that these overdetermined systems are compatible with the special case of the nonlinear shallow water equation and express explicitly the coefficients of the first two harmonics as polynomials of the Bessel functions of radius and of the trigonometric functions of angle. It gives a series of solutions to the nonlinear shallow water equation that are periodic in time and found with the same accuracy as the equation is derived
Linear fractional diffusion-wave equation for scientists and engineers
Povstenko, Yuriy
2015-01-01
This book systematically presents solutions to the linear time-fractional diffusion-wave equation. It introduces the integral transform technique and discusses the properties of the Mittag-Leffler, Wright, and Mainardi functions that appear in the solutions. The time-nonlocal dependence between the flux and the gradient of the transported quantity with the “long-tail” power kernel results in the time-fractional diffusion-wave equation with the Caputo fractional derivative. Time-nonlocal generalizations of classical Fourier’s, Fick’s and Darcy’s laws are considered and different kinds of boundary conditions for this equation are discussed (Dirichlet, Neumann, Robin, perfect contact). The book provides solutions to the fractional diffusion-wave equation with one, two and three space variables in Cartesian, cylindrical and spherical coordinates. The respective sections of the book can be used for university courses on fractional calculus, heat and mass transfer, transport processes in porous media and ...
Dark and composite rogue waves in the coupled Hirota equations
Chen, Shihua
2014-01-01
The intriguing dark and composite rogue wave dynamics in a coupled Hirota system are unveiled, based on the exact explicit rational solutions obtained under the assumption of equal background height. It is found that a dark rogue wave state would occur as a result of the strong coupling between two field components with large wavenumber difference, and there would appear plenty of composite structures that are attributed to the specific wavenumber difference and the free choice of three independent structural parameters. The coexistence of different fundamental rogue waves in such a coupled system is also demonstrated. - Highlights: • Exact rational rogue wave solutions under different parameter conditions are presented for the coupled Hirota equations. • The basic rogue wave features and hence the intriguing dark structures are unveiled. • We attributed the diversity of composite rogue wave dynamics to the free choice of three independent structural parameters. • The remarkable coexisting rogue wave behaviors in such a coupled system are demonstrated
Travelling wave solutions for a surface wave equation in fluid mechanics
Tian Yi
2016-01-01
Full Text Available This paper considers a non-linear wave equation arising in fluid mechanics. The exact traveling wave solutions of this equation are given by using G'/G-expansion method. This process can be reduced to solve a system of determining equations, which is large and difficult. To reduce this process, we used Wu elimination method. Example shows that this method is effective.
Persistence of travelling waves in a generalized Fisher equation
Kyrychko, Yuliya N.; Blyuss, Konstantin B.
2009-01-01
Travelling waves of the Fisher equation with arbitrary power of nonlinearity are studied in the presence of long-range diffusion. Using analogy between travelling waves and heteroclinic solutions of corresponding ODEs, we employ the geometric singular perturbation theory to prove the persistence of these waves when the influence of long-range effects is small. When the long-range diffusion coefficient becomes larger, the behaviour of travelling waves can only be studied numerically. In this case we find that starting with some values, solutions of the model lose monotonicity and become oscillatory
Numerical Simulation of Freak Waves Based on the Four-Order Nonlinear Schr(o)dinger Equation
ZHANG Yun-qiu; ZHANG Ning-chuan; PEI Yu-guo
2007-01-01
A numerical wave model based on the modified four-order nonlinear Schrodinger (NLS) equation in deep water is developed to simulate freak waves. A standard split-step, pseudo-spectral method is used to solve NLS equation. The validation of the model is firstly verified, and then the simulation of freak waves is performed by changing sideband conditions. Results show that freak waves entirely consistent with the definition in the evolution of wave trains are obtained. The possible occurrence mechanism of freak waves is discussed and the relevant characteristics are also analyzed.
Nonlinear wave equations, formation of singularities
John, Fritz
1990-01-01
This is the second volume in the University Lecture Series, designed to make more widely available some of the outstanding lectures presented in various institutions around the country. Each year at Lehigh University, a distinguished mathematical scientist presents the Pitcher Lectures in the Mathematical Sciences. This volume contains the Pitcher lectures presented by Fritz John in April 1989. The lectures deal with existence in the large of solutions of initial value problems for nonlinear hyperbolic partial differential equations. As is typical with nonlinear problems, there are many results and few general conclusions in this extensive subject, so the author restricts himself to a small portion of the field, in which it is possible to discern some general patterns. Presenting an exposition of recent research in this area, the author examines the way in which solutions can, even with small and very smooth initial data, "blow up" after a finite time. For various types of quasi-linear equations, this time de...
Mathematical aspects of surface water waves
Craig, Walter; Wayne, Clarence E
2007-01-01
The theory of the motion of a free surface over a body of water is a fascinating subject, with a long history in both applied and pure mathematical research, and with a continuing relevance to the enterprises of mankind having to do with the sea. Despite the recent advances in the field (some of which we will hear about during this Workshop on Mathematical Hydrodynamics at the Steklov Institute), and the current focus of the mathematical community on the topic, many fundamental mathematical questions remain. These have to do with the evolution of surface water waves, their approximation by model equations and by computer simulations, the detailed dynamics of wave interactions, such as would produce rogue waves in an open ocean, and the theory (partially probabilistic) of approximating wave fields over large regions by averaged 'macroscopic' quantities which satisfy essentially kinetic equations of motion. In this note we would like to point out open problems and some of the directions of current research in the field. We believe that the introduction of new analytical techniques and novel points of view will play an important role in the future development of the area.
Analiticity in fourth-order wave equations
Bollini, C.G.; Giambiagi, J.J.
1987-01-01
In this paper it is presented, through a familiar example (δ-function potential in one dimension), the analytic properties of Jost functions associated with fourth-order equations. It is shown how to construct the Jost functions and the two discontinuity matrices associated with the line of singularities. The latter divide the complex k-plane in eight regions of analiticity. One of these matrices is related to the asymptotic behaviour of the scattering state. The other is not. Both are necessary to solve the inverse problem. Besides the usual poles related to bound states there are also other poles associated with total reflexion
Analiticity in fourth order wave equations
Bollini, C.G.
1987-01-01
Through a familiar example (δ-function potential in one dimension) the analytic properties of Jost functions associated with fourth order equations are presented. It is shown how to construct the Jost functions and the two discontinuities matrices associated to the line of singularities. The latter divide the complex k-plane in eight regions of analiticity. One of these matrices is related to the asymptotic behaviour of scattering state. The other is not. Both being necessary to solve the inverse problem. Besides the usual poles related to bound states there are also other poles associated with total reflexion. (Author) [pt
Finite element and discontinuous Galerkin methods for transient wave equations
Cohen, Gary
2017-01-01
This monograph presents numerical methods for solving transient wave equations (i.e. in time domain). More precisely, it provides an overview of continuous and discontinuous finite element methods for these equations, including their implementation in physical models, an extensive description of 2D and 3D elements with different shapes, such as prisms or pyramids, an analysis of the accuracy of the methods and the study of the Maxwell’s system and the important problem of its spurious free approximations. After recalling the classical models, i.e. acoustics, linear elastodynamics and electromagnetism and their variational formulations, the authors present a wide variety of finite elements of different shapes useful for the numerical resolution of wave equations. Then, they focus on the construction of efficient continuous and discontinuous Galerkin methods and study their accuracy by plane wave techniques and a priori error estimates. A chapter is devoted to the Maxwell’s system and the important problem ...
Asymptotic solutions and spectral theory of linear wave equations
Adam, J.A.
1982-01-01
This review contains two closely related strands. Firstly the asymptotic solution of systems of linear partial differential equations is discussed, with particular reference to Lighthill's method for obtaining the asymptotic functional form of the solution of a scalar wave equation with constant coefficients. Many of the applications of this technique are highlighted. Secondly, the methods and applications of the theory of the reduced (one-dimensional) wave equation - particularly spectral theory - are discussed. While the breadth of application and power of the techniques is emphasised throughout, the opportunity is taken to present to a wider readership, developments of the methods which have occured in some aspects of astrophysical (particularly solar) and geophysical fluid dynamics. It is believed that the topics contained herein may be of relevance to the applied mathematician or theoretical physicist interest in problems of linear wave propagation in these areas. (orig./HSI)
N-body bound state relativistic wave equations
Sazdjian, H.
1988-06-01
The manifestly covariant formalism with constraints is used for the construction of relativistic wave equations to describe the dynamics of N interacting spin 0 and/or spin 1/2 particles. The total and relative time evolutions of the system are completely determined by means of kinematic type wave equations. The internal dynamics of the system is 3 N-1 dimensional, besides the contribution of the spin degrees of freedom. It is governed by a single dynamical wave equation, that determines the eigenvalue of the total mass squared of the system. The interaction is introduced in a closed form by means of two-body potentials. The system satisfies an approximate form of separability
Nonlinear electrostatic wave equations for magnetized plasmas - II
Dysthe, K. B.; Mjølhus, E.; Pécseli, H. L.
1985-01-01
For pt.I see ibid., vol.26, p.443-7 (1984). The problem of extending the high frequency part of the Zakharov equations for nonlinear electrostatic waves to magnetized plasmas, is considered. Weak electromagnetic and thermal effects are retained on an equal footing. Direction dependent (electrosta......For pt.I see ibid., vol.26, p.443-7 (1984). The problem of extending the high frequency part of the Zakharov equations for nonlinear electrostatic waves to magnetized plasmas, is considered. Weak electromagnetic and thermal effects are retained on an equal footing. Direction dependent...... (electrostatic) cut-off implies that various cases must be considered separately, leading to equations with rather different properties. Various equations encountered previously in the literature are recovered as limiting cases....
An Unconditionally Stable Method for Solving the Acoustic Wave Equation
Zhi-Kai Fu
2015-01-01
Full Text Available An unconditionally stable method for solving the time-domain acoustic wave equation using Associated Hermit orthogonal functions is proposed. The second-order time derivatives in acoustic wave equation are expanded by these orthogonal basis functions. By applying Galerkin temporal testing procedure, the time variable can be eliminated from the calculations. The restriction of Courant-Friedrichs-Levy (CFL condition in selecting time step for analyzing thin layer can be avoided. Numerical results show the accuracy and the efficiency of the proposed method.
Solitary Wave Solutions of the Boussinesq Equation and Its Improved Form
Reza Abazari
2013-01-01
Full Text Available This paper presents the general case study of previous works on generalized Boussinesq equations, (Abazari, 2011 and (Kılıcman and Abazari, 2012, that focuses on the application of G′/G-expansion method with the aid of Maple to construct more general exact solutions for the coupled Boussinesq equations. In this work, the mentioned method is applied to construct more general exact solutions of Boussinesq equation and improved Boussinesq equation, which the French scientist Joseph Valentin Boussinesq (1842–1929 described in the 1870s model equations for the propagation of long waves on the surface of water with small amplitude. Our work is motivated by the fact that the G′/G-expansion method provides not only more general forms of solutions but also periodic, solitary waves and rational solutions. The method appears to be easier and faster by means of a symbolic computation.
Periodic solutions for one dimensional wave equation with bounded nonlinearity
Ji, Shuguan
2018-05-01
This paper is concerned with the periodic solutions for the one dimensional nonlinear wave equation with either constant or variable coefficients. The constant coefficient model corresponds to the classical wave equation, while the variable coefficient model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. For finding the periodic solutions of variable coefficient wave equation, it is usually required that the coefficient u (x) satisfies ess infηu (x) > 0 with ηu (x) = 1/2 u″/u - 1/4 (u‧/u)2, which actually excludes the classical constant coefficient model. For the case ηu (x) = 0, it is indicated to remain an open problem by Barbu and Pavel (1997) [6]. In this work, for the periods having the form T = 2p-1/q (p , q are positive integers) and some types of boundary value conditions, we find some fundamental properties for the wave operator with either constant or variable coefficients. Based on these properties, we obtain the existence of periodic solutions when the nonlinearity is monotone and bounded. Such nonlinearity may cross multiple eigenvalues of the corresponding wave operator. In particular, we do not require the condition ess infηu (x) > 0.
An acoustic wave equation for pure P wave in 2D TTI media
Zhan, Ge; Pestana, Reynam C.; Stoffa, Paul L.
2011-01-01
In this paper, a pure P wave equation for an acoustic 2D TTI media is derived. Compared with conventional TTI coupled equations, the resulting equation is unconditionally stable due to the complete isolation of the SV wave mode. To avoid numerical dispersion and produce high quality images, the rapid expansion method REM is employed for numerical implementation. Synthetic results validate the proposed equation and show that it is a stable algorithm for modeling and reverse time migration RTM in a TTI media for any anisotropic parameter values. © 2011 Society of Exploration Geophysicists.
Resolution limits for wave equation imaging
Huang, Yunsong
2014-08-01
Formulas are derived for the resolution limits of migration-data kernels associated with diving waves, primary reflections, diffractions, and multiple reflections. They are applicable to images formed by reverse time migration (RTM), least squares migration (LSM), and full waveform inversion (FWI), and suggest a multiscale approach to iterative FWI based on multiscale physics. That is, at the early stages of the inversion, events that only generate low-wavenumber resolution should be emphasized relative to the high-wavenumber resolution events. As the iterations proceed, the higher-resolution events should be emphasized. The formulas also suggest that inverting multiples can provide some low- and intermediate-wavenumber components of the velocity model not available in the primaries. Finally, diffractions can provide twice or better the resolution than specular reflections for comparable depths of the reflector and diffractor. The width of the diffraction-transmission wavepath is approximately λ at the diffractor location for the diffraction-transmission wavepath. © 2014 Elsevier B.V.
Ginzburg-Landau equation as a heuristic model for generating rogue waves
Lechuga, Antonio
2016-04-01
Envelope equations have many applications in the study of physical systems. Particularly interesting is the case 0f surface water waves. In steady conditions, laboratory experiments are carried out for multiple purposes either for researches or for practical problems. In both cases envelope equations are useful for understanding qualitative and quantitative results. The Ginzburg-Landau equation provides an excellent model for systems of that kind with remarkable patterns. Taking into account the above paragraph the main aim of our work is to generate waves in a water tank with almost a symmetric spectrum according to Akhmediev (2011) and thus, to produce a succession of rogue waves. The envelope of these waves gives us some patterns whose model is a type of Ginzburg-Landau equation, Danilov et al (1988). From a heuristic point of view the link between the experiment and the model is achieved. Further, the next step consists of changing generating parameters on the water tank and also the coefficients of the Ginzburg-Landau equation, Lechuga (2013) in order to reach a sufficient good approach.
Simulation of Irregular Waves and Wave Induced Loads on Wind Power Plants in Shallow Water
Trumars, Jenny [Chalmers Univ. of Technology, Goeteborg (Sweden). Dept. of Water Environment Transport
2004-05-01
The essay gives a short introduction to waves and discusses the problem with non-linear waves in shallow water and how they effect an offshore wind energy converter. The focus is on the realisation of non-linear waves in the time domain from short-term statistics in the form of a variance density spectrum of the wave elevation. For this purpose the wave transformation from deep water to the near to shore site of a wind energy farm at Bockstigen has been calculated with the use of SWAN (Simulating Waves Near Shore). The result is a wave spectrum, which can be used as input to the realisation. The realisation of waves is done by perturbation theory to the first and second-order. The properties calculated are the wave elevation, water particle velocity and acceleration. The wave heights from the second order perturbation equations are higher than those from the first order perturbation equations. This is also the case for the water particle kinematics. The increase of variance is significant between the first order and the second order realisation. The calculated wave elevation exhibits non-linear features as the peaks become sharper and the troughs flatter. The resulting forces are calculated using Morison's equation. For second order force and base moment there is an increase in the maximum values. The force and base moment are largest approximately at the zero up and down crossing of the wave elevation. This indicates an inertia dominated wave load. So far the flexibility and the response of the structure have not been taken into account. They are, however, of vital importance. For verification of the wave model the results will later on be compared with measurements at Bockstigen off the coast of Gotland in the Baltic Sea.
Nonlinear evolution equations for waves in random media
Pelinovsky, E.; Talipova, T.
1994-01-01
The scope of this paper is to highlight the main ideas of asymptotical methods applying in modern approaches of description of nonlinear wave propagation in random media. We start with the discussion of the classical conception of ''mean field''. Then an exactly solvable model describing nonlinear wave propagation in the medium with fluctuating parameters is considered in order to demonstrate that the ''mean field'' method is not correct. We develop new asymptotic procedures of obtaining the nonlinear evolution equations for the wave fields in random media. (author). 16 refs
Statistical approach to LHCD modeling using the wave kinetic equation
Kupfer, K.; Moreau, D.; Litaudon, X.
1993-04-01
Recent work has shown that for parameter regimes typical of many present day current drive experiments, the orbits of the launched LH rays are chaotic (in the Hamiltonian sense), so that wave energy diffuses through the stochastic layer and fills the spectral gap. We have analyzed this problem using a statistical approach, by solving the wave kinetic equation for the coarse-grained spectral energy density. An interesting result is that the LH absorption profile is essentially independent of both the total injected power and the level of wave stochastic diffusion
Study of nonlinear waves described by the cubic Schroedinger equation
Walstead, A.E.
1980-01-01
The cubic Schroedinger equation (CSE) is ubiquitous as a model equation for the long-time evolution of finite-amplitude near-monochromatic dispersive waves. It incorporates the effects of the radiation field pressure on the constitutive properties of the supporting medium in a self-consistent manner. The properties of the uniformly transiating periodic wave solutions of the one-dimensional CSE are studied here. These (so-called cnoidal) waves are characterized by the values of four parameters. Whitham's averaged variational principle is used to derive a system of quasilinear evolution equations (the modulational equations) for the values of these parameters when they are slowly varying in space and time. Explicit expressions for the characteristic velocities of the modulational equations are obtained for the full set of cnoidal waves. Riemann invariants are obtained for several limits for the stable case, and growth rates are obtained for several limits, including the solitary wave chain, for the unstable case. The results for several nontrivial limiting cases agree with those obtained by independent methods by others. The dynamics of the CSE generalized to two spatial dimensions are studied for the unstable case. A large class of similarity solutions with cylindrical symmetry are obtained systematically using infinitesimal transformation group techniques. The methods are adapted to obtain the symmetries of the action functional of the CSE and to deduce nine integral invariants. A numerical study of the self-similar solutions reveals that they are modulationally unstable and that singularities dominate the dynamics of the CSE in two dimensions. The CSE is derived using perturbation theory for a specific problem in plasma physics: the evolution of the envelope of a near-monochromatic electromagnetic wave in a cold magnetized plasma. 13 figures, 2 tables
Study of nonlinear waves described by the cubic Schroedinger equation
Walstead, A.E.
1980-03-12
The cubic Schroedinger equation (CSE) is ubiquitous as a model equation for the long-time evolution of finite-amplitude near-monochromatic dispersive waves. It incorporates the effects of the radiation field pressure on the constitutive properties of the supporting medium in a self-consistent manner. The properties of the uniformly transiating periodic wave solutions of the one-dimensional CSE are studied here. These (so-called cnoidal) waves are characterized by the values of four parameters. Whitham's averaged variational principle is used to derive a system of quasilinear evolution equations (the modulational equations) for the values of these parameters when they are slowly varying in space and time. Explicit expressions for the characteristic velocities of the modulational equations are obtained for the full set of cnoidal waves. Riemann invariants are obtained for several limits for the stable case, and growth rates are obtained for several limits, including the solitary wave chain, for the unstable case. The results for several nontrivial limiting cases agree with those obtained by independent methods by others. The dynamics of the CSE generalized to two spatial dimensions are studied for the unstable case. A large class of similarity solutions with cylindrical symmetry are obtained systematically using infinitesimal transformation group techniques. The methods are adapted to obtain the symmetries of the action functional of the CSE and to deduce nine integral invariants. A numerical study of the self-similar solutions reveals that they are modulationally unstable and that singularities dominate the dynamics of the CSE in two dimensions. The CSE is derived using perturbation theory for a specific problem in plasma physics: the evolution of the envelope of a near-monochromatic electromagnetic wave in a cold magnetized plasma. 13 figures, 2 tables.
The wave equation: From eikonal to anti-eikonal approximation
Luis Vázquez
2016-06-01
Full Text Available When the refractive index changes very slowly compared to the wave-length we may use the eikonal approximation to the wave equation. In the opposite case, when the refractive index highly variates over the distance of one wave-length, we have what can be termed as the anti-eikonal limit. This situation is addressed in this work. The anti-eikonal limit seems to be a relevant tool in the modelling and design of new optical media. Besides, it describes a basic universal behaviour, independent of the actual values of the refractive index and, thus, of the media, for the components of a wave with wave-length much greater than the characteristic scale of the refractive index.
Wave-equation Migration Velocity Analysis Using Plane-wave Common Image Gathers
Guo, Bowen; Schuster, Gerard T.
2017-01-01
Wave-equation migration velocity analysis (WEMVA) based on subsurface-offset, angle domain or time-lag common image gathers (CIGs) requires significant computational and memory resources because it computes higher dimensional migration images
Weiguo Rui
2014-01-01
Full Text Available By using the integral bifurcation method together with factoring technique, we study a water wave model, a high-order nonlinear wave equation of KdV type under some newly solvable conditions. Based on our previous research works, some exact traveling wave solutions such as broken-soliton solutions, periodic wave solutions of blow-up type, smooth solitary wave solutions, and nonsmooth peakon solutions within more extensive parameter ranges are obtained. In particular, a series of smooth solitary wave solutions and nonsmooth peakon solutions are obtained. In order to show the properties of these exact solutions visually, we plot the graphs of some representative traveling wave solutions.
Plane waves and spherical means applied to partial differential equations
John, Fritz
2004-01-01
Elementary and self-contained, this heterogeneous collection of results on partial differential equations employs certain elementary identities for plane and spherical integrals of an arbitrary function, showing how a variety of results on fairly general differential equations follow from those identities. The first chapter deals with the decomposition of arbitrary functions into functions of the type of plane waves. Succeeding chapters introduce the first application of the Radon transformation and examine the solution of the initial value problem for homogeneous hyperbolic equations with con
New solutions of the generalized ellipsoidal wave equation
Harold Exton
1999-10-01
Full Text Available Certain aspects and a contribution to the theory of new forms of solutions of an algebraic form of the generalized ellipsoidal wave equation are deduced by considering the Laplace transform of a soluble system of linear differential equations. An ensuing system of non-linear algebraic equations is shown to be consistent and is numerically implemented by means of the computer algebra package MAPLE V. The main results are presented as series of hypergeometric type of there and four variables which readily lend themselves to numerical handling although this does not indicate all of the detailedanalytic properties of the solutions under consideration.
Solution of wave-like equation based on Haar wavelet
Naresh Berwal
2012-11-01
Full Text Available Wavelet transform and wavelet analysis are powerful mathematical tools for many problems. Wavelet also can be applied in numerical analysis. In this paper, we apply Haar wavelet method to solve wave-like equation with initial and boundary conditions known. The fundamental idea of Haar wavelet method is to convert the differential equations into a group of algebraic equations, which involves a finite number or variables. The results and graph show that the proposed way is quite reasonable when compared to exact solution.
Lecture Notes for the Course in Water Wave Mechanics
Andersen, Thomas Lykke; Frigaard, Peter; Burcharth, Hans F.
knowledge. The course is at the same time an introduction to the course in coastal hydraulics on the 8th semester. The notes cover the first four lectures of the course: • Definitions. Governing equations and boundary conditions. • Derivation of velocity potential for linear waves. Dispersion relationship...... Particle velocities and accelerations. • Particle paths, pressure variation, deep and shallow water waves, wave energy and group velocity. • Shoaling, refraction, diffraction and wave breaking. The last part of the course is on analysis of irregular waves and was included in the first two editions...
The Appell transformation for the paraxial wave equation
Torre, A
2011-01-01
Some issues related to the 1D heat equation are revisited and framed within the context of the free-space paraxial propagation, formally accounted for by the 2D paraxial wave equation. In particular, the Appell transformation, which is well known in the theory of the heat equation, is reformulated in optical terms, and accordingly interpreted in the light of the propagation of given source functions, which are in a definite relation with the source functions of the original wavefunctions. Basic to the discussion is the Lie-algebra-based approach, as developed in a series of seminal papers by Kalnins, Miller and Boyer, to evolutionary-type equations, ruled by Hamiltonian operators underlying a harmonic oscillator-like symmetry algebra. Indeed, both the heat equation and the paraxial wave equation are particular cases of this kind of equation. When interpreting such an approach in terms of the propagation of assigned 'source' functions, the transformations between wavefunctions may be traced back to definite relations between the respective source functions. Thus, the optical Appell transformation is seen to be a manifestation of the correspondence between wavefunctions generated by eigenstates of operators, which are linked through a Fourier-similarity transformation. As a mere consequence, one can introduce the fractional Appell transformation, thus displaying a family of symmetry transformations parameterized by a continuous parameter
Real time wave measurements and wave hindcasting in deep waters
Anand, N.M.; Mandal, S.; SanilKumar, V.; Nayak, B.U.
Deep water waves off Karwar (lat. 14~'45.1'N, long. 73~'34.8'E) at 75 m water depth pertaining to peak monsoon period have been measured using a Datawell waverider buoy. Measured wave data show that the significant wave height (Hs) predominantly...
Semilinear damped wave equation in locally uniform spaces
Michálek, Martin; Pražák, D.; Slavík, J.
2017-01-01
Roč. 16, č. 5 (2017), s. 1673-1695 ISSN 1534-0392 EU Projects: European Commission(XE) 320078 - MATHEF Institutional support: RVO:67985840 Keywords : damped wave equations * nonlinear damping * unbounded domains Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 0.801, year: 2016 http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=14110
''Localized'' tachyonic wavelet-solutions of the wave equation
Barut, A.O.; Chandola, H.C.
1993-05-01
Localized-nonspreading, wavelet-solutions of the wave equation □φ=0 with group velocity v>c and phase velocity u=c 2 /v< c are constructed explicitly by two different methods. Some recent experiments seem to find evidence for superluminal group velocities. (author). 7 refs, 2 figs
Invariant Solutions for a Class of Perturbed Nonlinear Wave Equations
Waheed A. Ahmed
2017-11-01
Full Text Available Approximate symmetries of a class of perturbed nonlinear wave equations are computed using two newly-developed methods. Invariant solutions associated with the approximate symmetries are constructed for both methods. Symmetries and solutions are compared through discussing the advantages and disadvantages of each method.
The scalar wave equation in a Schwarzschild spacetime
Stewart, J.M.; Schmidt, B.G.
1978-09-01
This paper studies the asymptotic behaviour of solutions of the zero rest mass scalar wave equation in the Schwarzschild spacetime in a neighbourhood of spatial infinity, which includes parts of future and past null infinity. The behaviour of such fields is essentially different from that which accurs in a flat spacetime. (orig.) [de
The wave equation on a curved space-time
Friedlander, F.G.
1975-01-01
It is stated that chapters on differential geometry, distribution theory, and characteristics and the propagation of discontinuities are preparatory. The main matter is in three chapters, entitled: fundamental solutions, representation theorems, and wave equations on n-dimensional space-times. These deal with general construction of fundamental solutions and their application to the Cauchy problem. (U.K.)
Exponential decay for solutions to semilinear damped wave equation
Gerbi, Stéphane
2011-10-01
This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Intro- ducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in [4].
An inhomogeneous wave equation and non-linear Diophantine approximation
Beresnevich, V.; Dodson, M. M.; Kristensen, S.
2008-01-01
A non-linear Diophantine condition involving perfect squares and arising from an inhomogeneous wave equation on the torus guarantees the existence of a smooth solution. The exceptional set associated with the failure of the Diophantine condition and hence of the existence of a smooth solution...
Rarita-Schwinger field and multicomponent wave equation
Kaloshin, A.E.; Lomov, V.P.
2011-01-01
We suggest a simple method to solve a wave equation for Rarita-Schwinger field without additional constraints. This method based on the use of off-shell projection operators allows one to diagonalize spin-1/2 sector of the field
Travelling wave solutions to the perturbed Π4 equation
Geicke, J.
1985-01-01
Exact travelling wave solutions to the Π 4 equation, perturbed by a dissipative force and a constant external field η, are presented. For |η| 3 -λ 2 and λ 2 -λ 1 where λ 1 2 3 are the real roots of λ 3 -λ+η=O. The class with |v/ 3 -λ 1 . The stability of the solutions is discussed. (author) [pt
The shallow water equations in Lagrangian coordinates
Mead, J.L.
2004-01-01
Recent advances in the collection of Lagrangian data from the ocean and results about the well-posedness of the primitive equations have led to a renewed interest in solving flow equations in Lagrangian coordinates. We do not take the view that solving in Lagrangian coordinates equates to solving on a moving grid that can become twisted or distorted. Rather, the grid in Lagrangian coordinates represents the initial position of particles, and it does not change with time. We apply numerical methods traditionally used to solve differential equations in Eulerian coordinates, to solve the shallow water equations in Lagrangian coordinates. The difficulty with solving in Lagrangian coordinates is that the transformation from Eulerian coordinates results in solving a highly nonlinear partial differential equation. The non-linearity is mainly due to the Jacobian of the coordinate transformation, which is a precise record of how the particles are rotated and stretched. The inverse Jacobian must be calculated, thus Lagrangian coordinates cannot be used in instances where the Jacobian vanishes. For linear (spatial) flows we give an explicit formula for the Jacobian and describe the two situations where the Lagrangian shallow water equations cannot be used because either the Jacobian vanishes or the shallow water assumption is violated. We also prove that linear (in space) steady state solutions of the Lagrangian shallow water equations have Jacobian equal to one. In the situations where the shallow water equations can be solved in Lagrangian coordinates, accurate numerical solutions are found with finite differences, the Chebyshev pseudospectral method, and the fourth order Runge-Kutta method. The numerical results shown here emphasize the need for high order temporal approximations for long time integrations
Closed form solutions of two time fractional nonlinear wave equations
Akbar, M. Ali; Ali, Norhashidah Hj. Mohd.; Roy, Ripan
2018-06-01
In this article, we investigate the exact traveling wave solutions of two nonlinear time fractional wave equations. The fractional derivatives are described in the sense of conformable fractional derivatives. In addition, the traveling wave solutions are accomplished in the form of hyperbolic, trigonometric, and rational functions involving free parameters. To investigate such types of solutions, we implement the new generalized (G‧ / G) -expansion method. The extracted solutions are reliable, useful and suitable to comprehend the optimal control problems, chaotic vibrations, global and local bifurcations and resonances, furthermore, fission and fusion phenomena occur in solitons, the relativistic energy-momentum relation, scalar electrodynamics, quantum relativistic one-particle theory, electromagnetic interactions etc. The results reveal that the method is very fruitful and convenient for exploring nonlinear differential equations of fractional order treated in theoretical physics.
Pierantozzi, T.; Vazquez, L.
2005-01-01
Through fractional calculus and following the method used by Dirac to obtain his well-known equation from the Klein-Gordon equation, we analyze a possible interpolation between the Dirac and the diffusion equations in one space dimension. We study the transition between the hyperbolic and parabolic behaviors by means of the generalization of the D'Alembert formula for the classical wave equation and the invariance under space and time inversions of the interpolating fractional evolution equations Dirac like. Such invariance depends on the values of the fractional index and is related to the nonlocal property of the time fractional differential operator. For this system of fractional evolution equations, we also find an associated conserved quantity analogous to the Hamiltonian for the classical Dirac case
Parsimonious wave-equation travel-time inversion for refraction waves
Fu, Lei; Hanafy, Sherif M.; Schuster, Gerard T.
2017-01-01
We present a parsimonious wave-equation travel-time inversion technique for refraction waves. A dense virtual refraction dataset can be generated from just two reciprocal shot gathers for the sources at the endpoints of the survey line, with N
Knezevic, David; Patera, Anthony T.; Huynh, Dinh Bao Phuong
2010-01-01
We present a certified reduced basis (RB) method for the heat equation and wave equation. The critical ingredients are certified RB approximation of the Laplace transform; the inverse Laplace transform to develop the time-domain RB output approximation and rigorous error bound; a (Butterworth) filter in time to effect the necessary “modal” truncation; RB eigenfunction decomposition and contour integration for Offline–Online decomposition. We present numerical results to demonstrate the accura...
Turbulent wind waves on a water current
M. V. Zavolgensky
2008-05-01
Full Text Available An analytical model of water waves generated by the wind over the water surface is presented. A simple modeling method of wind waves is described based on waves lengths diagram, azimuthal hodograph of waves velocities and others. Properties of the generated waves are described. The wave length and wave velocity are obtained as functions on azimuth of wave propagation and growth rate. Motionless waves dynamically trapped into the general picture of three dimensional waves are described. The gravitation force does not enter the three dimensional of turbulent wind waves. That is why these waves have turbulent and not gravitational nature. The Langmuir stripes are naturally modeled and existence of the rogue waves is theoretically proved.
THE FUNDAMENTAL SOLUTIONS FOR MULTI-TERM MODIFIED POWER LAW WAVE EQUATIONS IN A FINITE DOMAIN
Jiang, H.; Liu, F.; Meerschaert, M. M.; McGough, R. J.
2013-01-01
Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development.
Inverse Schroedinger equation and the exact wave function
Nakatsuji, Hiroshi
2002-01-01
Using the inverse of the Hamiltonian, we introduce the inverse Schroedinger equation (ISE) that is equivalent to the ordinary Schroedinger equation (SE). The ISE has the variational principle and the H-square group of equations as the SE has. When we use a positive Hamiltonian, shifting the energy origin, the inverse energy becomes monotonic and we further have the inverse Ritz variational principle and cross-H-square equations. The concepts of the SE and the ISE are combined to generalize the theory for calculating the exact wave function that is a common eigenfunction of the SE and ISE. The Krylov sequence is extended to include the inverse Hamiltonian, and the complete Krylov sequence is introduced. The iterative configuration interaction (ICI) theory is generalized to cover both the SE and ISE concepts and four different computational methods of calculating the exact wave function are presented in both analytical and matrix representations. The exact wave-function theory based on the inverse Hamiltonian can be applied to systems that have singularities in the Hamiltonian. The generalized ICI theory is applied to the hydrogen atom, giving the exact solution without any singularity problem
Scattering of quantized solitary waves in the cubic Schrodinger equation
Dolan, L.
1976-01-01
The quantum mechanics for N particles interacting via a delta-function potential in one space dimension and one time dimension is known. The second-quantized description of this system has for its Euler-Lagrange equations of motion the cubic Schrodinger equation. This nonlinear differential equation supports solitary wave solutions. A quantization of these solitons reproduces the weak-coupling limit to the known quantum mechanics. The phase shift for two-body scattering and the energy of the N-body bound state is derived in this approximation. The nonlinear Schrodinger equation is contrasted with the sine-Gordon theory in respect to the ideas which the classical solutions play in the description of the quantum states
Reduced-order prediction of rogue waves in two-dimensional deep-water waves
Sapsis, Themistoklis; Farazmand, Mohammad
2017-11-01
We consider the problem of large wave prediction in two-dimensional water waves. Such waves form due to the synergistic effect of dispersive mixing of smaller wave groups and the action of localized nonlinear wave interactions that leads to focusing. Instead of a direct simulation approach, we rely on the decomposition of the wave field into a discrete set of localized wave groups with optimal length scales and amplitudes. Due to the short-term character of the prediction, these wave groups do not interact and therefore their dynamics can be characterized individually. Using direct numerical simulations of the governing envelope equations we precompute the expected maximum elevation for each of those wave groups. The combination of the wave field decomposition algorithm, which provides information about the statistics of the system, and the precomputed map for the expected wave group elevation, which encodes dynamical information, allows (i) for understanding of how the probability of occurrence of rogue waves changes as the spectrum parameters vary, (ii) the computation of a critical length scale characterizing wave groups with high probability of evolving to rogue waves, and (iii) the formulation of a robust and parsimonious reduced-order prediction scheme for large waves. T.S. has been supported through the ONR Grants N00014-14-1-0520 and N00014-15-1-2381 and the AFOSR Grant FA9550-16-1-0231. M.F. has been supported through the second Grant.
Quaternion wave equations in curved space-time
Edmonds, J. D., Jr.
1974-01-01
The quaternion formulation of relativistic quantum theory is extended to include curvilinear coordinates and curved space-time in order to provide a framework for a unified quantum/gravity theory. Six basic quaternion fields are identified in curved space-time, the four-vector basis quaternions are identified, and the necessary covariant derivatives are obtained. Invariant field equations are derived, and a general invertable coordinate transformation is developed. The results yield a way of writing quaternion wave equations in curvilinear coordinates and curved space-time as well as a natural framework for solving the problem of second quantization for gravity.
Rogue waves in a water tank: Experiments and modeling
Lechuga, Antonio
2013-04-01
Recently many rogue waves have been reported as the main cause of ship incidents on the sea. One of the main characteristics of rogue waves is its elusiveness: they present unexpectedly and disappear in the same wave. Some authors (Zakharov and al.2010) are attempting to find the probability of their appearances apart from studyingthe mechanism of the formation. As an effort on this topic we tried the generation of rogue waves in a water wave tank using a symmetric spectrum(Akhmediev et al. 2011) as input on the wave maker. The produced waves were clearly rogue waves with a rate (maximum wave height/ Significant wave height) of 2.33 and a kurtosis of 4.77 (Janssen 2003, Onorato 2006). These results were already presented (Lechuga 2012). Similar waves (in pattern aspect, but without being extreme waves) were described as crossing waves in a water tank(Shemer and Lichter1988). To go on further the next step has been to apply a theoretical model to the envelope of these waves. After some considerations the best model has been an analogue of the Ginzburg-Landau equation. This apparently amazing result is easily explained: We know that the Ginzburg-Landau model is related to some regular structures on the surface of a liquid and also in plasmas, electric and magnetic fields and other media. Another important characteristic of the model is that their solutions are invariants with respectto the translation group. The main aim of this presentation is to extract conclusions of the model and the comparison with the measured waves in the water tank.The nonlinear structure of waves and their regularity make suitable the use of the Ginzburg-Landau model to the envelope of generated waves in the tank,so giving us a powerful tool to cope with the results of our experiment.
Dirac equation and optical wave propagation in one dimension
Gonzalez, Gabriel [Catedras CONACYT, Universidad Autonoma de San Luis Potosi (Mexico); Coordinacion para la Innovacion y la Aplicacion de la Ciencia y la Tecnologia, Universidad Autonoma de San Luis Potosi (Mexico)
2018-02-15
We show that the propagation of transverse electric (TE) polarized waves in one-dimensional inhomogeneous settings can be written in the form of the Dirac equation in one space dimension with a Lorentz scalar potential, and consequently perform photonic simulations of the Dirac equation in optical structures. In particular, we propose how the zero energy state of the Jackiw-Rebbi model can be generated in an optical set-up by controlling the refractive index landscape, where TE-polarized waves mimic the Dirac particles and the soliton field can be tuned by adjusting the refractive index. (copyright 2017 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Control Operator for the Two-Dimensional Energized Wave Equation
Sunday Augustus REJU
2006-07-01
Full Text Available This paper studies the analytical model for the construction of the two-dimensional Energized wave equation. The control operator is given in term of space and time t independent variables. The integral quadratic objective cost functional is subject to the constraint of two-dimensional Energized diffusion, Heat and a source. The operator that shall be obtained extends the Conjugate Gradient method (ECGM as developed by Hestenes et al (1952, [1]. The new operator enables the computation of the penalty cost, optimal controls and state trajectories of the two-dimensional energized wave equation when apply to the Conjugate Gradient methods in (Waziri & Reju, LEJPT & LJS, Issues 9, 2006, [2-4] to appear in this series.
Hidden regularity for a strongly nonlinear wave equation
Rivera, J.E.M.
1988-08-01
The nonlinear wave equation u''-Δu+f(u)=v in Q=Ωx]0,T[;u(0)=u 0 ,u'(0)=u 1 in Ω; u(x,t)=0 on Σ= Γx]0,T[ where f is a continuous function satisfying, lim |s| sup →+∞ f(s)/s>-∞, and Ω is a bounded domain of R n with smooth boundary Γ, is analysed. It is shown that there exist a solution for the presented nonlinear wave equation that satisfies the regularity condition: |∂u/∂ η|ε L 2 (Σ). Moreover, it is shown that there exist a constant C>0 such that, |∂u/∂ η|≤c{ E(0)+|v| 2 Q }. (author) [pt
Family of electrovac colliding wave solutions of Einstein's equations
Li, W.; Ernst, F.J.
1989-01-01
Beginning with any colliding wave solution of the vacuum Einstein equations, a corresponding electrified colliding wave solution can be generated through the use of a transformation due to Harrison [J. Math. Phys. 9, 1744 (1968)]. The method, long employed in the context of stationary axisymmetric fields, is equally applicable to colliding wave solutions. Here it is applied to a large family of vacuum metrics derived by applying a generalized Ehlers transformation to solutions published recently by Ernst, Garcia, and Hauser (EGH) [J. Math. Phys. 28, 2155, 2951 (1987); 29, 681 (1988)]. Those EGH solutions were themselves a generalization of solutions first derived by Ferrari, Ibanez, and Bruni [Phys. Rev. D 36, 1053 (1987)]. Among the electrovac solutions that are obtained is a charged version of the Nutku--Halil [Phys. Rev. Lett. 39, 1379 (1977)] metric that possesses an arbitrary complex charge parameter
Wave-equation Migration Velocity Analysis Using Plane-wave Common Image Gathers
Guo, Bowen
2017-06-01
Wave-equation migration velocity analysis (WEMVA) based on subsurface-offset, angle domain or time-lag common image gathers (CIGs) requires significant computational and memory resources because it computes higher dimensional migration images in the extended image domain. To mitigate this problem, a WEMVA method using plane-wave CIGs is presented. Plane-wave CIGs reduce the computational cost and memory storage because they are directly calculated from prestack plane-wave migration, and the number of plane waves is often much smaller than the number of shots. In the case of an inaccurate migration velocity, the moveout of plane-wave CIGs is automatically picked by a semblance analysis method, which is then linked to the migration velocity update by a connective function. Numerical tests on two synthetic datasets and a field dataset validate the efficiency and effectiveness of this method.
A delay differential equation model of follicle waves in women.
Panza, Nicole M; Wright, Andrew A; Selgrade, James F
2016-01-01
This article presents a mathematical model for hormonal regulation of the menstrual cycle which predicts the occurrence of follicle waves in normally cycling women. Several follicles of ovulatory size that develop sequentially during one menstrual cycle are referred to as follicle waves. The model consists of 13 nonlinear, delay differential equations with 51 parameters. Model simulations exhibit a unique stable periodic cycle and this menstrual cycle accurately approximates blood levels of ovarian and pituitary hormones found in the biological literature. Numerical experiments illustrate that the number of follicle waves corresponds to the number of rises in pituitary follicle stimulating hormone. Modifications of the model equations result in simulations which predict the possibility of two ovulations at different times during the same menstrual cycle and, hence, the occurrence of dizygotic twins via a phenomenon referred to as superfecundation. Sensitive parameters are identified and bifurcations in model behaviour with respect to parameter changes are discussed. Studying follicle waves may be helpful for improving female fertility and for understanding some aspects of female reproductive ageing.
Visco-acoustic wave-equation traveltime inversion and its sensitivity to attenuation errors
Yu, Han; Chen, Yuqing; Hanafy, Sherif M.; Huang, Jiangping
2018-01-01
A visco-acoustic wave-equation traveltime inversion method is presented that inverts for the shallow subsurface velocity distribution. Similar to the classical wave equation traveltime inversion, this method finds the velocity model that minimizes
Relativistic wave equations without the Velo-Zwanziger pathology
Khalil, M.A.K.
1976-06-01
For particles described by relativistic wave equations of the form: (-iGAMMA x delta + m) psi(x) = 0 interacting with an external field B(x) it is known that the ''noncausal'' propagation characteristics are not present when (1) GAMMA 0 is diagonalizable and (2) B(x) = -eGAMMA/sub mu/A/sup mu/(x) (Amar--Dozzio). The ''noncausality''difficulties arise for the Rarita--Schwinger spin 3 / 2 equation, with nondiagonalizable GAMMA 0 , in minimal coupling (i.e., B(x) = -eGAMMA x A(x)) and the PDK spin 1 equation, with diagonalizable GAMMA 0 , in a quadrupole coupling (Velo--Zwanziger) where either (1) or (2) of the Amar--Dozzio (sufficient) conditions are violated. Some sufficient conditions are derived and explored where the Velo--Zwanziger ''noncausality'' pathology can be avoided, even though one, or the other, or both of the conditions (1) and (2) are violated. Examples with both reducible and irreducible wave equations are included
On a class of nonlocal wave equations from applications
Beyer, Horst Reinhard; Aksoylu, Burak; Celiker, Fatih
2016-06-01
We study equations from the area of peridynamics, which is a nonlocal extension of elasticity. The governing equations form a system of nonlocal wave equations. We take a novel approach by applying operator theory methods in a systematic way. On the unbounded domain ℝn, we present three main results. As main result 1, we find that the governing operator is a bounded function of the governing operator of classical elasticity. As main result 2, a consequence of main result 1, we prove that the peridynamic solutions strongly converge to the classical solutions by utilizing, for the first time, strong resolvent convergence. In addition, main result 1 allows us to incorporate local boundary conditions, in particular, into peridynamics. This avenue of research is developed in companion papers, providing a remedy for boundary effects. As main result 3, employing spherical Bessel functions, we give a new practical series representation of the solution which allows straightforward numerical treatment with symbolic computation.
Assessing Tsunami Vulnerabilities of Geographies with Shallow Water Equations
Aras, Rifat; Shen, Yuzhong
2012-01-01
Tsunami preparedness is crucial for saving human lives in case of disasters that involve massive water movement. In this work, we develop a framework for visual assessment of tsunami preparedness of geographies. Shallow water equations (also called Saint Venant equations) are a set of hyperbolic partial differential equations that are derived by depth-integrating the Navier-Stokes equations and provide a great abstraction of water masses that have lower depths compared to their free surface area. Our specific contribution in this study is to use Microsoft's XNA Game Studio to import underwater and shore line geographies, create different tsunami scenarios, and visualize the propagation of the waves and their impact on the shore line geography. Most importantly, we utilized the computational power of graphical processing units (GPUs) as HLSL based shader files and delegated all of the heavy computations to the GPU. Finally, we also conducted a validation study, in which we have tested our model against a controlled shallow water experiment. We believe that such a framework with an easy to use interface that is based on readily available software libraries, which are widely available and easily distributable, would encourage not only researchers, but also educators to showcase ideas.
DG-FEM solution for nonlinear wave-structure interaction using Boussinesq-type equations
Engsig-Karup, Allan Peter; Hesthaven, Jan; Bingham, Harry B.
2008-01-01
equations in complex and curvilinear geometries which amends the application range of previous numerical models that have been based on structured Cartesian grids. The Boussinesq method provides the basis for the accurate description of fully nonlinear and dispersive water waves in both shallow and deep...... waters within the breaking limit. To demonstrate the current applicability of the model both linear and mildly nonlinear test cases are considered in two horizontal dimensions where the water waves interact with bottom-mounted fully reflecting structures. It is established that, by simple symmetry...... considerations combined with a mirror principle, it is possible to impose weak slip boundary conditions for both structured and general curvilinear wall boundaries while maintaining the accuracy of the scheme. As is standard for current high-order Boussinesq-type models, arbitrary waves can be generated...
Parsimonious wave-equation travel-time inversion for refraction waves
Fu, Lei
2017-02-14
We present a parsimonious wave-equation travel-time inversion technique for refraction waves. A dense virtual refraction dataset can be generated from just two reciprocal shot gathers for the sources at the endpoints of the survey line, with N geophones evenly deployed along the line. These two reciprocal shots contain approximately 2N refraction travel times, which can be spawned into O(N2) refraction travel times by an interferometric transformation. Then, these virtual refraction travel times are used with a source wavelet to create N virtual refraction shot gathers, which are the input data for wave-equation travel-time inversion. Numerical results show that the parsimonious wave-equation travel-time tomogram has about the same accuracy as the tomogram computed by standard wave-equation travel-time inversion. The most significant benefit is that a reciprocal survey is far less time consuming than the standard refraction survey where a source is excited at each geophone location.
Wang, T.
2017-05-26
Elastic full waveform inversion (EFWI) provides high-resolution parameter estimation of the subsurface but requires good initial guess of the true model. The traveltime inversion only minimizes traveltime misfits which are more sensitive and linearly related to the low-wavenumber model perturbation. Therefore, building initial P and S wave velocity models for EFWI by using elastic wave-equation reflections traveltime inversion (WERTI) would be effective and robust, especially for the deeper part. In order to distinguish the reflection travletimes of P or S-waves in elastic media, we decompose the surface multicomponent data into vector P- and S-wave seismogram. We utilize the dynamic image warping to extract the reflected P- or S-wave traveltimes. The P-wave velocity are first inverted using P-wave traveltime followed by the S-wave velocity inversion with S-wave traveltime, during which the wave mode decomposition is applied to the gradients calculation. Synthetic example on the Sigbee2A model proves the validity of our method for recovering the long wavelength components of the model.
Fu Jing-Li; He Yu-Fang; Hong Fang-Yu; Song Duan; Fu Hao
2013-01-01
In this paper, we present a new method to obtain the Lie symmetries and conserved quantities of the discrete wave equation with the Ablowitz—Ladik—Lattice equations. Firstly, the wave equation is transformed into a simple difference equation with the Ablowitz—Ladik—Lattice method. Secondly, according to the invariance of the discrete wave equation and the Ablowitz—Ladik—Lattice equations under infinitesimal transformation of dependent and independent variables, we derive the discrete determining equation and the discrete restricted equations. Thirdly, a series of the discrete analogs of conserved quantities, the discrete analogs of Lie groups, and the characteristic equations are obtained for the wave equation. Finally, we study a model of a biological macromolecule chain of mechanical behaviors, the Lie symmetry theory of discrete wave equation with the Ablowitz—Ladik—Lattice method is verified. (general)
Metamaterials, from electromagnetic waves to water waves, bending waves and beyond
Dupont, G.
2015-08-04
We will review our recent work on metamaterials for different types of waves. Transposition of transform optics to water waves and bending waves on plates will be considered with potential applications of cloaking to water waves protection and anti-vibrating systems.
Singular solitons and other solutions to a couple of nonlinear wave equations
Inc Mustafa; Ulutaş Esma; Biswas Anjan
2013-01-01
This paper addresses the extended (G'/G)-expansion method and applies it to a couple of nonlinear wave equations. These equations are modified the Benjamin—Bona—Mahoney equation and the Boussinesq equation. This extended method reveals several solutions to these equations. Additionally, the singular soliton solutions are revealed, for these two equations, with the aid of the ansatz method
Electromagnetic interactions in relativistic infinite component wave equations
Gerry, C.C.
1979-01-01
The electromagnetic interactions of a composite system described by relativistic infinite-component wave equations are considered. The noncompact group SO(4,2) is taken as the dynamical group of the systems, and its unitary irreducible representations, which are infinite dimensional, are used to find the energy spectra and to specify the states of the systems. First the interaction mechanism is examined in the nonrelativistic SO(4,2) formulation of the hydrogen atom as a heuristic guide. A way of making a minimal relativistic generalization of the minimal ineractions in the nonrelativistic equation for the hydrogen atom is proposed. In order to calculate the effects of the relativistic minimal interactions, a covariant perturbation theory suitable for infinite-component wave equations, which is an algebraic and relativistic version of the Rayleigh-Schroedinger perturbation theory, is developed. The electric and magnetic polarizabilities for the ground state of the hydrogen atom are calculated. The results have the correct nonrelativistic limits. Next, the relativistic cross section of photon absorption by the atom is evaluated. A relativistic expression for the cross section of light scattering corresponding to the seagull diagram is derived. The Born amplitude is combusted and the role of spacelike solutions is discussed. Finally, internal electromagnetic interactions that give rise to the fine structure splittings, the Lamb shifts and the hyperfine splittings are considered. The spin effects are introduced by extending the dynamical group
TRAVELING WAVE SOLUTIONS OF SOME FRACTIONAL DIFFERENTIAL EQUATIONS
SERIFE MUGE EGE
2016-07-01
Full Text Available The modified Kudryashov method is powerful, efficient and can be used as an alternative to establish new solutions of different type of fractional differential equations applied in mathematical physics. In this article, we’ve constructed new traveling wave solutions including symmetrical Fibonacci function solutions, hyperbolic function solutions and rational solutions of the space-time fractional Cahn Hillihard equation D_t^α u − γD_x^α u − 6u(D_x^α u^2 − (3u^2 − 1D_x^α (D_x^α u + D_x^α(D_x^α(D_x^α(D_x^α u = 0 and the space-time fractional symmetric regularized long wave (SRLW equation D_t^α(D_t^α u + D_x^α(D_x^α u + uD_t^α(D_x^α u + D_x^α u D_t^α u + D_t^α(D_t^α(D_x^α(D_x^α u = 0 via modified Kudryashov method. In addition, some of the solutions are described in the figures with the help of Mathematica.
A new iterative solver for the time-harmonic wave equation
Riyanti, C.D.; Erlangga, Y.A.; Plessix, R.E.; Mulder, W.A.; Vuik, C.; Oosterlee, C.
2006-01-01
The time-harmonic wave equation, also known as the Helmholtz equation, is obtained if the constant-density acoustic wave equation is transformed from the time domain to the frequency domain. Its discretization results in a large, sparse, linear system of equations. In two dimensions, this system can
Water waves generated by underwater explosion
Mehaute, Bernard Le
1996-01-01
This is the first book on explosion-generated water waves. It presents the theoretical foundations and experimental results of the generation and propagation of impulsively generated waves resulting from underwater explosions. Many of the theories and concepts presented herein are applicable to other types of water waves, in particular, tsunamis and waves generated by the fall of a meteorite. Linear and nonlinear theories, as well as experimental calibrations, are presented for cases of deep and shallow water explosions. Propagation of transient waves on dissipative, nonuniform bathymetries to
To the complete integrability of long-wave short-wave interaction equations
Roy Chowdhury, A.; Chanda, P.K.
1984-10-01
We show that the non-linear partial differential equations governing the interaction of long and short waves are completely integrable. The methodology we use is that of Ablowitz et al. though in the last section of our paper we have discussed the problem also in the light of the procedure due to Weiss et al. and have obtained a Baecklund transformation. (author)
Shock wave convergence in water with parabolic wall boundaries
Yanuka, D.; Shafer, D.; Krasik, Ya.
2015-01-01
The convergence of shock waves in water, where the cross section of the boundaries between which the shock wave propagates is either straight or parabolic, was studied. The shock wave was generated by underwater electrical explosions of planar Cu wire arrays using a high-current generator with a peak output current of ∼45 kA and rise time of ∼80 ns. The boundaries of the walls between which the shock wave propagates were symmetric along the z axis, which is defined by the direction of the exploding wires. It was shown that with walls having a parabolic cross section, the shock waves converge faster and the pressure in the vicinity of the line of convergence, calculated by two-dimensional hydrodynamic simulations coupled with the equations of state of water and copper, is also larger
Shang Yadong
2008-01-01
The extended hyperbolic functions method for nonlinear wave equations is presented. Based on this method, we obtain a multiple exact explicit solutions for the nonlinear evolution equations which describe the resonance interaction between the long wave and the short wave. The solutions obtained in this paper include (a) the solitary wave solutions of bell-type for S and L, (b) the solitary wave solutions of kink-type for S and bell-type for L, (c) the solitary wave solutions of a compound of the bell-type and the kink-type for S and L, (d) the singular travelling wave solutions, (e) periodic travelling wave solutions of triangle function types, and solitary wave solutions of rational function types. The variety of structure to the exact solutions of the long-short wave equation is illustrated. The methods presented here can also be used to obtain exact solutions of nonlinear wave equations in n dimensions
Explicit solution for a wave equation with nonlocal condition
Bazhlekova, Emilia; Dimovski, Ivan
2012-11-01
An initial-boundary value problem with a nonlocal boundary condition for one-dimensional wave equation is studied. Applying spectral projections, we find a series solution of the problem. The character of the solution found shows that the oscillation amplitude of the system described by this equation increases with time at any fixed x in absence of external forces. To find a representation of the solution more convenient for numerical calculation we develop a two-dimensional operational calculus for the problem. The solution is expressed as a sum of non-classical convolution products of particular solutions and the arbitrary initial functions. This result is an extension of the classical Duhamel principle for the space variable. The representation is used successfully for numerical computation and visualization of the solution. Numerical results obtained for specific test problems with known exact solutions indicate that the present technique provides accurate numerical solutions.
Gao, Yingjie; Zhang, Jinhai; Yao, Zhenxing
2015-12-01
The complex frequency shifted perfectly matched layer (CFS-PML) can improve the absorbing performance of PML for nearly grazing incident waves. However, traditional PML and CFS-PML are based on first-order wave equations; thus, they are not suitable for second-order wave equation. In this paper, an implementation of CFS-PML for second-order wave equation is presented using auxiliary differential equations. This method is free of both convolution calculations and third-order temporal derivatives. As an unsplit CFS-PML, it can reduce the nearly grazing incidence. Numerical experiments show that it has better absorption than typical PML implementations based on second-order wave equation.
The scalar wave equation in a Schwarzschild space-time
Schmidt, B.G.; Stewart, J.M.
1979-01-01
This paper studies the asymptotic behaviour of solutions of the zero rest mass scalar wave equation in the Schwarzschild space-time in a neighbourhood of spatial infinity which includes parts of future and pass null infinity. The behaviour of such fields is essentially different from that which occurs in a flat space-time. In particular fields which have a Bondi-type expansion in powers of 'r(-1)' near past null infinity do not have such an expansion near future null infinity. Further solutions which have physically reasonable Cauchy data probably fail to have Bondi-type expansions near null infinity. (author)
Controllability for a Wave Equation with Moving Boundary
Lizhi Cui
2014-01-01
Full Text Available We investigate the controllability for a one-dimensional wave equation in domains with moving boundary. This model characterizes small vibrations of a stretched elastic string when one of the two endpoints varies. When the speed of the moving endpoint is less than 1-1/e, by Hilbert uniqueness method, sidewise energy estimates method, and multiplier method, we get partial Dirichlet boundary controllability. Moreover, we will give a sharper estimate on controllability time that only depends on the speed of the moving endpoint.
Scully, M O
2008-01-01
The time dependent Schrodinger equation is frequently 'derived' by postulating the energy E → i h-bar (∂/∂t) and momentum p-vector → ( h-bar /i)∇ operator relations. In the present paper we review the quantum field theoretic route to the Schrodinger wave equation which treats time and space as parameters, not operators. Furthermore, we recall that a classical (nonlinear) wave equation can be derived from the classical action via Hamiltonian-Jacobi theory. By requiring the wave equation to be linear we again arrive at the Schrodinger equation, without postulating operator relations. The underlying philosophy is operational: namely 'a particle is what a particle detector detects.' This leads us to a useful physical picture combining the wave (field) and particle paradigms which points the way to the time-dependent Schrodinger equation
New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod
Seadawy, Aly R.; Manafian, Jalil
2018-03-01
This paper examines the effectiveness of an integration scheme which called the extended trial equation method (ETEM) in exactly solving a well-known nonlinear equation of partial differential equations (PDEs). In this respect, the longitudinal wave equation (LWE) that arises in mathematical physics with dispersion caused by the transverse Poisson's effect in a magneto-electro-elastic (MEE) circular rod, which a series of exact traveling wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of the longitudinal wave equation. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomena of this longitudinal wave equation. Many other such types of nonlinear equations arising in non-destructive evaluation of structures made of the advanced MEE material can also be solved by this method.
Carter, B.; McLenaghan, R.G.
1982-01-01
It is shown how previous general formulae for the separated radial and angular parts of the massive, charged scalar (Klein, Gordon) wave equation on one hand, and of the zero mass, neutral, but higher spin (neutrino, electromagnetic and gravitational) wave equations on the other hand may be combined in a more general formula which also covers the case of the full massive charged Dirac equation in a Kerr or Kerr-Newman background space. (Auth.)
Yomba, Emmanuel
2005-01-01
By using a modified extended Fan's sub-equation method, we have obtained new and more general solutions including a series of non-travelling wave and coefficient function solutions namely: soliton-like solutions, triangular-like solutions, single and combined non-degenerative Jacobi elliptic wave function-like solutions for the (2 + 1)-dimensional dispersive long wave equation. The most important achievement of this method lies on the fact that, we have succeeded in one move to give all the solutions which can be previously obtained by application of at least four methods (method using Riccati equation, or first kind elliptic equation, or auxiliary ordinary equation, or generalized Riccati equation as mapping equation)
Rotating magnetic shallow water waves and instabilities in a sphere
Márquez-Artavia, X.; Jones, C. A.; Tobias, S. M.
2017-07-01
Waves in a thin layer on a rotating sphere are studied. The effect of a toroidal magnetic field is considered, using the shallow water ideal MHD equations. The work is motivated by suggestions that there is a stably stratified layer below the Earth's core mantle boundary, and the existence of stable layers in stellar tachoclines. With an azimuthal background field known as the Malkus field, ?, ? being the co-latitude, a non-diffusive instability is found with azimuthal wavenumber ?. A necessary condition for instability is that the Alfvén speed exceeds ? where ? is the rotation rate and ? the sphere radius. Magneto-inertial gravity waves propagating westward and eastward occur, and become equatorially trapped when the field is strong. Magneto-Kelvin waves propagate eastward at low field strength, but a new westward propagating Kelvin wave is found when the field is strong. Fast magnetic Rossby waves travel westward, whilst the slow magnetic Rossby waves generally travel eastward, except for some ? modes at large field strength. An exceptional very slow westward ? magnetic Rossby wave mode occurs at all field strengths. The current-driven instability occurs for ? when the slow and fast magnetic Rossby waves interact. With strong field the magnetic Rossby waves become trapped at the pole. An asymptotic analysis giving the wave speed and wave form in terms of elementary functions is possible both in polar trapped and equatorially trapped cases.
Hydraulic jump and Bernoulli equation in nonlinear shallow water model
Sun, Wen-Yih
2018-06-01
A shallow water model was applied to study the hydraulic jump and Bernoulli equation across the jump. On a flat terrain, when a supercritical flow plunges into a subcritical flow, discontinuity develops on velocity and Bernoulli function across the jump. The shock generated by the obstacle may propagate downstream and upstream. The latter reflected from the inflow boundary, moves downstream and leaves the domain. Before the reflected wave reaching the obstacle, the short-term integration (i.e., quasi-steady) simulations agree with Houghton and Kasahara's results, which may have unphysical complex solutions. The quasi-steady flow is quickly disturbed by the reflected wave, finally, flow reaches steady and becomes critical without complex solutions. The results also indicate that Bernoulli function is discontinuous but the potential of mass flux remains constant across the jump. The latter can be used to predict velocity/height in a steady flow.
Wave-equation dispersion inversion of surface waves recorded on irregular topography
Li, Jing
2017-08-17
Significant topographic variations will strongly influence the amplitudes and phases of propagating surface waves. Such effects should be taken into account, otherwise the S-velocity model inverted from the Rayleigh dispersion curves will contain significant inaccuracies. We now show that the recently developed wave-equation dispersion inversion (WD) method naturally takes into account the effects of topography to give accurate S-velocity tomograms. Application of topographic WD to demonstrates that WD can accurately invert dispersion curves from seismic data recorded over variable topography. We also apply this method to field data recorded on the crest of mountainous terrain and find with higher resolution than the standard WD tomogram.
Wave-equation dispersion inversion of surface waves recorded on irregular topography
Li, Jing; Schuster, Gerard T.; Lin, Fan-Chi; Alam, Amir
2017-01-01
Significant topographic variations will strongly influence the amplitudes and phases of propagating surface waves. Such effects should be taken into account, otherwise the S-velocity model inverted from the Rayleigh dispersion curves will contain significant inaccuracies. We now show that the recently developed wave-equation dispersion inversion (WD) method naturally takes into account the effects of topography to give accurate S-velocity tomograms. Application of topographic WD to demonstrates that WD can accurately invert dispersion curves from seismic data recorded over variable topography. We also apply this method to field data recorded on the crest of mountainous terrain and find with higher resolution than the standard WD tomogram.
Stability of negative solitary waves for an integrable modified Camassa-Holm equation
Yin Jiuli; Tian Lixin; Fan Xinghua
2010-01-01
In this paper, we prove that the modified Camassa-Holm equation is Painleve integrable. We also study the orbital stability problem of negative solitary waves for this integrable equation. It is shown that the negative solitary waves are stable for arbitrary wave speed of propagation.
Destrade, Michel; Goriely, Alain; Saccomandi, Giuseppe
2011-01-01
We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent, and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy-Green strain). However, we show that the Z equation c...
Chuang, Wei-Liang; Chang, Kuang-An; Mercier, Richard
2018-06-01
Green water kinematics and dynamics due to wave impingements on a simplified geometry, fixed platform were experimentally investigated in a large, deep-water wave basin. Both plane focusing waves and random waves were employed in the generation of green water. The focusing wave condition was designed to create two consecutive plunging breaking waves with one impinging on the frontal vertical wall of the fixed platform, referred as wall impingement, and the other directly impinging on the deck surface, referred as deck impingement. The random wave condition was generated using the JONSWAP spectrum with a significant wave height approximately equal to the freeboard. A total of 179 green water events were collected in the random wave condition. By examining the green water events in random waves, three different flow types are categorized: collapse of overtopping wave, fall of bulk water, and breaking wave crest. The aerated flow velocity was measured using bubble image velocimetry, while the void fraction was measured using fiber optic reflectometry. For the plane focusing wave condition, measurements of impact pressure were synchronized with the flow velocity and void fraction measurements. The relationship between the peak pressures and the pressure rise times is examined. For the high-intensity impact in the deck impingement events, the peak pressures are observed to be proportional to the aeration levels. The maximum horizontal velocities in the green water events in random waves are well represented by the lognormal distribution. Ritter's solution is shown to quantitatively describe the green water velocity distributions under both the focusing wave condition and the random wave condition. A prediction equation for green water velocity distribution under random waves is proposed.
On an Acoustic Wave Equation Arising in Non-Equilibrium Gasdynamics. Classroom Notes
Chandran, Pallath
2004-01-01
The sixth-order wave equation governing the propagation of one-dimensional acoustic waves in a viscous, heat conducting gaseous medium subject to relaxation effects has been considered. It has been reduced to a system of lower order equations corresponding to the finite speeds occurring in the equation, following a method due to Whitham. The lower…
Collisional drift fluid equations and implications for drift waves
Pfirsch, Dieter; Correa-Restrepo, Dario
1996-01-01
The usual theoretical description of drift-wave turbulence (considered to be one possible cause of anomalous transport in a plasma), e.g. the Hasegawa-Wakatani theory, makes use of various approximations, the effects of which are extremely difficult to assess. This concerns in particular the conservation laws for energy and momentum. The latter law is important in relation to charge separation and the resulting electric fields, which are possibly related to the L-H transition. Energy conservation is crucial to the stability behaviour, it will be discussed by means of an example. New collisional multi-species drift-fluid equations were derived by a new method which yields, in a transparent way, conservation of energy and total angular momentum and the law for energy dissipation. Both electrostatic and electromagnetic field variations are considered. The only restriction involved is the validity of the drift approximation; in particular, there are no assumptions restricting the geometry of the system. The method is based primarily on a Lagrangian for dissipationless fluids in the drift approximation with isotropic pressures. The dissipative terms are introduced by adding corresponding terms to the ideal equations of motion and of the pressures. The equations of motion, of course, no longer result from a Lagrangian via Hamilton's principle. However, their relation to the ideal equations also implies a relation to the ideal Lagrangian, which can be used to advantage. Instead of introducing heat conduction one can also assume isothermal behaviour, e.g. T v (x) = constant. Assumptions of this kind are often made in the literature. The new method of introducing dissipation is not restricted to the present kind of theory; it can equally well be applied to theories such as multi-fluid theories without using the drift approximation of the present paper. (author)
Initial-value problem for the Gardner equation applied to nonlinear internal waves
Rouvinskaya, Ekaterina; Kurkina, Oxana; Kurkin, Andrey; Talipova, Tatiana; Pelinovsky, Efim
2017-04-01
The Gardner equation is a fundamental mathematical model for the description of weakly nonlinear weakly dispersive internal waves, when cubic nonlinearity cannot be neglected. Within this model coefficients of quadratic and cubic nonlinearity can both be positive as well as negative, depending on background conditions of the medium, where waves propagate (sea water density stratification, shear flow profile) [Rouvinskaya et al., 2014, Kurkina et al., 2011, 2015]. For the investigation of weakly dispersive behavior in the framework of nondimensional Gardner equation with fixed (positive) sign of quadratic nonlinearity and positive or negative cubic nonlinearity {eq1} partial η/partial t+6η( {1± η} )partial η/partial x+partial ^3η/partial x^3=0, } the series of numerical experiments of initial-value problem was carried out for evolution of a bell-shaped impulse of negative polarity (opposite to the sign of quadratic nonlinear coefficient): {eq2} η(x,t=0)=-asech2 ( {x/x0 } ), for which amplitude a and width x0 was varied. Similar initial-value problem was considered in the paper [Trillo et al., 2016] for the Korteweg - de Vries equation. For the Gardner equation with different signs of cubic nonlinearity the initial-value problem for piece-wise constant initial condition was considered in detail in [Grimshaw et al., 2002, 2010]. It is widely known, for example, [Pelinovsky et al., 2007], that the Gardner equation (1) with negative cubic nonlinearity has a family of classic solitary wave solutions with only positive polarity,and with limiting amplitude equal to 1. Therefore evolution of impulses (2) of negative polarity (whose amplitudes a were varied from 0.1 to 3, and widths at the level of a/2 were equal to triple width of solitons with the same amplitude for a 1) was going on a universal scenario with the generation of nonlinear Airy wave. For the Gardner equation (1) with the positive cubic nonlinearity coefficient there exist two one-parametric families of
Numerical study of the Kadomtsev-Petviashvili equation and dispersive shock waves
Grava, T.; Klein, C.; Pitton, G.
2018-02-01
A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev-Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrödinger equation in the semiclassical limit.
CFD Analysis of Water Solitary Wave Reflection
K. Smida
2011-12-01
Full Text Available A new numerical wave generation method is used to investigate the head-on collision of two solitary waves. The reflection at vertical wall of a solitary wave is also presented. The originality of this model, based on the Navier-Stokes equations, is the specification of an internal inlet velocity, defined as a source line within the computational domain for the generation of these non linear waves. This model was successfully implemented in the PHOENICS (Parabolic Hyperbolic Or Elliptic Numerical Integration Code Series code. The collision of two counter-propagating solitary waves is similar to the interaction of a soliton with a vertical wall. This wave generation method allows the saving of considerable time for this collision process since the counter-propagating wave is generated directly without reflection at vertical wall. For the collision of two solitary waves, numerical results show that the run-up phenomenon can be well explained, the solution of the maximum wave run-up is almost equal to experimental measurement. The simulated wave profiles during the collision are in good agreement with experimental results. For the reflection at vertical wall, the spatial profiles of the wave at fixed instants show that this problem is equivalent to the collision process.
Mostafa M.A. Khater
Full Text Available In this article and for the first time, we introduce and describe Khater method which is a new technique for solving nonlinear partial differential equations (PDEs.. We apply this method for each of the following models Bogoyavlenskii equation, couple Boiti-Leon-Pempinelli system and Time-fractional Cahn-Allen equation. Khater method is very powerful, Effective, felicitous and fabulous method to get exact and solitary wave solution of (PDEs.. Not only just like that but it considers too one of the general methods for solving that kind of equations since it involves some methods as we will see in our discuss of the results. We make a comparison between the results of this new method and another method. Keywords: Bogoyavlenskii equations system, Couple Boiti-Leon-Pempinelli equations system, Time-fractional Cahn-Allen equation, Khater method, Traveling wave solutions, Solitary wave solutions
Zou, Li; Tian, Shou-Fu; Feng, Lian-Li
2017-12-01
In this paper, we consider the (2+1)-dimensional breaking soliton equation, which describes the interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis. By virtue of the truncated Painlevé expansion method, we obtain the nonlocal symmetry, Bäcklund transformation and Schwarzian form of the equation. Furthermore, by using the consistent Riccati expansion (CRE), we prove that the breaking soliton equation is solvable. Based on the consistent tan-function expansion, we explicitly derive the interaction solutions between solitary waves and cnoidal periodic waves.
Lu, Dianchen; Seadawy, Aly R.; Ali, Asghar
2018-06-01
In this current work, we employ novel methods to find the exact travelling wave solutions of Modified Liouville equation and the Symmetric Regularized Long Wave equation, which are called extended simple equation and exp(-Ψ(ξ))-expansion methods. By assigning the different values to the parameters, different types of the solitary wave solutions are derived from the exact traveling wave solutions, which shows the efficiency and precision of our methods. Some solutions have been represented by graphical. The obtained results have several applications in physical science.
Characteristics of phase-averaged equations for modulated wave groups
Klopman, G.; Petit, H.A.H.; Battjes, J.A.
2000-01-01
The project concerns the influence of long waves on coastal morphology. The modelling of the combined motion of the long waves and short waves in the horizontal plane is done by phase-averaging over the short wave motion and using intra-wave modelling for the long waves, see e.g. Roelvink (1993).
Chiral symmetry breaking and confinement - solutions of relativistic wave equations
Murugesan, P.
1983-01-01
In this thesis, an attempt is made to explore the question whether confinement automatically leads to chiral symmetry breaking. While it should be accepted that chiral symmetry breaking manifests in nature in the absence of scalar partners of pseudoscalar mesons, it does not necessarily follow that confinement should lead to chiral symmetry breaking. If chiral conserving forces give rise to observed spectrum of hadrons, then the conjuncture that confinement is responsible for chiral symmetry breaking is not valid. The method employed to answer the question whether confinement leads to chiral symmetry breaking or not is to solve relativistic wave equations by introducing chiral conserving as well as chiral breaking confining potentials and compare the results with experimental observations. It is concluded that even though chiral symmetry is broken in nature, confinement of quarks need not be the cause of it
Time evolution of the wave equation using rapid expansion method
Pestana, Reynam C.; Stoffa, Paul L.
2010-01-01
Forward modeling of seismic data and reverse time migration are based on the time evolution of wavefields. For the case of spatially varying velocity, we have worked on two approaches to evaluate the time evolution of seismic wavefields. An exact solution for the constant-velocity acoustic wave equation can be used to simulate the pressure response at any time. For a spatially varying velocity, a one-step method can be developed where no intermediate time responses are required. Using this approach, we have solved for the pressure response at intermediate times and have developed a recursive solution. The solution has a very high degree of accuracy and can be reduced to various finite-difference time-derivative methods, depending on the approximations used. Although the two approaches are closely related, each has advantages, depending on the problem being solved. © 2010 Society of Exploration Geophysicists.
Time evolution of the wave equation using rapid expansion method
Pestana, Reynam C.
2010-07-01
Forward modeling of seismic data and reverse time migration are based on the time evolution of wavefields. For the case of spatially varying velocity, we have worked on two approaches to evaluate the time evolution of seismic wavefields. An exact solution for the constant-velocity acoustic wave equation can be used to simulate the pressure response at any time. For a spatially varying velocity, a one-step method can be developed where no intermediate time responses are required. Using this approach, we have solved for the pressure response at intermediate times and have developed a recursive solution. The solution has a very high degree of accuracy and can be reduced to various finite-difference time-derivative methods, depending on the approximations used. Although the two approaches are closely related, each has advantages, depending on the problem being solved. © 2010 Society of Exploration Geophysicists.
Lipschitz Metrics for a Class of Nonlinear Wave Equations
Bressan, Alberto; Chen, Geng
2017-12-01
The nonlinear wave equation {u_{tt}-c(u)(c(u)u_x)_x=0} determines a flow of conservative solutions taking values in the space {H^1(R)}. However, this flow is not continuous with respect to the natural H 1 distance. The aim of this paper is to construct a new metric which renders the flow uniformly Lipschitz continuous on bounded subsets of {H^1(R)}. For this purpose, H 1 is given the structure of a Finsler manifold, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, one can carefully estimate how the weighted length grows in time. By the generic regularity result proved in [7], these piecewise regular paths are dense and can be used to construct a geodesic distance with the desired Lipschitz property.
New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod
Aly R. Seadawy
2018-03-01
Full Text Available This paper examines the effectiveness of an integration scheme which called the extended trial equation method (ETEM in exactly solving a well-known nonlinear equation of partial differential equations (PDEs. In this respect, the longitudinal wave equation (LWE that arises in mathematical physics with dispersion caused by the transverse Poisson’s effect in a magneto-electro-elastic (MEE circular rod, which a series of exact traveling wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of the longitudinal wave equation. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomena of this longitudinal wave equation. Many other such types of nonlinear equations arising in non-destructive evaluation of structures made of the advanced MEE material can also be solved by this method. Keywords: Extended trial equation method, Longitudinal wave equation in a MEE circular rod, Dark solitons, Bright solitons, Solitary wave, Periodic solitary wave
Limiting Behavior of Travelling Waves for the Modified Degasperis-Procesi Equation
Jiuli Yin
2014-01-01
Full Text Available Using an improved qualitative method which combines characteristics of several methods, we classify all travelling wave solutions of the modified Degasperis-Procesi equation in specified regions of the parametric space. Besides some popular exotic solutions including peaked waves, and looped and cusped waves, this equation also admits some very particular waves, such as fractal-like waves, double stumpons, double kinked waves, and butterfly-like waves. The last three types of solutions have not been reported in the literature. Furthermore, we give the limiting behavior of all periodic solutions as the parameters trend to some special values.
Uesaka, S [Kyoto University, Kyoto (Japan). Faculty of Engineering; Watanabe, T; Sassa, K [Kyoto University, Kyoto (Japan)
1997-05-27
Algorithm is constructed and a program developed for a full-wave inversion (FWI) method utilizing the elastic wave equation in seismic exploration. The FWI method is a method for obtaining a physical property distribution using the whole observed waveforms as the data. It is capable of high resolution which is several times smaller than the wavelength since it can handle such phenomena as wave reflection and dispersion. The method for determining the P-wave velocity structure by use of the acoustic wave equation does not provide information about the S-wave velocity since it does not consider S-waves or converted waves. In an analysis using the elastic wave equation, on the other hand, not only P-wave data but also S-wave data can be utilized. In this report, under such circumstances, an inverse analysis algorithm is constructed on the basis of the elastic wave equation, and a basic program is developed. On the basis of the methods of Mora and of Luo and Schuster, the correction factors for P-wave and S-wave velocities are formulated directly from the elastic wave equation. Computations are performed and the effects of the hypocenter frequency and vibration transmission direction are examined. 6 refs., 8 figs.
Wave forces on cylinder submerged horizontally in shallow water
Mitani, H; Sasaki, K; Kobayashi, T; Nomura, N; Kawabe, H; Sugimoto, H
1976-12-01
To estimate the wave forces on offshore and/or coastal structures, the ideal method is undoubtedly to obtain the more accurate solution of hydrodynamic equations under suitable boundary conditions. However, in practice, it is difficult to introduce precise solutions under present technical levels because some important problems still remain. Among them is the unsteady boundary layers with separation around the objects. Consequently, every effort is being made in this field to approximate these conditions. Among these approximations, the Diffraction Wave Theory and the Morrison's Method are the most famous means in practice, although both still have some problems. Some problems with the traditional Finite Amplitude Wave Theories such as Stokes and Cnoidal Wave Theories are examined, and by applying additional computed results to the Morrison's formula, the estimated formula for wave forces on a cylinder submerged horizontally in shallow water is introduced. Subsequently, the applicability of the formula and also the specific characteristics of wave forces on a horizontally settled cylinder are investigated in detail, attaching first importance to the distinctions from the vertically settled cylinder, based on the comparison of computed results with experimental results. The experiments were carried out on two different diameters of cylinder, 70 mm and 140 mm, and bottom slopes of the experimental tanks, /sup 1///sub 100/ and /sup 1///sub 30/, under various conditions varying water depth, wave period, wave height and also setting position of cylinder.
Advanced computational simulations of water waves interacting with wave energy converters
Pathak, Ashish; Freniere, Cole; Raessi, Mehdi
2017-03-01
Wave energy converter (WEC) devices harness the renewable ocean wave energy and convert it into useful forms of energy, e.g. mechanical or electrical. This paper presents an advanced 3D computational framework to study the interaction between water waves and WEC devices. The computational tool solves the full Navier-Stokes equations and considers all important effects impacting the device performance. To enable large-scale simulations in fast turnaround times, the computational solver was developed in an MPI parallel framework. A fast multigrid preconditioned solver is introduced to solve the computationally expensive pressure Poisson equation. The computational solver was applied to two surface-piercing WEC geometries: bottom-hinged cylinder and flap. Their numerically simulated response was validated against experimental data. Additional simulations were conducted to investigate the applicability of Froude scaling in predicting full-scale WEC response from the model experiments.
A stochastic collocation method for the second order wave equation with a discontinuous random speed
Motamed, Mohammad; Nobile, Fabio; Tempone, Raul
2012-01-01
In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical
New traveling wave solutions to AKNS and SKdV equations
Ozer, Teoman
2009-01-01
We analyze the traveling wave solutions of Ablowitz-Kaup-Newell-Segur (AKNS) and Schwarz-Korteweg-de Vries (SKdV) equations. As the solution method for differential equations we consider the improved tanh approach. This approach provides to transform the partial differential equation into the ordinary differential equation and then obtain the new families of exact solutions based on the solutions of the Riccati equation. The different values of the coefficients of the Riccati equation allow us to obtain new type of traveling wave solutions to AKNS and SKdV equations.
Zhang, Zhendong; Schuster, Gerard T.; Liu, Yike; Hanafy, Sherif M.; Li, Jing
2016-01-01
We present a surface-wave inversion method that inverts for the S-wave velocity from the Rayleigh wave dispersion curve using a difference approximation to the gradient of the misfit function. We call this wave equation inversion of skeletonized
Bofill, Josep Maria; Quapp, Wolfgang; Caballero, Marc
2012-12-11
The potential energy surface (PES) of a molecule can be decomposed into equipotential hypersurfaces. We show in this article that the hypersurfaces are the wave fronts of a certain hyperbolic partial differential equation, a wave equation. It is connected with the gradient lines, or the steepest descent, or the steepest ascent lines of the PES. The energy seen as a reaction coordinate plays the central role in this treatment.
Wave-equation Q tomography and least-squares migration
Dutta, Gaurav
2016-03-01
This thesis designs new methods for Q tomography and Q-compensated prestack depth migration when the recorded seismic data suffer from strong attenuation. A motivation of this work is that the presence of gas clouds or mud channels in overburden structures leads to the distortion of amplitudes and phases in seismic waves propagating inside the earth. If the attenuation parameter Q is very strong, i.e., Q<30, ignoring the anelastic effects in imaging can lead to dimming of migration amplitudes and loss of resolution. This, in turn, adversely affects the ability to accurately predict reservoir properties below such layers. To mitigate this problem, I first develop an anelastic least-squares reverse time migration (Q-LSRTM) technique. I reformulate the conventional acoustic least-squares migration problem as a viscoacoustic linearized inversion problem. Using linearized viscoacoustic modeling and adjoint operators during the least-squares iterations, I show with numerical tests that Q-LSRTM can compensate for the amplitude loss and produce images with better balanced amplitudes than conventional migration. To estimate the background Q model that can be used for any Q-compensating migration algorithm, I then develop a wave-equation based optimization method that inverts for the subsurface Q distribution by minimizing a skeletonized misfit function ε. Here, ε is the sum of the squared differences between the observed and the predicted peak/centroid-frequency shifts of the early-arrivals. Through numerical tests on synthetic and field data, I show that noticeable improvements in the migration image quality can be obtained from Q models inverted using wave-equation Q tomography. A key feature of skeletonized inversion is that it is much less likely to get stuck in a local minimum than a standard waveform inversion method. Finally, I develop a preconditioning technique for least-squares migration using a directional Gabor-based preconditioning approach for isotropic
Gaussian solitary waves for the logarithmic-KdV and the logarithmic-KP equations
Wazwaz, Abdul-Majid
2014-01-01
We investigate the logarithmic-KdV equation for more Gaussian solitary waves. We extend this work to derive the logarithmic-KP (Kadomtsev–Petviashvili) equation. We show that both logarithmic models are characterized by their Gaussian solitons. (paper)
Application of perturbation theory to a P-wave eikonal equation in orthorhombic media
Stovas, Alexey; Masmoudi, Nabil; Alkhalifah, Tariq Ali
2016-01-01
The P-wave eikonal equation for orthorhombic (ORT) anisotropic media is a highly nonlinear partial differential equation requiring the solution of a sixth-order polynomial to obtain traveltimes, resulting in complex and time-consuming numerical
Exact solitary and periodic wave solutions for a generalized nonlinear Schroedinger equation
Sun Chengfeng; Gao Hongjun
2009-01-01
The generalized nonlinear Schroedinger equation (GNLS) iu t + u xx + β | u | 2 u + γ | u | 4 u + iα (| u | 2 u) x + iτ(| u | 2 ) x u = 0 is studied. Using the bifurcation of travelling waves of this equation, some exact solitary wave solutions were obtained in [Wang W, Sun J,Chen G, Bifurcation, Exact solutions and nonsmooth behavior of solitary waves in the generalized nonlinear Schroedinger equation. Int J Bifucat Chaos 2005:3295-305.]. In this paper, more explicit exact solitary wave solutions and some new smooth periodic wave solutions are obtained.
Turbulence beneath finite amplitude water waves
Beya, J.F. [Universidad de Valparaiso, Escuela de Ingenieria Civil Oceanica, Facultad de Ingenieria, Valparaiso (Chile); The University of New South Wales, Water Research Laboratory, School of Civil and Environmental Engineering, Sydney, NSW (Australia); Peirson, W.L. [The University of New South Wales, Water Research Laboratory, School of Civil and Environmental Engineering, Sydney, NSW (Australia); Banner, M.L. [The University of New South Wales, School of Mathematics and Statistics, Sydney, NSW (Australia)
2012-05-15
Babanin and Haus (J Phys Oceanogr 39:2675-2679, 2009) recently presented evidence of near-surface turbulence generated below steep non-breaking deep-water waves. They proposed a threshold wave parameter a {sup 2}{omega}/{nu} = 3,000 for the spontaneous occurrence of turbulence beneath surface waves. This is in contrast to conventional understanding that irrotational wave theories provide a good approximation of non-wind-forced wave behaviour as validated by classical experiments. Many laboratory wave experiments were carried out in the early 1960s (e.g. Wiegel 1964). In those experiments, no evidence of turbulence was reported, and steep waves behaved as predicted by the high-order irrotational wave theories within the accuracy of the theories and experimental techniques at the time. This contribution describes flow visualisation experiments for steep non-breaking waves using conventional dye techniques in the wave boundary layer extending above the wave trough level. The measurements showed no evidence of turbulent mixing up to a value of a {sup 2}{omega}/{nu} = 7,000 at which breaking commenced in these experiments. These present findings are in accord with the conventional understandings of wave behaviour. (orig.)
A boundary element model for diffraction of water waves on varying water depth
Poulin, Sanne
1997-12-31
In this thesis a boundary element model for calculating diffraction of water waves on varying water depth is presented. The varying water depth is approximated with a perturbed constant depth in the mild-slope wave equation. By doing this, the domain integral which is a result of the varying depth is no longer a function of the unknown wave potential but only a function of position and the constant depth wave potential. The number of unknowns is the resulting system of equations is thus reduced significantly. The integration procedures in the model are tested very thoroughly and it is found that a combination of analytical integration in the singular region and standard numerical integration outside works very well. The gradient of the wave potential is evaluated successfully using a hypersingular integral equation. Deviations from the analytical solution are only found on the boundary or very close to, but these deviations have no significant influence on the accuracy of the solution. The domain integral is evaluated using the dual reciprocity method. The results are compared with a direct integration of the integral, and the accuracy is quite satisfactory. The problem with irregular frequencies is taken care of by the CBIEM (or CHIEF-method) together with a singular value decomposition technique. This method is simple to implement and works very well. The model is verified using Homma`s island as a test case. The test cases are limited to shallow water since the analytical solution is only valid in this region. Several depth ratios are examined, and it is found that the accuracy of the model increases with increasing wave period and decreasing depth ratio. Short waves, e.g. wind generated waves, can allow depth variations up to approximately 2 before the error exceeds 10%, while long waves can allow larger depth ratios. It is concluded that the perturbation idea is highly usable. A study of (partially) absorbing boundary conditions is also conducted. (EG)
Destrade, M.
2010-12-08
We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy-Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then, we introduce dispersive and dissipative terms to deduce the scalar Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and Khokhlov- Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid mechanics. © 2010 The Royal Society.
Destrade, M.; Goriely, A.; Saccomandi, G.
2010-01-01
We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy-Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then, we introduce dispersive and dissipative terms to deduce the scalar Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and Khokhlov- Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid mechanics. © 2010 The Royal Society.
Wang, Xin; Chen, Yong; Cao, Jianli
2015-01-01
In this paper, we utilize generalized Darboux transformation to study higher-order rogue wave solutions of the three-wave resonant interaction equation, which describes the propagation and mixing of waves with different frequencies in weakly nonlinear dispersive media. A general Nth-order rogue wave solution with two characteristic velocities structural parameters and 3N independent parameters under a determined plane-wave background and a specific parameter condition is derived. As an application, we show that four fundamental rogue waves with fundamental, two kinds of line and quadrilateral patterns, or six fundamental rogue waves with fundamental, triangular, two kinds of quadrilateral and circular patterns can emerge in the second-order rogue waves. Moreover, several important wave characteristics including the maximum values, the corresponding coordinate positions of the humps, and the stability problem for some special higher-order rogue wave solutions such as the fundamental and quadrilateral cases are discussed. (paper)
WAVE-E: The WAter Vapour European-Explorer Mission
Jimenez-LLuva, David; Deiml, Michael; Pavesi, Sara
2017-04-01
In the last decade, stratosphere-troposphere coupling processes in the Upper Troposphere Lower Stratosphere (UTLS) have been increasingly recognized to severely impact surface climate and high-impact weather phenomena. Weakened stratospheric circumpolar jets have been linked to worldwide extreme temperature and high-precipitation events, while anomalously strong stratospheric jets can lead to an increase in surface winds and tropical cyclone intensity. Moreover, stratospheric water vapor has been identified as an important forcing for global decadal surface climate change. In the past years, operational weather forecast and climate models have adapted a high vertical resolution in the UTLS region in order to capture the dynamical processes occurring in this highly stratified region. However, there is an evident lack of available measurements in the UTLS region to consistently support these models and further improve process understanding. Consequently, both the IPCC fifth assessment report and the ESA-GEWEX report 'Earth Observation and Water Cycle Science Priorities' have identified an urgent need for long-term observations and improved process understanding in the UTLS region. To close this gap, the authors propose the 'WAter Vapour European - Explorer' (WAVE-E) space mission, whose primary goal is to monitor water vapor in the UTLS at 1 km vertical, 25 km horizontal and sub-daily temporal resolution. WAVE-E consists of three quasi-identical small ( 500 kg) satellites (WAVE-E 1-3) in a constellation of Sun-Synchronous Low Earth Orbits, each carrying a limb sounding and cross-track scanning mid-infrared passive spectrometer (824 cm-1 to 829 cm-1). The core of the instruments builds a monolithic, field-widened type of Michelson interferometer without any moving parts, rendering it rigid and fault tolerant. Synergistic use of WAVE-E and MetOp-NG operational satellites is identified, such that a data fusion algorithm could provide water vapour profiles from the
Explicit wave action conservation for water waves on vertically sheared flows
Quinn, Brenda; Toledo, Yaron; Shrira, Victor
2016-04-01
Water waves almost always propagate on currents with a vertical structure such as currents directed towards the beach accompanied by an under-current directed back toward the deep sea or wind-induced currents which change magnitude with depth due to viscosity effects. On larger scales they also change their direction due to the Coriolis force as described by the Ekman spiral. This implies that the existing wave models, which assume vertically-averaged currents, is an approximation which is far from realistic. In recent years, ocean circulation models have significantly improved with the capability to model vertically-sheared current profiles in contrast with the earlier vertically-averaged current profiles. Further advancements have coupled wave action models to circulation models to relate the mutual effects between the two types of motion. Restricting wave models to vertically-averaged non-turbulent current profiles is obviously problematic in these cases and the primary goal of this work is to derive and examine a general wave action equation which accounts for these shortcoming. The formulation of the wave action conservation equation is made explicit by following the work of Voronovich (1976) and using known asymptotic solutions of the boundary value problem which exploit the smallness of the current magnitude compared to the wave phase velocity and/or its vertical shear and curvature. The adopted approximations are shown to be sufficient for most of the conceivable applications. This provides correction terms to the group velocity and wave action definition accounting for the shear effects, which are fitting for application to operational wave models. In the limit of vanishing current shear, the new formulation reduces to the commonly used Bretherton & Garrett (1968) no-shear wave action equation where the invariant is calculated with the current magnitude taken at the free surface. It is shown that in realistic oceanic conditions, the neglect of the vertical
Li, Jing
2017-12-22
A robust imaging technology is reviewed that provide subsurface information in challenging environments: wave-equation dispersion inversion (WD) of surface waves for the shear velocity model. We demonstrate the benefits and liabilities of the method with synthetic seismograms and field data. The benefits of WD are that 1) there is no layered medium assumption, as there is in conventional inversion of dispersion curves, so that the 2D or 3D S-velocity model can be reliably obtained with seismic surveys over rugged topography, and 2) WD mostly avoids getting stuck in local minima. The synthetic and field data examples demonstrate that WD can accurately reconstruct the S-wave velocity distributions in laterally heterogeneous media if the dispersion curves can be identified and picked. The WD method is easily extended to anisotropic media and the inversion of dispersion curves associated with Love wave. The liability is that is almost as expensive as FWI and only recovers the Vs distribution to a depth no deeper than about 1/2~1/3 wavelength.
Simple functional-differential equations for the bound-state wave-function components
Kamuntavicius, G.P.
1986-01-01
The author presents a new method of a direct derivation of differential equations for the wave-function components of identical-particles systems. The method generates in a simple manner all the possible variants of these equations. In some cases they are the differential equations of Faddeev or Yakubovskii. It is shown that the case of the bound states allows to formulate very simple equations for the components which are equivalent to the Schroedinger equation for the complete wave function. The components with a minimal antisymmetry are defined and the corresponding equations are derived. (Auth.)
Zhang Weiguo; Dong Chunyan; Fan Engui
2006-01-01
In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travelling wave form satisfies some special conditions.
Higher-Order Hamiltonian Model for Unidirectional Water Waves
Bona, J. L.; Carvajal, X.; Panthee, M.; Scialom, M.
2018-04-01
Formally second-order correct, mathematical descriptions of long-crested water waves propagating mainly in one direction are derived. These equations are analogous to the first-order approximations of KdV- or BBM-type. The advantage of these more complex equations is that their solutions corresponding to physically relevant initial perturbations of the rest state may be accurate on a much longer timescale. The initial value problem for the class of equations that emerges from our derivation is then considered. A local well-posedness theory is straightforwardly established by a contraction mapping argument. A subclass of these equations possess a special Hamiltonian structure that implies the local theory can be continued indefinitely.
Some Further Results on Traveling Wave Solutions for the ZK-BBM( Equations
Shaoyong Li
2013-01-01
Full Text Available We investigate the traveling wave solutions for the ZK-BBM( equations by using bifurcation method of dynamical systems. Firstly, for ZK-BBM(2, 2 equation, we obtain peakon wave, periodic peakon wave, and smooth periodic wave solutions and point out that the peakon wave is the limit form of the periodic peakon wave. Secondly, for ZK-BBM(3, 2 equation, we obtain some elliptic function solutions which include periodic blow-up and periodic wave. Furthermore, from the limit forms of the elliptic function solutions, we obtain some trigonometric and hyperbolic function solutions which include periodic blow-up, blow-up, and smooth solitary wave. We also show that our work extends some previous results.
Mechanical Balance Laws for Boussinesq Models of Surface Water Waves
Ali, Alfatih; Kalisch, Henrik
2012-06-01
Depth-integrated long-wave models, such as the shallow-water and Boussinesq equations, are standard fare in the study of small amplitude surface waves in shallow water. While the shallow-water theory features conservation of mass, momentum and energy for smooth solutions, mechanical balance equations are not widely used in Boussinesq scaling, and it appears that the expressions for many of these quantities are not known. This work presents a systematic derivation of mass, momentum and energy densities and fluxes associated with a general family of Boussinesq systems. The derivation is based on a reconstruction of the velocity field and the pressure in the fluid column below the free surface, and the derivation of differential balance equations which are of the same asymptotic validity as the evolution equations. It is shown that all these mechanical quantities can be expressed in terms of the principal dependent variables of the Boussinesq system: the surface excursion η and the horizontal velocity w at a given level in the fluid.
On the wave equation with semilinear porous acoustic boundary conditions
Graber, Philip Jameson; Said-Houari, Belkacem
2012-01-01
The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function. © 2012 Elsevier Inc.
Energy decay of a viscoelastic wave equation with supercritical nonlinearities
Guo, Yanqiu; Rammaha, Mohammad A.; Sakuntasathien, Sawanya
2018-06-01
This paper presents a study of the asymptotic behavior of the solutions for the history value problem of a viscoelastic wave equation which features a fading memory term as well as a supercritical source term and a frictional damping term: u_{tt}- k(0) Δ u - \\int \\limits _0^{&infty } k'(s) Δ u(t-s) ds +|u_t|^{m-1}u_t =|u|^{p-1}u, { in } Ω × (0,T), u(x,t)=u_0(x,t), \\quad { in } Ω × (-∞,0]), where Ω is a bounded domain in R^3 with a Dirichlét boundary condition and u_0 represents the history value. A suitable notion of a potential well is introduced for the system, and global existence of solutions is justified, provided that the history value u_0 is taken from a subset of the potential well. Also, uniform energy decay rate is obtained which depends on the relaxation kernel -k'(s) as well as the growth rate of the damping term. This manuscript complements our previous work (Guo et al. in J Differ Equ 257:3778-3812, 2014, J Differ Equ 262:1956-1979, 2017) where Hadamard well-posedness and the singularity formulation have been studied for the system. It is worth stressing the special features of the model, namely the source term here has a supercritical growth rate and the memory term accounts to the full past history that goes back to -∞.
On the wave equation with semilinear porous acoustic boundary conditions
Graber, Philip Jameson
2012-05-01
The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function. © 2012 Elsevier Inc.
On the wave equations with memory in noncylindrical domains
Mauro de Lima Santos
2007-10-01
Full Text Available In this paper we prove the exponential and polynomial decays rates in the case $n > 2$, as time approaches infinity of regular solutions of the wave equations with memory $$ u_{tt}-Delta u+int^{t}_{0}g(t-sDelta u(sds=0 quad mbox{in } widehat{Q} $$ where $widehat{Q}$ is a non cylindrical domains of $mathbb{R}^{n+1}$, $(nge1$. We show that the dissipation produced by memory effect is strong enough to produce exponential decay of solution provided the relaxation function $g$ also decays exponentially. When the relaxation function decay polynomially, we show that the solution decays polynomially with the same rate. For this we introduced a new multiplier that makes an important role in the obtaining of the exponential and polynomial decays of the energy of the system. Existence, uniqueness and regularity of solutions for any $n ge 1$ are investigated. The obtained result extends known results from cylindrical to non-cylindrical domains.
Travelling Solitary Wave Solutions for Generalized Time-delayed Burgers-Fisher Equation
Deng Xijun; Han Libo; Li Xi
2009-01-01
In this paper, travelling wave solutions for the generalized time-delayed Burgers-Fisher equation are studied. By using the first-integral method, which is based on the ring theory of commutative algebra, we obtain a class of travelling solitary wave solutions for the generalized time-delayed Burgers-Fisher equation. A minor error in the previous article is clarified. (general)
无
2008-01-01
Using direct algebraic method,exact solitary wave solutions are performed for a class of third order nonlinear dispersive disipative partial differential equations. These solutions are obtained under certain conditions for the relationship between the coefficients of the equation. The exact solitary waves of this class are rational functions of real exponentials of kink-type solutions.
Travelling wave solutions of the generalized Benjamin-Bona-Mahony equation
Estevez, P.G.; Kuru, S.; Negro, J.; Nieto, L.M.
2009-01-01
A class of particular travelling wave solutions of the generalized Benjamin-Bona-Mahony equation is studied systematically using the factorization technique. Then, the general travelling wave solutions of Benjamin-Bona-Mahony equation, and of its modified version, are also recovered.
The periodic wave solutions for the (2 + 1)-dimensional Konopelchenko-Dubrovsky equations
Sheng Zhang
2006-01-01
More periodic wave solutions expressed by Jacobi elliptic functions for the (2 + 1)-dimensional Konopelchenko-Dubrovsky equations are obtained by using the extended F-expansion method. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained
Diffusion phenomenon for linear dissipative wave equations in an exterior domain
Ikehata, Ryo
Under the general condition of the initial data, we will derive the crucial estimates which imply the diffusion phenomenon for the dissipative linear wave equations in an exterior domain. In order to derive the diffusion phenomenon for dissipative wave equations, the time integral method which was developed by Ikehata and Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.
Fifth-order amplitude equation for traveling waves in isothermal double diffusive convection
Mendoza, S.; Becerril, R.
2009-01-01
Third-order amplitude equations for isothermal double diffusive convection are known to hold the tricritical condition all along the oscillatory branch, predicting that stable traveling waves exist Only at the onset of the instability. In order to properly describe stable traveling waves, we perform a fifth-order calculation and present explicitly the corresponding amplitude equation.
Erlangga, Mokhammad Puput [Geophysical Engineering, Institut Teknologi Bandung, Ganesha Street no.10 Basic Science B Buliding fl.2-3 Bandung, 40132, West Java Indonesia puput.erlangga@gmail.com (Indonesia)
2015-04-16
Separation between signal and noise, incoherent or coherent, is important in seismic data processing. Although we have processed the seismic data, the coherent noise is still mixing with the primary signal. Multiple reflections are a kind of coherent noise. In this research, we processed seismic data to attenuate multiple reflections in the both synthetic and real seismic data of Mentawai. There are several methods to attenuate multiple reflection, one of them is Radon filter method that discriminates between primary reflection and multiple reflection in the τ-p domain based on move out difference between primary reflection and multiple reflection. However, in case where the move out difference is too small, the Radon filter method is not enough to attenuate the multiple reflections. The Radon filter also produces the artifacts on the gathers data. Except the Radon filter method, we also use the Wave Equation Multiple Elimination (WEMR) method to attenuate the long period multiple reflection. The WEMR method can attenuate the long period multiple reflection based on wave equation inversion. Refer to the inversion of wave equation and the magnitude of the seismic wave amplitude that observed on the free surface, we get the water bottom reflectivity which is used to eliminate the multiple reflections. The WEMR method does not depend on the move out difference to attenuate the long period multiple reflection. Therefore, the WEMR method can be applied to the seismic data which has small move out difference as the Mentawai seismic data. The small move out difference on the Mentawai seismic data is caused by the restrictiveness of far offset, which is only 705 meter. We compared the real free multiple stacking data after processing with Radon filter and WEMR process. The conclusion is the WEMR method can more attenuate the long period multiple reflection than the Radon filter method on the real (Mentawai) seismic data.
New exact travelling wave solutions for the Ostrovsky equation
Kangalgil, Figen; Ayaz, Fatma
2008-01-01
In this Letter, auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equation with the aid of symbolic computation. In order to illustrate the validity and the advantages of the method we choose the Ostrovsky equation. As a result, many new and more general exact solutions have been obtained for the equation
Exact travelling wave solutions of the Whitham-Broer-Kaup and Broer-Kaup-Kupershmidt equations
Xu Guiqiong; Li Zhibin
2005-01-01
In this paper, an interesting fact is found that the auxiliary equation method is also applicable to a coupled system of two different equations involving both even-order and odd-order partial derivative terms. Furthermore, singular travelling wave solutions can also be obtained by considering other types of exact solutions of auxiliary equation. The Whitham-Broer-Kaup and the (2 + 1)-dimensional Broer-Kaup-Kupershmidt equations are chosen as examples to illustrate the effectiveness of the auxiliary equation method
Stumpons and fractal-like wave solutions to the Dullin-Gottwald-Holm equation
Yin Jiuli; Tian Lixin
2009-01-01
The traveling wave solutions to the Dullin-Gottwald-Holm equation (called DGH equation) are classified by an improved qualitative analysis method. Meanwhile, the influence of the parameters on the traveling wave forms is specifically considered. The equation is shown to admit more traveling wave forms solutions, especially new solutions such as stumpons and fractal-like waves are first given. We also point out that the smooth solutions can converge to non-smooth ones under certain conditions. Furthermore, the new explicit forms of peakons with period are obtained.
Johnson, Kennita A.; Vormohr, Hannah R.; Doinikov, Alexander A.; Bouakaz, Ayache; Shields, C. Wyatt; López, Gabriel P.; Dayton, Paul A.
2016-05-01
Acoustophoresis uses acoustic radiation force to remotely manipulate particles suspended in a host fluid for many scientific, technological, and medical applications, such as acoustic levitation, acoustic coagulation, contrast ultrasound imaging, ultrasound-assisted drug delivery, etc. To estimate the magnitude of acoustic radiation forces, equations derived for an inviscid host fluid are commonly used. However, there are theoretical predictions that, in the case of a traveling wave, viscous effects can dramatically change the magnitude of acoustic radiation forces, which make the equations obtained for an inviscid host fluid invalid for proper estimation of acoustic radiation forces. To date, experimental verification of these predictions has not been published. Experimental measurements of viscous effects on acoustic radiation forces in a traveling wave were conducted using a confocal optical and acoustic system and values were compared with available theories. Our results show that, even in a low-viscosity fluid such as water, the magnitude of acoustic radiation forces is increased manyfold by viscous effects in comparison with what follows from the equations derived for an inviscid fluid.
Wave power potential in Malaysian territorial waters
Asmida Mohd Nasir, Nor; Maulud, Khairul Nizam Abdul
2016-06-01
Up until today, Malaysia has used renewable energy technology such as biomass, solar and hydro energy for power generation and co-generation in palm oil industries and also for the generation of electricity, yet, we are still far behind other countries which have started to optimize waves for similar production. Wave power is a renewable energy (RE) transported by ocean waves. It is very eco-friendly and is easily reachable. This paper presents an assessment of wave power potential in Malaysian territorial waters including waters of Sabah and Sarawak. In this research, data from Malaysia Meteorology Department (MetMalaysia) is used and is supported by a satellite imaginary obtained from National Aeronautics and Space Administration (NASA) and Malaysia Remote Sensing Agency (ARSM) within the time range of the year 1992 until 2007. There were two types of analyses conducted which were mask analysis and comparative analysis. Mask analysis of a research area is the analysis conducted to filter restricted and sensitive areas. Meanwhile, comparative analysis is an analysis conducted to determine the most potential area for wave power generation. Four comparative analyses which have been carried out were wave power analysis, comparative analysis of wave energy power with the sea topography, hot-spot area analysis and comparative analysis of wave energy with the wind speed. These four analyses underwent clipping processes using Geographic Information System (GIS) to obtain the final result. At the end of this research, the most suitable area to develop a wave energy converter was found, which is in the waters of Terengganu and Sarawak. Besides that, it was concluded that the average potential energy that can be generated in Malaysian territorial waters is between 2.8kW/m to 8.6kW/m.
Abdou, M.A.
2008-01-01
The generalized F-expansion method with a computerized symbolic computation is used for constructing a new exact travelling wave solutions for the generalized nonlinear Schrodinger equation with a source. As a result, many exact travelling wave solutions are obtained which include new periodic wave solution, trigonometric function solutions and rational solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics
Yang Zonghang
2007-01-01
We find new exact travelling wave solutions for two potential KdV equations which are presented by Foursov [Foursov MV. J Math Phys 2000;41:6173-85]. Compared with the extended tanh-function method, the algorithm used in our paper can obtain some new kinds of exact travelling wave solutions. With the aid of symbolic computation, some novel exact travelling wave solutions of the potential KdV equations are constructed
The (′/-Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation
Hasibun Naher
2011-01-01
Full Text Available We construct the traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon (CDG equation by the (/-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the (/-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.
de Jong, G.
1975-01-01
With the aid of a two-dimensional integral equation formulation, the ground wave propagation of electromagnetic waves transmitted by a vertical electric dipole over an inhomogeneous flat earth is investigated. For the configuration in which a ground wave is propagating across an ''island'' on a flat earth, the modulus and argument of the attenuation function have been computed. The results for the two-dimensional treatment are significantly more accurate in detail than the calculations using a one-dimensional integral equation
Gauthier, Robert C.; Alzahrani, Mohammed A.; Jafari, Seyed Hamed
2015-02-01
The plane wave expansion (PWM) technique applied to Maxwell's wave equations provides researchers with a supply of information regarding the optical properties of dielectric structures. The technique is well suited for structures that display a linear periodicity. When the focus is directed towards optical resonators and structures that lack linear periodicity the eigen-process can easily exceed computational resources and time constraints. In the case of dielectric structures which display cylindrical or spherical symmetry, a coordinate system specific set of basis functions have been employed to cast Maxwell's wave equations into an eigen-matrix formulation from which the resonator states associated with the dielectric profile can be obtained. As for PWM, the inverse of the dielectric and field components are expanded in the basis functions (Fourier-Fourier-Bessel, FFB, in cylindrical and Fourier- Bessel-Legendre, BLF, in spherical) and orthogonality is employed to form the matrix expressions. The theoretical development details will be presented indicating how certain mathematical complications in the process have been overcome and how the eigen-matrix can be tuned to a specific mode type. The similarities and differences in PWM, FFB and BLF are presented. In the case of structures possessing axial cylindrical symmetry, the inclusion of the z axis component of propagation constant makes the technique applicable to photonic crystal fibers and other waveguide structures. Computational results will be presented for a number of different dielectric geometries including Bragg ring resonators, cylindrical space slot channel waveguides and bottle resonators. Steps to further enhance the computation process will be reported.
On the solution of the equations for nonlinear interaction of three damped waves
1976-01-01
Three-wave interactions are analyzed in a coherent wave description assuming different linear damping (or growth) of the individual waves. It is demonstrated that when two of the coefficients of dissipation are equal, the set of equations can be reduced to a single equivalent equation, which in the nonlinearly unstable case, where one wave is undamped, asymptotically takes the form of an equation defining the third Painleve transcendent. It is then possible to find an asymptotic expansion near the time of explosion. This solution is of principal interest since it indicates that the solution of the general three-wave system, where the waves undergo different individual dissipations, belongs to a higher class of functions, which reduces to Jacobian elliptic functions only in the case where all waves suffer the same damping [fr
Jacobian elliptic wave solutions for the Wadati-Segur-Ablowitz equation
Teh, C.G.R.; Koo, W.K.; Lee, B.S.
1996-07-01
Jacobian elliptic travelling wave solutions for a new Hamiltonian amplitude equation determining some instabilities of modulated wave train are obtained. By a mere variation of the Jacobian elliptic parameter k 2 from zero to one, these solutions are transformed from a trivial one to the known solitary wave solutions. (author). 9 refs
Numerical study of traveling-wave solutions for the Camassa-Holm equation
Kalisch, Henrik; Lenells, Jonatan
2005-01-01
We explore numerically different aspects of periodic traveling-wave solutions of the Camassa-Holm equation. In particular, the time evolution of some recently found new traveling-wave solutions and the interaction of peaked and cusped waves is studied
Wave-Breaking Phenomena and Existence of Peakons for a Generalized Compressible Elastic-Rod Equation
Xiaolian Ai
2014-01-01
Full Text Available Consideration in this paper is the Cauchy problem of a generalized hyperelastic-rod wave equation. We first derive a wave-breaking mechanism for strong solutions, which occurs in finite time for certain initial profiles. In addition, we determine the existence of some new peaked solitary wave solutions.
Explicit Solutions and Bifurcations for a Class of Generalized Boussinesq Wave Equation
Ma Zhi-Min; Sun Yu-Huai; Liu Fu-Sheng
2013-01-01
In this paper, the generalized Boussinesq wave equation u tt — u xx + a(u m ) xx + bu xxxx = 0 is investigated by using the bifurcation theory and the method of phase portraits analysis. Under the different parameter conditions, the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained. (general)
Tian Lixin; Yin Jiuli
2004-01-01
In this paper, we introduce the fully nonlinear generalized Camassa-Holm equation C(m,n,p) and by using four direct ansatzs, we obtain abundant solutions: compactons (solutions with the absence of infinite wings), solitary patterns solutions having infinite slopes or cups, solitary waves and singular periodic wave solutions and obtain kink compacton solutions and nonsymmetry compacton solutions. We also study other forms of fully nonlinear generalized Camassa-Holm equation, and their compacton solutions are governed by linear equations
Energy decay of a variable-coefficient wave equation with nonlinear time-dependent localized damping
Jieqiong Wu
2015-09-01
Full Text Available We study the energy decay for the Cauchy problem of the wave equation with nonlinear time-dependent and space-dependent damping. The damping is localized in a bounded domain and near infinity, and the principal part of the wave equation has a variable-coefficient. We apply the multiplier method for variable-coefficient equations, and obtain an energy decay that depends on the property of the coefficient of the damping term.
Classification of All Single Travelling Wave Solutions to Calogero-Degasperis-Focas Equation
Liu Chengshi
2007-01-01
Under the travelling wave transformation, Calogero-Degasperis-Focas equation is reduced to an ordinary differential equation. Using a symmetry group of one parameter, this ODE is reduced to a second-order linear inhomogeneous ODE. Furthermore, we apply the change of the variable and complete discrimination system for polynomial to solve the corresponding integrals and obtained the classification of all single travelling wave solutions to Calogero-Degasperis-Focas equation.
THE FUNDAMENTAL SOLUTIONS FOR MULTI-TERM MODIFIED POWER LAW WAVE EQUATIONS IN A FINITE DOMAIN.
Jiang, H; Liu, F; Meerschaert, M M; McGough, R J
2013-01-01
Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development. In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0, n ) ( n > 2), respectively. Analytical solutions of the multi-term modified power law wave equations are derived. These new techniques are based on Luchko's Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi-term time-space fractional models including fractional Laplacian.
Hossen, Md. Belal; Roshid, Harun-Or; Ali, M. Zulfikar
2018-05-01
Under inquisition in this paper is a (2 + 1)-dimensional Breaking Soliton equation, which can describe various nonlinear scenarios in fluid dynamics. Using the Bell polynomials, some proficient auxiliary functions are offered to apparently construct its bilinear form and corresponding soliton solutions which are different from the previous literatures. Moreover, a direct method is used to construct its rogue wave and solitary wave solutions using particular auxiliary function with the assist of bilinear formalism. Finally, the interactions between solitary waves and rogue waves are offered with a complete derivation. These results enhance the variety of the dynamics of higher dimensional nonlinear wave fields related to mathematical physics and engineering.
On the exact solutions of high order wave equations of KdV type (I)
Bulut, Hasan; Pandir, Yusuf; Baskonus, Haci Mehmet
2014-12-01
In this paper, by means of a proper transformation and symbolic computation, we study high order wave equations of KdV type (I). We obtained classification of exact solutions that contain soliton, rational, trigonometric and elliptic function solutions by using the extended trial equation method. As a result, the motivation of this paper is to utilize the extended trial equation method to explore new solutions of high order wave equation of KdV type (I). This method is confirmed by applying it to this kind of selected nonlinear equations.
Generalized internal long wave equations: construction, hamiltonian structure and conservation laws
Lebedev, D.R.
1982-01-01
Some aspects of the theory of the internal long-wave equations (ILW) are considered. A general class of the ILW type equations is constructed by means of the Zakharov-Shabat ''dressing'' method. Hamiltonian structure and infinite numbers of conservation laws are introduced. The considered equations are shown to be Hamiltonian in the so-called second Hamiltonian structu
Relating systems properties of the wave and the Schrödinger equation
Zwart, Heiko J.; Le Gorrec, Yann; Maschke, B.M.
In this article we show that systems properties of the systems governed by the second order differential equation d2wdt2=−A0w and the first order differential equation dzdt=iA0z are related. This can be used to show that, for instance, exact observability of the N-dimensional wave equation implies
Differential equation for Alfven ion cyclotron waves in finite-length plasma
Watson, D.C.; Fateman, R.J.; Baldwin, D.E.
1977-01-01
One finds the fourth-order differential equation describing an Alfven-ion-cyclotron wave propagating along a magnetic field of varying intensity. The equation is self-adjoint and possesses non-trivial turning points. The final form of the equation is checked using MACSYMA, a system for performing algebra on a computer
Travelling Wave Solutions to Stretched Beam's Equation: Phase Portraits Survey
Betchewe, Gambo; Victor, Kuetche Kamgang; Thomas, Bouetou Bouetou; Kofane, Timoleon Crepin
2011-01-01
In this paper, following the phase portraits analysis, we investigate the integrability of a system which physically describes the transverse oscillation of an elastic beam under end-thrust. As a result, we find that this system actually comprises two families of travelling waves: the sub- and super-sonic periodic waves of positive- and negative-definite velocities, respectively, and the localized sub-sonic loop-shaped waves of positive-definite velocity. Expressing the energy-like of this system while depicting its phase portrait dynamics, we show that these multivalued localized travelling waves appear as the boundary solutions to which the periodic travelling waves tend asymptotically. (general)
Zhang Zaiyun; Liu Zhenhai; Miao Xiujin; Chen Yuezhong
2011-01-01
In this Letter, we investigate the perturbed nonlinear Schroedinger's equation (NLSE) with Kerr law nonlinearity. All explicit expressions of the bounded traveling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded traveling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.
Zhang, Zhendong
2016-07-26
We present a surface-wave inversion method that inverts for the S-wave velocity from the Rayleigh wave dispersion curve using a difference approximation to the gradient of the misfit function. We call this wave equation inversion of skeletonized surface waves because the skeletonized dispersion curve for the fundamental-mode Rayleigh wave is inverted using finite-difference solutions to the multi-dimensional elastic wave equation. The best match between the predicted and observed dispersion curves provides the optimal S-wave velocity model. Our method can invert for lateral velocity variations and also can mitigate the local minimum problem in full waveform inversion with a reasonable computation cost for simple models. Results with synthetic and field data illustrate the benefits and limitations of this method. © 2016 Elsevier B.V.
Analysis of efficient preconditioned defect correction methods for nonlinear water waves
Engsig-Karup, Allan Peter
2014-01-01
Robust computational procedures for the solution of non-hydrostatic, free surface, irrotational and inviscid free-surface water waves in three space dimensions can be based on iterative preconditioned defect correction (PDC) methods. Such methods can be made efficient and scalable to enable...... prediction of free-surface wave transformation and accurate wave kinematics in both deep and shallow waters in large marine areas or for predicting the outcome of experiments in large numerical wave tanks. We revisit the classical governing equations are fully nonlinear and dispersive potential flow...... equations. We present new detailed fundamental analysis using finite-amplitude wave solutions for iterative solvers. We demonstrate that the PDC method in combination with a high-order discretization method enables efficient and scalable solution of the linear system of equations arising in potential flow...
Symmetries of the triple degenerate DNLS equations for weakly nonlinear dispersive MHD waves
Webb, G. M.; Brio, M.; Zank, G. P.
1996-01-01
A formulation of Hamiltonian and Lagrangian variational principles, Lie point symmetries and conservation laws for the triple degenerate DNLS equations describing the propagation of weakly nonlinear dispersive MHD waves along the ambient magnetic field, in β∼1 plasmas is given. The equations describe the interaction of the Alfven and magnetoacoustic modes near the triple umbilic point, where the fast magnetosonic, slow magnetosonic and Alfven speeds coincide and a g 2 =V A 2 where a g is the gas sound speed and V A is the Alfven speed. A discussion is given of the travelling wave similarity solutions of the equations, which include solitary wave and periodic traveling waves. Strongly compressible solutions indicate the necessity for the insertion of shocks in the flow, whereas weakly compressible, near Alfvenic solutions resemble similar, shock free travelling wave solutions of the DNLS equation
Deterministic and stochastic evolution equations for fully dispersive and weakly nonlinear waves
Eldeberky, Y.; Madsen, Per A.
1999-01-01
and stochastic formulations are solved numerically for the case of cross shore motion of unidirectional waves and the results are verified against laboratory data for wave propagation over submerged bars and over a plane slope. Outside the surf zone the two model predictions are generally in good agreement......This paper presents a new and more accurate set of deterministic evolution equations for the propagation of fully dispersive, weakly nonlinear, irregular, multidirectional waves. The equations are derived directly from the Laplace equation with leading order nonlinearity in the surface boundary...... is significantly underestimated for larger wave numbers. In the present work we correct this inconsistency. In addition to the improved deterministic formulation, we present improved stochastic evolution equations in terms of the energy spectrum and the bispectrum for multidirectional waves. The deterministic...
Numerical simulation of solitary waves on deep water with constant vorticity
Dosaev, A. S.; Shishina, M. I.; Troitskaya, Yu I.
2018-01-01
Characteristics of solitary deep water waves on a flow with constant vorticity are investigated by numerical simulation within the framework of fully nonlinear equations of motion (Euler equations) using the method of surface-tracking conformal coordinates. To ensure that solutions observed are stable, soliton formation as a result of disintegration of an initial pulse-like disturbance is modeled. Evidence is obtained that solitary waves with height above a certain threshold are unstable.
Analysis of the Numerical Solution of the Shallow Water Equations
Hamrick, Thomas
1997-01-01
.... The two schemes are finite difference method (FDM) and the finite element method (FEM). After presenting the shallow water equations in several formulations, some examples will be presented. The use of the Fourier transform to find the solution of a semidiscrete analog of the shallow water equations is also demonstrated.
Multiple periodic-soliton solutions of the (3+1)-dimensional generalised shallow water equation
Li, Ye-Zhou; Liu, Jian-Guo
2018-06-01
Based on the extended variable-coefficient homogeneous balance method and two new ansätz functions, we construct auto-Bäcklund transformation and multiple periodic-soliton solutions of (3 {+} 1)-dimensional generalised shallow water equations. Completely new periodic-soliton solutions including periodic cross-kink wave, periodic two-solitary wave and breather type of two-solitary wave are obtained. In addition, cross-kink three-soliton and cross-kink four-soliton solutions are derived. Furthermore, propagation characteristics and interactions of the obtained solutions are discussed and illustrated in figures.
On the Generalized Maxwell Equations and Their Prediction of Electroscalar Wave
Arbab A. I.
2009-04-01
Full Text Available We have formulated the basic laws of electromagnetic theory in quaternion form. The formalism shows that Maxwell equations and Lorentz force are derivable from just one quaternion equation that only requires the Lorentz gauge. We proposed a quaternion form of the continuity equation from which we have derived the ordinary continuity equation. We introduce new transformations that produces a scalar wave and generalize the continuity equation to a set of three equations. These equations imply that both current and density are waves. Moreover, we have shown that the current can not cir- culate around a point emanating from it. Maxwell equations are invariant under these transformations. An electroscalar wave propagating with speed of light is derived upon requiring the invariance of the energy conservation equation under the new transforma- tions. The electroscalar wave function is found to be proportional to the electric field component along the charged particle motion. This scalar wave exists with or without considering the Lorentz gauge. We have shown that the electromagnetic fields travel with speed of light in the presence or absence of free charges.
Kong Cuicui; Wang Dan; Song Lina; Zhang Hongqing
2009-01-01
In this paper, with the aid of symbolic computation and a general ansaetz, we presented a new extended rational expansion method to construct new rational formal exact solutions to nonlinear partial differential equations. In order to illustrate the effectiveness of this method, we apply it to the MKDV-Burgers equation and the (2 + 1)-dimensional dispersive long wave equation, then several new kinds of exact solutions are successfully obtained by using the new ansaetz. The method can also be applied to other nonlinear partial differential equations.
Letlhogonolo Daddy Moleleki
2014-01-01
Full Text Available We analyze the (3+1-dimensional Boussinesq equation, which has applications in fluid mechanics. We find exact solutions of the (3+1-dimensional Boussinesq equation by utilizing the Lie symmetry method along with the simplest equation method. The solutions obtained are traveling wave solutions. Moreover, we construct the conservation laws of the (3+1-dimensional Boussinesq equation using the new conservation theorem, which is due to Ibragimov.
Witten, Matthew
1983-01-01
Hyperbolic Partial Differential Equations, Volume 1: Population, Reactors, Tides and Waves: Theory and Applications covers three general areas of hyperbolic partial differential equation applications. These areas include problems related to the McKendrick/Von Foerster population equations, other hyperbolic form equations, and the numerical solution.This text is composed of 15 chapters and begins with surveys of age specific population interactions, populations models of diffusion, nonlinear age dependent population growth with harvesting, local and global stability for the nonlinear renewal eq
Forecasting Water Waves and Currents: A Space-time Approach
Ambati, V.R.
2008-01-01
Forecasting water waves and currents in near shore and off shore regions of the seas and oceans is essential to maintain and protect our environment and man made structures. In wave hydrodynamics, waves can be classified as shallow and deep water waves based on its water depth. The mathematical
Rogue waves and rational solutions of a (3+1)-dimensional nonlinear evolution equation
Zhaqilao,
2013-01-01
A simple symbolic computation approach for finding the rogue waves and rational solutions to the nonlinear evolution equation is proposed. It turns out that many rational solutions with real and complex forms of a (3+1)-dimensional nonlinear evolution equation are obtained. Some features of rogue waves and rational solutions are graphically discussed. -- Highlights: •A simple symbolic computation approach for finding the rational solutions to the NEE is proposed. •Some rogue waves and rational solutions with real and complex forms of a (3+1)-D NEE are obtained. •Some features of rogue waves are graphically discussed
Rogue waves and rational solutions of a (3+1)-dimensional nonlinear evolution equation
Zhaqilao,, E-mail: zhaqilao@imnu.edu.cn
2013-12-06
A simple symbolic computation approach for finding the rogue waves and rational solutions to the nonlinear evolution equation is proposed. It turns out that many rational solutions with real and complex forms of a (3+1)-dimensional nonlinear evolution equation are obtained. Some features of rogue waves and rational solutions are graphically discussed. -- Highlights: •A simple symbolic computation approach for finding the rational solutions to the NEE is proposed. •Some rogue waves and rational solutions with real and complex forms of a (3+1)-D NEE are obtained. •Some features of rogue waves are graphically discussed.
Smooth and non-smooth travelling waves in a nonlinearly dispersive Boussinesq equation
Shen Jianwei; Xu Wei; Lei Youming
2005-01-01
The dynamical behavior and special exact solutions of nonlinear dispersive Boussinesq equation (B(m,n) equation), u tt -u xx -a(u n ) xx +b(u m ) xxxx =0, is studied by using bifurcation theory of dynamical system. As a result, all possible phase portraits in the parametric space for the travelling wave system, solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions are obtained. It can be shown that the existence of singular straight line in the travelling wave system is the reason why smooth waves converge to cusp waves, finally. When parameter are varied, under different parametric conditions, various sufficient conditions guarantee the existence of the above solutions are given
Local-in-space blow-up criteria for a class of nonlinear dispersive wave equations
Novruzov, Emil
2017-11-01
This paper is concerned with blow-up phenomena for the nonlinear dispersive wave equation on the real line, ut -uxxt +[ f (u) ] x -[ f (u) ] xxx +[ g (u) + f″/(u) 2 ux2 ] x = 0 that includes the Camassa-Holm equation as well as the hyperelastic-rod wave equation (f (u) = ku2 / 2 and g (u) = (3 - k) u2 / 2) as special cases. We establish some a local-in-space blow-up criterion (i.e., a criterion involving only the properties of the data u0 in a neighborhood of a single point) simplifying and precising earlier blow-up criteria for this equation.
Guo, Hairun; Zeng, Xianglong; Zhou, Binbin
2013-01-01
We interpret the purely spectral forward Maxwell equation with up to third-order induced polarizations for pulse propagation and interactions in quadratic nonlinear crystals. The interpreted equation, also named the nonlinear wave equation in the frequency domain, includes quadratic and cubic...... nonlinearities, delayed Raman effects, and anisotropic nonlinearities. The full potential of this wave equation is demonstrated by investigating simulations of solitons generated in the process of ultrafast cascaded second-harmonic generation. We show that a balance in the soliton delay can be achieved due...
A One-Dimensional Wave Equation with White Noise Boundary Condition
Kim, Jong Uhn
2006-01-01
We discuss the Cauchy problem for a one-dimensional wave equation with white noise boundary condition. We also establish the existence of an invariant measure when the noise is additive. Similar problems for parabolic equations were discussed by several authors. To our knowledge, there is only one work which investigated the initial-boundary value problem for a wave equation with random noise at the boundary. We handle a more general case by a different method. Our result on the existence of an invariant measure relies on the author's recent work on a certain class of stochastic evolution equations
Speed ot travelling waves in reaction-diffusion equations
Benguria, R.D.; Depassier, M.C.; Mendez, V.
2002-01-01
Reaction diffusion equations arise in several problems of population dynamics, flame propagation and others. In one dimensional cases the systems may evolve into travelling fronts. Here we concentrate on a reaction diffusion equation which arises as a simple model for chemotaxis and present results for the speed of the travelling fronts. (Author)
New solitary wave solutions to the modified Kawahara equation
Wazwaz, Abdul-Majid
2007-01-01
In this work we use the sine-cosine method, the tanh method, the extended tanh method, and ansatze of hyperbolic functions for analytic treatment for the modified Kawahara equation. New solitons solutions and periodic solutions are formally derived. The change of the parameters, that will drastically change the characteristics of the equation, is examined. The employed approaches are reliable and manageable
Critical string wave equations and the QCD (U(N{sub c})) string. (Some comments)
Botelho, Luiz C.L. [Universidade Federal Fluminense (UFF), Niteroi, RJ (Brazil). Inst. de Matematica. Dept. de Matematica Aplicada], e-mail: botelho.luiz@superig.com.br
2009-07-01
We present a simple proof that self-avoiding fermionic strings solutions solve formally (in a Quantum Mechanical Framework) the QCD(U(N{sub c})) loop wave equation written in terms of random loops. (author)
Numerical calculation of the cross section by the solution of the wave equation
Drewko, J.
1982-01-01
A numerical method of solving of the wave equation is described for chosen vibrational eigenfunctions. A prepared program calculates the total cross sections for the resonant vibrational excitation for diatomic molecules on the basis of introduced molecular data. (author)
Analysis and computation of the elastic wave equation with random coefficients
Motamed, Mohammad; Nobile, Fabio; Tempone, Raul
2015-01-01
We consider the stochastic initial-boundary value problem for the elastic wave equation with random coefficients and deterministic data. We propose a stochastic collocation method for computing statistical moments of the solution or statistics
Jiang, Lijian; Efendiev, Yalchin; Ginting, Victor
2010-01-01
In this paper, we discuss a numerical multiscale approach for solving wave equations with heterogeneous coefficients. Our interest comes from geophysics applications and we assume that there is no scale separation with respect to spatial variables. To obtain the solution of these multiscale problems on a coarse grid, we compute global fields such that the solution smoothly depends on these fields. We present a Galerkin multiscale finite element method using the global information and provide a convergence analysis when applied to solve the wave equations. We investigate the relation between the smoothness of the global fields and convergence rates of the global Galerkin multiscale finite element method for the wave equations. Numerical examples demonstrate that the use of global information renders better accuracy for wave equations with heterogeneous coefficients than the local multiscale finite element method. © 2010 IMACS.
Spinor-electron wave guided modes in coupled quantum wells structures by solving the Dirac equation
Linares, Jesus; Nistal, Maria C.
2009-01-01
A quantum analysis based on the Dirac equation of the propagation of spinor-electron waves in coupled quantum wells, or equivalently coupled electron waveguides, is presented. The complete optical wave equations for Spin-Up (SU) and Spin-Down (SD) spinor-electron waves in these electron guides couplers are derived from the Dirac equation. The relativistic amplitudes and dispersion equations of the spinor-electron wave-guided modes in a planar quantum coupler formed by two coupled quantum wells, or equivalently by two coupled slab electron waveguides, are exactly derived. The main outcomes related to the spinor modal structure, such as the breaking of the non-relativistic degenerate spin states, the appearance of phase shifts associated with the spin polarization and so on, are shown.
THE EXPONENTIAL STABILIZATION FOR A SEMILINEAR WAVE EQUATION WITH LOCALLY DISTRIBUTED FEEDBACK
JIA CHAOHUA; FENG DEXING
2005-01-01
This paper considers the exponential decay of the solution to a damped semilinear wave equation with variable coefficients in the principal part by Riemannian multiplier method. A differential geometric condition that ensures the exponential decay is obtained.
Solution of the Bethe-Salpeter equation in the field of a plane electromagnetic wave
Starostin, V.S.
1988-01-01
A solution is obtained of the Bethe--Salpeter equation for positronium in the field of linearly and circularly polarized plane electromagnetic waves at frequencies much higher than atomic. It is not assumed that the field is weak
Angle-domain Migration Velocity Analysis using Wave-equation Reflection Traveltime Inversion
Zhang, Sanzong; Schuster, Gerard T.; Luo, Yi
2012-01-01
way as wave-equation transmission traveltime inversion. The residual movemout analysis in the angle-domain common image gathers provides a robust estimate of the depth residual which is converted to the reflection traveltime residual for the velocity
Extended common-image-point gathers for anisotropic wave-equation migration
Sava, Paul C.; Alkhalifah, Tariq Ali
2010-01-01
In regions characterized by complex subsurface structure, wave-equation depth migration is a powerful tool for accurately imaging the earth’s interior. The quality of the final image greatly depends on the quality of the model which includes
Study on monostable and bistable reaction-diffusion equations by iteration of travelling wave maps
Yi, Taishan; Chen, Yuming
2017-12-01
In this paper, based on the iterative properties of travelling wave maps, we develop a new method to obtain spreading speeds and asymptotic propagation for monostable and bistable reaction-diffusion equations. Precisely, for Dirichlet problems of monostable reaction-diffusion equations on the half line, by making links between travelling wave maps and integral operators associated with the Dirichlet diffusion kernel (the latter is NOT invariant under translation), we obtain some iteration properties of the Dirichlet diffusion and some a priori estimates on nontrivial solutions of Dirichlet problems under travelling wave transformation. We then provide the asymptotic behavior of nontrivial solutions in the space-time region for Dirichlet problems. These enable us to develop a unified method to obtain results on heterogeneous steady states, travelling waves, spreading speeds, and asymptotic spreading behavior for Dirichlet problem of monostable reaction-diffusion equations on R+ as well as of monostable/bistable reaction-diffusion equations on R.
Traveling Wave Solutions of ZK-BBM Equation Sine-Cosine Method
Sadaf Bibi
2014-03-01
Full Text Available Travelling wave solutions are obtained by using a relatively new technique which is called sine-cosine method for ZK-BBM equations. Solution procedure and obtained results re-confirm the efficiency of the proposed scheme.
Wave-packet revival for the Schroedinger equation with position-dependent mass
Schmidt, Alexandre G.M.
2006-01-01
We study the temporal evolution of solutions of 1D Schroedinger equation with position-dependent mass inside an infinite well. Revival of wave-packet is shown to exist and partial revivals are different from the usual ones
Jiang, Lijian
2010-08-01
In this paper, we discuss a numerical multiscale approach for solving wave equations with heterogeneous coefficients. Our interest comes from geophysics applications and we assume that there is no scale separation with respect to spatial variables. To obtain the solution of these multiscale problems on a coarse grid, we compute global fields such that the solution smoothly depends on these fields. We present a Galerkin multiscale finite element method using the global information and provide a convergence analysis when applied to solve the wave equations. We investigate the relation between the smoothness of the global fields and convergence rates of the global Galerkin multiscale finite element method for the wave equations. Numerical examples demonstrate that the use of global information renders better accuracy for wave equations with heterogeneous coefficients than the local multiscale finite element method. © 2010 IMACS.
Shock wave equation of state of powder material
Dijken, D.K.; Hosson, J.Th.M. De
1994-01-01
A model is proposed to predict the following quantities for powder materials compacted by shock waves: the pressure, the specific volume, the internal energy behind the shock wave, and the shock-wave velocity U-s. They are calculated as a function of flyerplate velocity u(p) and initial powder specific volume V-00. The model is tested on Cu, Al2024, and Fe. Calculated U-s vs u(p) curves agree well with experiments provided V-00 is smaller than about two times the solid specific volume. The mo...
Inaccuracy caused by the use of thermodynamic equation inside shock wave front
Sano, Yukio; Abe, Akihisa; Tokushima, Koji; Arathoon, P.
1998-01-01
The aim of this study is to examine the difference between shock temperatures predicted by an equation for temperature inside a steady wave front and the Walsh-Christian equation. Calculations are for yttria-doped tetragonal zirconia, which shows an elastic-plastic and a phase transition: Thus the shock waves treated are multiple structure waves composed of one to three steady wave fronts. The evaluated temperature was 3350K at the minimum specific volume of 0.1175 cm 3 /g (or maximum Hugoniot shock pressure of 140GPa) considered in the present examination, while the temperature predicted by the Walsh-Christian equation under identical conditions was 2657K. The cause of the large temperature discrepancy is considered to be that the present model treats nonequilibrium states inside steady waves
Tarikul Islam
2018-03-01
Full Text Available In this article, the analytical solutions to the space-time fractional foam drainage equation and the space-time fractional symmetric regularized long wave (SRLW equation are successfully examined by the recently established rational (G′/G-expansion method. The suggested equations are reduced into the nonlinear ordinary differential equations with the aid of the fractional complex transform. Consequently, the theories of the ordinary differential equations are implemented effectively. Three types closed form traveling wave solutions, such as hyperbolic function, trigonometric function and rational, are constructed by using the suggested method in the sense of conformable fractional derivative. The obtained solutions might be significant to analyze the depth and spacing of parallel subsurface drain and small-amplitude long wave on the surface of the water in a channel. It is observed that the performance of the rational (G′/G-expansion method is reliable and will be used to establish new general closed form solutions for any other NPDEs of fractional order.
Said-Houari, Belkacem
2012-03-01
In this paper, we consider a viscoelastic wave equation with an absorbing term and space-time dependent damping term. Based on the weighted energy method, and by assuming that the kernel decaying exponentially, we obtain the L2 decay rates of the solutions. More precisely, we show that the decay rates are the same as those obtained in Lin et al. (2010) [15] for the semilinear wave equation with absorption term. © 2011 Elsevier Inc.
Exact traveling wave solutions of the bbm and kdv equations using (G'/G)-expansion method
Saddique, I.; Nazar, K.
2009-01-01
In this paper, we construct the traveling wave solutions involving parameters of the Benjamin Bona-Mahony (BBM) and KdV equations in terms of the hyperbolic, trigonometric and rational functions by using the (G'/G)-expansion method, where G = G(zeta) satisfies a second order linear ordinary differential equation. When the parameters are taken special values, the Solitary was are derived from the traveling waves. (author)
Said-Houari, Belkacem
2012-01-01
In this paper, we consider a viscoelastic wave equation with an absorbing term and space-time dependent damping term. Based on the weighted energy method, and by assuming that the kernel decaying exponentially, we obtain the L2 decay rates of the solutions. More precisely, we show that the decay rates are the same as those obtained in Lin et al. (2010) [15] for the semilinear wave equation with absorption term. © 2011 Elsevier Inc.
On the relativistic transport equation for a multiple discontinuity wave
Giambo, Sebastiano
1980-01-01
The theory of singular hypersurfaces is combined with the ray theory to study propagation of weak discontinuities of solutions of quasi-linear hyperbolic system in the context of special relativity. The case of a multiple wave is considered [fr
Traveling waves in lattice differential equations with distributed maturation delay
Hui-Ling Niu
2013-07-01
Full Text Available In this paper we derive a lattice model with infinite distributed delay to describe the growth of a single-species population in a 2D patchy environment with infinite number of patches connected locally by diffusion and global interaction. We consider the existence of traveling wave solutions when the birth rate is large enough that each patch can sustain a positive equilibrium. When the birth function is monotone, we prove that there exists a traveling wave solution connecting two equilibria with wave speed $c>c^*(\\theta$ by using the monotone iterative method and super and subsolution technique, where $\\theta\\in [0,2\\pi]$ is any fixed direction of propagation. When the birth function is non-monotone, we prove the existence of non-trivial traveling wave solutions by constructing two auxiliary systems satisfying quasi-monotonicity.
Relativistic transport equation for a multiple discontinuity wave
Giambo, S [Istituto di Matematica, Universita degli Studi, Messina (Italy)
1980-09-29
The theory of singular hypersurfaces is combined with the ray theory to study propagation of weak discontinuities of solutions of a quasi-linear hyperbolic system in the context of special relativity. The case of a multiple wave is considered.
Sound field prediction of ultrasonic lithotripsy in water with spheroidal beam equations
Zhang Lue; Wang Xiang-Da; Liu Xiao-Zhou; Gong Xiu-Fen
2015-01-01
With converged shock wave, extracorporeal shock wave lithotripsy (ESWL) has become a preferable way to crush human calculi because of its advantages of efficiency and non-intrusion. Nonlinear spheroidal beam equations (SBE) are employed to illustrate the acoustic wave propagation for transducers with a wide aperture angle. To predict the acoustic field distribution precisely, boundary conditions are obtained for the SBE model of the monochromatic wave when the source is located on the focus of an ESWL transducer. Numerical results of the monochromatic wave propagation in water are analyzed and the influences of half-angle, fundamental frequency, and initial pressure are investigated. According to our results, with optimization of these factors, the pressure focal gain of ESWL can be enhanced and the effectiveness of treatment can be improved. (paper)
Wapenaar, Kees
2017-06-01
A unified scalar wave equation is formulated, which covers three-dimensional (3D) acoustic waves, 2D horizontally-polarised shear waves, 2D transverse-electric EM waves, 2D transverse-magnetic EM waves, 3D quantum-mechanical waves and 2D flexural waves. The homogeneous Green's function of this wave equation is a combination of the causal Green's function and its time-reversal, such that their singularities at the source position cancel each other. A classical representation expresses this homogeneous Green's function as a closed boundary integral. This representation finds applications in holographic imaging, time-reversed wave propagation and Green's function retrieval by cross correlation. The main drawback of the classical representation in those applications is that it requires access to a closed boundary around the medium of interest, whereas in many practical situations the medium can be accessed from one side only. Therefore, a single-sided representation is derived for the homogeneous Green's function of the unified scalar wave equation. Like the classical representation, this single-sided representation fully accounts for multiple scattering. The single-sided representation has the same applications as the classical representation, but unlike the classical representation it is applicable in situations where the medium of interest is accessible from one side only.
The solution of the dam-break problem in the Porous Shallow water Equations
Cozzolino, Luca; Pepe, Veronica; Cimorelli, Luigi; D'Aniello, Andrea; Della Morte, Renata; Pianese, Domenico
2018-04-01
The Porous Shallow water Equations are commonly used to evaluate the propagation of flooding waves in the urban environment. These equations may exhibit not only classic shocks, rarefactions, and contact discontinuities, as in the ordinary two-dimensional Shallow water Equations, but also special discontinuities at abrupt porosity jumps. In this paper, an appropriate parameterization of the stationary weak solutions of one-dimensional Porous Shallow water Equations supplies the inner structure of the porosity jumps. The exact solution of the corresponding dam-break problem is presented, and six different wave configurations are individuated, proving that the solution exists and it is unique for given initial conditions and geometric characteristics. These results can be used as a benchmark in order to validate one- and two-dimensional numerical models for the solution of the Porous Shallow water Equations. In addition, it is presented a novel Finite Volume scheme where the porosity jumps are taken into account by means of a variables reconstruction approach. The dam-break results supplied by this numerical scheme are compared with the exact dam-break results, showing the promising capabilities of this numerical approach. Finally, the advantages of the novel porosity jump definition are shown by comparison with other definitions available in the literature, demonstrating its advantages, and the issues raising in real world applications are discussed.
Relativistic wave equations for particles in electromagnetic fields
Good, R.H. Jr.
1989-01-01
A new type of generalization of the Dirac equation of higher spin particles and antiparticles is given, in case only the terms proportional to the external fields need to be retained. copyright 1989 Academic Press, Inc
Physical dynamics of quasi-particles in nonlinear wave equations
Christov, Ivan [Department of Mathematics, Texas A and M University, College Station, TX 77843-3368 (United States)], E-mail: christov@alum.mit.edu; Christov, C.I. [Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010 (United States)], E-mail: christov@louisiana.edu
2008-02-04
By treating the centers of solitons as point particles and studying their discrete dynamics, we demonstrate a new approach to the quantization of the soliton solutions of the sine-Gordon equation, one of the first model nonlinear field equations. In particular, we show that a linear superposition of the non-interacting shapes of two solitons offers a qualitative (and to a good approximation quantitative) description of the true two-soliton solution, provided that the trajectories of the centers of the superimposed solitons are considered unknown. Via variational calculus, we establish that the dynamics of the quasi-particles obey a pseudo-Newtonian law, which includes cross-mass terms. The successful identification of the governing equations of the (discrete) quasi-particles from the (continuous) field equation shows that the proposed approach provides a basis for the passage from the continuous to a discrete description of the field.
Physical dynamics of quasi-particles in nonlinear wave equations
Christov, Ivan; Christov, C.I.
2008-01-01
By treating the centers of solitons as point particles and studying their discrete dynamics, we demonstrate a new approach to the quantization of the soliton solutions of the sine-Gordon equation, one of the first model nonlinear field equations. In particular, we show that a linear superposition of the non-interacting shapes of two solitons offers a qualitative (and to a good approximation quantitative) description of the true two-soliton solution, provided that the trajectories of the centers of the superimposed solitons are considered unknown. Via variational calculus, we establish that the dynamics of the quasi-particles obey a pseudo-Newtonian law, which includes cross-mass terms. The successful identification of the governing equations of the (discrete) quasi-particles from the (continuous) field equation shows that the proposed approach provides a basis for the passage from the continuous to a discrete description of the field
Solitary wave solutions to nonlinear evolution equations in ...
1Computer Engineering Technique Department, Al-Rafidain University College, Baghdad, ... applied to extract solutions are tan–cot method and functional variable approaches. ... Consider the nonlinear partial differential equation in the form.
The two-wave X-ray field calculated by means of integral-equation methods
Bremer, J.
1984-01-01
The problem of calculating the two-wave X-ray field on the basis of the Takagi-Taupin equations is discussed for the general case of curved lattice planes. A two-dimensional integral equation which incorporates the nature of the incoming radiation, the form of the crystal/vacuum boundary, and the curvature of the structure, is deduced. Analytical solutions for the symmetrical Laue case with incoming plane waves are obtained directly for perfect crystals by means of iteration. The same method permits a simple derivation of the narrow-wave Laue and Bragg cases. Modulated wave fronts are discussed, and it is shown that a cut-off in the width of an incoming plane wave leads to lateral oscillations which are superimposed on the Pendelloesung fringes. Bragg and Laue shadow fields are obtained. The influence of a non-zero kernel is discussed and a numerical procedure for calculating wave amplitudes in curved crystals is presented. (Auth.)
Asymptotic Behavior of Periodic Wave Solution to the Hirota—Satsuma Equation
Wu Yong-Qi
2011-01-01
The one- and two-periodic wave solutions for the Hirota—Satsuma (HS) equation are presented by using the Hirota derivative and Riemann theta function. The rigorous proofs on asymptotic behaviors of these two solutions are given such that soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure. (general)
Bifurcations of Exact Traveling Wave Solutions for (2+1)-Dimensional HNLS Equation
Xu Yuanfen
2012-01-01
For the (2+1)-Dimensional HNLS equation, what are the dynamical behavior of its traveling wave solutions and how do they depend on the parameters of the systems? This paper will answer these questions by using the methods of dynamical systems. Ten exact explicit parametric representations of the traveling wave solutions are given. (general)
Solitons and cnoidal waves of the Klein–Gordon–Zakharov equation ...
In (3), κ represents the wave number of the soliton while ω represents ... integration constant to be zero, since the search is for soliton solutions only, gives ..... and also using relations (3)–(5) gives the following rational travelling wave ... In future, the plan is to study the numerical simulations for this equation along with.
A membrane wave equation for Q.C.D. (SU(infinity))
Botelho, L.C.L.
1988-01-01
It is proposed a quantum membrane wave functional describing the interaction between a colored SU(N c ) membrane and a quantized Yang-Mills field. Additionally, its associated wave equation in the t'Hooft N c ->infinity limit is deduced. (A.C.A.S.) [pt
New Exact Travelling Wave and Periodic Solutions of Discrete Nonlinear Schroedinger Equation
Yang Qin; Dai Chaoqing; Zhang Jiefang
2005-01-01
Some new exact travelling wave and period solutions of discrete nonlinear Schroedinger equation are found by using a hyperbolic tangent function approach, which was usually presented to find exact travelling wave solutions of certain nonlinear partial differential models. Now we can further extend the new algorithm to other nonlinear differential-different models.
Instability of traveling waves of the convective-diffusive Cahn-Hilliard equation
Gao Hongjun; Liu Changchun
2004-01-01
In this paper we study the instability of the traveling waves of the convective-diffusive Cahn-Hilliard equation. We prove that it is nonlinearly unstable under H 2 perturbations, for some traveling wave solution that is asymptotic to a constant as x→∞
Nonlinear water waves: introduction and overview
Constantin, A.
2017-12-01
For more than two centuries progress in the study of water waves proved to be interdependent with innovative and deep developments in theoretical and experimental directions of investigation. In recent years, considerable progress has been achieved towards the understanding of waves of large amplitude. Within this setting one cannot rely on linear theory as nonlinearity becomes an essential feature. Various analytic methods have been developed and adapted to come to terms with the challenges encountered in settings where approximations (such as those provided by linear or weakly nonlinear theory) are ineffective. Without relying on simpler models, progress becomes contingent upon the discovery of structural properties, the exploitation of which requires a combination of creative ideas and state-of-the-art technical tools. The successful quest for structure often reveals unexpected patterns and confers aesthetic value on some of these studies. The topics covered in this issue are both multi-disciplinary and interdisciplinary: there is a strong interplay between mathematical analysis, numerical computation and experimental/field data, interacting with each other via mutual stimulation and feedback. This theme issue reflects some of the new important developments that were discussed during the programme `Nonlinear water waves' that took place at the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) from 31st July to 25th August 2017. A cross-section of the experts in the study of water waves who participated in the programme authored the collected papers. These papers illustrate the diversity, intensity and interconnectivity of the current research activity in this area. They offer new insight, present emerging theoretical methodologies and computational approaches, and describe sophisticated experimental results. This article is part of the theme issue 'Nonlinear water waves'.
Fierros Palacios, Angel [Instituto de Investigaciones Electricas, Temixco, Morelos (Mexico)
2001-02-01
In this work the complete set of differential field equations which describes the dynamic state of a continuos conducting media which flow in presence of a perturbed magnetic field is obtained. Then, the thermic equation of state, the wave equation and the conservation law of energy for the Alfven MHD waves are obtained. [Spanish] Es este trabajo se obtiene el conjunto completo de ecuaciones diferenciales de campo que describen el estado dinamico de un medio continuo conductor que se mueve en presencia de un campo magnetico externo perturbado. Asi, se obtiene la ecuacion termica de estado, la ecuacion de onda y la ley de la conservacion de la energia para las ondas de Alfven de la MHD.
Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets
Ai-Min Yang
2013-01-01
Full Text Available We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.
Application of the Exp-function method to the equal-width wave equation
Biazar, J; Ayati, Z
2008-01-01
In this paper, the Exp-function method is used to find an exact solution of the equal-width wave (EW) equation. The method is straightforward and concise, and its applications are promising. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving the EW equation.
Boundary Observability and Stabilization for Westervelt Type Wave Equations without Interior Damping
Kaltenbacher, Barbara
2010-01-01
In this paper we show boundary observability and boundary stabilizability by linear feedbacks for a class of nonlinear wave equations including the undamped Westervelt model used in nonlinear acoustics. We prove local existence for undamped generalized Westervelt equations with homogeneous Dirichlet boundary conditions as well as global existence and exponential decay with absorbing type boundary conditions.
Dandan Guo
2017-08-01
Full Text Available In this article we consider the boundary stabilization of a wave equation with variable coefficients. This equation has an acceleration term and a delayed velocity term on the boundary. Under suitable geometric conditions, we obtain the exponential decay for the solutions. Our proof relies on the geometric multiplier method and the Lyapunov approach.
Werby, M.F.; Broadhead, M.K.; Strayer, M.R.; Bottcher, C.
1992-01-01
The Helmholtz-Poincarf Wave Equation (H-PWE) arises in many areas of classical wave scattering theory. In particular it can be found for the cases of acoustical scattering from submerged bounded objects and electromagnetic scattering from objects. The extended boundary integral equations (EBIE) method is derived from considering both the exterior and interior solutions of the H-PWECs. This coupled set of expressions has the advantage of not only offering a prescription for obtaining a solution for the exterior scattering problem, but it also obviates the problem of irregular values corresponding to fictitious interior eigenvalues. Once the coupled equations are derived, they can be obtained in matrix form by expanding all relevant terms in partial wave expansions, including a bi-orthogonal expansion of the Green's function. However some freedom in the choice of the surface expansion is available since the unknown surface quantities may be expanded in a variety of ways so long as closure is obtained. Out of many possible choices, we develop an optimal method to obtain such expansions which is based on the optimum eigenfunctions related to the surface of the object. In effect, we convert part of the problem (that associated with the Fredholms integral equation of the first kind) an eigenvalue problem of a related Hermitian operator. The methodology will be explained in detail and examples will be presented
Several Dynamical Properties for a Nonlinear Shallow Water Equation
Ls Yong
2014-01-01
Full Text Available A nonlinear third order dispersive shallow water equation including the Degasperis-Procesi model is investigated. The existence of weak solutions for the equation is proved in the space L1(R∩BV (R under certain assumptions. The Oleinik type estimate and L2N(R (N is a natural number estimate for the solution are obtained.
Global existence and decay of solutions of a nonlinear system of wave equations
Said-Houari, Belkacem
2012-01-01
This work is concerned with a system of two wave equations with nonlinear damping and source terms acting in both equations. Under some restrictions on the nonlinearity of the damping and the source terms, we show that our problem has a unique local solution. Also, we prove that, for some restrictions on the initial data, the rate of decay of the total energy is exponential or polynomial depending on the exponents of the damping terms in both equations.
Exact traveling wave solution of nonlinear variants of the RLW and the PHI-four equations
Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish), Suez Canal University, AL-Arish 45111 (Egypt); Department of Mathematics, Teacher' s College, Bisha, P.O. Box 551 (Saudi Arabia)], E-mail: asoliman_99@yahoo.com
2007-08-27
By means of the modified extended tanh-function (METF) method the multiple traveling wave solutions of some different kinds of nonlinear partial differential equations are presented and implemented in a computer algebraic system. The solutions for the nonlinear equations such as variants of the RLW and variant of the PHI-four equations are exactly obtained and so the efficiency of the method can be demonstrated.
Integrable discretizations for the short-wave model of the Camassa-Holm equation
Feng Baofeng; Maruno, Ken-ichi; Ohta, Yasuhiro
2010-01-01
The link between the short-wave model of the Camassa-Holm equation (SCHE) and bilinear equations of the two-dimensional Toda lattice equation is clarified. The parametric form of the N-cuspon solution of the SCHE in Casorati determinant is then given. Based on the above finding, integrable semi-discrete and full-discrete analogues of the SCHE are constructed. The determinant solutions of both semi-discrete and fully discrete analogues of the SCHE are also presented.
Jun-Mao, Wang; Miao, Zhang; Wen-Liang, Zhang; Rui, Zhang; Jia-Hua, Han
2008-01-01
We present a new method to find the exact travelling wave solutions of nonlinear evolution equations, with the aid of the symbolic computation. Based on this method, we successfully solve the modified Benjamin–Bona–Mahoney equation, and obtain some new solutions which can be expressed by trigonometric functions and hyperbolic functions. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics. (general)
Conservation Laws and Traveling Wave Solutions of a Generalized Nonlinear ZK-BBM Equation
Khadijo Rashid Adem
2014-01-01
Full Text Available We study a generalized two-dimensional nonlinear Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM equation, which is in fact Benjamin-Bona-Mahony equation formulated in the ZK sense. Conservation laws for this equation are constructed by using the new conservation theorem due to Ibragimov and the multiplier method. Furthermore, traveling wave solutions are obtained by employing the (G'/G-expansion method.
Traveling Wave Solutions of Reaction-Diffusion Equations Arising in Atherosclerosis Models
Narcisa Apreutesei
2014-05-01
Full Text Available In this short review article, two atherosclerosis models are presented, one as a scalar equation and the other one as a system of two equations. They are given in terms of reaction-diffusion equations in an infinite strip with nonlinear boundary conditions. The existence of traveling wave solutions is studied for these models. The monostable and bistable cases are introduced and analyzed.
Global existence and decay of solutions of a nonlinear system of wave equations
Said-Houari, Belkacem
2012-03-01
This work is concerned with a system of two wave equations with nonlinear damping and source terms acting in both equations. Under some restrictions on the nonlinearity of the damping and the source terms, we show that our problem has a unique local solution. Also, we prove that, for some restrictions on the initial data, the rate of decay of the total energy is exponential or polynomial depending on the exponents of the damping terms in both equations.
On shallow water waves in a medium with time-dependent
Hamdy I. Abdel-Gawad
2015-07-01
Full Text Available In this paper, we studied the progression of shallow water waves relevant to the variable coefficient Korteweg–de Vries (vcKdV equation. We investigated two kinds of cases: when the dispersion and nonlinearity coefficients are proportional, and when they are not linearly dependent. In the first case, it was shown that the progressive waves have some geometric structures as in the case of KdV equation with constant coefficients but the waves travel with time dependent speed. In the second case, the wave structure is maintained when the nonlinearity balances the dispersion. Otherwise, water waves collapse. The objectives of the study are to find a wide class of exact solutions by using the extended unified method and to present a new algorithm for treating the coupled nonlinear PDE’s.
Dutta, Gaurav
2013-08-20
Attenuation leads to distortion of amplitude and phase of seismic waves propagating inside the earth. Conventional acoustic and least-squares reverse time migration do not account for this distortion which leads to defocusing of migration images in highly attenuative geological environments. To account for this distortion, we propose to use the visco-acoustic wave equation for least-squares reverse time migration. Numerical tests on synthetic data show that least-squares reverse time migration with the visco-acoustic wave equation corrects for this distortion and produces images with better balanced amplitudes compared to the conventional approach. © 2013 SEG.
Non-linear wave equations:Mathematical techniques
1978-01-01
An account of certain well-established mathematical methods, which prove useful to deal with non-linear partial differential equations is presented. Within the strict framework of Functional Analysis, it describes Semigroup Techniques in Banach Spaces as well as variational approaches towards critical points. Detailed proofs are given of the existence of local and global solutions of the Cauchy problem and of the stability of stationary solutions. The formal approach based upon invariance under Lie transformations deserves attention due to its wide range of applicability, even if the explicit solutions thus obtained do not allow for a deep analysis of the equations. A compre ensive introduction to the inverse scattering approach and to the solution concept for certain non-linear equations of physical interest are also presented. A detailed discussion is made about certain convergence and stability problems which arise in importance need not be emphasized. (author) [es
Optical tsunamis: shoaling of shallow water rogue waves in nonlinear fibers with normal dispersion
Wabnitz, Stefan
2013-01-01
In analogy with ocean waves running up towards the beach, shoaling of pre-chirped optical pulses may occur in the normal group-velocity dispersion regime of optical fibers. We present exact Riemann wave solutions of the optical shallow water equations and show that they agree remarkably well with the numerical solutions of the nonlinear Schrödinger equation, at least up to the point where a vertical pulse front develops. We also reveal that extreme wave events or optical tsunamis may be generated in dispersion tapered fibers in the presence of higher-order dispersion. (paper)
Variable coefficient Korteweg-de Vries equations and travelling waves in an inhomogeneous medium
Baby, B.V.
1987-04-01
The well-known Korteweg-de Vries equations with the coefficients as two arbitrary functions of the time variable, is studied in this paper. The Painleve property analysis provides the conditions on the two variable coefficients, in order to form the Lax pairs associated with this equation. The similarity analysis shows the non-existence of travelling wave solutions when the equation has variable coefficients. These results are used to show the non-existence of travelling waves in an inhomogeneous medium. (author). 33 refs
New binary travelling-wave periodic solutions for the modified KdV equation
Yan Zhenya
2008-01-01
In this Letter, the modified Korteweg-de Vries (mKdV) equations with the focusing (+) and defocusing (-) branches are investigated, respectively. Many new types of binary travelling-wave periodic solutions are obtained for the mKdV equation in terms of Jacobi elliptic functions such as sn(ξ,m)cn(ξ,m)dn(ξ,m) and their extensions. Moreover, we analyze asymptotic properties of some solutions. In addition, with the aid of the Miura transformation, we also give the corresponding binary travelling-wave periodic solutions of KdV equation
Rani, Monika; Bhatti, Harbax S.; Singh, Vikramjeet
2017-11-01
In optical communication, the behavior of the ultrashort pulses of optical solitons can be described through nonlinear Schrodinger equation. This partial differential equation is widely used to contemplate a number of physically important phenomena, including optical shock waves, laser and plasma physics, quantum mechanics, elastic media, etc. The exact analytical solution of (1+n)-dimensional higher order nonlinear Schrodinger equation by He's variational iteration method has been presented. Our proposed solutions are very helpful in studying the solitary wave phenomena and ensure rapid convergent series and avoid round off errors. Different examples with graphical representations have been given to justify the capability of the method.
Nonlinear and linear wave equations for propagation in media with frequency power law losses
Szabo, Thomas L.
2003-10-01
The Burgers, KZK, and Westervelt wave equations used for simulating wave propagation in nonlinear media are based on absorption that has a quadratic dependence on frequency. Unfortunately, most lossy media, such as tissue, follow a more general frequency power law. The authors first research involved measurements of loss and dispersion associated with a modification to Blackstock's solution to the linear thermoviscous wave equation [J. Acoust. Soc. Am. 41, 1312 (1967)]. A second paper by Blackstock [J. Acoust. Soc. Am. 77, 2050 (1985)] showed the loss term in the Burgers equation for plane waves could be modified for other known instances of loss. The authors' work eventually led to comprehensive time-domain convolutional operators that accounted for both dispersion and general frequency power law absorption [Szabo, J. Acoust. Soc. Am. 96, 491 (1994)]. Versions of appropriate loss terms were developed to extend the standard three nonlinear wave equations to these more general losses. Extensive experimental data has verified the predicted phase velocity dispersion for different power exponents for the linear case. Other groups are now working on methods suitable for solving wave equations numerically for these types of loss directly in the time domain for both linear and nonlinear media.
The non-local Fisher–KPP equation: travelling waves and steady states
Berestycki, Henri; Nadin, Grégoire; Perthame, Benoit; Ryzhik, Lenya
2009-01-01
We consider the Fisher–KPP equation with a non-local saturation effect defined through an interaction kernel φ(x) and investigate the possible differences with the standard Fisher–KPP equation. Our first concern is the existence of steady states. We prove that if the Fourier transform φ-circumflex(ξ) is positive or if the length σ of the non-local interaction is short enough, then the only steady states are u ≡ 0 and u ≡ 1. Next, we study existence of the travelling waves. We prove that this equation admits travelling wave solutions that connect u = 0 to an unknown positive steady state u ∞ (x), for all speeds c ≥ c * . The travelling wave connects to the standard state u ∞ (x) ≡ 1 under the aforementioned conditions: φ-circumflex(ξ) > 0 or σ is sufficiently small. However, the wave is not monotonic for σ large
Evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations
Zhang, Linghai
2017-10-01
The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations ut =uxx + α [ βH (u - θ) - u ] - w, wt = ε (u - γw) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ], under different conditions on the model constants. To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0 1; the existence and instability of two standing wave fronts if 2 (1 + αγ) θ = αβγ and 0 traveling wave front as well as the existence and instability of a standing pulse solution if 0 traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0 traveling wave back of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ] -w0, where w0 = α (β - 2 θ) > 0 is a positive constant, if 0 motivation to study the existence, stability, instability and bifurcations of the nonlinear waves is to study the existence and stability/instability of infinitely many fast/slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations. The existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience.
Periodic and solitary-wave solutions of the Degasperis-Procesi equation
Vakhnenko, V.O.; Parkes, E.J.
2004-01-01
Travelling-wave solutions of the Degasperis-Procesi equation are investigated. The solutions are characterized by two parameters. For propagation in the positive x-direction, hump-like, inverted loop-like and coshoidal periodic-wave solutions are found; hump-like, inverted loop-like and peakon solitary-wave solutions are obtained as well. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. A transformed version of the Degasperis-Procesi equation, which is a generalization of the Vakhnenko equation, is also considered. For propagation in the positive x-direction, hump-like, loop-like, inverted loop-like, bell-like and coshoidal periodic-wave solutions are found; loop-like, inverted loop-like and kink-like solitary-wave solutions are obtained as well. For propagation in the negative x-direction, well-like and inverted coshoidal periodic-wave solutions are found; well-like and inverted peakon solitary-wave solutions are obtained as well. In an appropriate limit, the previously known solutions of the Vakhnenko equation are recovered
Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue
2018-01-01
This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0 . Furthermore, we prove the global existence and uniqueness of C^{α ,β } -solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1 -space. The exponential convergence rate is also derived.
Huang, Rui; Jin, Chunhua; Mei, Ming; Yin, Jingxue
2018-06-01
This paper deals with the existence and stability of traveling wave solutions for a degenerate reaction-diffusion equation with time delay. The degeneracy of spatial diffusion together with the effect of time delay causes us the essential difficulty for the existence of the traveling waves and their stabilities. In order to treat this case, we first show the existence of smooth- and sharp-type traveling wave solutions in the case of c≥c^* for the degenerate reaction-diffusion equation without delay, where c^*>0 is the critical wave speed of smooth traveling waves. Then, as a small perturbation, we obtain the existence of the smooth non-critical traveling waves for the degenerate diffusion equation with small time delay τ >0. Furthermore, we prove the global existence and uniqueness of C^{α ,β }-solution to the time-delayed degenerate reaction-diffusion equation via compactness analysis. Finally, by the weighted energy method, we prove that the smooth non-critical traveling wave is globally stable in the weighted L^1-space. The exponential convergence rate is also derived.
Smooth, cusped, and discontinuous traveling waves in the periodic fluid resonance equation
Kruse, Matthew Thomas
The principal motivation for this dissertation is to extend the study of small amplitude high frequency wave propagation in solutions for hyperbolic conservation laws begun by A. Majda and R. Rosales in 1984. It was then that Majda and Rosales obtained equations governing the leading order wave amplitudes of resonantly interacting weakly nonlinear high frequency wave trains in the compressible Euler equations. The equations were obtained through systematic application of multiple scales and result in a pair of nonlinear acoustic wave equations coupled through a convolution operator. The extended solutions satisfy a pair of inviscid Burgers' equations coupled via a spatial convolution operator. Since then, many mathematicians have used this technique to extend the time validity of solutions to systems of equations other than the Euler equations and have arrived at similar nonlinear non-local systems. This work attempts to look at some of the basic features of the linear and nonlinear coupled and decoupled non- local equations, offering some analytic solutions and numerical insight into the phenomena associated with these equations. We do so by examining a single non-local linear equation, and then a single equation coupling a Burgers' nonlinearity with a linear convolution operator. The linear case is completely solvable. Analytic solutions are provided along with numerical results showing the fundamental properties of the linear non- local equations. In the nonlinear case some analytic solutions, including steady state profiles and traveling wave solutions, are provided along with a battery of numerical simulations. Evidence indicates the existence of attractors for solutions of the single equation with a single mode kernel. Provided resonant interaction takes place, the profile of the attractor is uniquely dependent on the kernel alone. Hamiltonian equations are obtained for both the linear and nonlinear equations with the condition that the resonant kernel must
''Free-space'' boundary conditions for the time-dependent wave equation
Lindman, E.L.
1975-01-01
Boundary conditions for the discrete wave equation which act like an infinite region of free space in contact with the computational region can be constructed using projection operators. Propagating and evanescent waves coming from within the computational region generate no reflected waves as they cross the boundary. At the same time arbitrary waves may be launched into the computational region. Well known projection operators for one-dimensional waves may be used for this purpose in one dimension. Extensions of these operators to higher dimensions along with numerically efficient approximations to them are described for higher-dimensional problems. The separation of waves into ingoing and outgoing waves inherent in these boundary conditions greatly facilitates diagnostics
Watanabe, T; Sassa, K [Kyoto University, Kyoto (Japan); Uesaka, S [Kyoto University, Kyoto (Japan). Faculty of Engineering
1996-10-01
The effect of initial models on full-wave inversion (FWI) analysis based on acoustic wave-equation was studied for elastic wave tomography of underground structures. At present, travel time inversion using initial motion travel time is generally used, and inverse analysis is conducted using the concept `ray,` assuming very high wave frequency. Although this method can derive stable solutions relatively unaffected by initial model, it uses only the data of initial motion travel time. FWI calculates theoretical waveform at each receiver using all of observed waveforms as data by wave equation modeling where 2-D underground structure is calculated by difference calculus under the assumption that wave propagation is described by wave equation of P wave. Although it is a weak point that FWI is easily affected by noises in an initial model and data, it is featured by high resolution of solutions. This method offers very excellent convergence as a proper initial model is used, resulting in sufficient performance, however, it is strongly affected by initial model. 2 refs., 7 figs., 1 tab.
Orbital stability of periodic traveling-wave solutions for the log-KdV equation
Natali, Fábio; Pastor, Ademir; Cristófani, Fabrício
2017-09-01
In this paper we establish the orbital stability of periodic waves related to the logarithmic Korteweg-de Vries equation. Our motivation is inspired in the recent work [3], in which the authors established the well-posedness and the linear stability of Gaussian solitary waves. By using the approach put forward recently in [20] to construct a smooth branch of periodic waves as well as to get the spectral properties of the associated linearized operator, we apply the abstract theories in [13] and [25] to deduce the orbital stability of the periodic traveling waves in the energy space.
Stability of Planar Rarefaction Wave to 3D Full Compressible Navier-Stokes Equations
Li, Lin-an; Wang, Teng; Wang, Yi
2018-05-01
We prove time-asymptotic stability toward the planar rarefaction wave for the three-dimensional full, compressible Navier-Stokes equations with the heat-conductivities in an infinite long flat nozzle domain {R × T^2} . Compared with one-dimensional case, the proof here is based on our new observations on the cancellations on the flux terms and viscous terms due to the underlying wave structures, which are crucial for overcoming the difficulties due to the wave propagation in the transverse directions x 2 and x 3 and its interactions with the planar rarefaction wave in x 1 direction.
SIMULATION OF NEGATIVE PRESSURE WAVE PROPAGATION IN WATER PIPE NETWORK
Tang Van Lam
2017-11-01
Full Text Available Subject: factors such as pipe wall roughness, mechanical properties of pipe materials, physical properties of water affect the pressure surge in the water supply pipes. These factors make it difficult to analyze the transient problem of pressure evolution using simple programming language, especially in the studies that consider only the magnitude of the positive pressure surge with the negative pressure phase being neglected. Research objectives: determine the magnitude of the negative pressure in the pipes on the experimental model. The propagation distance of the negative pressure wave will be simulated by the valve closure scenarios with the help of the HAMMER software and it is compared with an experimental model to verify the quality the results. Materials and methods: academic version of the Bentley HAMMER software is used to simulate the pressure surge wave propagation due to closure of the valve in water supply pipe network. The method of characteristics is used to solve the governing equations of transient process of pressure change in the pipeline. This method is implemented in the HAMMER software to calculate the pressure surge value in the pipes. Results: the method has been applied for water pipe networks of experimental model, the results show the affected area of negative pressure wave from valve closure and thereby we assess the largest negative pressure that may appear in water supply pipes. Conclusions: the experiment simulates the water pipe network with a consumption node for various valve closure scenarios to determine possibility of appearance of maximum negative pressure value in the pipes. Determination of these values in real-life network is relatively costly and time-consuming but nevertheless necessary for identification of the risk of pipe failure, and therefore, this paper proposes using the simulation model by the HAMMER software. Initial calibration of the model combined with the software simulation results and
Quasiclassical deformation in KP hierarchy and Benney's long wave equations
Kolokol'tsov, V.N.; Lebedev, D.R.
1987-01-01
In the paper by means of the formal variant of Zakharov-Shabat ''dressing'' method various formulas are obtained for the generating functions of the conservation laws of Kadomtsev-Petvias hierarchy which turn into analogous formulas for Benney hierarchy in the quasiclassical limit. The generating fucntion of the conservation laws of Miura type is constructed for higher Benney equations and the simple proof of the related identities is given
Hypersonic Shock Wave Computations Using the Generalized Boltzmann Equation
Agarwal, Ramesh; Chen, Rui; Cheremisin, Felix G.
2006-11-01
Hypersonic shock structure in diatomic gases is computed by solving the Generalized Boltzmann Equation (GBE), where the internal and translational degrees of freedom are considered in the framework of quantum and classical mechanics respectively [1]. The computational framework available for the standard Boltzmann equation [2] is extended by including both the rotational and vibrational degrees of freedom in the GBE. There are two main difficulties encountered in computation of high Mach number flows of diatomic gases with internal degrees of freedom: (1) a large velocity domain is needed for accurate numerical description of the distribution function resulting in enormous computational effort in calculation of the collision integral, and (2) about 50 energy levels are needed for accurate representation of the rotational spectrum of the gas. Our methodology addresses these problems, and as a result the efficiency of calculations has increased by several orders of magnitude. The code has been validated by computing the shock structure in Nitrogen for Mach numbers up to 25 including the translational and rotational degrees of freedom. [1] Beylich, A., ``An Interlaced System for Nitrogen Gas,'' Proc. of CECAM Workshop, ENS de Lyon, France, 2000. [2] Cheremisin, F., ``Solution of the Boltzmann Kinetic Equation for High Speed Flows of a Rarefied Gas,'' Proc. of the 24th Int. Symp. on Rarefied Gas Dynamics, Bari, Italy, 2004.
Traveling wave solutions for two nonlinear evolution equations with nonlinear terms of any order
Feng Qing-Hua; Zhang Yao-Ming; Meng Fan-Wei
2011-01-01
In this paper, based on the known first integral method and the Riccati sub-ordinary differential equation (ODE) method, we try to seek the exact solutions of the general Gardner equation and the general Benjamin—Bona—Mahoney equation. As a result, some traveling wave solutions for the two nonlinear equations are established successfully. Also we make a comparison between the two methods. It turns out that the Riccati sub-ODE method is more effective than the first integral method in handling the proposed problems, and more general solutions are constructed by the Riccati sub-ODE method. (general)
On "new travelling wave solutions" of the KdV and the KdV-Burgers equations
Kudryashov, Nikolai A.
The Korteweg-de Vries and the Korteweg-de Vries-Burgers equations are considered. Using the travelling wave the general solutions of these equations are presented. "New travelling wave solutions" of the KdV and the KdV-Burgers equations by Wazzan [Wazzan L Commun Nonlinear Sci Numer Simulat
Quantum-Wave Equation and Heisenberg Inequalities of Covariant Quantum Gravity
Claudio Cremaschini
2017-07-01
Full Text Available Key aspects of the manifestly-covariant theory of quantum gravity (Cremaschini and Tessarotto 2015–2017 are investigated. These refer, first, to the establishment of the four-scalar, manifestly-covariant evolution quantum wave equation, denoted as covariant quantum gravity (CQG wave equation, which advances the quantum state ψ associated with a prescribed background space-time. In this paper, the CQG-wave equation is proved to follow at once by means of a Hamilton–Jacobi quantization of the classical variational tensor field g ≡ g μ ν and its conjugate momentum, referred to as (canonical g-quantization. The same equation is also shown to be variational and to follow from a synchronous variational principle identified here with the quantum Hamilton variational principle. The corresponding quantum hydrodynamic equations are then obtained upon introducing the Madelung representation for ψ , which provides an equivalent statistical interpretation of the CQG-wave equation. Finally, the quantum state ψ is proven to fulfill generalized Heisenberg inequalities, relating the statistical measurement errors of quantum observables. These are shown to be represented in terms of the standard deviations of the metric tensor g ≡ g μ ν and its quantum conjugate momentum operator.
Rational solutions to the KPI equation and multi rogue waves
Gaillard, Pierre
2016-04-01
We construct here rational solutions to the Kadomtsev-Petviashvili equation (KPI) as a quotient of two polynomials in x, y and t depending on several real parameters. This method provides an infinite hierarchy of rational solutions written in terms of polynomials of degrees 2 N(N + 1) in x, y and t depending on 2 N - 2 real parameters for each positive integer N. We give explicit expressions of the solutions in the simplest cases N = 1 and N = 2 and we study the patterns of their modulus in the (x , y) plane for different values of time t and parameters.
A four-equation friction model for water hammer calculation in quasi-rigid pipelines
Ghodhbani, Abdelaziz; Haj Taïeb, Ezzeddine
2017-01-01
Friction coupling affects water hammer evolution in pipelines according to the initial flow regime. Unsteady friction models are only validated with uncoupled formulation. On the other hand, coupled models such as four-equation model, provide more accurate prediction of water hammer since fluid-structure interaction (FSI) is taken into account, but they are limited to steady-state friction formulation. This paper deals with the creation of the “four-equation friction model” which is based on the incorporation of the unsteady head loss given by an unsteady friction model into the four-equation model. For transient laminar flow cases, the Zielke model is considered. The proposed model is applied to a quasi-rigid pipe with axial moving valve, and then calculated by the method of characteristics (MOC). Damping and shape of the numerical solution are in good agreement with experimental data. Thus, the proposed model can be incorporated into a new computer code. - Highlights: • Both Zielke model and four-equation model are insufficient to predict water hammer. • The four-equation friction model proposed is obtained by incorporating the unsteady head loss in the four-equation model. • The solution obtained by the proposed model is in good agreement with experimental data. • The wave-speed adjustment scheme is more efficient than interpolations schemes.
On a Shallow Water Equation Perturbed in Schwartz Class
Zhu Xiang’ou
2012-01-01
We discuss the Camassa-Holm equation perturbed in Schwartz class around a suitable constant. This paper is concerned with the wave breaking mechanism for periodic case where two special classes of initial data were involved. The asymptotic behavior of solutions is also analyzed in the following sense: the corresponding solution to initial data with algebraic decay at infinity will retain this property at infinity in its lifespan.
Visco-acoustic wave-equation traveltime inversion and its sensitivity to attenuation errors
Yu, Han
2018-02-23
A visco-acoustic wave-equation traveltime inversion method is presented that inverts for the shallow subsurface velocity distribution. Similar to the classical wave equation traveltime inversion, this method finds the velocity model that minimizes the squared sum of the traveltime residuals. Even though, wave-equation traveltime inversion can partly avoid the cycle skipping problem, a good initial velocity model is required for the inversion to converge to a reasonable tomogram with different attenuation profiles. When Q model is far away from the real model, the final tomogram is very sensitive to the starting velocity model. Nevertheless, a minor or moderate perturbation of the Q model from the true one does not strongly affect the inversion if the low wavenumber information of the initial velocity model is mostly correct. These claims are validated with numerical tests on both the synthetic and field data sets.
Visco-acoustic wave-equation traveltime inversion and its sensitivity to attenuation errors
Yu, Han; Chen, Yuqing; Hanafy, Sherif M.; Huang, Jiangping
2018-04-01
A visco-acoustic wave-equation traveltime inversion method is presented that inverts for the shallow subsurface velocity distribution. Similar to the classical wave equation traveltime inversion, this method finds the velocity model that minimizes the squared sum of the traveltime residuals. Even though, wave-equation traveltime inversion can partly avoid the cycle skipping problem, a good initial velocity model is required for the inversion to converge to a reasonable tomogram with different attenuation profiles. When Q model is far away from the real model, the final tomogram is very sensitive to the starting velocity model. Nevertheless, a minor or moderate perturbation of the Q model from the true one does not strongly affect the inversion if the low wavenumber information of the initial velocity model is mostly correct. These claims are validated with numerical tests on both the synthetic and field data sets.
Smooth and non-smooth traveling wave solutions of a class of nonlinear dispersive equation
Zhao Xiaoshan; Wu Aidi; He Wenzhang
2009-01-01
There is the widespread existence of wave phenomena in physics, mechanics. This clearly necessitates a study of traveling waves in depth and of the modeling and analysis involved. In this paper, we study a nonlinear dispersive K(n,-n,2n) equation, which can be regarded as a generalized K(n,n) equation. Applying the bifurcation theory and the method of phase portraits analysis, we obtain the dynamical behavior and special exact solutions of the K(n,-n,2n) equation. As a result, the conditions under which peakon and compacton solutions appear are also given and the analytic expressions of peakon solutions, compacton and periodic cusp wave solutions are obtained.
On the interaction of deep water waves and exponential shear currents
Cheng, Jun; Cang, Jie; Liao, Shi-Jun
2009-05-01
A train of periodic deep-water waves propagating on a steady shear current with a vertical distribution of vorticity is investigated by an analytic method, namely the homotopy analysis method (HAM). The magnitude of the vorticity varies exponentially with the magnitude of the stream function, while remaining constant on a particular streamline. The so-called Dubreil-Jacotin transformation is used to transfer the original exponentially nonlinear boundary-value problem in an unknown domain into an algebraically nonlinear boundary-value problem in a known domain. Convergent series solutions are obtained not only for small amplitude water waves on a weak current but also for large amplitude waves on a strong current. The nonlinear wave-current interaction is studied in detail. It is found that an aiding shear current tends to enlarge the wave phase speed, sharpen the wave crest, but shorten the maximum wave height, while an opposing shear current has the opposite effect. Besides, the amplitude of waves and fluid velocity decay over the depth more quickly on an aiding shear current but more slowly on an opposing shear current than that of waves on still water. Furthermore, it is found that Stokes criteria of wave breaking is still valid for waves on a shear current: a train of propagating waves on a shear current breaks as the fiuid velocity at crest equals the wave phase speed. Especially, it is found that the highest waves on an opposing shear current are even higher and steeper than that of waves on still water. Mathematically, this analytic method is rather general in principle and can be employed to solve many types of nonlinear partial differential equations with variable coefficients in science, finance and engineering.
Stability properties of solitary waves for fractional KdV and BBM equations
Angulo Pava, Jaime
2018-03-01
This paper sheds new light on the stability properties of solitary wave solutions associated with Korteweg-de Vries-type models when the dispersion is very low. Using a compact, analytic approach and asymptotic perturbation theory, we establish sufficient conditions for the existence of exponentially growing solutions to the linearized problem and so a criterium of spectral instability of solitary waves is obtained for both models. Moreover, the nonlinear stability and spectral instability of the ground state solutions for both models is obtained for some specific regimen of parameters. Via a Lyapunov strategy and a variational analysis, we obtain the stability of the blow-up of solitary waves for the critical fractional KdV equation. The arguments presented in this investigation show promise for use in the study of the instability of traveling wave solutions of other nonlinear evolution equations.
Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria
Frieman, E.A.; Chen, L.
1981-10-01
A nonlinear gyrokinetic formalism for low-frequency (less than the cyclotron frequency) microscopic electromagnetic perturbations in general magnetic field configurations is developed. The nonlinear equations thus derived are valid in the strong-turbulence regime and contain effects due to finite Larmor radius, plasma inhomogeneities, and magentic field geometries. The specific case of axisymmetric tokamaks is then considered, and a model nonlinear equation is derived for electrostatic drift waves. Also, applying the formalism to the shear Alfven wave heating sceme, it is found that nonlinear ion Landau damping of kinetic shear-Alfven waves is modified, both qualitatively and quantitatively, by the diamagnetic drift effects. In particular, wave energy is found to cascade in wavenumber instead of frequency
Relativistic n-body wave equations in scalar quantum field theory
Emami-Razavi, Mohsen
2006-01-01
The variational method in a reformulated Hamiltonian formalism of Quantum Field Theory (QFT) is used to derive relativistic n-body wave equations for scalar particles (bosons) interacting via a massive or massless mediating scalar field (the scalar Yukawa model). Simple Fock-space variational trial states are used to derive relativistic n-body wave equations. The equations are shown to have the Schroedinger non-relativistic limits, with Coulombic interparticle potentials in the case of a massless mediating field and Yukawa interparticle potentials in the case of a massive mediating field. Some examples of approximate ground state solutions of the n-body relativistic equations are obtained for various strengths of coupling, for both massive and massless mediating fields
Shock formation in small-data solutions to 3D quasilinear wave equations
Speck, Jared
2016-01-01
In 1848 James Challis showed that smooth solutions to the compressible Euler equations can become multivalued, thus signifying the onset of a shock singularity. Today it is known that, for many hyperbolic systems, such singularities often develop. However, most shock-formation results have been proved only in one spatial dimension. Serge Alinhac's groundbreaking work on wave equations in the late 1990s was the first to treat more than one spatial dimension. In 2007, for the compressible Euler equations in vorticity-free regions, Demetrios Christodoulou remarkably sharpened Alinhac's results and gave a complete description of shock formation. In this monograph, Christodoulou's framework is extended to two classes of wave equations in three spatial dimensions. It is shown that if the nonlinear terms fail to satisfy the null condition, then for small data, shocks are the only possible singularities that can develop. Moreover, the author exhibits an open set of small data whose solutions form a shock, and he prov...
Lamb, George L
1995-01-01
INTRODUCTORY APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS. With Emphasis on Wave Propagation and Diffusion. This is the ideal text for students and professionals who have some familiarity with partial differential equations, and who now wish to consolidate and expand their knowledge. Unlike most other texts on this topic, it interweaves prior knowledge of mathematics and physics, especially heat conduction and wave motion, into a presentation that demonstrates their interdependence. The result is a superb teaching text that reinforces the reader's understanding of both mathematics and physic
The Multi-Wave Method for Exact Solutions of Nonlinear Partial Differential Equations
Yusuf Pandir
2018-02-01
Full Text Available In this research, we use the multi-wave method to obtain new exact solutions for generalized forms of 5th order KdV equation and fth order KdV (fKdV equation with power law nonlinearity. Computations are performed with the help of the mathematics software Mathematica. Then, periodic wave solutions, bright soliton solutions and rational function solutions with free parameters are obtained by this approach. It is shown that this method is very useful and effective.
Perfectly Matched Layer for the Wave Equation Finite Difference Time Domain Method
Miyazaki, Yutaka; Tsuchiya, Takao
2012-07-01
The perfectly matched layer (PML) is introduced into the wave equation finite difference time domain (WE-FDTD) method. The WE-FDTD method is a finite difference method in which the wave equation is directly discretized on the basis of the central differences. The required memory of the WE-FDTD method is less than that of the standard FDTD method because no particle velocity is stored in the memory. In this study, the WE-FDTD method is first combined with the standard FDTD method. Then, Berenger's PML is combined with the WE-FDTD method. Some numerical demonstrations are given for the two- and three-dimensional sound fields.
Stochastic wave-function unravelling of the generalized Lindblad equation using correlated states
Moodley, Mervlyn; Nsio Nzundu, T; Paul, S
2012-01-01
We perform a stochastic wave-function unravelling of the generalized Lindblad master equation using correlated states, a combination of the system state vectors and the environment population. The time-convolutionless projection operator method using correlated projection superoperators is applied to a two-state system, a qubit, that is coupled to an environment consisting of two energy bands which are both populated. These results are compared to the data obtained from Monte Carlo wave-function simulations based on the unravelling of the master equation. We also show a typical quantum trajectory and the average time evolution of the state vector on the Bloch sphere. (paper)
NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION.
Liu, F; Meerschaert, M M; McGough, R J; Zhuang, P; Liu, Q
2013-03-01
In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.
NUMERICAL METHODS FOR SOLVING THE MULTI-TERM TIME-FRACTIONAL WAVE-DIFFUSION EQUATION
Liu, F.; Meerschaert, M.M.; McGough, R.J.; Zhuang, P.; Liu, Q.
2013-01-01
In this paper, the multi-term time-fractional wave-diffusion equations are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and technique...
Baumeister, K. J.
1983-01-01
A time-dependent finite difference formulation to the inhomogeneous wave equation is derived for plane wave propagation with harmonic noise sources. The difference equation and boundary conditions are developed along with the techniques to simulate the Dirac delta function associated with a concentrated noise source. Example calculations are presented for the Green's function and distributed noise sources. For the example considered, the desired Fourier transformed acoustic pressures are determined from the transient pressures by use of a ramping function and an integration technique, both of which eliminates the nonharmonic pressure associated with the initial transient.
Baumeiste, K. J.
1983-01-01
A time-dependent finite difference formulation to the inhomogeneous wave equation is derived for plane wave propagation with harmonic noise sources. The difference equation and boundary conditions are developed along with the techniques to simulate the Dirac delta function associated with a concentrated noise source. Example calculations are presented for the Green's function and distributed noise sources. For the example considered, the desired Fourier transformed acoustic pressures are determined from the transient pressures by use of a ramping function and an integration technique, both of which eliminates the nonharmonic pressure associated with the initial transient.
Multiple travelling wave solutions of nonlinear evolution equations using a unified algebraic method
Fan Engui
2002-01-01
A new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system. Compared with most of the existing tanh methods, the Jacobi elliptic function method or other sophisticated methods, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the travelling wave solutions according to the values of some parameters. The solutions obtained in this paper include (a) kink-shaped and bell-shaped soliton solutions, (b) rational solutions, (c) triangular periodic solutions and (d) Jacobi and Weierstrass doubly periodic wave solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. The efficiency of the method can be demonstrated on a large variety of nonlinear evolution equations such as those considered in this paper, KdV-MKdV, Ito's fifth MKdV, Hirota, Nizhnik-Novikov-Veselov, Broer-Kaup, generalized coupled Hirota-Satsuma, coupled Schroedinger-KdV, (2+1)-dimensional dispersive long wave, (2+1)-dimensional Davey-Stewartson equations. In addition, as an illustrative sample, the properties of the soliton solutions and Jacobi doubly periodic solutions for the Hirota equation are shown by some figures. The links among our proposed method, the tanh method, extended tanh method and the Jacobi elliptic function method are clarified generally. (author)
Three-dimensional wave-induced current model equations and radiation stresses
Xia, Hua-yong
2017-08-01
After the approach by Mellor (2003, 2008), the present paper reports on a repeated effort to derive the equations for three-dimensional wave-induced current. Via the vertical momentum equation and a proper coordinate transformation, the phase-averaged wave dynamic pressure is well treated, and a continuous and depth-dependent radiation stress tensor, rather than the controversial delta Dirac function at the surface shown in Mellor (2008), is provided. Besides, a phase-averaged vertical momentum flux over a sloping bottom is introduced. All the inconsistencies in Mellor (2003, 2008), pointed out by Ardhuin et al. (2008) and Bennis and Ardhuin (2011), are overcome in the presently revised equations. In a test case with a sloping sea bed, as shown in Ardhuin et al. (2008), the wave-driving forces derived in the present equations are in good balance, and no spurious vertical circulation occurs outside the surf zone, indicating that Airy's wave theory and the approach of Mellor (2003, 2008) are applicable for the derivation of the wave-induced current model.
Soliton and periodic solutions for higher order wave equations of KdV type (I)
Khuri, S.A.
2005-01-01
The aim of the paper is twofold. First, a new ansaetze is introduced for the construction of exact solutions for higher order wave equations of KdV type (I). We show the existence of a class of discontinuous soliton solutions with infinite spikes. Second, the projective Riccati technique is implemented as an alternate approach for obtaining new exact solutions, solitary solutions, and periodic wave solutions
Travelling water waves with compactly supported vorticity
Shatah, Jalal; Walsh, Samuel; Zeng, Chongchun
2013-01-01
In this paper, we prove the existence of two-dimensional, travelling, capillary-gravity, water waves with compactly supported vorticity. Specifically, we consider the cases where the vorticity is a δ-function (a point vortex), or has small compact support (a vortex patch). Using a global bifurcation theoretic argument, we construct a continuum of finite-amplitude, finite-vorticity solutions for the periodic point vortex problem. For the non-periodic case, with either a vortex point or patch, we prove the existence of a continuum of small-amplitude, small-vorticity solutions. (paper)
Angel, Jordan B.; Banks, Jeffrey W.; Henshaw, William D.
2018-01-01
High-order accurate upwind approximations for the wave equation in second-order form on overlapping grids are developed. Although upwind schemes are well established for first-order hyperbolic systems, it was only recently shown by Banks and Henshaw [1] how upwinding could be incorporated into the second-order form of the wave equation. This new upwind approach is extended here to solve the time-domain Maxwell's equations in second-order form; schemes of arbitrary order of accuracy are formulated for general curvilinear grids. Taylor time-stepping is used to develop single-step space-time schemes, and the upwind dissipation is incorporated by embedding the exact solution of a local Riemann problem into the discretization. Second-order and fourth-order accurate schemes are implemented for problems in two and three space dimensions, and overlapping grids are used to treat complex geometry and problems with multiple materials. Stability analysis of the upwind-scheme on overlapping grids is performed using normal mode theory. The stability analysis and computations confirm that the upwind scheme remains stable on overlapping grids, including the difficult case of thin boundary grids when the traditional non-dissipative scheme becomes unstable. The accuracy properties of the scheme are carefully evaluated on a series of classical scattering problems for both perfect conductors and dielectric materials in two and three space dimensions. The upwind scheme is shown to be robust and provide high-order accuracy.
Travelling wave solutions for some time-delayed equations through factorizations
Fahmy, E.S.
2008-01-01
In this work, we use factorization method to find explicit particular travelling wave solutions for the following important nonlinear second-order partial differential equations: The generalized time-delayed Burgers-Huxley, time-delayed convective Fishers, and the generalized time-delayed Burgers-Fisher. Using the particular solutions for these equations we find the general solutions, two-parameter solution, as special cases
Combined solitary-wave solution for coupled higher-order nonlinear Schroedinger equations
Tian Jinping; Tian Huiping; Li Zhonghao; Zhou Guosheng
2004-01-01
Coupled nonlinear Schroedinger equations model several interesting physical phenomena. We used a trigonometric function transform method based on a homogeneous balance to solve the coupled higher-order nonlinear Schroedinger equations. We obtained four pairs of exact solitary-wave solutions including a dark and a bright-soliton pair, a bright- and a dark-soliton pair, a bright- and a bright-soliton pair, and the last pair, a combined bright-dark-soliton pair
THE NEW SOLUTION OF TIME FRACTIONAL WAVE EQUATION WITH CONFORMABLE FRACTIONAL DERIVATIVE DEFINITION
Çenesiz, Yücel; Kurt, Ali
2015-01-01
– In this paper, we used new fractional derivative definition, the conformable fractional derivative, for solving two and three dimensional time fractional wave equation. This definition is simple and very effective in the solution procedures of the fractional differential equations that have complicated solutions with classical fractional derivative definitions like Caputo, Riemann-Liouville and etc. The results show that conformable fractional derivative definition is usable and convenient ...
Abundant families of new traveling wave solutions for the coupled Drinfel'd-Sokolov-Wilson equation
Yao Yuqin
2005-01-01
The generalized Jacobi elliptic function method is further improved by introducing an elliptic function φ(ξ) as a new independent variable and it is easy to calculate the over-determined equations. Abundant new traveling wave solutions of the coupled Drinfel'd-Sokolov-Wilson equation are obtained. The solutions obtained include the kink-shaped solutions, bell-shaped solutions, singular solutions and periodic solutions
On exact solitary wave solutions of the nonlinear Schroedinger equation with a source
Raju, T Solomon; Kumar, C Nagaraja; Panigrahi, Prasanta K
2005-01-01
We use a fractional transformation to connect the travelling wave solutions of the nonlinear Schroedinger equation (NLSE), phase locked with a source, to the elliptic equations satisfying, f-Prime ± af ± λf 3 = 0. The solutions are necessarily of rational form, containing both trigonometric and hyperbolic types as special cases. Bright and dark solitons, as well as singular solitons, are obtained in a suitable range of parameter values. (letter to the editor)
Perturbation theory for the Bethe-Salpeter equation in the field of a plane electromagnetic wave
Starostin, V.S.; Litskevich, I.K.
1990-01-01
The completeness and orthogonality of the solutions of the Bethe-Salpeter equation is proven. A correct derivation of perturbation-theory equations is given. A generalization that includes the field of a plane electromagnetic wave is proposed. The rate of one-photon annihilation of positronium in this field is calculated. If the one-photon decay is allowed, the stationary states of the system are found (states of light-positronium)
2013-09-30
Abstracts, Vol. 15, EGU2013- 5284-1, 2013, EGU General Assembly 2013 [published] [P6] S. I. Badulin and V. G. Grigorieva, Weak turbulence theory as... generalization of compact 1D water waves equation for 2D situation; study of the implications of modulational instability on solitons, rogue waves and air...integrability of 1-D Zakharov equation and generalization of compact equation for almost 1-D waves • Study of modulational instability and its
Boundary value problems in time for wave equations on RN
M. W. Smiley
1990-01-01
Full Text Available Let Lλ denote the linear operator associated with the radially symmetric form of the wave operator ∂t2−Δ+λ together with the side conditions of decay to zero as r=‖x‖→+∞ and T-periodicity in time. Thus Lλω=ωtt−(ωrr+N−1rωr+λω, when there are N space variables. For δ,R,T>0 let DT,R=(0,T×(R,+∞ and Lδ2(D denote the weighted L2 space with weight function exp(δr. It is shown that Lλ is a Fredholm operator from dom(Lλ⊂L2(D onto Lδ2(D with non-negative index depending on λ. If [2πj/T]2<λ≤[2π(j+1/T]2 then the index is 2j+1. In addition it is shown that Lλ has a bounded partial inverse Kλ:Lδ2(D→Hδ1(D⋂Lδ∞(D, with all spaces weighted by the function exp(δr. This provides a key ingredient for the analysis of nonlinear problems via the method of alternative problems.
Multi-wave solutions of the space–time fractional Burgers and Sharma–Tasso–Olver equations
Emad A.-B. Abdel-Salam
2016-03-01
Full Text Available Based on the improved generalized exp-function method, the space–time fractional Burgers and Sharma–Tasso–Olver equations were studied. The single-wave, double-wave, three-wave and four-wave solution discussed. With the best of our knowledge, some of the results are obtained for the first time. The improved generalized exp-function method can be applied to other fractional differential equations.
Multi-wave solutions of the space–time fractional Burgers and Sharma–Tasso–Olver equations
Emad A.-B. Abdel-Salam; Gamal F. Hassan
2016-01-01
Based on the improved generalized exp-function method, the space–time fractional Burgers and Sharma–Tasso–Olver equations were studied. The single-wave, double-wave, three-wave and four-wave solution discussed. With the best of our knowledge, some of the results are obtained for the first time. The improved generalized exp-function method can be applied to other fractional differential equations.
Favrie, N.; Gavrilyuk, S.
2017-07-01
A new numerical method for solving the Serre-Green-Naghdi (SGN) equations describing dispersive waves on shallow water is proposed. From the mathematical point of view, the SGN equations are the Euler-Lagrange equations for a ‘master’ lagrangian submitted to a differential constraint which is the mass conservation law. One major numerical challenge in solving the SGN equations is the resolution of an elliptic problem at each time instant. This is the most time-consuming part of the numerical method. The idea is to replace the ‘master’ lagrangian by a one-parameter family of ‘augmented’ lagrangians, depending on a greater number of variables, for which the corresponding Euler-Lagrange equations are hyperbolic. In such an approach, the ‘master’ lagrangian is recovered by the augmented lagrangian in some limit (for example, when the corresponding parameter is large). The choice of such a family of augmented lagrangians is proposed and discussed. The corresponding hyperbolic system is numerically solved by a Godunov type method. Numerical solutions are compared with exact solutions to the SGN equations. It appears that the computational time in solving the hyperbolic system is much lower than in the case where the elliptic operator is inverted. The new method is applied, in particular, to the study of ‘Favre waves’ representing non-stationary undular bores produced after reflection of the fluid flow with a free surface at an immobile wall.
Travelling wave solutions in a class of generalized Korteweg-de Vries equation
Shen Jianwei; Xu Wei
2007-01-01
In this paper, we consider a new generalization of KdV equation u t = u x u l-2 + α[2u xxx u p + 4pu p-1 u x u xx + p(p - 1)u p-2 (u x ) 3 ] and investigate its bifurcation of travelling wave solutions. From the above analysis, we know that there exists compacton and cusp waves in the system. We explain the reason that these non-smooth travelling wave solution arise by using the bifurcation theory
Four-dimensional integral equations for the MHD diffraction waves in plasma
Alexandrova, A.A.; Khizhnyak, N.A.
2000-01-01
The superficial analysis of the boundary-value nonstationary problem for Alfven wave has shown the principal possibility of using the method of evolutionary integral equations of non-stationary macroscopic electrodynamical in a case of MHD description of waves in plasma. With the importance of strict mathematical solutions obtained for simple model problems that is the diffraction of one separately taken Alfven wave is that it can be the basis for construction of the approximate solutions of more complex boundary-value problems
Arevalo, Edward, E-mail: arevalo@temf.tu-darmstadt.d [Technische Universitaet Darmstadt, Institut fuer Theorie elektromagnetischer Felder, TEMF, Schlossgartenstr. 8, D-64289 Darmstadt (Germany)
2009-09-21
The effect of instability on the propagation of solitary waves along one-dimensional discrete nonlinear Schroedinger equation with cubic nonlinearity is revisited. A self-contained quasicontinuum approximation is developed to derive closed-form expressions for small-amplitude solitary waves. The notion that the existence of nonlinear solitary waves in discrete systems is a signature of the modulation instability is used. With the help of this notion we conjecture that instability effects on moving solitons can be qualitative estimated from the analytical solutions. Results from numerical simulations are presented to support this conjecture.
Classification of homoclinic rogue wave solutions of the nonlinear Schrödinger equation
Osborne, A. R.
2014-01-01
Certain homoclinic solutions of the nonlinear Schrödinger (NLS) equation, with spatially periodic boundary conditions, are the most common unstable wave packets associated with the phenomenon of oceanic rogue waves. Indeed the homoclinic solutions due to Akhmediev, Peregrine and Kuznetsov-Ma are almost exclusively used in scientific and engineering applications. Herein I investigate an infinite number of other homoclinic solutions of NLS and show that they reduce to the above three classical homoclinic solutions for particular spectral values in the periodic inverse scattering transform. Furthermore, I discuss another infinity of solutions to the NLS equation that are not classifiable as homoclinic solutions. These latter are the genus-2N theta function solutions of the NLS equation: they are the most general unstable spectral solutions for periodic boundary conditions. I further describe how the homoclinic solutions of the NLS equation, for N = 1, can be derived directly from the theta functions in a particular limit. The solutions I address herein are actual spectral components in the nonlinear Fourier transform theory for the NLS equation: The periodic inverse scattering transform. The main purpose of this paper is to discuss a broader class of rogue wave packets1 for ship design, as defined in the Extreme Seas program. The spirit of this research came from D. Faulkner (2000) who many years ago suggested that ship design procedures, in order to take rogue waves into account, should progress beyond the use of simple sine waves. 1An overview of other work in the field of rogue waves is given elsewhere: Osborne 2010, 2012 and 2013. See the books by Olagnon and colleagues 2000, 2004 and 2008 for the Brest meetings. The books by Kharif et al. (2008) and Pelinovsky et al. (2010) are excellent references.
Stability of plane wave solutions of the two-space-dimensional nonlinear Schroedinger equation
Martin, D.U.; Yuen, H.C.; Saffman, P.G.
1980-01-01
The stability of plane, periodic solutions of the two-dimensional nonlinear Schroedinger equation to infinitesimal, two-dimensional perturbation has been calculated and verified numerically. For standing wave disturbances, instability is found for both odd and even modes; as the period of the unperturbed solution increases, the instability associated with the odd modes remains but that associated with the even mode disappears, which is consistent with the results of Zakharov and Rubenchik, Saffman and Yuen and Ablowitz and Segur on the stability of solitons. In addition, we have identified travelling wave instabilities for the even mode perturbations which are absent in the long-wave limit. Extrapolation to the case of an unperturbed solution with infinite period suggests that these instabilities may also be present for the soliton. In other words, the soliton is unstable to odd, standing-wave perturbations, and very likely also to even, travelling-wave perturbations. (orig.)
Soliton on a cnoidal wave background in the coupled nonlinear Schroedinger equation
Shin, H J
2004-01-01
An application of the Darboux transformation on a cnoidal wave background in the coupled nonlinear Schroedinger equation gives a new solution which describes a soliton moving on a cnoidal wave. This is a generalized version of the previously known soliton solutions of dark-bright pair. Here a dark soliton resides on a cnoidal wave instead of on a constant background. It also exhibits a new type of soliton solution in a self-focusing medium, which describes a breakup of a generalized dark-bright pair into another generalized dark-bright pair and an 'oscillating' soliton. We calculate the shift of the crest of the cnoidal wave along a soliton and the moving direction of the soliton on a cnoidal wave
Complex dynamical behaviors of compact solitary waves in the perturbed mKdV equation
Yin Jiu-Li; Xing Qian-Qian; Tian Li-Xin
2014-01-01
In this paper, we give a detailed discussion about the dynamical behaviors of compact solitary waves subjected to the periodic perturbation. By using the phase portrait theory, we find one of the nonsmooth solitary waves of the mKdV equation, namely, a compact solitary wave, to be a weak solution, which can be proved. It is shown that the compact solitary wave easily turns chaotic from the Melnikov theory. We focus on the sufficient conditions by keeping the system stable through selecting a suitable controller. Furthermore, we discuss the chaotic threshold for a perturbed system. Numerical simulations including chaotic thresholds, bifurcation diagrams, the maximum Lyapunov exponents, and phase portraits demonstrate that there exists a special frequency which has a great influence on our system; with the increase of the controller strength, chaos disappears in the perturbed system. But if the controller strength is sufficiently large, the solitary wave vibrates violently. (general)
Modified wave operators for nonlinear Schrodinger equations in one and two dimensions
Nakao Hayashi
2004-04-01
Full Text Available We study the asymptotic behavior of solutions, in particular the scattering theory, for the nonlinear Schr"{o}dinger equations with cubic and quadratic nonlinearities in one or two space dimensions. The nonlinearities are summation of gauge invariant term and non-gauge invariant terms. The scattering problem of these equations belongs to the long range case. We prove the existence of the modified wave operators to those equations for small final data. Our result is an improvement of the previous work [13
Traveling waves and the renormalization group improvedBalitsky-Kovchegov equation
Enberg, Rikard
2006-12-01
I study the incorporation of renormalization group (RG)improved BFKL kernels in the Balitsky-Kovchegov (BK) equation whichdescribes parton saturation. The RG improvement takes into accountimportant parts of the next-to-leading and higher order logarithmiccorrections to the kernel. The traveling wave front method for analyzingthe BK equation is generalized to deal with RG-resummed kernels,restricting to the interesting case of fixed QCD coupling. The resultsshow that the higher order corrections suppress the rapid increase of thesaturation scale with increasing rapidity. I also perform a "diffusive"differential equation approximation, which illustrates that someimportant qualitative properties of the kernel change when including RGcorrections.
Travelling wave solutions of the Schamel–Korteweg–de Vries and the Schamel equations
Figen Kangalgil
2016-10-01
Full Text Available In this paper, the extended (G′/G-expansion method has been suggested for constructing travelling wave solutions of the Schamel–Korteweg–de Vries (s-KdV and the Schamel equations with aid of computer systems like Maple or Mathematica. The hyperbolic function solutions and the trigonometric function solutions with free parameters of these equations have been obtained. Moreover, it has been shown that the suggested method is elementary, effective and has been used to solve nonlinear evolution equations in applied mathematics, engineering and mathematical physics.
New multidimensional partially integrable generalization of S-integrable N-wave equation
Zenchuk, A. I.
2007-01-01
This paper develops a modification of the dressing method based on the inhomogeneous linear integral equation with integral operator having nonempty kernel. The method allows one to construct the systems of multidimensional partial differential equations having differential polynomial structure in any dimension n. The associated solution space is not full, although it is parametrized by certain number of arbitrary functions of (n-1) variables. We consider four-dimensional generalization of the classical (2+1)-dimensional S-integrable N-wave equation as an example
Solving the KPI wave equation with a moving adaptive FEM grid
Granville Sewell
2013-04-01
Full Text Available The Kadomtsev-Petviashvili I (KPI equation is the difficult nonlinear wave equation $U_{xt} + 6U_x^2 + 6UU_{xx} + U_{xxxx} = 3U_{yy}.$ We solve this equation using PDE2D (www.pde2d.com with initial conditions consisting of two lump solitons, which collide and reseparate. Since the solution has steep, moving, peaks, an adaptive finite element grid is used with a grading which moves with the peaks.
Janssen, T.T.; Herbers, T.H.C.; Battjes, J.A.
2008-01-01
We present a stochastic model for the evolution of random ocean surface waves in coastal waters with complex seafloor topography. First, we derive a deterministic coupled-mode model based on a forward scattering approximation of the nonlinear mild slope equation; this model describes the evolution
Andrews, P.L.; Perkins, F.W.
1983-01-01
The investigation of the scattering of lower-hybrid waves by density fluctuations arising from drift waves in tokamaks is distinguished by the presence in the wave equation of a large, random, derivative-coupling term. The propagation of the lower-hybrid waves is well represented by a radiative transfer equation when the scale size of the density fluctuations is small compared to the overall plasma size. The radiative transfer equation is solved in two limits: first, the forward scattering limit, where the scale size of density fluctuations is large compared to the lower-hybrid perpendicular wavelength, and second, the large-angle scattering limit, where this inequality is reversed. The most important features of these solutions are well represented by analytical formulas derived by simple arguments. Based on conventional estimates for density fluctuations arising from drift waves and a parabolic density profile, the optical depth tau for scattering through a significant angle, is given by tauroughly-equal(2/N 2 /sub parallel/) (#betta#/sub p/i0/#betta#) 2 (m/sub e/c 2 /2T/sub i/)/sup 1/2/ [c/α(Ω/sub i/Ω/sub e/)/sup 1/2/ ], where #betta#/sub p/i0 is the central ion plasma frequency and T/sub i/ denotes the ion temperature near the edge of the plasma. Most of the scattering occurs near the surface. The transmission through the scattering region scales as tau - 1 and the emerging intensity has an angular spectrum proportional to cos theta, where sin theta = k/sub perpendicular/xB/sub p//(k/sub perpendicular/B/sub p/), and B/sub p/ is the poloidal field
Antisymmetrized molecular dynamics of wave packets with stochastic incorporation of Vlasov equation
Ono, Akira; Horiuchi, Hisashi.
1996-01-01
The first purpose of this report is to present an extended AMD model which can generally describe such minor branching processes by removing the restriction on the one-body distribution function. This is done not by generalizing the wave packets to arbitrary single-particle wave functions but by representing the diffused and/or deformed wave packet as an ensemble of Gaussian wave packets. In other words, stochastic displacements are given to the wave packets in phase space so that the ensemble-average of the time evolution of the one-body distribution function is essentially equivalent to the solution of Vlasov equation which does not have any restriction on the shape of wave packets. This new model is called AMD-V. Although AMD-V is equivalent to Vlasov equation in the instantaneous time evolution of the one-body distribution function for an AMD wave function, AMD-V describes the branching into channels and the fluctuation of the mean field which are caused by the spreading or the splitting of the single-particle wave function. The second purpose of this report is to show the drastic effect of this new stochastic process of wave packet splitting on the dynamics of heavy ion collisions, especially in the fragmentation mechanism. We take the 40 Ca + 40 Ca system at the incident energy 35 MeV/nucleon. It will be shown that the reproduction of data by the AMD-V calculation is surprisingly good. We will see that the effect of the wave packet diffusion is crucially important to remove the spurious binary feature of the AMD calculation and to enable the multi-fragment final state. (J.P.N.)
Analysis of a time fractional wave-like equation with the homotopy analysis method
Xu Hang; Cang Jie
2008-01-01
The time fractional wave-like differential equation with a variable coefficient is studied analytically. By using a simple transformation, the governing equation is reduced to two fractional ordinary differential equations. Then the homotopy analysis method is employed to derive the solutions of these equations. The accurate series solutions are obtained. Especially, when h f =h g =-1, these solutions are exactly the same as those results given by the Adomian decomposition method. The present work shows the validity and great potential of the homotopy analysis method for solving nonlinear fractional differential equations. The basic idea described in this Letter is expected to be further employed to solve other similar nonlinear problems in fractional calculus
Asymmetric systems described by a pair of local covariant wave equations
Mallik, S [Bern Univ. (Switzerland). Inst. fuer Theoretische Physik
1979-07-16
A class of asymmetric solutions of the integrability conditions for systems obeying the Leutwyler-Stern pair of covariant wave equations is obtained. The class of unequal-mass systems described by these solutions does not embed the particle-antiparticle system behaving as a relativistic harmonic oscillator.
Poincare group and relativistic wave equations in 2+1 dimensions
Gitman, Dmitri M. [Instituto de Fisica, Universidade de Sao Paulo, Sao Paulo, SP (Brazil); Shelepin, A.L. [Moscow Institute of Radio Engenering, Electronics and Automation, Moscow (Russian Federation)
1997-09-07
Using the generalized regular representation, an explicit construction of the unitary irreducible representations of the (2+1)-Poincare group is presented. A detailed description of the angular momentum and spin in 2+1 dimensions is given. On this base the relativistic wave equations for all spins (including fractional) are constructed. (author)
Spectral Approach to Derive the Representation Formulae for Solutions of the Wave Equation
Gusein Sh. Guseinov
2012-01-01
Full Text Available Using spectral properties of the Laplace operator and some structural formula for rapidly decreasing functions of the Laplace operator, we offer a novel method to derive explicit formulae for solutions to the Cauchy problem for classical wave equation in arbitrary dimensions. Among them are the well-known d'Alembert, Poisson, and Kirchhoff representation formulae in low space dimensions.
A surface-integral-equation approach to the propagation of waves in EBG-based devices
Lancellotti, V.; Tijhuis, A.G.
2012-01-01
We combine surface integral equations with domain decomposition to formulate and (numerically) solve the problem of electromagnetic (EM) wave propagation inside finite-sized structures. The approach is of interest for (but not limited to) the analysis of devices based on the phenomenon of
Stabilization of the Wave Equation with Boundary Time-Varying Delay
Hao Li
2014-01-01
Full Text Available We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying delay term in the time-varying, weakly nonlinear boundary feedbacks. By the Riemannian geometry methods and a suitable assumption of nonlinearity, we obtain the uniform decay of the energy of the closed loop system.
On the relativistic transport equation for a discontinuity wave of multiplicity one
Giambo, Sebastiano; Palumbo, Annunziata
1980-01-01
In the framework of the theory of the singular hypersurfaces, the transport equation for the amplitude of a discontinuity wave, corresponding to a simple characteristic of a quasi-linear hyperbolic system, is established in the context of special relativity [fr
Exact traveling wave solutions for a new nonlinear heat transfer equation
Gao Feng
2017-01-01
Full Text Available In this paper, we propose a new non-linear partial differential equation to de-scribe the heat transfer problems at the extreme excess temperatures. Its exact traveling wave solutions are obtained by using Cornejo-Perez and Rosu method.
Exact explicit travelling wave solutions for (n + 1)-dimensional Klein-Gordon-Zakharov equations
Li Jibin
2007-01-01
Using the methods of dynamical systems for the (n + 1)-dimensional KGS nonlinear wave equations, five classes of exact explicit parametric representations of the bounded travelling solutions are obtained. To guarantee the existence of the above solutions, all parameter conditions are given
Initial boundary value problems of nonlinear wave equations in an exterior domain
Chen Yunmei.
1987-06-01
In this paper, we investigate the existence and uniqueness of the global solutions to the initial boundary value problems of nonlinear wave equations in an exterior domain. When the space dimension n >= 3, the unique global solution of the above problem is obtained for small initial data, even if the nonlinear term is fully nonlinear and contains the unknown function itself. (author). 10 refs
Shock-wave structure based on the Navier-Stokes-Fourier equations
Uribe, F. J.; Velasco, R. M.
2018-04-01
We use the Navier-Stokes-Fourier constitutive equations to study plane shock waves in dilute gases. It is shown that the experimental information on the normalized density profiles can be fit by using the so-called soft sphere model, in which the viscosity and thermal conductivity are proportional to a power of the temperature.
An Integral Spectral Representation of the Propagator for the Wave Equation in the Kerr Geometry
Finster, F.; Kamran, N.; Smoller, J.; Yau, S.-T.
2005-12-01
We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the solution as a superposition of solutions of the radial and angular ODEs which arise in the separation of variables. In particular, we prove completeness of the solutions of the separated ODEs.
Global existence of solutions for semilinear damped wave equation in 2-D exterior domain
Ikehata, Ryo
We consider a mixed problem of a damped wave equation utt-Δ u+ ut=| u| p in the two dimensional exterior domain case. Small global in time solutions can be constructed in the case when the power p on the nonlinear term | u| p satisfies p ∗=2Japon. 55 (2002) 33) plays an effective role.
From Fermat principle to wave equation - quantization of 'particle mechanics of light'
Ogawa, Naohisa
2004-01-01
The Fermat principle states that light chooses the temporally shortest path. The action for this 'motion' is the observed time, and it has no Lorentz invariance. In this Letter we show how this action can be obtained from a relativistic action of massive particle, and how the classical wave equation of light can be obtained from this action
Shang Yadong
2005-01-01
In this paper, the evolution equations with strong nonlinear term describing the resonance interaction between the long wave and the short wave are studied. Firstly, based on the qualitative theory and bifurcation theory of planar dynamical systems, all of the explicit and exact solutions of solitary waves are obtained by qualitative seeking the homoclinic and heteroclinic orbits for a class of Lienard equations. Then the singular travelling wave solutions, periodic travelling wave solutions of triangle functions type are also obtained on the basis of the relationships between the hyperbolic functions and that between the hyperbolic functions with the triangle functions. The varieties of structure of exact solutions of the generalized long-short wave equation with strong nonlinear term are illustrated. The methods presented here also suitable for obtaining exact solutions of nonlinear wave equations in multidimensions
Zhang, Yu; Li, Yan; Shao, Hao; Zhong, Yaozhao; Zhang, Sai; Zhao, Zongxi
2012-06-01
Band structure and wave localization are investigated for sea surface water waves over large-scale sand wave topography. Sand wave height, sand wave width, water depth, and water width between adjacent sand waves have significant impact on band gaps. Random fluctuations of sand wave height, sand wave width, and water depth induce water wave localization. However, random water width produces a perfect transmission tunnel of water waves at a certain frequency so that localization does not occur no matter how large a disorder level is applied. Together with theoretical results, the field experimental observations in the Taiwan Bank suggest band gap and wave localization as the physical mechanism of sea surface water wave propagating over natural large-scale sand waves.
Experimental Observation of Negative Effective Gravity in Water Waves
Hu, Xinhua; Yang, Jiong; Zi, Jian; Chan, C. T.; Ho, Kai-Ming
2013-01-01
The gravity of Earth is responsible for the formation of water waves and usually difficult to change. Although negative effective gravity was recently predicted theoretically in water waves, it has not yet been observed in experiments and remains a mathematical curiosity which is difficult to understand. Here we experimentally demonstrate that close to the resonant frequency of purposely-designed resonating units, negative effective gravity can occur for water waves passing through an array of resonators composing of bottom-mounted split tubes, resulting in the prohibition of water wave propagation. It is found that when negative gravity occurs, the averaged displacement of water surface in a unit cell of the array has a phase difference of π to that along the boundary of the unit cell, consistent with theoretical predictions. Our results provide a mechanism to block water waves and may find applications in wave energy conversion and coastal protection. PMID:23715132
The two-fermion relativistic wave equations of Constraint Theory in the Pauli-Schroedinger form
Mourad, J.; Sazdjian, H.
1994-01-01
The two-fermion relativistic wave equations of Constraint Theory are reduced, after expressing the components of the 4x4 matrix wave function in terms of one of the 2x2 components, to a single equation of the Pauli-Schroedinger type, valid for all sectors of quantum numbers. The potentials that are present belong to the general classes of scalar, pseudoscalar and vector interactions and are calculable in perturbation theory from Feynman diagrams. In the limit when one of the masses becomes infinite, the equation reduces to the two-component form of the one-particle Dirac equation with external static potentials. The Hamiltonian, to order 1/c 2 , reproduces most of the known theoretical results obtained by other methods. The gauge invariance of the wave equation is checked, to that order, in the case of QED. The role of the c.m. energy dependence of the relativistic interquark confining potential is emphasized and the structure of the Hamiltonian, to order 1/c 2 , corresponding to confining scalar potentials, is displayed. (authors). 32 refs., 2 figs
Finite-dimensional attractor for a composite system of wave/plate equations with localized damping
Bucci, Francesca; Toundykov, Daniel
2010-01-01
The long-term behaviour of solutions to a model for acoustic–structure interactions is addressed; the system consists of coupled semilinear wave (3D) and plate equations with nonlinear damping and critical sources. The questions of interest are the existence of a global attractor for the dynamics generated by this composite system as well as dimensionality and regularity of the attractor. A distinct and challenging feature of the problem is the geometrically restricted dissipation on the wave component of the system. It is shown that the existence of a global attractor of finite fractal dimension—established in a previous work by Bucci et al (2007 Commun. Pure Appl. Anal. 6 113–40) only in the presence of full-interior acoustic damping—holds even in the case of localized dissipation. This nontrivial generalization is inspired by, and consistent with, the recent advances in the study of wave equations with nonlinear localized damping
Application of perturbation theory to a P-wave eikonal equation in orthorhombic media
Stovas, Alexey
2016-10-12
The P-wave eikonal equation for orthorhombic (ORT) anisotropic media is a highly nonlinear partial differential equation requiring the solution of a sixth-order polynomial to obtain traveltimes, resulting in complex and time-consuming numerical solutions. To alleviate this complexity, we approximate the solution of this equation by applying a multiparametric perturbation approach. We also investigated the sensitivity of traveltime surfaces inORT mediawith respect to three anelliptic parameters. As a result, a simple and accurate P-wave traveltime approximation valid for ORT media was derived. Two different possible anelliptic parameterizations were compared. One of the parameterizations includes anelliptic parameters defined at zero offset: η1, η2, and ηxy. Another parameterization includes anelliptic parameters defined for all symmetry planes: η1, η2, and η3. The azimuthal behavior of sensitivity coefficients with different parameterizations was used to analyze the crosstalk between anelliptic parameters. © 2016 Society of Exploration Geophysicists.
An Inverse Source Problem for a One-dimensional Wave Equation: An Observer-Based Approach
Asiri, Sharefa M.
2013-05-25
Observers are well known in the theory of dynamical systems. They are used to estimate the states of a system from some measurements. However, recently observers have also been developed to estimate some unknowns for systems governed by Partial differential equations. Our aim is to design an observer to solve inverse source problem for a one dimensional wave equation. Firstly, the problem is discretized in both space and time and then an adaptive observer based on partial field measurements (i.e measurements taken form the solution of the wave equation) is applied to estimate both the states and the source. We see the effectiveness of this observer in both noise-free and noisy cases. In each case, numerical simulations are provided to illustrate the effectiveness of this approach. Finally, we compare the performance of the observer approach with Tikhonov regularization approach.
van Duijn, C. J.; Mitra, K.; Pop, I. S.
2018-01-01
The Richards equation is a mathematical model for unsaturated flow through porous media. This paper considers an extension of the Richards equation, where non-equilibrium effects like hysteresis and dynamic capillarity are incorporated in the relationship that relates the water pressure and the
Kyprianou, A.E.
2000-01-01
Recently Harris using probabilistic methods alone has given new proofs for the known existence asymptotics and unique ness of travelling wave solutions to the KPP equation Following in this vein we outline alternative probabilistic proofs for wave speeds exceeding the critical minimal wave speed
New chorus wave properties near the equator from Van Allen Probes wave observations
Li, W.; Santolík, Ondřej; Bortnik, J.; Thorne, R. M.; Kletzing, C. A.; Kurth, W. S.; Hospodarsky, G. B.
2016-01-01
Roč. 43, č. 10 (2016), s. 4725-4735 ISSN 0094-8276 R&D Projects: GA MŠk(CZ) LH15304 Grant - others:AV ČR(CZ) AP1401 Program:Akademická prémie - Praemium Academiae Institutional support: RVO:68378289 Keywords : chorus wave * wave normal angles * oblique * quasi-parallel * quasi-electrostatic Subject RIV: BL - Plasma and Gas Discharge Physics Impact factor: 4.253, year: 2016 http://onlinelibrary.wiley.com/doi/10.1002/2016GL068780/abstract
LeMesurier, Brenton
2012-01-01
A new approach is described for generating exactly energy-momentum conserving time discretizations for a wide class of Hamiltonian systems of DEs with quadratic momenta, including mechanical systems with central forces; it is well-suited in particular to the large systems that arise in both spatial discretizations of nonlinear wave equations and lattice equations such as the Davydov System modeling energetic pulse propagation in protein molecules. The method is unconditionally stable, making it well-suited to equations of broadly “Discrete NLS form”, including many arising in nonlinear optics. Key features of the resulting discretizations are exact conservation of both the Hamiltonian and quadratic conserved quantities related to continuous linear symmetries, preservation of time reversal symmetry, unconditional stability, and respecting the linearity of certain terms. The last feature allows a simple, efficient iterative solution of the resulting nonlinear algebraic systems that retain unconditional stability, avoiding the need for full Newton-type solvers. One distinction from earlier work on conservative discretizations is a new and more straightforward nearly canonical procedure for constructing the discretizations, based on a “discrete gradient calculus with product rule” that mimics the essential properties of partial derivatives. This numerical method is then used to study the Davydov system, revealing that previously conjectured continuum limit approximations by NLS do not hold, but that sech-like pulses related to NLS solitons can nevertheless sometimes arise.
Yuan, Na
2018-04-01
With the aid of the symbolic computation, we present an improved ( G ‧ / G ) -expansion method, which can be applied to seek more types of exact solutions for certain nonlinear evolution equations. In illustration, we choose the (3 + 1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation to demonstrate the validity and advantages of the method. As a result, abundant explicit and exact nontraveling wave solutions are obtained including two solitary waves solutions, nontraveling wave solutions and dromion soliton solutions. Some particular localized excitations and the interactions between two solitary waves are researched. The method can be also applied to other nonlinear partial differential equations.
Shallow water equations: viscous solutions and inviscid limit
Chen, Gui-Qiang; Perepelitsa, Mikhail
2012-12-01
We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C 2 test-functions, are confined in a compact set in H -1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.
Two modified symplectic partitioned Runge-Kutta methods for solving the elastic wave equation
Su, Bo; Tuo, Xianguo; Xu, Ling
2017-08-01
Based on a modified strategy, two modified symplectic partitioned Runge-Kutta (PRK) methods are proposed for the temporal discretization of the elastic wave equation. The two symplectic schemes are similar in form but are different in nature. After the spatial discretization of the elastic wave equation, the ordinary Hamiltonian formulation for the elastic wave equation is presented. The PRK scheme is then applied for time integration. An additional term associated with spatial discretization is inserted into the different stages of the PRK scheme. Theoretical analyses are conducted to evaluate the numerical dispersion and stability of the two novel PRK methods. A finite difference method is used to approximate the spatial derivatives since the two schemes are independent of the spatial discretization technique used. The numerical solutions computed by the two new schemes are compared with those computed by a conventional symplectic PRK. The numerical results, which verify the new method, are superior to those generated by traditional conventional methods in seismic wave modeling.
The Liouville equation for flavour evolution of neutrinos and neutrino wave packets
Hansen, Rasmus Sloth Lundkvist; Smirnov, Alexei Yu., E-mail: rasmus@mpi-hd.mpg.de, E-mail: smirnov@mpi-hd.mpg.de [Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg (Germany)
2016-12-01
We consider several aspects related to the form, derivation and applications of the Liouville equation (LE) for flavour evolution of neutrinos. To take into account the quantum nature of neutrinos we derive the evolution equation for the matrix of densities using wave packets instead of Wigner functions. The obtained equation differs from the standard LE by an additional term which is proportional to the difference of group velocities. We show that this term describes loss of the propagation coherence in the system. In absence of momentum changing collisions, the LE can be reduced to a single derivative equation over a trajectory coordinate. Additional time and spatial dependence may stem from initial (production) conditions. The transition from single neutrino evolution to the evolution of a neutrino gas is considered.
Wave equations on a de Sitter fiber bundle. [Semiclassical wave function, bundle space, L-S coupling
Drechsler, W [Max-Planck-Institut fuer Physik und Astrophysik, Muenchen (F.R. Germany)
1975-01-01
A gauge theory of strong interaction is developed based on fields defined on a fiber bundle. The structural group of the bundle is taken to be the Lsub(4,1) de Sitter group. An internal variable xi, varying in the fiber over a space-time point x, is introduced as a means to describe - with the help of a semiclassical wave function psi(x,xi) defined on the bundle space - the internal structure of extended hadrons in a framework using differential geometric techniques. Three basic nonlinear wave equations for psi(x,xi) are established which are of integro-differential type. The nonlinear coupling terms in these de Sitter gauge invariant equations represent physically a generalized spin orbit coupling or a generalized spin coupling for the motion taking place in the fiber. The motivation for using a bigger space for the definition of hadronic matter wave functions as well as the implications of this geometric approach to strong interaction physics is discussed in detail, in particular with respect to the problem of hadronic constituents. The proposed fiber bundle formalism allows a dynamical description of extended structures for hadrons without implying the necessity of introducing any constituents.
Seadawy, A. R.; El-Rashidy, K.
2018-03-01
The Kadomtsev-Petviashvili (KP) and modified KP equations are two of the most universal models in nonlinear wave theory, which arises as a reduction of system with quadratic nonlinearity which admit weakly dispersive waves. The generalized extended tanh method and the F-expansion method are used to derive exact solitary waves solutions of KP and modified KP equations. The region of solutions are displayed graphically.
Zhou Yubin; Li Chao
2009-01-01
A modified G'/G-expansion method is presented to derive traveling wave solutions for a class of nonlinear partial differential equations called Whitham-Broer-Kaup-Like equations. As a result, the hyperbolic function solutions, trigonometric function solutions, and rational solutions with parameters to the equations are obtained. When the parameters are taken as special values the solitary wave solutions can be obtained. (general)
Laser driven shock wave experiments for equation of state studies at megabar pressures
Pant, H C; Senecha, V K; Bandyopadhyay, S; Rai, V N; Khare, P; Bhat, R K; Gupta, N K; Godwal, B K
2002-01-01
We present the results from laser driven shock wave experiments for equation of state (EOS) studies of gold metal. An Nd:YAG laser chain (2 J, 1.06 mu m wavelength, 200 ps pulse FWHM) is used to generate shocks in planar Al foils and Al + Au layered targets. The EOS of gold in the pressure range of 9-13 Mbar is obtained using the impedance matching technique. The numerical simulations performed using the one-dimensional radiation hydrodynamic code support the experimental results. The present experimental data show remarkable agreement with the existing standard EOS models and with other experimental data obtained independently using laser driven shock wave experiments.
Artoun, Ojenie; David-Rus, Diana; Emmett, Matthew; Fishman, Lou; Fital, Sandra; Hogan, Chad; Lim, Jisun; Lushi, Enkeleida; Marinov, Vesselin
2006-05-01
In this report we summarize an extension of Fourier analysis for the solution of the wave equation with a non-constant coefficient corresponding to an inhomogeneous medium. The underlying physics of the problem is exploited to link pseudodifferential operators and phase space path integrals to obtain a marching algorithm that incorporates the backward scattering into the evolution of the wave. This allows us to successfully apply single-sweep, one-way marching methods in inherently two-way environments, which was not achieved before through other methods for this problem.
Blow-up of solutions to the rotation b-family system modeling equatorial water waves
Min Zhu
2018-03-01
Full Text Available We consider the blow-up mechanism to the periodic generalized rotation b-family system (R-b-family system. This model can be derived from the f-plane governing equations for the geographical water waves with a constant underlying current in the equatorial water waves with effect of the Coriolis force. When b=2, it is a rotation two-component Camassa-Holm (R2CH system. We consider the periodic R2CH system when linear dispersion is absent (which model is called r2CH system and derive two finite-time blow-up results.
A study of wave forces on an offshore platform by direct CFD and Morison equation
Zhang D.
2015-01-01
The next step is the presentation of 3D multiphase RANS simulation of the wind-turbine platform in single-harmonic regular waves. Simulation results from full 3D simulation will be compared to the results from Morison’s equation. We are motivated by the challenges of a floating platform which has complex underwater geometry (e.g. tethered semi-submersible. In cases like this, our hypothesis is that Morison’s equation will result in inaccurate prediction of forces, due to the limitations of 2D coefficients of simple geometries, and that 3D multiphase RANS CFD will be required to generate reliable predictions of platform loads and motions.
Representations and Classification of Traveling Wave Solutions to sinh-Goerdon Equation
Liu Chengshi
2008-01-01
Two concepts named atom solution and combinatory solution are defined. The classification of all single traveling wave atom solutions to sinh-Goerdon equation is obtained, and qualitative properties of solutions are discussed. In particular, we point out that some qualitative properties derived intuitively from dynamic system method are not true. Finally, we prove that our solutions to sinh-Goerdon equation include all solutions obtained in the paper [Z.T. Fu, et al., Commun. Theor. Phys. (Beijing, China) 45 (2006) 55]. Through an example, we show how to give some new identities on Jacobian elliptic functions.
An Adaptive Observer-Based Algorithm for Solving Inverse Source Problem for the Wave Equation
Asiri, Sharefa M.; Zayane, Chadia; Laleg-Kirati, Taous-Meriem
2015-01-01
Observers are well known in control theory. Originally designed to estimate the hidden states of dynamical systems given some measurements, the observers scope has been recently extended to the estimation of some unknowns, for systems governed by partial differential equations. In this paper, observers are used to solve inverse source problem for a one-dimensional wave equation. An adaptive observer is designed to estimate the state and source components for a fully discretized system. The effectiveness of the algorithm is emphasized in noise-free and noisy cases and an insight on the impact of measurements’ size and location is provided.
An Adaptive Observer-Based Algorithm for Solving Inverse Source Problem for the Wave Equation
Asiri, Sharefa M.
2015-08-31
Observers are well known in control theory. Originally designed to estimate the hidden states of dynamical systems given some measurements, the observers scope has been recently extended to the estimation of some unknowns, for systems governed by partial differential equations. In this paper, observers are used to solve inverse source problem for a one-dimensional wave equation. An adaptive observer is designed to estimate the state and source components for a fully discretized system. The effectiveness of the algorithm is emphasized in noise-free and noisy cases and an insight on the impact of measurements’ size and location is provided.
A class of periodic solutions of nonlinear wave and evolution equations
Kashcheev, V.N.
1987-01-01
For the case of 1+1 dimensions a new heuristic method is proposed for deriving dels-similar solutions to nonlinear autonomous differential equations. If the differential function f is a polynomial, then: (i) in the case of even derivatives in f the solution is the ratio of two polynomials from the Weierstrass elliptic functions; (ii) in the case of any order derivatives in f the solution is the ratio of two polynomials from simple exponents. Numerous examples are given constructing such periodic solutions to the wave and evolution equations
Numerical simulation of the regularized long wave equation by He's homotopy perturbation method
Inc, Mustafa [Department of Mathematics, Firat University, 23119 Elazig (Turkey)], E-mail: minc@firat.edu.tr; Ugurlu, Yavuz [Department of Mathematics, Firat University, 23119 Elazig (Turkey)
2007-09-17
In this Letter, we present the homotopy perturbation method (shortly HPM) for obtaining the numerical solution of the RLW equation. We obtain the exact and numerical solutions of the Regularized Long Wave (RLW) equation for certain initial condition. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Comparison of the results with those of other methods have led us to significant consequences. The numerical solutions are compared with the known analytical solutions.
Numerical simulation of the regularized long wave equation by He's homotopy perturbation method
Inc, Mustafa; Ugurlu, Yavuz
2007-01-01
In this Letter, we present the homotopy perturbation method (shortly HPM) for obtaining the numerical solution of the RLW equation. We obtain the exact and numerical solutions of the Regularized Long Wave (RLW) equation for certain initial condition. The initial approximation can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Comparison of the results with those of other methods have led us to significant consequences. The numerical solutions are compared with the known analytical solutions
Wave-induced extreme water levels in the Puerto Morelos fringing reef lagoon
A. Torres-Freyermuth
2012-12-01
Full Text Available Wave-induced extreme water levels in the Puerto Morelos fringing reef lagoon are investigated by means of a phase-resolving non-hydrostatic wave model (SWASH. This model solves the nonlinear shallow water equations including non-hydrostatic pressure. The one-dimensional version of the model is implemented in order to investigate wave transformation in fringing reefs. Firstly, the numerical model is validated with (i laboratory experiments conducted on a physical model (Demirbilek et al., 2007and (ii field observations (Coronado et al., 2007. Numerical results show good agreement with both experimental and field data. The comparison against the physical model results, for energetic wave conditions, indicates that high- and low-frequency wave transformation is well reproduced. Moreover, extreme water-level conditions measured during the passage of Hurricane Ivan in Puerto Morelos are also estimated by the numerical tool. Subsequently, the model is implemented at different along-reef locations in Puerto Morelos. Extreme water levels, wave-induced setup, and infragravity wave energy are estimated inside the reef lagoon for different storm wave conditions (H_{s} >2 m. The numerical results revealed a strong correlation between the offshore sea-swell wave energy and the setup. In contrast, infragravity waves are shown to be the result of a more complex pattern which heavily relies on the reef geometry. Indeed, the southern end of the reef lagoon provides evidence of resonance excitation, suggesting that the reef barrier may act as either a natural flood protection morphological feature, or as an inundation hazard enhancer depending on the incident wave conditions.
REFLECT: a program to integrate the wave equation through a plane stratified plasma
Greene, J.W.
1975-01-01
A program was developed to integrate the wave equation through a plane stratified plasma with a general density distribution. The reflection and transmission of a plane wave are computed as a function of the angle of incidence. The polarization of the electric vector is assumed to be perpendicular to the plane of incidence. The model for absorption by classical inverse bremsstrahlung avoids the improper extrapolation of underdense formulae that are singular at the plasma critical surface. Surprisingly good agreement with the geometric-optics analysis of a linear layer was found. The system of ordinary differential equations is integrated by the variable-step, variable-order Adams method in the Lawrence Livermore Laboratory Gear package. Parametric studies of the absorption are summarized, and some possibilities for further development of the code are discussed. (auth)
Ma Hongcai; Ge Dongjie; Yu Yaodong
2008-01-01
Based on the Bäcklund method and the multilinear variable separation approach (MLVSA), this paper nds a general solution including two arbitrary functions for the (2+1)-dimensional Burgers equations. Then a class of new doubly periodic wave solutions for (2+1)-dimensional Burgers equations is obtained by introducing appropriate Jacobi elliptic functions, Weierstrass elliptic functions and their combination in the general solutions (which contains two arbitrary functions). Two types of limit cases are considered. Firstly, taking one of the moduli to be unity and the other zero, it obtains particular wave (called semi-localized) patterns, which is periodic in one direction, but localized in the other direction. Secondly, if both moduli are tending to 1 as a limit, it derives some novel localized excitations (two-dromion solution). (general)
Abundant general solitary wave solutions to the family of KdV type equations
Md. Azmol Huda
2017-03-01
Full Text Available This work explores the construction of more general exact traveling wave solutions of some nonlinear evolution equations (NLEEs through the application of the (G′/G, 1/G-expansion method. This method is allied to the widely used (G′/G-method initiated by Wang et al. and can be considered as an extension of the (G′/G-expansion method. For effectiveness, the method is applied to the family of KdV type equations. Abundant general form solitary wave solutions as well as periodic solutions are successfully obtained through this method. Moreover, in the obtained wider set of solutions, if we set special values of the parameters, some previously known solutions are revived. The approach of this method is simple and elegantly standard. Having been computerized it is also powerful, reliable and effective.
Heat-flow equation motivated by the ideal-gas shock wave.
Holian, Brad Lee; Mareschal, Michel
2010-08-01
We present an equation for the heat-flux vector that goes beyond Fourier's Law of heat conduction, in order to model shockwave propagation in gases. Our approach is motivated by the observation of a disequilibrium among the three components of temperature, namely, the difference between the temperature component in the direction of a planar shock wave, versus those in the transverse directions. This difference is most prominent near the shock front. We test our heat-flow equation for the case of strong shock waves in the ideal gas, which has been studied in the past and compared to Navier-Stokes solutions. The new heat-flow treatment improves the agreement with nonequilibrium molecular-dynamics simulations of hard spheres under strong shockwave conditions.
Interactions of Soliton Waves for a Generalized Discrete KdV Equation
Zhou Tong; Zhu Zuo-Nong
2017-01-01
It is well known that soliton interactions in discrete integrable systems often possess new properties which are different from the continuous integrable systems, e.g., we found that there are such discrete solitons in a semidiscrete integrable system (the time variable is continuous and the space one is discrete) that the shorter solitary waves travel faster than the taller ones. Very recently, this kind of soliton was also observed in a full discrete generalized KdV system (the both of time and space variables are discrete) introduced by Kanki et al. In this paper, for the generalized discrete KdV (gdKdV) equation, we describe its richer structures of one-soliton solutions. The interactions of two-soliton waves to the gdKdV equation are studied. Some new features of the soliton interactions are proposed by rigorous theoretical analysis. (paper)
Method for the Direct Solve of the Many-Body Schrödinger Wave Equation
Jerke, Jonathan; Tymczak, C. J.; Poirier, Bill
We report on theoretical and computational developments towards a computationally efficient direct solve of the many-body Schrödinger wave equation for electronic systems. This methodology relies on two recent developments pioneered by the authors: 1) the development of a Cardinal Sine basis for electronic structure calculations; and 2) the development of a highly efficient and compact representation of multidimensional functions using the Canonical tensor rank representation developed by Belykin et. al. which we have adapted to electronic structure problems. We then show several relevant examples of the utility and accuracy of this methodology, scaling with system size, and relevant convergence issues of the methodology. Method for the Direct Solve of the Many-Body Schrödinger Wave Equation.
Viscoacoustic wave-equation traveltime inversion with correct and incorrect attenuation profiles
Yu, Han
2017-08-17
A visco-acoustic wave-equation traveltime inversion method is presented that inverts for a shallow subsurface velocity distribution with correct and incorrect attenuation profiles. Similar to the classical wave equation traveltime inversion, this method applies the misfit functional that minimizes the first break differences between the observed and predicted data. Although, WT can partly avoid the cycle skipping problem, an initial velocity model approaches to the right or wrong velocity models under different setups of the attenuation profiles. However, with a Q model far away from the real model, the inverted tomogram is obviously different from the true velocity model while a small change of the Q model does not improve the inversion quality in a strong manner if low frequency information is not lost.
Sound field prediction of ultrasonic lithotripsy in water with spheroidal beam equations
Zhang, Lue; Wang, Xiang-Da; Liu, Xiao-Zhou; Gong, Xiu-Fen
2015-01-01
With converged shock wave, extracorporeal shock wave lithotripsy (ESWL) has become a preferable way to crush human calculi because of its advantages of efficiency and non-intrusion. Nonlinear spheroidal beam equations (SBE) are employed to illustrate the acoustic wave propagation for transducers with a wide aperture angle. To predict the acoustic field distribution precisely, boundary conditions are obtained for the SBE model of the monochromatic wave when the source is located on the focus of an ESWL transducer. Numerical results of the monochromatic wave propagation in water are analyzed and the influences of half-angle, fundamental frequency, and initial pressure are investigated. According to our results, with optimization of these factors, the pressure focal gain of ESWL can be enhanced and the effectiveness of treatment can be improved. Project supported by the National Basic Research Program of China (Grant Nos. 2012CB921504 and 2011CB707902), the National Natural Science Foundation of China (Grant No. 11274166), the State Key Laboratory of Acoustics, Chinese Academy of Sciences (Grant No. SKLA201401), and the China Postdoctoral Science Foundation (Grant No. 2013M531313).
Solutions to Time-Fractional Diffusion-Wave Equation in Cylindrical Coordinates
Povstenko YZ
2011-01-01
Full Text Available Nonaxisymmetric solutions to time-fractional diffusion-wave equation with a source term in cylindrical coordinates are obtained for an infinite medium. The solutions are found using the Laplace transform with respect to time , the Hankel transform with respect to the radial coordinate , the finite Fourier transform with respect to the angular coordinate , and the exponential Fourier transform with respect to the spatial coordinate . Numerical results are illustrated graphically.
Well-posedness and exponential stability for a wave equation with nonlocal time-delay condition
Carlos Alberto Raposo
2017-11-01
Full Text Available Well-posedness and exponential stability of nonlocal time-delayed of a wave equation with a integral conditions of the 1st kind forms the center of this work. Through semigroup theory we prove the well-posedness by the Hille-Yosida theorem and the exponential stability exploring the dissipative properties of the linear operator associated to damped model using the Gearhart-Huang-Pruss theorem.
Gerbi, Sté phane; Said-Houari, Belkacem
2011-01-01
In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the KelvinVoigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained. © 2011 Elsevier Ltd. All rights reserved.
Gerbi, Stéphane
2011-12-01
In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the KelvinVoigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained. © 2011 Elsevier Ltd. All rights reserved.
Variational Approach to the Orbital Stability of Standing Waves of the Gross-Pitaevskii Equation
Hadj Selem, Fouad
2014-08-26
This paper is concerned with the mathematical analysis of a masssubcritical nonlinear Schrödinger equation arising from fiber optic applications. We show the existence and symmetry of minimizers of the associated constrained variational problem. We also prove the orbital stability of such solutions referred to as standing waves and characterize the associated orbit. In the last section, we illustrate our results with few numerical simulations. © 2014 Springer Basel.
Kylling, A.
1991-01-01
The transfer equations for normal waves in finite, inhomogeneous and plane-parallel magnetoactive media are solved using the discrete ordinate method. The physical process of absorption, emission, and multiple scattering are accounted for, and the medium may be forced both at the top and bottom boundary by anisotropic radiation as well as by internal anisotropic sources. The computational procedure is numerically stable for arbitrarily large optical depths, and the computer time is independent of optical thickness.
Optimization method for identifying the source term in an inverse wave equation
Arumugam Deiveegan
2017-08-01
Full Text Available In this work, we investigate the inverse problem of identifying a space-wise dependent source term of wave equation from the measurement on the boundary. On the basis of the optimal control framework, the inverse problem is transformed into an optimization problem. The existence and necessary condition of the minimizer for the cost functional are obtained. The projected gradient method and two-parameter model function method are applied to the minimization problem and numerical results are illustrated.
Soliton solutions and chaotic motions of the Zakharov equations for the Langmuir wave in the plasma
Zhen, Hui-Ling; Tian, Bo, E-mail: tian-bupt@163.com; Wang, Yu-Feng; Liu, De-Yin [State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876 (China)
2015-03-15
For the interaction between the high-frequency Langmuir waves and low-frequency ion-acoustic waves in the plasma, the Zakharov equations are studied in this paper. Via the Hirota method, we obtain the soliton solutions, based on which the soliton propagation is presented. It is found that with λ increasing, the amplitude of u decreases, whereas that of v remains unchanged, where λ is the ion-acoustic speed, u is the slowly-varying envelope of the Langmuir wave, and v is the fluctuation of the equilibrium ion density. Both the head-on and bound-state interactions between the two solitons are displayed. We observe that with λ decreasing, the interaction period of u decreases, while that of v keeps unchanged. It is found that the Zakharov equations cannot admit any chaotic motions. With the external perturbations taken into consideration, the perturbed Zakharov equations are studied for us to see the associated chaotic motions. Both the weak and developed chaotic motions are investigated, and the difference between them roots in the relative magnitude of the nonlinearities and perturbations. The chaotic motions are weakened with λ increasing, or else, strengthened. Periodic motion appears when the nonlinear terms and external perturbations are balanced. With such a balance kept, one period increases with λ increasing.
N. N. Romanova
1998-01-01
Full Text Available The dynamics of weakly nonlinear wave trains in unstable media is studied. This dynamics is investigated in the framework of a broad class of dynamical systems having a Hamiltonian structure. Two different types of instability are considered. The first one is the instability in a weakly supercritical media. The simplest example of instability of this type is the Kelvin-Helmholtz instability. The second one is the instability due to a weak linear coupling of modes of different nature. The simplest example of a geophysical system where the instability of this and only of this type takes place is the three-layer model of a stratified shear flow with a continuous velocity profile. For both types of instability we obtain nonlinear evolution equations describing the dynamics of wave trains having an unstable spectral interval of wavenumbers. The transformation to appropriate canonical variables turns out to be different for each case, and equations we obtained are different for the two types of instability we considered. Also obtained are evolution equations governing the dynamics of wave trains in weakly subcritical media and in media where modes are coupled in a stable way. Presented results do not depend on a specific physical nature of a medium and refer to a broad class of dynamical systems having the Hamiltonian structure of a special form.
Test of a new heat-flow equation for dense-fluid shock waves.
Holian, Brad Lee; Mareschal, Michel; Ravelo, Ramon
2010-09-21
Using a recently proposed equation for the heat-flux vector that goes beyond Fourier's Law of heat conduction, we model shockwave propagation in the dense Lennard-Jones fluid. Disequilibrium among the three components of temperature, namely, the difference between the kinetic temperature in the direction of a planar shock wave and those in the transverse directions, particularly in the region near the shock front, gives rise to a new transport (equilibration) mechanism not seen in usual one-dimensional heat-flow situations. The modification of the heat-flow equation was tested earlier for the case of strong shock waves in the ideal gas, which had been studied in the past and compared to Navier-Stokes-Fourier solutions. Now, the Lennard-Jones fluid, whose equation of state and transport properties have been determined from independent calculations, allows us to study the case where potential, as well as kinetic contributions are important. The new heat-flow treatment improves the agreement with nonequilibrium molecular-dynamics simulations under strong shock wave conditions, compared to Navier-Stokes.