Chaoticons described by nonlocal nonlinear Schrödinger equation
Zhong, Lanhua; Li, Yuqi; Chen, Yong; Hong, Weiyi; Hu, Wei; Guo, Qi
2017-01-01
It is shown that the unstable evolutions of the Hermite-Gauss-type stationary solutions for the nonlocal nonlinear Schrödinger equation with the exponential-decay response function can evolve into chaotic states. This new kind of entities are referred to as chaoticons because they exhibit not only chaotic properties (with positive Lyapunov exponents and spatial decoherence) but also soliton-like properties (with invariant statistic width and interaction of quasi-elastic collisions). PMID:28134268
Energy Technology Data Exchange (ETDEWEB)
Ibragimov, Nail H [Department of Mathematics and Science, Blekinge Institute of Technology, SE-371 79 Karlskrona (Sweden); Meleshko, Sergey V [School of Mathematics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima 30000 (Thailand); Rudenko, Oleg V, E-mail: nib@bth.se, E-mail: sergey@math.sut.ac.th, E-mail: rudenko@acs366.phys.msu.ru [Department of Physics, Moscow State University, 119991 Moscow (Russian Federation)
2011-08-05
The paper deals with an evolutionary integro-differential equation describing nonlinear waves. A particular choice of the kernel in the integral leads to well-known equations such as the Khokhlov-Zabolotskaya equation, the Kadomtsev-Petviashvili equation and others. Since the solutions of these equations describe many physical phenomena, the analysis of the general model studied in this paper is important. One of the methods for obtaining solutions of differential equations is provided by the Lie group analysis. However, this method is not applicable to integro-differential equations. Therefore, we discuss new approaches developed in modern group analysis and apply them to the general model considered in this paper. Reduced equations and exact solutions are also presented.
Localization in physical systems described by discrete nonlinear Schrodinger-type equations.
Bishop, A R; Kalosakas, G; Rasmussen, K O; Kevrekidis, P G
2003-06-01
Following a short introduction on localized modes in a model system, namely the discrete nonlinear Schrodinger equation, we present explicit results pertaining to three different physical systems described by similar equations. The applications range from the Raman scattering spectra of a complex electronic material through intrinsic localized vibrational modes, to the manifestation of an abrupt and irreversible delocalizing transition of Bose-Einstein condensates trapped in two-dimensional optical lattices, and to the instabilities of localized modes in coupled arrays of optical waveguides.
On realization of nonlinear systems described by higher-order differential equations
Schaft, van der A.J.
1987-01-01
We consider systems of smooth nonlinear differential and algebraic equations in which some of the variables are distinguished as “external variables.” The realization problem is to replace the higher-order implicit differential equations by first-order explicit differential equations and the algebra
Gupta, A. K.; Ray, S. Saha
2014-09-01
In this paper, KdV-Burger-Kuramoto equation involving instability, dissipation, and dispersion parameters is solved numerically. The numerical solution for the fractional order KdV-Burger-Kuramoto (KBK) equation has been presented using two-dimensional Legendre wavelet method. The approximate solutions of nonlinear fractional KBK equation thus obtained by Legendre wavelet method are compared with the exact solutions. The present scheme is very simple, effective and convenient for obtaining numerical solution of the KBK equation.
Directory of Open Access Journals (Sweden)
A. K. Gupta
2014-09-01
Full Text Available In this paper, KdV-Burger-Kuramoto equation involving instability, dissipation, and dispersion parameters is solved numerically. The numerical solution for the fractional order KdV-Burger-Kuramoto (KBK equation has been presented using two-dimensional Legendre wavelet method. The approximate solutions of nonlinear fractional KBK equation thus obtained by Legendre wavelet method are compared with the exact solutions. The present scheme is very simple, effective and convenient for obtaining numerical solution of the KBK equation.
Directory of Open Access Journals (Sweden)
Gui Mu
2013-01-01
Full Text Available The existence of the exponential attractors for coupled Ginzburg-Landau equations describing Bose-Einstein condensates and nonlinear optical waveguides and cavities with periodic initial boundary is obtained by showing Lipschitz continuity and the squeezing property.
Directory of Open Access Journals (Sweden)
G. Wunner
2011-01-01
Full Text Available The coalescence of two eigenfunctions with the same energy eigenvalue is not possible in Hermitian Hamiltonians. It is, however, a phenomenon well known from non-hermitian quantum mechanics. It can appear, e.g., for resonances in open systems, with complex energy eigenvalues. If two eigenvalues of a quantum mechanical system which depends on two or more parameters pass through such a branch point singularity at a critical set of parameters, the point in the parameter space is called an exceptional point. We will demonstrate that exceptional points occur not only for non-hermitean Hamiltonians but also in the nonlinear Schroedinger equations which describe Bose-Einstein condensates, i.e., the Gross-Pitaevskii equation for condensates with a short-range contact interaction, and with additional long-range interactions. Typically, in these condensates the exceptional points are also found to be bifurcation points in parameter space. For condensates with a gravity-like interaction between the atoms, these findings can be confirmed in an analytical way.
Energy Technology Data Exchange (ETDEWEB)
Mihalache, D.; Panoiu, N.-C.; Moldoveanu, F.; Baboiu, D.-M. [Dept. of Theor. Phys., Inst. of Atomic Phys., Bucharest (Romania)
1994-09-21
We used the Riemann problem method with a 3*3 matrix system to find the femtosecond single soliton solution for a perturbed nonlinear Schroedinger equation which describes bright ultrashort pulse propagation in properly tailored monomode optical fibres. Compared with the Gel'fand-Levitan-Marchenko approach, the major advantage of the Riemann problem method is that it provides the general single soliton solution in a simple and compact form. Unlike the standard nonlinear Schroedinger equation, here the single soliton solution exhibits periodic evolution patterns. (author)
Nonlinear waves described by a fifth-order equation derived from the Fermi-Pasta-Ulam system
Volkov, A. K.; Kudryashov, N. A.
2016-04-01
Nonlinear wave processes described by a fifth-order generalized KdV equation derived from the Fermi-Pasta-Ulam (FPU) model are considered. It is shown that, in contrast to the KdV equation, which demonstrates the recurrence of initial states and explains the FPU paradox, the fifthorder equation fails to pass the Painlevé test, is not integrable, and does not exhibit the recurrence of the initial state. The results of this paper show that the FPU paradox occurs only at an initial stage of a numerical experiment, which is explained by the existence of KdV solitons only on a bounded initial time interval.
Directory of Open Access Journals (Sweden)
T. D. Frank
2016-12-01
Full Text Available In physics, several attempts have been made to apply the concepts and tools of physics to the life sciences. In this context, a thermostatistic framework for active Nambu systems is proposed. The so-called free energy Fokker–Planck equation approach is used to describe stochastic aspects of active Nambu systems. Different thermostatistic settings are considered that are characterized by appropriately-defined entropy measures, such as the Boltzmann–Gibbs–Shannon entropy and the Tsallis entropy. In general, the free energy Fokker–Planck equations associated with these generalized entropy measures correspond to nonlinear partial differential equations. Irrespective of the entropy-related nonlinearities occurring in these nonlinear partial differential equations, it is shown that semi-analytical solutions for the stationary probability densities of the active Nambu systems can be obtained provided that the pumping mechanisms of the active systems assume the so-called canonical-dissipative form and depend explicitly only on Nambu invariants. Applications are presented both for purely-dissipative and for active systems illustrating that the proposed framework includes as a special case stochastic equilibrium systems.
On the non-linearity of the master equation describing spin-selective radical-ion-pair reactions
Kominis, I. K.
2010-01-01
We elaborate on the physical meaning of the non-linear master equation that was recently derived to account for spin-selective radical-ion-pair reactions. Based on quite general arguments, we show that such a non-linear master equation is indeed to be expected.
Guo, Bang-Xing; Gao, Zhan-Jie; Lin, Ji
2016-12-01
The consistent tanh expansion (CTE) method is applied to the (2+1)-dimensional Boussinesq equation which describes the propagation of ultrashort pulse in quadratic nonlinear medium. The interaction solutions are explicitly given, such as the bright soliton-periodic wave interaction solution, variational amplitude periodic wave solution, and kink-periodic wave interaction solution. We also obtain the bright soliton solution, kind bright soliton solution, double well dark soliton solution and kink-bright soliton interaction solution by using Painlevé truncated expansion method. And we investigate interactive properties of solitons and periodic waves. Supported by the National Natural Science Foundation of Zhejiang Province under Grant No. LZ15A050001 and the National Natural Science Foundation of China under Grant No. 11675164
Bingzhen, Xu; Wenzheng, Wang
1995-02-01
We give a traveling-wave method for obtaining exact solutions of the modified nonlinear Schrödinger equation iut+ɛuxx+2p||u||2u +2iq(||u||2u)x=0, describing the propagation of light pulses in optical fibers, where u represents a normalized complex amplitude of a pulse envelope, t is the normalized distance along a fiber, and x is the normalized time within the frame of reference moving along the fiber at the group velocity. With the help of the ``potential function'' we obtained by this method, we find a family of solutions that are finite everywhere, particularly including periodic solutions expressed in terms of Jacobi elliptic functions, stationary periodic solutions, and ``algebraic'' soliton solutions. Compared with previous work [D. Mihalache and N. C. Panoiu, J. Math. Phys. 33, 2323 (1992)] in which two kinds of the simplest solution were given, the physical meaning of the integration constants in the potential function we give is clearer and more easily fixed with the initial parameters of the light pulse.
Energy Technology Data Exchange (ETDEWEB)
Wang, Pan [State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876 (China); School of Science, Beijing University of Posts and Telecommunications, P.O. Box 122, Beijing 100876 (China); Tian, Bo, E-mail: tian.bupt@yahoo.com.cn [State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876 (China); School of Science, Beijing University of Posts and Telecommunications, P.O. Box 122, Beijing 100876 (China); Jiang, Yan; Wang, Yu-Feng [State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876 (China); School of Science, Beijing University of Posts and Telecommunications, P.O. Box 122, Beijing 100876 (China)
2013-02-15
For describing the dynamics of alpha helical proteins with internal molecular excitations, nonlinear couplings between lattice vibrations and molecular excitations, and spin excitations in one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin with the octupole–dipole interactions, we consider an inhomogeneous generalized fourth-order nonlinear Schrödinger equation. Based on the Ablowitz–Kaup–Newell–Segur system, infinitely many conservation laws for the equation are derived. Through the auxiliary function, bilinear forms and N-soliton solutions for the equation are obtained. Interactions of solitons are discussed by means of the asymptotic analysis. Effects of linear inhomogeneity on the interactions of solitons are also investigated graphically and analytically. Since the inhomogeneous coefficient of the equation h=α x+β, the soliton takes on the parabolic profile during the evolution. Soliton velocity is related to the parameter α, distance scale coefficient and biquadratic exchange coefficient, but has no relation with the parameter β. Soliton amplitude and width are only related to α. Soliton position is related to β.
Directory of Open Access Journals (Sweden)
Wei Khim Ng
2009-02-01
Full Text Available We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.
Cherevko, A. A.; Bord, E. E.; Khe, A. K.; Panarin, V. A.; Orlov, K. J.; Chupakhin, A. P.
2016-06-01
This article considers method of describing the behaviour of hemodynamic parameters near vascular pathologies. We study the influence of arterial aneurysms and arteriovenous malformations on the vascular system. The proposed method involves using generalized model of Van der Pol-Duffing to find out the characteristic behaviour of blood flow parameters. These parameters are blood velocity and pressure in the vessel. The velocity and pressure are obtained during the neurosurgery measurements. It is noted that substituting velocity on the right side of the equation gives good pressure approximation. Thus, the model reproduces clinical data well enough. In regard to the right side of the equation, it means external impact on the system. The harmonic functions with various frequencies and amplitudes are substituted on the right side of the equation to investigate its properties. Besides, variation of the right side parameters provides additional information about pressure. Non-linear analogue of Nyquist diagrams is used to find out how the properties of solution depend on the parameter values. We have analysed 60 cases with aneurysms and 14 cases with arteriovenous malformations. It is shown that the diagrams are divided into classes. Also, the classes are replaced by another one in the definite order with increasing of the right side amplitude.
Wu Zhuo Qun; Li Hui Lai; Zhao Jun Ning
2001-01-01
Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which
Asymptotics for dissipative nonlinear equations
Hayashi, Nakao; Kaikina, Elena I; Shishmarev, Ilya A
2006-01-01
Many of problems of the natural sciences lead to nonlinear partial differential equations. However, only a few of them have succeeded in being solved explicitly. Therefore different methods of qualitative analysis such as the asymptotic methods play a very important role. This is the first book in the world literature giving a systematic development of a general asymptotic theory for nonlinear partial differential equations with dissipation. Many typical well-known equations are considered as examples, such as: nonlinear heat equation, KdVB equation, nonlinear damped wave equation, Landau-Ginzburg equation, Sobolev type equations, systems of equations of Boussinesq, Navier-Stokes and others.
Auxiliary equation method for solving nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Sirendaoreji,; Jiong, Sun
2003-03-31
By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation.
On Differential Equations Describing 3-Dimensional Hyperbolic Spaces
Institute of Scientific and Technical Information of China (English)
WU Jun-Yi; DING Qing; Keti Tenenblat
2006-01-01
In this paper, we introduce the notion of a (2+1)-dimensional differential equation describing threedimensional hyperbolic spaces (3-h.s.). The (2+1)-dimensional coupled nonlinear Schrodinger equation and its sister equation, the (2+1)-dimensional coupled derivative nonlinear Schrodinger equation, are shown to describe 3-h.s. The (2+1)-dimensional generalized HF model: St = (1/2i [S, Sy] + 2iσS)x, σx = 1-4itr(SSxSy), in which S ∈ GLC(2)/GLC(1)×GLC(1),provides another example of (2+1)-dimensional differential equations describing 3-h.s. As a direct consequence, the geometric construction of an infinite number of conservation laws of such equations is illustrated. Furthermore we display a new infinite number of conservation laws of the (2+1)-dimensional nonlinear Schrodinger equation and the (2+1)odimensional derivative nonlinear Schrodinger equation by a geometric way.
Using fundamental equations to describe basic phenomena
DEFF Research Database (Denmark)
Jakobsen, Arne; Rasmussen, Bjarne D.
1999-01-01
constraining the total charge of refrigerant in the system, which is missing.In traditional mathematical modelling of a refrigeration cycle/system, the influence from the total charge of refrigerant on the system behaviour is normally not modelled explicitly. Instead, parameters such as superheat......When the fundamental thermodynamic balance equations (mass, energy, and momentum) are used to describe the processes in a simple refrigeration system, then one finds that the resulting equation system will have a degree of freedom equal to one. Further investigations reveal that it is the equation...... and the before mentioned parameters. In doing so, a systematic use of control volumes for modelling a refrigeration system is outlined....
On Gaussian Beams Described by Jacobi's Equation
Smith, Steven Thomas
2013-01-01
Gaussian beams describe the amplitude and phase of rays and are widely used to model acoustic propagation. This paper describes four new results in the theory of Gaussian beams. (1) It is shown that the \\v{C}erven\\'y equations for the amplitude and phase are equivalent to the classical Jacobi Equation of differential geometry. The \\v{C}erven\\'y equations describe Gaussian beams using Hamilton-Jacobi theory, whereas the Jacobi Equation expresses how Gaussian and Riemannian curvature determine geodesic flow on a Riemannian manifold. Thus the paper makes a fundamental connection between Gaussian beams and an acoustic channel's so-called intrinsic Gaussian curvature from differential geometry. (2) A new formula $\\pi(c/c")^{1/2}$ for the distance between convergence zones is derived and applied to several well-known profiles. (3) A class of "model spaces" are introduced that connect the acoustics of ducting/divergence zones with the channel's Gaussian curvature $K=cc"-(c')^2$. The "model" SSPs yield constant Gauss...
Some geometrical iteration methods for nonlinear equations
Institute of Scientific and Technical Information of China (English)
LU Xing-jiang; QIAN Chun
2008-01-01
This paper describes geometrical essentials of some iteration methods (e.g. Newton iteration,secant line method,etc.) for solving nonlinear equations and advances some geomet-rical methods of iteration that are flexible and efficient.
Solving Nonlinear Wave Equations by Elliptic Equation
Institute of Scientific and Technical Information of China (English)
FU Zun-Tao; LIU Shi-Da; LIU Shi-Kuo
2003-01-01
The elliptic equation is taken as a transformation and applied to solve nonlinear wave equations. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wave solutions,periodic wave solutions and so on, so it can be taken as a generalized method.
Second order evolution equations which describe pseudospherical surfaces
Catalano Ferraioli, D.; de Oliveira Silva, L. A.
2016-06-01
Second order evolution differential equations that describe pseudospherical surfaces are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature K = - 1, and can be seen as the compatibility condition of an associated sl (2 , R) -valued linear problem, also referred to as a zero curvature representation. Under the assumption that the linear problem is defined by 1-forms ωi =fi1 dx +fi2 dt, i = 1 , 2 , 3, with fij depending on (x , t , z ,z1 ,z2) and such that f21 = η, η ∈ R, we give a complete and explicit classification of equations of the form zt = A (x , t , z) z2 + B (x , t , z ,z1) . According to the classification, these equations are subdivided in three main classes (referred to as Types I-III) together with the corresponding linear problems. Explicit examples of differential equations of each type are determined by choosing certain arbitrary differentiable functions. Svinolupov-Sokolov equations admitting higher weakly nonlinear symmetries, Boltzmann equation and reaction-diffusion equations like Murray equation are some known examples of such equations. Other explicit examples are presented, as well.
Introduction to nonlinear dispersive equations
Linares, Felipe
2015-01-01
This textbook introduces the well-posedness theory for initial-value problems of nonlinear, dispersive partial differential equations, with special focus on two key models, the Korteweg–de Vries equation and the nonlinear Schrödinger equation. A concise and self-contained treatment of background material (the Fourier transform, interpolation theory, Sobolev spaces, and the linear Schrödinger equation) prepares the reader to understand the main topics covered: the initial-value problem for the nonlinear Schrödinger equation and the generalized Korteweg–de Vries equation, properties of their solutions, and a survey of general classes of nonlinear dispersive equations of physical and mathematical significance. Each chapter ends with an expert account of recent developments and open problems, as well as exercises. The final chapter gives a detailed exposition of local well-posedness for the nonlinear Schrödinger equation, taking the reader to the forefront of recent research. The second edition of Introdu...
Nonlinear evolution equations in QCD
Stasto, A. M.
2004-01-01
The following lectures are an introduction to the phenomena of partonic saturation and nonlinear evolution equations in Quantum Chromodynamics. After a short introduction to the linear evolution, the problems of unitarity bound and parton saturation are discussed. The nonlinear Balitsky-Kovchegov evolution equation in the high energy limit is introduced, and the progress towards the understanding of the properties of its solution is reviewed. We discuss the concepts of the saturation scale, g...
Saaty, Thomas L
1981-01-01
Covers major types of classical equations: operator, functional, difference, integro-differential, and more. Suitable for graduate students as well as scientists, technologists, and mathematicians. "A welcome contribution." - Math Reviews. 1964 edition.
Stochastic nonlinear differential equations. I
Heilmann, O.J.; Kampen, N.G. van
1974-01-01
A solution method is developed for nonlinear differential equations having the following two properties. Their coefficients are stochastic through their dependence on a Markov process. The magnitude of the fluctuations, multiplied with their auto-correlation time, is a small quantity. Under these co
Steen-Ermakov-Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation
Prykarpatskyy, Yarema
2017-01-01
The paper deals with nonlinear one-dimensional Dirac equation. We describe its invariants set by means of the deformed linear Dirac equation, using the fact that two ordinary differential equations are equivalent if their sets of invariants coincide.
Approximating parameters in nonlinear reaction diffusion equations
Directory of Open Access Journals (Sweden)
Robert R. Ferdinand
2001-07-01
Full Text Available We present a model describing population dynamics in an environment. The model is a nonlinear, nonlocal, reaction diffusion equation with Neumann boundary conditions. An inverse method, involving minimization of a least-squares cost functional, is developed to identify unknown model parameters. Finally, numerical results are presented which display estimates of these parameters using computationally generated data.
Standing waves for discrete nonlinear Schrodinger equations
Ming Jia
2016-01-01
The discrete nonlinear Schrodinger equation is a nonlinear lattice system that appears in many areas of physics such as nonlinear optics, biomolecular chains and Bose-Einstein condensates. By using critical point theory, we establish some new sufficient conditions on the existence results of standing waves for the discrete nonlinear Schrodinger equations. We give an appropriate example to illustrate the conclusion obtained.
Discrete nonlinear Schrodinger equations with arbitrarily high-order nonlinearities
DEFF Research Database (Denmark)
Khare, A.; Rasmussen, Kim Ø; Salerno, M.
2006-01-01
A class of discrete nonlinear Schrodinger equations with arbitrarily high-order nonlinearities is introduced. These equations are derived from the same Hamiltonian using different Poisson brackets and include as particular cases the saturable discrete nonlinear Schrodinger equation and the Ablowi......-Ladik equation. As a common property, these equations possess three kinds of exact analytical stationary solutions for which the Peierls-Nabarro barrier is zero. Several properties of these solutions, including stability, discrete breathers, and moving solutions, are investigated....
Quasi self-adjoint nonlinear wave equations
Energy Technology Data Exchange (ETDEWEB)
Ibragimov, N H [Department of Mathematics and Science, Blekinge Institute of Technology, SE-371 79 Karlskrona (Sweden); Torrisi, M; Tracina, R, E-mail: nib@bth.s, E-mail: torrisi@dmi.unict.i, E-mail: tracina@dmi.unict.i [Dipartimento di Matematica e Informatica, University of Catania (Italy)
2010-11-05
In this paper we generalize the classification of self-adjoint second-order linear partial differential equation to a family of nonlinear wave equations with two independent variables. We find a class of quasi self-adjoint nonlinear equations which includes the self-adjoint linear equations as a particular case. The property of a differential equation to be quasi self-adjoint is important, e.g. for constructing conservation laws associated with symmetries of the differential equation. (fast track communication)
Solution and Positive Solution to Nonlinear Cantilever Beam Equations
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
Using the decomposition technique of equation and the fixed point theorem, the existence of solution and positive solution is studied for a nonlinear cantilever beam equation. The equation describes the deformation of the elastic beam with a fixed end and a free end. The main results show that the equation has at least one solution or positive solution, provided that the "height" of nonlinear term is appropriate on a bounded set.
The myth about nonlinear differential equations
Radhakrishnan, C.
2002-01-01
Taking the example of Koretweg--de Vries equation, it is shown that soliton solutions need not always be the consequence of the trade-off between the nonlinear terms and the dispersive term in the nonlinear differential equation. Even the ordinary one dimensional linear partial differential equation can produce a soliton.
Standing waves for discrete nonlinear Schrodinger equations
Directory of Open Access Journals (Sweden)
Ming Jia
2016-07-01
Full Text Available The discrete nonlinear Schrodinger equation is a nonlinear lattice system that appears in many areas of physics such as nonlinear optics, biomolecular chains and Bose-Einstein condensates. By using critical point theory, we establish some new sufficient conditions on the existence results of standing waves for the discrete nonlinear Schrodinger equations. We give an appropriate example to illustrate the conclusion obtained.
Solitonlike solutions of the generalized discrete nonlinear Schrödinger equation
DEFF Research Database (Denmark)
Rasmussen, Kim; Henning, D.; Gabriel, H.
1996-01-01
We investigate the solution properties oi. a generalized discrete nonlinear Schrodinger equation describing a nonlinear lattice chain. The generalized equation interpolates between the integrable discrete Ablowitz-Ladik equation and the nonintegrable discrete Schrodinger equation. Special interest...... nonlinear Schrodinger equation. In this way eve are able to construct coherent solitonlike structures of profile determined by the map parameters....
Alternative Forms of Enhanced Boussinesq Equations with Improved Nonlinearity
Directory of Open Access Journals (Sweden)
Kezhao Fang
2013-01-01
Full Text Available We propose alternative forms of the Boussinesq equations which extend the equations of Madsen and Schäffer by introducing extra nonlinear terms during enhancement. Theoretical analysis shows that nonlinear characteristics are considerably improved. A numerical implementation of one-dimensional equations is described. Three tests involving strongly nonlinear evolution, namely, regular waves propagating over an elevated bar feature in a tank with an otherwise constant depth, wave group transformation over constant water depth, and nonlinear shoaling of unsteady waves over a sloping beach, are simulated by the model. The model is found to be effective.
Nonlinear Elliptic Differential Equations with Multivalued Nonlinearities
Indian Academy of Sciences (India)
Antonella Fiacca; Nikolaos Matzakos; Nikolaos S Papageorgiou; Raffaella Servadei
2001-11-01
In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all $\\mathbb{R}$. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of $\\mathbb{R}$. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).
Symmetrized solutions for nonlinear stochastic differential equations
Directory of Open Access Journals (Sweden)
G. Adomian
1981-01-01
Full Text Available Solutions of nonlinear stochastic differential equations in series form can be put into convenient symmetrized forms which are easily calculable. This paper investigates such forms for polynomial nonlinearities, i.e., equations of the form Ly+ym=x where x is a stochastic process and L is a linear stochastic operator.
Directory of Open Access Journals (Sweden)
E. M. E. Zayed
2014-01-01
Full Text Available We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.
Generalized solutions of nonlinear partial differential equations
Rosinger, EE
1987-01-01
During the last few years, several fairly systematic nonlinear theories of generalized solutions of rather arbitrary nonlinear partial differential equations have emerged. The aim of this volume is to offer the reader a sufficiently detailed introduction to two of these recent nonlinear theories which have so far contributed most to the study of generalized solutions of nonlinear partial differential equations, bringing the reader to the level of ongoing research.The essence of the two nonlinear theories presented in this volume is the observation that much of the mathematics concernin
Lectures on nonlinear evolution equations initial value problems
Racke, Reinhard
2015-01-01
This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behavior of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial-boundary value p...
Thermoviscous Model Equations in Nonlinear Acoustics
DEFF Research Database (Denmark)
Rasmussen, Anders Rønne
Four nonlinear acoustical wave equations that apply to both perfect gasses and arbitrary fluids with a quadratic equation of state are studied. Shock and rarefaction wave solutions to the equations are studied. In order to assess the accuracy of the wave equations, their solutions are compared...... to solutions of the basic equations from which the wave equations are derived. A straightforward weakly nonlinear equation is the most accurate for shock modeling. A higher order wave equation is the most accurate for modeling of smooth disturbances. Investigations of the linear stability properties...... of solutions to the wave equations, reveal that the solutions may become unstable. Such instabilities are not found in the basic equations. Interacting shocks and standing shocks are investigated....
Elliptic Equation and New Solutions to Nonlinear Wave Equations
Institute of Scientific and Technical Information of China (English)
FU Zun-Tao; LIU Shi-Kuo; LIU Shi-Da
2004-01-01
The new solutions to elliptic equation are shown, and then the elliptic equation is taken as a transformationand is applied to solve nonlinear wave equations. It is shown that more kinds of solutions are derived, such as periodicsolutions of rational form, solitary wave solutions of rational form, and so on.
Explicit Traveling Wave Solutions to Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
Linghai ZHANG
2011-01-01
First of all,some technical tools are developed. Then the author studies explicit traveling wave solutions to nonlinear dispersive wave equations,nonlinear dissipative dispersive wave equations,nonlinear convection equations,nonlinear reaction diffusion equations and nonlinear hyperbolic equations,respectively.
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.
EXACT SOLUTIONS TO NONLINEAR WAVE EQUATION
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
In this paper,we use an invariant set to construct exact solutions to a nonlinear wave equation with a variable wave speed. Moreover,we obtain conditions under which the equation admits a nonclassical symmetry. Several different nonclassical symmetries for equations with different diffusion terms are presented.
Approximate solution of a nonlinear partial differential equation
Vajta, M.
2007-01-01
Nonlinear partial differential equations (PDE) are notorious to solve. In only a limited number of cases can we find an analytic solution. In most cases, we can only apply some numerical scheme to simulate the process described by a nonlinear PDE. Therefore, approximate solutions are important for t
Modified Homotopy Analysis Method for Nonlinear Fractional Partial Differential Equations
Directory of Open Access Journals (Sweden)
D. Ziane
2017-05-01
Full Text Available In this paper, a combined form of natural transform with homotopy analysis method is proposed to solve nonlinear fractional partial differential equations. This method is called the fractional homotopy analysis natural transform method (FHANTM. The FHANTM can easily be applied to many problems and is capable of reducing the size of computational work. The fractional derivative is described in the Caputo sense. The results show that the FHANTM is an appropriate method for solving nonlinear fractional partial differentia equation.
ANALYTICAL SOLUTION OF NONLINEAR BAROTROPIC VORTICITY EQUATION
Institute of Scientific and Technical Information of China (English)
WANG Yue-peng; SHI Wei-hui
2008-01-01
The stability of nonlinear barotropic vorticity equation was proved. The necessary and sufficient conditions for the initial value problem to be well-posed were presented. Under the conditions of well-posedness, the corresponding analytical solution was also gained.
GLOBAL SOLUTIONS OF NONLINEAR SCHRODINGER EQUATIONS
Institute of Scientific and Technical Information of China (English)
Ye Yaojun
2005-01-01
In this paper we study the existence of global solutions to the Cauchy problem of nonlinear Schrodinger equation by establishing time weight function spaces and using the contraction mapping principle.
Homogenization of a nonlinear degenerate parabolic equation
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
The homogenization of one kind of nonlinear parabolic equation is studied. The weak convergence and corrector results are obtained by combining carefully the compactness method and two-scale convergence method in the homogenization theory.
Kinetic equation for nonlinear resonant wave-particle interaction
Artemyev, A. V.; Neishtadt, A. I.; Vasiliev, A. A.; Mourenas, D.
2016-09-01
We investigate the nonlinear resonant wave-particle interactions including the effects of particle (phase) trapping, detrapping, and scattering by high-amplitude coherent waves. After deriving the relationship between probability of trapping and velocity of particle drift induced by nonlinear scattering (phase bunching), we substitute this relation and other characteristic equations of wave-particle interaction into a kinetic equation for the particle distribution function. The final equation has the form of a Fokker-Planck equation with peculiar advection and collision terms. This equation fully describes the evolution of particle momentum distribution due to particle diffusion, nonlinear drift, and fast transport in phase-space via trapping. Solutions of the obtained kinetic equation are compared with results of test particle simulations.
Linearized oscillation theory for a nonlinear delay impulsive equation
Berezansky, Leonid; Braverman, Elena
2003-12-01
For a scalar nonlinear impulsive delay differential equationwith rk(t)≥0,hk(t)≤t, limj-->∞ τj=∞, such an auxiliary linear impulsive delay differential equationis constructed that oscillation (nonoscillation) of the nonlinear equation can be deduced from the corresponding properties of the linear equation. Coefficients rk(t) and delays are not assumed to be continuous. Explicit oscillation and nonoscillation conditions are established for some nonlinear impulsive models of population dynamics, such as the impulsive logistic equation and the impulsive generalized Lasota-Wazewska equation which describes the survival of red blood cells. It is noted that unlike nonimpulsive delay logistic equations a solution of a delay impulsive logistic equation may become negative.
Nonlinear Electrostatic Wave Equations for Magnetized Plasmas
DEFF Research Database (Denmark)
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 elliptic equations of the second order
Han, Qing
2016-01-01
Nonlinear elliptic differential equations are a diverse subject with important applications to the physical and social sciences and engineering. They also arise naturally in geometry. In particular, much of the progress in the area in the twentieth century was driven by geometric applications, from the Bernstein problem to the existence of Kähler-Einstein metrics. This book, designed as a textbook, provides a detailed discussion of the Dirichlet problems for quasilinear and fully nonlinear elliptic differential equations of the second order with an emphasis on mean curvature equations and on Monge-Ampère equations. It gives a user-friendly introduction to the theory of nonlinear elliptic equations with special attention given to basic results and the most important techniques. Rather than presenting the topics in their full generality, the book aims at providing self-contained, clear, and "elementary" proofs for results in important special cases. This book will serve as a valuable resource for graduate stu...
Conjugated Molecules Described by a One-Dimensional Dirac Equation.
Ernzerhof, Matthias; Goyer, Francois
2010-06-08
Starting from the Hückel Hamiltonian of conjugated hydrocarbon chains (ethylene, allyl radical, butadiene, pentadienyl radical, hexatriene, etc.), we perform a simple unitary transformation and obtain a Dirac matrix Hamiltonian. Thus already small molecules are described exactly in terms of a discrete Dirac equation, the continuum limit of which yields a one-dimensional Dirac Hamiltonian. Augmenting this Hamiltonian with specially adapted boundary conditions, we find that all the orbitals of the unsaturated hydrocarbon chains are reproduced by the continuous Dirac equation. However, only orbital energies close to the highest occupied molecular orbital/lowest unoccupied molecular orbital energy are accurately predicted by the Dirac equation. Since it is known that a continuous Dirac equation describes the electronic structure of graphene around the Fermi energy, our findings answer the question to what extent this peculiar electronic structure is already developed in small molecules containing a delocalized π-electron system. We illustrate how the electronic structure of small polyenes carries over to a certain class of rectangular graphene sheets and eventually to graphene itself. Thus the peculiar electronic structure of graphene extends to a large degree to the smallest unsaturated molecule (ethylene).
The Nonlinear Analytical Envelope Equation in quadratic nonlinear crystals
Bache, Morten
2016-01-01
We here derive the so-called Nonlinear Analytical Envelope Equation (NAEE) inspired by the work of Conforti et al. [M. Conforti, A. Marini, T. X. Tran, D. Faccio, and F. Biancalana, "Interaction between optical fields and their conjugates in nonlinear media," Opt. Express 21, 31239-31252 (2013)], whose notation we follow. We present a complete model that includes $\\chi^{(2)}$ terms [M. Conforti, F. Baronio, and C. De Angelis, "Nonlinear envelope equation for broadband optical pulses in quadratic media," Phys. Rev. A 81, 053841 (2010)], $\\chi^{(3)}$ terms, and then extend the model to delayed Raman effects in the $\\chi^{(3)}$ term. We therefore get a complete model for ultrafast pulse propagation in quadratic nonlinear crystals similar to the Nonlinear Wave Equation in Frequency domain [H. Guo, X. Zeng, B. Zhou, and M. Bache, "Nonlinear wave equation in frequency domain: accurate modeling of ultrafast interaction in anisotropic nonlinear media," J. Opt. Soc. Am. B 30, 494-504 (2013)], but where the envelope is...
Nonlinear Schrodinger equation with chaotic, random, and nonperiodic nonlinearity
Cardoso, W B; Avelar, A T; Bazeia, D; Hussein, M S
2009-01-01
In this paper we deal with a nonlinear Schr\\"{o}dinger equation with chaotic, random, and nonperiodic cubic nonlinearity. Our goal is to study the soliton evolution, with the strength of the nonlinearity perturbed in the space and time coordinates and to check its robustness under these conditions. Comparing with a real system, the perturbation can be related to, e.g., impurities in crystalline structures, or coupling to a thermal reservoir which, on the average, enhances the nonlinearity. We also discuss the relevance of such random perturbations to the dynamics of Bose-Einstein Condensates and their collective excitations and transport.
Extended Trial Equation Method for Nonlinear Partial Differential Equations
Gepreel, Khaled A.; Nofal, Taher A.
2015-04-01
The main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber-Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.
Wave equation with concentrated nonlinearities
Noja, Diego; Posilicano, Andrea
2004-01-01
In this paper we address the problem of wave dynamics in presence of concentrated nonlinearities. Given a vector field $V$ on an open subset of $\\CO^n$ and a discrete set $Y\\subset\\RE^3$ with $n$ elements, we define a nonlinear operator $\\Delta_{V,Y}$ on $L^2(\\RE^3)$ which coincides with the free Laplacian when restricted to regular functions vanishing at $Y$, and which reduces to the usual Laplacian with point interactions placed at $Y$ when $V$ is linear and is represented by an Hermitean m...
The Homoclinic Orbits in Nonlinear Schroedinger Equation
Institute of Scientific and Technical Information of China (English)
PengchengXU; BolingGUO; 等
1998-01-01
The persistence of Homoclinic orbits for perturbed nonlinear Schroedinger equation with five degree term under een periodic boundary conditions is considered.The exstences of the homoclinic orbits for the truncation equation is established by Melnikov's analysis and geometric singular perturbation theory.
Linearization of Systems of Nonlinear Diffusion Equations
Institute of Scientific and Technical Information of China (English)
KANG Jing; QU Chang-Zheng
2007-01-01
We investigate the linearization of systems of n-component nonlinear diffusion equations; such systems have physical applications in soil science, mathematical biology and invariant curve flows. Equivalence transformations of their auxiliary systems are used to identify the systems that can be linearized. We also provide several examples of systems with two-component equations, and show how to linearize them by nonlocal mappings.
The approximate solutions of nonlinear Boussinesq equation
Lu, Dianhen; Shen, Jie; Cheng, Yueling
2016-04-01
The homotopy analysis method (HAM) is introduced to solve the generalized Boussinesq equation. In this work, we establish the new analytical solution of the exponential function form. Applying the homotopy perturbation method to solve the variable coefficient Boussinesq equation. The results indicate that this method is efficient for the nonlinear models with variable coefficients.
SEMICLASSICAL LIMIT OF NONLINEAR SCHRODINGER EQUATION (Ⅱ)
Institute of Scientific and Technical Information of China (English)
张平
2002-01-01
In this paper, we use the Wigner measure approach to study the semiclassical limit of nonlinear Schrodinger equation in small time. We prove that: the limits of the quantum density: pε =: |ψε|2, and the quantum momentum: Jε =: εIm(ψεψε) satisfy the compressible Euler equations before the formation of singularities in the limit system.
Nonlinear second order elliptic equations involving measures
Marcus, Moshe
2013-01-01
This book presents a comprehensive study of boundary value problems for linear and semilinear second order elliptic equations with measure data,especially semilinear equations with absorption. The interactions between the diffusion operator and the absorption term give rise to a large class of nonlinear phenomena in the study of which singularities and boundary trace play a central role.
Extended nonlinear feedback model for describing episodes of high inflation
Szybisz, M A; Szybisz, L.
2016-01-01
An extension of the nonlinear feedback (NLF) formalism to describe regimes of hyper- and high-inflation in economy is proposed in the present work. In the NLF model the consumer price index (CPI) exhibits a finite time singularity of the type $1/(t_c -t)^{(1- \\beta)/\\beta}$, with $\\beta>0$, predicting a blow up of the economy at a critical time $t_c$. However, this model fails in determining $t_c$ in the case of weak hyperinflation regimes like, e.g., that occurred in Israel. To overcome this...
Nonlinear differentiation equation and analytic function spaces
Li, Hao; Li, Songxiao
2015-01-01
In this paper we consider the nonlinear complex differential equation $$(f^{(k)})^{n_{k}}+A_{k-1}(z)(f^{(k-1)})^{n_{k-1}}+\\cdot\\cdot\\cdot+A_{1}(z)(f')^{n_{1}}+A_{0}(z)f^{n_{0}}=0, $$where $ A_{j}(z)$, $ j=0, \\cdots, k-1 $, are analytic in the unit disk $ \\mathbb{D} $, $ n_{j}\\in R^{+} $ for all $ j=0, \\cdots, k $. We investigate this nonlinear differential equation from two aspects. On one hand, we provide some sufficient conditions on coefficients such that all solutions of this equation bel...
The nonlinear Schroedinger equation on a disordered chain
Energy Technology Data Exchange (ETDEWEB)
Scharf, R.; Bishop, A.R.
1990-01-01
The integrable lattice nonlinear Schroedinger equation is a unique model with which to investigate the effects of disorder on a discrete integrable dynamics, and its interplay with nonlinearity. We first review some features of the lattice nonlinear Schroedinger equation in the absence of disorder and introduce a 1- and 2-soliton collective variable approximation. Then we describe the effect of different types of disorder: attractive and repulsive isolated impurities, spatially periodic potentials, random potentials, and time dependent (kicked) long wavelength perturbations. 18 refs., 15 figs.
Advances in iterative methods for nonlinear equations
Busquier, Sonia
2016-01-01
This book focuses on the approximation of nonlinear equations using iterative methods. Nine contributions are presented on the construction and analysis of these methods, the coverage encompassing convergence, efficiency, robustness, dynamics, and applications. Many problems are stated in the form of nonlinear equations, using mathematical modeling. In particular, a wide range of problems in Applied Mathematics and in Engineering can be solved by finding the solutions to these equations. The book reveals the importance of studying convergence aspects in iterative methods and shows that selection of the most efficient and robust iterative method for a given problem is crucial to guaranteeing a good approximation. A number of sample criteria for selecting the optimal method are presented, including those regarding the order of convergence, the computational cost, and the stability, including the dynamics. This book will appeal to researchers whose field of interest is related to nonlinear problems and equations...
Exponential Attractor for a Nonlinear Boussinesq Equation
Institute of Scientific and Technical Information of China (English)
Ahmed Y. Abdallah
2006-01-01
This paper is devoted to prove the existence of an exponential attractor for the semiflow generated by a nonlinear Boussinesq equation. We formulate the Boussinesq equation as an abstract equation in the Hilbert space H20(0, 1) × L2(0, 1). The main step in this research is to show that there exists an absorbing set for the solution semiflow in the Hilbert space H03(0, 1) × H10(0, 1).
The Nonlinear Convection—Reaction—Diffusion Equation
Institute of Scientific and Technical Information of China (English)
ShiminTANG; MaochangCUI; 等
1996-01-01
A nonlinear convection-reaction-diffusion equation is used as a model equation of the El Nino events.In this model,the effects of convection,turbulent diffusion,linear feed-back and nolinear radiation on the anomaly of Sea Surface Temperature(SST) are considered.In the case of constant convection,this equation has exact kink-like travelling wave solutions,which can be used to explain the history of an El Nino event.
The Swift-Hohenberg equation with a nonlocal nonlinearity
2013-01-01
It is well known that aspects of the formation of localised states in a one-dimensional Swift--Hohenberg equation can be described by Ginzburg--Landau-type envelope equations. This paper extends these multiple scales analyses to cases where an additional nonlinear integral term, in the form of a convolution, is present. The presence of a kernel function introduces a new lengthscale into the problem, and this results in additional complexity in both the derivation of envelope equations and in ...
SPHERICAL NONLINEAR PULSES FOR THE SOLUTIONS OF NONLINEAR WAVE EQUATIONS Ⅱ, NONLINEAR CAUSTIC
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
This article discusses spherical pulse like solutions of the system of semilinear wave equations with the pulses focusing at a point and emerging outgoing in three space variables. In small initial data case, it shows that the nonlinearities have a strong effect at the focal point. Scattering operator is introduced to describe the caustic crossing. With the aid of the L∞ norms, it analyzes the relative errors in approximate solutions.
The Duffing Equation Nonlinear Oscillators and their Behaviour
Kovacic, Ivana
2011-01-01
The Duffing Equation: Nonlinear Oscillators and their Behaviour brings together the results of a wealth of disseminated research literature on the Duffing equation, a key engineering model with a vast number of applications in science and engineering, summarizing the findings of this research. Each chapter is written by an expert contributor in the field of nonlinear dynamics and addresses a different form of the equation, relating it to various oscillatory problems and clearly linking the problem with the mathematics that describe it. The editors and the contributors explain the mathematical
DEFF Research Database (Denmark)
Oster, Michael; Gaididei, Yuri B.; Johansson, Magnus
2004-01-01
We study the continuum limit of a nonlinear Schrodinger lattice model with both on-site and inter-site nonlinearities, describing weakly coupled optical waveguides or Bose-Einstein condensates. The resulting continuum nonlinear Schrodinger-type equation includes both nonlocal and nonlinear...
Entropy and convexity for nonlinear partial differential equations.
Ball, John M; Chen, Gui-Qiang G
2013-12-28
Partial differential equations are ubiquitous in almost all applications of mathematics, where they provide a natural mathematical description of many phenomena involving change in physical, chemical, biological and social processes. The concept of entropy originated in thermodynamics and statistical physics during the nineteenth century to describe the heat exchanges that occur in the thermal processes in a thermodynamic system, while the original notion of convexity is for sets and functions in mathematics. Since then, entropy and convexity have become two of the most important concepts in mathematics. In particular, nonlinear methods via entropy and convexity have been playing an increasingly important role in the analysis of nonlinear partial differential equations in recent decades. This opening article of the Theme Issue is intended to provide an introduction to entropy, convexity and related nonlinear methods for the analysis of nonlinear partial differential equations. We also provide a brief discussion about the content and contributions of the papers that make up this Theme Issue.
The numerical dynamic for highly nonlinear partial differential equations
Lafon, A.; Yee, H. C.
1992-01-01
Problems associated with the numerical computation of highly nonlinear equations in computational fluid dynamics are set forth and analyzed in terms of the potential ranges of spurious behaviors. A reaction-convection equation with a nonlinear source term is employed to evaluate the effects related to spatial and temporal discretizations. The discretization of the source term is described according to several methods, and the various techniques are shown to have a significant effect on the stability of the spurious solutions. Traditional linearized stability analyses cannot provide the level of confidence required for accurate fluid dynamics computations, and the incorporation of nonlinear analysis is proposed. Nonlinear analysis based on nonlinear dynamical systems complements the conventional linear approach and is valuable in the analysis of hypersonic aerodynamics and combustion phenomena.
Numerical methods for nonlinear partial differential equations
Bartels, Sören
2015-01-01
The description of many interesting phenomena in science and engineering leads to infinite-dimensional minimization or evolution problems that define nonlinear partial differential equations. While the development and analysis of numerical methods for linear partial differential equations is nearly complete, only few results are available in the case of nonlinear equations. This monograph devises numerical methods for nonlinear model problems arising in the mathematical description of phase transitions, large bending problems, image processing, and inelastic material behavior. For each of these problems the underlying mathematical model is discussed, the essential analytical properties are explained, and the proposed numerical method is rigorously analyzed. The practicality of the algorithms is illustrated by means of short implementations.
On implicit abstract neutral nonlinear differential equations
Energy Technology Data Exchange (ETDEWEB)
Hernández, Eduardo, E-mail: lalohm@ffclrp.usp.br [Universidade de São Paulo, Departamento de Computação e Matemática, Faculdade de Filosofia Ciências e Letras de Ribeirão Preto (Brazil); O’Regan, Donal, E-mail: donal.oregan@nuigalway.ie [National University of Ireland, School of Mathematics, Statistics and Applied Mathematics (Ireland)
2016-04-15
In this paper we continue our developments in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial “nonlinear” neutral differential equations. Some applications involving partial neutral differential equations are presented.
The constructive technique and its application in solving a nonlinear reaction diffusion equation
Institute of Scientific and Technical Information of China (English)
Lai Shao-Yong; Guo Yun-Xi; Qing Yin; Wu Yong-Hong
2009-01-01
A mathematical technique based on the consideration of a nonlinear partial differential equation together with an additional condition in the form of an ordinary differential equation is employed to study a nonlinear reaction diffusion equation which describes a real process in physics and in chemistry. Several exact solutions for the equation are acquired under certain circumstances.
TAYLOR EXPANSION METHOD FOR NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
HE Yin-nian
2005-01-01
A new numerical method of integrating the nonlinear evolution equations, namely the Taylor expansion method, was presented. The standard Galerkin method can be viewed as the 0-th order Taylor expansion method; while the nonlinear Galerkin method can be viewed as the 1-st order modified Taylor expansion method. Moreover, the existence of the numerical solution and its convergence rate were proven. Finally, a concrete example,namely, the two-dimensional Navier-Stokes equations with a non slip boundary condition,was provided. The result is that the higher order Taylor expansion method is of the higher convergence rate under some assumptions about the regularity of the solution.
Numerical study of fractional nonlinear Schrodinger equations
Klein, Christian
2014-10-08
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.
Explicit solutions of nonlinear wave equation systems
Institute of Scientific and Technical Information of China (English)
Ahmet Bekir; Burcu Ayhan; M.Naci (O)zer
2013-01-01
We apply the (G'/G)-expansion method to solve two systems of nonlinear differential equations and construct traveling wave solutions expressed in terms of hyperbolic functions,trigonometric functions,and rational functions with arbitrary parameters.We highlight the power of the (G'/G)-expansion method in providing generalized solitary wave solutions of different physical structures.It is shown that the (G'/G)-expansion method is very effective and provides a powerful mathematical tool to solve nonlinear differential equation systems in mathematical physics.
Exact solutions for nonlinear foam drainage equation
Zayed, E. M. E.; Al-Nowehy, Abdul-Ghani
2016-09-01
In this paper, the modified simple equation method, the exp-function method, the soliton ansatz method, the Riccati equation expansion method and the ( G^' }/G) -expansion method are used to construct exact solutions with parameters of the nonlinear foam drainage equation. When these parameters are taken to be special values, the solitary wave solutions and the trigonometric function solutions are derived from the exact solutions. The obtained results confirm that the proposed methods are efficient techniques for analytic treatments of a wide variety of nonlinear partial differential equations in mathematical physics. We compare our results together with each other yielding from these integration tools. Also, our results have been compared with the well-known results of others.
Exact solutions for nonlinear foam drainage equation
Zayed, E. M. E.; Al-Nowehy, Abdul-Ghani
2017-02-01
In this paper, the modified simple equation method, the exp-function method, the soliton ansatz method, the Riccati equation expansion method and the ( G^' }/G)-expansion method are used to construct exact solutions with parameters of the nonlinear foam drainage equation. When these parameters are taken to be special values, the solitary wave solutions and the trigonometric function solutions are derived from the exact solutions. The obtained results confirm that the proposed methods are efficient techniques for analytic treatments of a wide variety of nonlinear partial differential equations in mathematical physics. We compare our results together with each other yielding from these integration tools. Also, our results have been compared with the well-known results of others.
Global attractivity in a nonlinear difference equation
Directory of Open Access Journals (Sweden)
Chuanxi Qian
2007-02-01
Full Text Available In this paper, we study the asymptotic behavior of positive solutions of the nonlinear difference equation $$ x_{n+1}=x_n f(x_{n-k}, $$ where $f:[0,inftyo(0, infty$ is a unimodal function, and $k$ is a nonnegative integer. Sufficient conditions for the positive equilibrium to be a global attractor of all positive solutions are established. Our results can be applied to to some difference equations derived from mathematical biology.
Extended nonlinear feedback model for describing episodes of high inflation
Szybisz, Martín A.; Szybisz, Leszek
2017-01-01
An extension of the nonlinear feedback (NLF) formalism to describe regimes of hyper- and high-inflation in economy is proposed in the present work. In the NLF model the consumer price index (CPI) exhibits a finite time singularity of the type 1 /(tc - t) (1 - β) / β, with β > 0, predicting a blow up of the economy at a critical time tc. However, this model fails in determining tc in the case of weak hyperinflation regimes like, e.g., that occurred in Israel. To overcome this trouble, the NLF model is extended by introducing a parameter γ, which multiplies all terms with past growth rate index (GRI). In this novel approach the solution for CPI is also analytic being proportional to the Gaussian hypergeometric function 2F1(1 / β , 1 / β , 1 + 1 / β ; z) , where z is a function of β, γ, and tc. For z → 1 this hypergeometric function diverges leading to a finite time singularity, from which a value of tc can be determined. This singularity is also present in GRI. It is shown that the interplay between parameters β and γ may produce phenomena of multiple equilibria. An analysis of the severe hyperinflation occurred in Hungary proves that the novel model is robust. When this model is used for examining data of Israel a reasonable tc is got. High-inflation regimes in Mexico and Iceland, which exhibit weaker inflations than that of Israel, are also successfully described.
Energy Technology Data Exchange (ETDEWEB)
Zhang Huiqun [College of Mathematical Science, Qingdao University, Qingdao, Shandong 266071 (China)], E-mail: hellozhq@yahoo.com.cn
2009-02-15
By using some exact solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact complex solutions for nonlinear partial differential equations. The method is implemented for the NLS equation, a new Hamiltonian amplitude equation, the coupled Schrodinger-KdV equations and the Hirota-Maccari equations. New exact complex solutions are obtained.
Anomalous wave structure in magnetized materials described by non-convex equations of state
Energy Technology Data Exchange (ETDEWEB)
Serna, Susana, E-mail: serna@mat.uab.es [Departament de Matematiques, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona (Spain); Marquina, Antonio, E-mail: marquina@uv.es [Departamento de Matematicas, Universidad de Valencia, 46100 Burjassot, Valencia (Spain)
2014-01-15
We analyze the anomalous wave structure appearing in flow dynamics under the influence of magnetic field in materials described by non-ideal equations of state. We consider the system of magnetohydrodynamics equations closed by a general equation of state (EOS) and propose a complete spectral decomposition of the fluxes that allows us to derive an expression of the nonlinearity factor as the mathematical tool to determine the nature of the wave phenomena. We prove that the possible formation of non-classical wave structure is determined by both the thermodynamic properties of the material and the magnetic field as well as its possible rotation. We demonstrate that phase transitions induced by material properties do not necessarily imply the loss of genuine nonlinearity of the wavefields as is the case in classical hydrodynamics. The analytical expression of the nonlinearity factor allows us to determine the specific amount of magnetic field necessary to prevent formation of complex structure induced by phase transition in the material. We illustrate our analytical approach by considering two non-convex EOS that exhibit phase transitions and anomalous behavior in the evolution. We present numerical experiments validating the analysis performed through a set of one-dimensional Riemann problems. In the examples we show how to determine the appropriate amount of magnetic field in the initial conditions of the Riemann problem to transform a thermodynamic composite wave into a simple nonlinear wave.
Advances in nonlinear partial differential equations and stochastics
Kawashima, S
1998-01-01
In the past two decades, there has been great progress in the theory of nonlinear partial differential equations. This book describes the progress, focusing on interesting topics in gas dynamics, fluid dynamics, elastodynamics etc. It contains ten articles, each of which discusses a very recent result obtained by the author. Some of these articles review related results.
Exact solutions for nonlinear partial fractional differential equations
Institute of Scientific and Technical Information of China (English)
Khaled A.Gepreel; Saleh Omran
2012-01-01
In this article,we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations.We use the improved (G’/G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation.This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.
Nonlinear Acoustics -- Perturbation Theory and Webster's Equation
Jorge, Rogério
2013-01-01
Webster's horn equation (1919) offers a one-dimensional approximation for low-frequency sound waves along a rigid tube with a variable cross-sectional area. It can be thought as a wave equation with a source term that takes into account the nonlinear geometry of the tube. In this document we derive this equation using a simplified fluid model of an ideal gas. By a simple change of variables, we convert it to a Schr\\"odinger equation and use the well-known variational and perturbative methods to seek perturbative solutions. As an example, we apply these methods to the Gabriel's Horn geometry, deriving the first order corrections to the linear frequency. An algorithm to the harmonic modes in any order for a general horn geometry is derived.
The Lie algebra of infinitesimal symmetries of nonlinear diffusion equations
Kersten, Paul H.M.; Gragert, Peter K.H.
1983-01-01
By using developed software for solving overdetermined systems of partial differential equations, the authors establish the complete Lie algebra of infinitesimal symmetries of nonlinear diffusion equations.
Lattice Boltzmann model for nonlinear convection-diffusion equations.
Shi, Baochang; Guo, Zhaoli
2009-01-01
A lattice Boltzmann model for convection-diffusion equation with nonlinear convection and isotropic-diffusion terms is proposed through selecting equilibrium distribution function properly. The model can be applied to the common real and complex-valued nonlinear evolutionary equations, such as the nonlinear Schrödinger equation, complex Ginzburg-Landau equation, Burgers-Fisher equation, nonlinear heat conduction equation, and sine-Gordon equation, by using a real and complex-valued distribution function and relaxation time. Detailed simulations of these equations are performed, and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies.
Exact Solutions for Nonlinear Differential Difference Equations in Mathematical Physics
Directory of Open Access Journals (Sweden)
Khaled A. Gepreel
2013-01-01
Full Text Available We modified the truncated expansion method to construct the exact solutions for some nonlinear differential difference equations in mathematical physics via the general lattice equation, the discrete nonlinear Schrodinger with a saturable nonlinearity, the quintic discrete nonlinear Schrodinger equation, and the relativistic Toda lattice system. Also, we put a rational solitary wave function method to find the rational solitary wave solutions for some nonlinear differential difference equations. The proposed methods are more effective and powerful to obtain the exact solutions for nonlinear difference differential equations.
Nonsmooth analysis of doubly nonlinear evolution equations
Mielke, Alexander; Savare', Giuseppe
2011-01-01
In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional,for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme with variational techniques. Finally, we discuss an application to a material model in finite-strain elasticity.
Solving Nonlinear Euler Equations with Arbitrary Accuracy
Dyson, Rodger W.
2005-01-01
A computer program that efficiently solves the time-dependent, nonlinear Euler equations in two dimensions to an arbitrarily high order of accuracy has been developed. The program implements a modified form of a prior arbitrary- accuracy simulation algorithm that is a member of the class of algorithms known in the art as modified expansion solution approximation (MESA) schemes. Whereas millions of lines of code were needed to implement the prior MESA algorithm, it is possible to implement the present MESA algorithm by use of one or a few pages of Fortran code, the exact amount depending on the specific application. The ability to solve the Euler equations to arbitrarily high accuracy is especially beneficial in simulations of aeroacoustic effects in settings in which fully nonlinear behavior is expected - for example, at stagnation points of fan blades, where linearizing assumptions break down. At these locations, it is necessary to solve the full nonlinear Euler equations, and inasmuch as the acoustical energy is of the order of 4 to 5 orders of magnitude below that of the mean flow, it is necessary to achieve an overall fractional error of less than 10-6 in order to faithfully simulate entropy, vortical, and acoustical waves.
A nonlinear Schroedinger wave equation with linear quantum behavior
Energy Technology Data Exchange (ETDEWEB)
Richardson, Chris D.; Schlagheck, Peter; Martin, John; Vandewalle, Nicolas; Bastin, Thierry [Departement de Physique, University of Liege, 4000 Liege (Belgium)
2014-07-01
We show that a nonlinear Schroedinger wave equation can reproduce all the features of linear quantum mechanics. This nonlinear wave equation is obtained by exploring, in a uniform language, the transition from fully classical theory governed by a nonlinear classical wave equation to quantum theory. The classical wave equation includes a nonlinear classicality enforcing potential which when eliminated transforms the wave equation into the linear Schroedinger equation. We show that it is not necessary to completely cancel this nonlinearity to recover the linear behavior of quantum mechanics. Scaling the classicality enforcing potential is sufficient to have quantum-like features appear and is equivalent to scaling Planck's constant.
Identification of parameters in amplitude equations describing coupled wakes
Fullana, J M; Zaleski, S; Le Gal, P; Fullana, Jose Maria; Rossi, Maurice; Zaleski, Stephane; Le Gal, Patrice
1996-01-01
We study the flow behind an array of equally spaced parallel cylinders. A system of Stuart-Landau equations with complex parameters is used to model the oscillating wakes. Our purpose is to identify the 6 scalar parameters which most accurately reproduce the experimental data of Chauve and Le Gal [{Physica D {\\bf 58}}, pp 407--413, (1992)]. To do so, we perform a computational search for the minimum of a distance \\calj. We define \\calj as the sum-square difference of the data and amplitudes reconstructed using coupled equations. The search algorithm is made more efficient through the use of a partially analytical expression for the gradient \
Filippov, A T
2016-01-01
The dynamics of any spherical cosmology with a scalar field (`scalaron') coupling to gravity is described by the nonlinear second-order differential equations for two metric functions and the scalaron depending on the `time' parameter. The equations depend on the scalaron potential and on arbitrary gauge function that describes time parameterizations. This dynamical system can be integrated for flat, isotropic models with very special potentials. But, somewhat unexpectedly, replacing the independent variable $t$ by one of the metric functions allows us to completely integrate the general spherical theory in any gauge and with arbitrary potentials. In this approach, inflationary solutions can be easily identified, explicitly derived, and compared to the standard approximate expressions. This approach is also applicable to intrinsically anisotropic models with a massive vector field (`vecton') as well as to some non-inflationary models.
Explicit Integration of Friedmann's Equation with Nonlinear Equations of State
Chen, Shouxin; Yang, Yisong
2015-01-01
This paper is a continuation of our earlier study on the integrability of the Friedmann equations in the light of the Chebyshev theorem. Our main focus will be on a series of important, yet not previously touched, problems when the equation of state for the perfect-fluid universe is nonlinear. These include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born--Infeld, and two-fluid models. We show that some of these may be integrated using Chebyshev's result while other are out of reach by the theorem but may be integrated explicitly by other methods. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution. For example, in the Chaplygin gas universe, it is seen that, as far as there is a tiny presence of nonlinear matter, linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics ...
The tanh-coth method combined with the Riccati equation for solving non-linear equation
Energy Technology Data Exchange (ETDEWEB)
Bekir, Ahmet [Dumlupinar University, Art-Science Faculty, Department of Mathematics, Kuetahya (Turkey)], E-mail: abekir@dumlupinar.edu.tr
2009-05-15
In this work, we established abundant travelling wave solutions for some non-linear evolution equations. This method was used to construct solitons and traveling wave solutions of non-linear evolution equations. The tanh-coth method combined with Riccati equation presents a wider applicability for handling non-linear wave equations.
ACCELERATION METHODS OF NONLINEAR ITERATION FOR NONLINEAR PARABOLIC EQUATIONS
Institute of Scientific and Technical Information of China (English)
Guang-wei Yuan; Xu-deng Hang
2006-01-01
This paper discusses the accelerating iterative methods for solving the implicit scheme of nonlinear parabolic equations. Two new nonlinear iterative methods named by the implicit-explicit quasi-Newton (IEQN) method and the derivative free implicit-explicit quasi-Newton (DFIEQN) method are introduced, in which the resulting linear equations from the linearization can preserve the parabolic characteristics of the original partial differential equations. It is proved that the iterative sequence of the iteration method can converge to the solution of the implicit scheme quadratically. Moreover, compared with the Jacobian Free Newton-Krylov (JFNK) method, the DFIEQN method has some advantages, e.g., its implementation is easy, and it gives a linear algebraic system with an explicit coefficient matrix, so that the linear (inner) iteration is not restricted to the Krylov method. Computational results by the IEQN, DFIEQN, JFNK and Picard iteration meth-ods are presented in confirmation of the theory and comparison of the performance of these methods.
An Efficient Numerical Approach for Nonlinear Fokker-Planck equations
Otten, Dustin; Vedula, Prakash
2009-03-01
Fokker-Planck equations which are nonlinear with respect to their probability densities that occur in many nonequilibrium systems relevant to mean field interaction models, plasmas, classical fermions and bosons can be challenging to solve numerically. To address some underlying challenges in obtaining numerical solutions, we propose a quadrature based moment method for efficient and accurate determination of transient (and stationary) solutions of nonlinear Fokker-Planck equations. In this approach the distribution function is represented as a collection of Dirac delta functions with corresponding quadrature weights and locations, that are in turn determined from constraints based on evolution of generalized moments. Properties of the distribution function can be obtained by solution of transport equations for quadrature weights and locations. We will apply this computational approach to study a wide range of problems, including the Desai-Zwanzig Model (for nonlinear muscular contraction) and multivariate nonlinear Fokker-Planck equations describing classical fermions and bosons, and will also demonstrate good agreement with results obtained from Monte Carlo and other standard numerical methods.
General Symmetry Approach to Solve Variable-Coefficient Nonlinear Equations
Institute of Scientific and Technical Information of China (English)
RUAN HangYu; CHEN YiXin; LOU SenYue
2001-01-01
After considering the variable coefficient of a nonlinear equation as a new dependent variable, some special types of variable-coefficient equation can be solved from the corresponding constant-coefficient equations by using the general classical Lie approach. Taking the nonlinear Schrodinger equation as a concrete example, the method is recommended in detail.``
Modulational instability in fractional nonlinear Schrödinger equation
Zhang, Lifu; He, Zenghui; Conti, Claudio; Wang, Zhiteng; Hu, Yonghua; Lei, Dajun; Li, Ying; Fan, Dianyuan
2017-07-01
Fractional calculus is entering the field of nonlinear optics to describe unconventional regimes, as disorder biological media and soft-matter. Here we investigate spatiotemporal modulational instability (MI) in a fractional nonlinear Schrödinger equation. We derive the MI gain spectrum in terms of the Lévy indexes and a varying number of spatial dimensions. We show theoretically and numerically that the Lévy indexes affect fastest growth frequencies and MI bandwidth and gain. Our results unveil a very rich scenario that may occur in the propagation of ultrashort pulses in random media and metamaterials, and may sustain novel kinds of propagation invariant optical bullets.
Use of the 4/5 Kolmogorov equation for describing some characteristics of fully developed turbulence
Tatarskii, V. I.
2005-03-01
The Kolmogorov equation, which relates the second-order structure function Dll(r ) and the third-ordrer structure function Dlll(r ), may be presented as a closed linear integrodifferential equation for the probability distribution function W(U ,r) of longitudinal velocity difference U. In general, without any restrictions, the function of two variables W(U ,r) may be presented as [Dll(r )]-1/2F(U/√Dll(r ) ,r). As a first approximation, we neglect dependence of F on the second argument r. In this approximation (self-similarity of the probability density function), the integrodifferential equation for W reduces to the ordinary nonlinear differential equation for Dll(r ). It follows from this equation that for r →∞ the function Dll(r )˜r2/3. This consideration does not use the 1941 Kolmogorov hypothesis that is based on dimension analysis. Any deviations from the 2/3 law, including intermittency effects, must be related to the violation of the above-mentioned self-similarity of W, i.e., with the additional dependence of F on the second argument r. On the basis of experimental data, we suggest a simple model of W, which allows us to describe deviations from the 2/3 law, caused by intermittency, and describe the local exponents κ in the structure functions ⟨∣U(r)∣ρ⟩˜rκ(ρ ) for moderate ρ. The so called "bottleneck effect" also can be described by 4/5 Kolmogorov equation.
DYNAMIC BIFURCATION OF NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
MA TIAN; WANG SHOUHONG
2005-01-01
The authors introduce a notion of dynamic bifurcation for nonlinear evolution equations, which can be called attractor bifurcation. It is proved that as the control parameter crosses certain critical value, the system bifurcates from a trivial steady state solution to an attractor with dimension between m and m + 1, where m + 1 is the number of eigenvalues crossing the imaginary axis. The attractor bifurcation theory presented in this article generalizes the existing steady state bifurcations and the Hopf bifurcations. It provides a unified point of view on dynamic bifurcation and can be applied to many problems in physics and mechanics.
REITERATED HOMOGENIZATION OF DEGENERATE NONLINEAR ELLIPTIC EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The authors study homogenization of some nonlinear partial differential equations of the form -div (a (hx,h2x,Duh)) = f,where a is periodic in the first two arguments and monotone in the third.In particular the case where a satisfies degenerated structure conditions is studied.It is proved that uh converges weakly in Wo1.1 (Ω) to the unique solution of a limit problem as h →∞.Moreover,explicit expressions for the limit problem are obtained.
Can Maxwell's Equations Describe Elementary Waves and Charges?
Bulyzhenkov, I E
2009-01-01
The steady Poynting flow defines the elementary Maxwell-Heaviside wave which spirality replicates the helicity of photons in quantum physics. The angular momentum of this classical wave requires non-point classical charges for elementary emission - absorption events. Analytical field solutions of Maxwell's equations can specify the nonlocal radial source for EM fields instead of the localized point charge. The Maxwell-Mie (astro)electron has the monotonous radial astrodistribution of only negative charge densities, while the positive elementary (astro)charge contains a sharp radial front between positive and negative densities. Non-empty space around visible macroscopic frames of laboratory bodies is filled by only negative fractions of continuous (astro)protons as well as of (astro)electrons.
Coupled Nonlinear Schr\\"{o}dinger equation and Toda equation (the Root of Integrability)
Hisakado, Masato
1997-01-01
We consider the relation between the discrete coupled nonlinear Schr\\"{o}dinger equation and Toda equation. Introducing complex times we can show the intergability of the discrete coupled nonlinear Schr\\"{o}dinger equation. In the same way we can show the integrability in coupled case of dark and bright equations. Using this method we obtain several integrable equations.
Asymptotic behavior of nonlinear semigroup describing a model of selective cell growth regulation.
Arino, O; Kimmel, M
1991-01-01
A new scheme of regulation of cell population growth is considered, called the selective growth regulation. The principle is that cells are withdrawn from proliferation depending on their contents of certain biochemical species. The dynamics of the cell population structured by the contents of this species is described by the functional integral equation model, previously introduced by the authors. The solutions of the model equations generate a semigroup of nonlinear positive operators. The main problem solved in this paper concerns stability of the equilibria of the model. This requires stating and proving of an original abstract result on the spectral radius of a perturbation of a semigroup of positive linear operators. Biological applications are discussed.
Exact Solution of a Generalized Nonlinear Schrodinger Equation Dimer
DEFF Research Database (Denmark)
Christiansen, Peter Leth; Maniadis, P.; Tsironis, G.P.
1998-01-01
We present exact solutions for a nonlinear dimer system defined throught a discrete nonlinear Schrodinger equation that contains also an integrable Ablowitz-Ladik term. The solutions are obtained throught a transformation that maps the dimer into a double Sine-Gordon like ordinary nonlinear...... differential equation....
Nonlinear Terms of MHD Equations for Homogeneous Magnetized Shear Flow
Dimitrov, Z D; Hristov, T S; Mishonov, T M
2011-01-01
We have derived the full set of MHD equations for incompressible shear flow of a magnetized fluid and considered their solution in the wave-vector space. The linearized equations give the famous amplification of slow magnetosonic waves and describe the magnetorotational instability. The nonlinear terms in our analysis are responsible for the creation of turbulence and self-sustained spectral density of the MHD (Alfven and pseudo-Alfven) waves. Perspectives for numerical simulations of weak turbulence and calculation of the effective viscosity of accretion disks are shortly discussed in k-space.
Soliton states of Maxwell’s equations and nonlinear Schrodinger equation
Institute of Scientific and Technical Information of China (English)
陈翼强
1997-01-01
Similarities and fundamental differences between Maxwell’s equations and nonlinear Schrodinger equation in predicting a soliton evolution in a uniform nonlinear anisotropic medium are analyzed.It is found that in some cases,the soliton solutions to the nonlinear Schrodinger equation cannot be recovered from Maxwell’s equations while in others the soliton solutions to Maxwell’s equations are lost from the nonlinear Schrodinger equation through approximation,although there are cases where the soliton solutions to the two sets of the equations demonstrate only quantitative difference.The origin of the differences is also discussed.
Chaos and Intermittency in the DNLS Equation Describing the Parallel Alfvén Wave Propagation
Directory of Open Access Journals (Sweden)
Gustavo Krause
2014-01-01
Full Text Available When the Hall effect is included in the magnetohydrodynamics equations (Hall-MHD model the wave propagation modes become coupled, but for propagation parallel to the ambient magnetic field the Alfvén mode decouples from the magnetosonic ones, resulting in circularly polarized waves that are described by the derivative nonlinear Schrödinger (DNLS equation. In this paper, the DNLS equation is numerically solved using spectral methods for the spatial derivatives and a fourth order Runge-Kutta scheme for time integration. Firstly, the nondiffusive DNLS equation is considered to test the validity of the method by verifying the analytical condition of modulational stability. Later, diffusive and excitatory effects are incorporated to compare the numerical results with those obtained by a three-wave truncation model. The results show that different types of attractors can exist depending on the diffusion level: for relatively large damping, there are fixed points for which the truncation model is a good approximation; for low damping, chaotic solutions appear and the three-wave truncation model fails due to the emergence of new nonnegligible modes.
A new application of Riccati equation to some nonlinear evolution equations
Energy Technology Data Exchange (ETDEWEB)
Geng Tao [School of Science, PO Box 122, Beijing University of Posts and Telecommunications, Beijing 100876 (China)], E-mail: taogeng@yahoo.com.cn; Shan Wenrui [School of Science, PO Box 122, Beijing University of Posts and Telecommunications, Beijing 100876 (China)
2008-03-03
By means of symbolic computation, a new application of Riccati equation is presented to obtain novel exact solutions of some nonlinear evolution equations, such as nonlinear Klein-Gordon equation, generalized Pochhammer-Chree equation and nonlinear Schroedinger equation. Comparing with the existing tanh methods and the proposed modifications, we obtain the exact solutions in the form as a non-integer power polynomial of tanh (or tan) functions by using this method, and the availability of symbolic computation is demonstrated.
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
A trial equation method to nonlinear evolution equation with rank inhomogeneous is given. As applications, the exact traveling wave solutions to some higher-order nonlinear equations such as generalized Boussinesq equation,generalized Pochhammer-Chree equation, KdV-Burgers equation, and KS equation and so on, are obtained. Among these, some results are new. The proposed method is based on the idea of reduction of the order of ODE. Some mathematical details of the proposed method are discussed.
Hyperbolic function method for solving nonlinear differential-different equations
Institute of Scientific and Technical Information of China (English)
Zhu Jia-Min
2005-01-01
An algorithm is devised to obtained exact travelling wave solutions of differential-different equations by means of hyperbolic function. For illustration, we apply the method to solve the discrete nonlinear (2+1)-dimensional Toda lattice equation and the discretized nonlinear mKdV lattice equation, and successfully constructed some explicit and exact travelling wave solutions.
Extension of Variable Separable Solutions for Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
JIA Hua-Bing; ZHANG Shun-Li; XU Wei; ZHU Xiao-Ning; WANG Yong-Mao; LOU Sen-Yue
2008-01-01
We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separablecation, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.
Almost Periodic Viscosity Solutions of Nonlinear Parabolic Equations
Directory of Open Access Journals (Sweden)
Zhang Shilin
2009-01-01
Full Text Available We generalize the comparison result 2007 on Hamilton-Jacobi equations to nonlinear parabolic equations, then by using Perron's method to study the existence and uniqueness of time almost periodic viscosity solutions of nonlinear parabolic equations under usual hypotheses.
Positive periodic solutions for third-order nonlinear differential equations
Directory of Open Access Journals (Sweden)
Jingli Ren
2011-05-01
Full Text Available For several classes of third-order constant coefficient linear differential equations we obtain existence and uniqueness of periodic solutions utilizing explicit Green's functions. We discuss an iteration method for constant coefficient nonlinear differential equations and provide new conditions for the existence of periodic positive solutions for third-order time-varying nonlinear and neutral differential equations.
Energy Technology Data Exchange (ETDEWEB)
Xie, Xi-Yang; Tian, Bo, E-mail: tian_bupt@163.com; Wang, Yu-Feng; Sun, Ya; Jiang, Yan
2015-11-15
In this paper, we investigate a generalized nonautonomous nonlinear equation which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions for the generalized nonautonomous nonlinear equation are obtained, under some variable–coefficient constraints. Properties of the first- and second-order rogue waves are graphically presented and analyzed: When the coefficients are all chosen as the constants, we can observe the some functions, the shapes of wave crests and troughs for the first- and second-order rogue waves change. Oscillating behaviors of the first- and second-order rogue waves are observed when the coefficients are the trigonometric functions.
Exact periodic wave solutions for some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
El-Wakil, S.A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt); Elgarayhi, A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)]. E-mail: elgarayhi@yahoo.com; Elhanbaly, A. [Theoretical Physics Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)
2006-08-15
The periodic wave solutions for some nonlinear partial differential equations, including generalized Klein-Gordon equation, Kadomtsev-Petviashvili (KP) equation and Boussinesq equations, are obtained by using the solutions of Jacobi elliptic equation. Under limit conditions, exact solitary wave solutions, shock wave solutions and triangular periodic wave solutions have been recovered.
Single and multi-solitary wave solutions to a class of nonlinear evolution equations
Wang, Deng-Shan; Li, Hongbo
2008-07-01
In this paper, an effective discrimination algorithm is presented to deal with equations arising from physical problems. The aim of the algorithm is to discriminate and derive the single traveling wave solutions of a large class of nonlinear evolution equations. Many examples are given to illustrate the algorithm. At the same time, some factorization technique are presented to construct the traveling wave solutions of nonlinear evolution equations, such as Camassa-Holm equation, Kolmogorov-Petrovskii-Piskunov equation, and so on. Then a direct constructive method called multi-auxiliary equations expansion method is described to derive the multi-solitary wave solutions of nonlinear evolution equations. Finally, a class of novel multi-solitary wave solutions of the (2+1)-dimensional asymmetric version of the Nizhnik-Novikov-Veselov equation are given by three direct methods. The algorithm proposed in this paper can be steadily applied to some other nonlinear problems.
Exact solitary wave solutions of nonlinear wave equations
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
The hyperbolic function method for nonlinear wave equations ispresented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Grbner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.
The Jeffcott equations in nonlinear rotordynamics
Zalik, R. A.
1989-01-01
The solutions of the Jeffcott equations describing the behavior of a rotating shaft are investigated analytically, with a focus on the case where deadband is taken into account. Bounds on the solutions are obtained from those for the linearized equations, and the onset of destructive vibrations is predicted by analyzing the Fourier transforms of the solutions; good agreement with numerical solutions and power-spectrum density plots is demonstrated. It is suggested that the present analytical approach could be applied to determine cryogenic-pump stability margins in flight and hot-fire ground testing of launch vehicles such as the Space Shuttle.
Some new solutions of nonlinear evolution equations with variable coefficients
Virdi, Jasvinder Singh
2016-05-01
We construct the traveling wave solutions of nonlinear evolution equations (NLEEs) with variable coefficients arising in physics. Some interesting nonlinear evolution equations are investigated by traveling wave solutions which are expressed by the hyperbolic functions, the trigonometric functions and rational functions. The applied method will be used in further works to establish more entirely new solutions for other kinds of such nonlinear evolution equations with variable coefficients arising in physics.
Logarithmic singularities of solutions to nonlinear partial differential equations
Tahara, Hidetoshi
2007-01-01
We construct a family of singular solutions to some nonlinear partial differential equations which have resonances in the sense of a paper due to T. Kobayashi. The leading term of a solution in our family contains a logarithm, possibly multiplied by a monomial. As an application, we study nonlinear wave equations with quadratic nonlinearities. The proof is by the reduction to a Fuchsian equation with singular coefficients.
New travelling wave solutions for nonlinear stochastic evolution equations
Indian Academy of Sciences (India)
Hyunsoo Kim; Rathinasamy Sakthivel
2013-06-01
The nonlinear stochastic evolution equations have a wide range of applications in physics, chemistry, biology, economics and finance from various points of view. In this paper, the (′/)-expansion method is implemented for obtaining new travelling wave solutions of the nonlinear (2 + 1)-dimensional stochastic Broer–Kaup equation and stochastic coupled Korteweg–de Vries (KdV) equation. The study highlights the significant features of the method employed and its capability of handling nonlinear stochastic problems.
Nonclassical Symmetries for Nonlinear Partial Differential Equations via Compatibility
Institute of Scientific and Technical Information of China (English)
Mostafa F. El-Sabbagh; Ahmad T. Ali
2011-01-01
The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The （2＋1）-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples i11ustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.
Homoclinic orbits of second-order nonlinear difference equations
Directory of Open Access Journals (Sweden)
Haiping Shi
2015-06-01
Full Text Available We establish existence criteria for homoclinic orbits of second-order nonlinear difference equations by using the critical point theory in combination with periodic approximations.
Bifurcation and stability for a nonlinear parabolic partial differential equation
Chafee, N.
1973-01-01
Theorems are developed to support bifurcation and stability of nonlinear parabolic partial differential equations in the solution of the asymptotic behavior of functions with certain specified properties.
On the exact controllability of a nonlinear stochastic heat equation
Directory of Open Access Journals (Sweden)
Bui An Ton
2006-01-01
Full Text Available The exact controllability of a nonlinear stochastic heat equation with null Dirichlet boundary conditions, nonzero initial and target values, and an interior control is established.
SOLVABILITY FOR NONLINEAR ELLIPTIC EQUATION WITH BOUNDARY PERTURBATION
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
The solvability of nonlinear elliptic equation with boundary perturbation is considered. The perturbed solution of original problem is obtained and the uniformly valid expansion of solution is proved.
Residual models for nonlinear partial differential equations
Directory of Open Access Journals (Sweden)
Garry Pantelis
2005-11-01
Full Text Available Residual terms that appear in nonlinear PDEs that are constructed to generate filtered representations of the variables of the fully resolved system are examined by way of a consistency condition. It is shown that certain commonly used empirical gradient models for the residuals fail the test of consistency and therefore cannot be validated as approximations in any reliable sense. An alternate method is presented for computing the residuals. These residual models are independent of free or artificial parameters and there direct link with the functional form of the system of PDEs which describe the fully resolved system are established.
OSCILLATION OF NONLINEAR IMPULSIVE PARABOLIC DIFFERENTIAL EQUATIONS WITH SEVERAL DELAYS
Institute of Scientific and Technical Information of China (English)
CuiChenpei; ZouMin; LiuAnping; XiaoLi
2005-01-01
In this paper, oscillatory properties for solutions of certain nonlinear impulsive parabolic equations with several delays are investigated and a series of new sufficient conditions for oscillations of the equation are established.
ALMOST PERIODIC SOLUTIONS TO SOME NONLINEAR DELAY DIFFERENTIAL EQUATION
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
The existence of an almost periodic solutions to a nonlinear delay diffierential equation is considered in this paper. A set of sufficient conditions for the existence and uniqueness of almost periodic solutions to some delay diffierential equations is obtained.
Generalized Nonlinear Proca Equation and its Free-Particle Solutions
Nobre, F D
2016-01-01
We introduce a non-linear extension of Proca's field theory for massive vector (spin $1$) bosons. The associated relativistic nonlinear wave equation is related to recently advanced nonlinear extensions of the Schroedinger, Dirac, and Klein-Gordon equations inspired on the non-extensive generalized thermostatistics. This is a theoretical framework that has been applied in recent years to several problems in nuclear and particle physics, gravitational physics, and quantum field theory. The nonlinear Proca equation investigated here has a power-law nonlinearity characterized by a real parameter $q$ (formally corresponding to the Tsallis entropic parameter) in such a way that the standard linear Proca wave equation is recovered in the limit $q \\rightarrow 1$. We derive the nonlinear Proca equation from a Lagrangian that, besides the usual vectorial field $\\Psi^{\\mu}(\\vec{x},t)$, involves an additional field $\\Phi^{\\mu}(\\vec{x},t)$. We obtain exact time dependent soliton-like solutions for these fields having the...
Directory of Open Access Journals (Sweden)
Elsayed Mohamed Elsayed ZAYED
2014-07-01
Full Text Available In this article, many new exact solutions of the (2+1-dimensional nonlinear Boussinesq-Kadomtsev-Petviashvili equation and the (1+1-dimensional nonlinear heat conduction equation are constructed using the Riccati equation mapping method. By means of this method, many new exact solutions are successfully obtained. This method can be applied to many other nonlinear evolution equations in mathematical physics.doi:10.14456/WJST.2014.14
A nonlinear wave equation with a nonlinear integral equation involving the boundary value
Directory of Open Access Journals (Sweden)
Thanh Long Nguyen
2004-09-01
Full Text Available We consider the initial-boundary value problem for the nonlinear wave equation $$displaylines{ u_{tt}-u_{xx}+f(u,u_{t}=0,quad xin Omega =(0,1,; 0
Fuzzy Modeling for Uncertainty Nonlinear Systems with Fuzzy Equations
Directory of Open Access Journals (Sweden)
Raheleh Jafari
2017-01-01
Full Text Available The uncertain nonlinear systems can be modeled with fuzzy equations by incorporating the fuzzy set theory. In this paper, the fuzzy equations are applied as the models for the uncertain nonlinear systems. The nonlinear modeling process is to find the coefficients of the fuzzy equations. We use the neural networks to approximate the coefficients of the fuzzy equations. The approximation theory for crisp models is extended into the fuzzy equation model. The upper bounds of the modeling errors are estimated. Numerical experiments along with comparisons demonstrate the excellent behavior of the proposed method.
Power Series Solution for Solving Nonlinear Burgers-Type Equations
Directory of Open Access Journals (Sweden)
E. López-Sandoval
2015-01-01
Full Text Available Power series solution method has been traditionally used to solve ordinary and partial linear differential equations. However, despite their usefulness the application of this method has been limited to this particular kind of equations. In this work we use the method of power series to solve nonlinear partial differential equations. The method is applied to solve three versions of nonlinear time-dependent Burgers-type differential equations in order to demonstrate its scope and applicability.
A NONLINEAR LAVRENTIEV-BITSADZE MIXED TYPE EQUATION
Institute of Scientific and Technical Information of China (English)
Chen Shuxing
2011-01-01
In this paper the Tricomi problem for a nonlinear mixed type equation is studied.The coefficients of the mixed type equation are discontinuous on the line,where the equation changes its type.The existence of solution to this problem is proved.The method developed in this paper can be applied to study more difficult problems for nonlinear mixed type equations arising in gas dynamics.
Power Series Solution for Solving Nonlinear Burgers-Type Equations
López-Sandoval, E.; Mello, A.; Godina-Nava, J. J.; Samana, A. R.
2015-01-01
Power series solution method has been traditionally used to solve ordinary and partial linear differential equations. However, despite their usefulness the application of this method has been limited to this particular kind of equations. In this work we use the method of power series to solve nonlinear partial differential equations. The method is applied to solve three versions of nonlinear time-dependent Burgers-type differential equations in order to demonstrate its scope and applicability.
Exact solutions for some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Peng, Yan-Ze
2003-08-11
Exact solutions to some nonlinear partial differential equations, including (2+1)-dimensional breaking soliton equation, sine-Gordon equation and double sine-Gordon equation, are studied by means of the mapping method proposed by the author recently. Many new results are presented. A simple review of the method is finally given.
Simple equation method for nonlinear partial differential equations and its applications
Directory of Open Access Journals (Sweden)
Taher A. Nofal
2016-04-01
Full Text Available In this article, we focus on the exact solution of the some nonlinear partial differential equations (NLPDEs such as, Kodomtsev–Petviashvili (KP equation, the (2 + 1-dimensional breaking soliton equation and the modified generalized Vakhnenko equation by using the simple equation method. In the simple equation method the trial condition is the Bernoulli equation or the Riccati equation. It has been shown that the method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering problems.
DEFF Research Database (Denmark)
Backi, Christoph Josef; Bendtsen, Jan Dimon; Leth, John-Josef
2014-01-01
In this work the stability properties of a nonlinear partial differential equation (PDE) with state–dependent parameters is investigated. Among other things, the PDE describes freezing of foodstuff, and is closely related to the (Potential) Burgers’ Equation. We show that for certain forms...
ANALYTICAL SOLUTIONS FOR SOME NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
胡建兰; 张汉林
2003-01-01
The following partial differential equations are studied: generaliz ed fifth-orderKdV equation, water wave equation, Kupershmidt equation, couples KdV equation. Theanalytical solutions to these problems via using various ansaiz es by introducing a second-order ordinary differential equation are found out.
DIFFERENCE METHODS FOR A NON-LINEAR ELLIPTIC SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS,
DIFFERENCE EQUATIONS, ITERATIONS), (*ITERATIONS, DIFFERENCE EQUATIONS), (* PARTIAL DIFFERENTIAL EQUATIONS , BOUNDARY VALUE PROBLEMS), EQUATIONS, FUNCTIONS(MATHEMATICS), SEQUENCES(MATHEMATICS), NONLINEAR DIFFERENTIAL EQUATIONS
Exact Travelling Wave Solutions to a Coupled Nonlinear Evolution Equation[
Institute of Scientific and Technical Information of China (English)
HUANGDing-Jiang; ZHANGHong-Qing
2004-01-01
By using an improved hyperbola function method, several types of exact travelling wave solutions to a coupled nonlinear evolution equation are obtained, which include kink-shaped soliton solutions, bell-shaped soliton solutions, envelop solitary wave solutions, and new solitary waves. The method can be applied to other nonlinear evolution equations in mathematical physics.
Exact Travelling Wave Solutions to a Coupled Nonlinear Evolution Equation
Institute of Scientific and Technical Information of China (English)
HUANG Ding-Jiang; ZHANG Hong-Qing
2004-01-01
By using an improved hyperbola function method, several types of exact travelling wave solutions to a coupled nonlinear evolution equation are obtained, which include kink-shaped soliton solutions, bell-shaped soliton solutions, envelop solitary wave solutions, and new solitary waves. The method can be applied to other nonlinear evolution equations in mathematical physics.
A variational approach to nonlinear evolution equations in optics
Indian Academy of Sciences (India)
D Anderson; M Lisak; A Berntson
2001-11-01
A tutorial review is presented of the use of direct variational methods based on RayleighRitz optimization for ﬁnding approximate solutions to various nonlinear evolution equations. The practical application of the approach is demonstrated by some illustrative examples in connection with the nonlinear Schrödinger equation.
Collapse in a forced three-dimensional nonlinear Schrodinger equation
DEFF Research Database (Denmark)
Lushnikov, P.M.; Saffman, M.
2000-01-01
We derive sufficient conditions for the occurrence of collapse in a forced three-dimensional nonlinear Schrodinger equation without dissipation. Numerical studies continue the results to the case of finite dissipation.......We derive sufficient conditions for the occurrence of collapse in a forced three-dimensional nonlinear Schrodinger equation without dissipation. Numerical studies continue the results to the case of finite dissipation....
Prolongation Structure of Semi-discrete Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
Based on noncommutative differential calculus, we present a theory of prolongation structure for semi-discrete nonlinear evolution equations. As an illustrative example, a semi-discrete model of the nonlinear Schr(o)dinger equation is discussed in terms of this theory and the corresponding Lax pairs are also given.
Exact solutions for some nonlinear systems of partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Darwish, A.A. [Department of Mathematics, Faculty of Science, Helwan University (Egypt)], E-mail: profdarwish@yahoo.com; Ramady, A. [Department of Mathematics, Faculty of Science, Beni-Suef University (Egypt)], E-mail: aramady@yahoo.com
2009-04-30
A direct and unified algebraic method for constructing multiple travelling wave solutions of nonlinear systems of partial differential equations (PDEs) is used and implemented in a computer algebraic system. New solutions for some nonlinear partial differential equations (NLPDEs) are obtained. Graphs of the solutions are displayed.
DEFF Research Database (Denmark)
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...
Marchenko Equation for the Derivative Nonlinear Schr(o)dinger Equation
Institute of Scientific and Technical Information of China (English)
HUANG Nian-Ning
2007-01-01
A simple derivation of the Marchenko equation is given for the derivative nonlinear Schr(o)dinger equation.The kernel of the Marchenko equation is demanded to satisfy the conditions given by the compatibility equations.the soliton solutions to the Marchenko equation are verified.The derivation is not concerned with the revisions of Kaup and Newell.
Estimation of saturation and coherence effects in the KGBJS equation - a non-linear CCFM equation
Deak, Michal
2012-01-01
We solve the modified non-linear extension of the CCFM equation - KGBJS equation - numerically for certain initial conditions and compare the resulting gluon Green functions with those obtained from solving the original CCFM equation and the BFKL and BK equations for the same initial conditions. We improve the low transversal momentum behaviour of the KGBJS equation by a small modification.
Exploring the Nonlinear Cloud and Rain Equation
Koren, Ilan; Feingold, Graham
2016-01-01
Marine stratocumulus cloud decks are regarded as the reflectors of the climate system, returning back to space a significant part of the income solar radiation, thus cooling the atmosphere. Such clouds can exist in two stable modes, open and closed cells, for a wide range of environmental conditions. This emergent behavior of the system, and its sensitivity to aerosol and environmental properties, is captured by a set of nonlinear equations. Here, using linear stability analysis, we express the transition from steady to a limit-cycle state analytically, showing how it depends on the model parameters. We show that the control of the droplet concentration (N) the environmental carrying-capacity (H0) and the cloud recovery parameter (tau) can be linked by a single nondimensional parameter mu=N/(alfa*tau*H0), suggesting that for deeper clouds the transition from open (oscillating) to closed (stable fixed point) cells will occur for higher droplet concentration (i.e. higher aerosol loading). The analytical calcula...
Lienard Equation and Exact Solutions for Some Soliton-Producing Nonlinear Equations
Institute of Scientific and Technical Information of China (English)
ZHANG Wei-Guo; CHANG Qian-Shun; ZHANG Qi-Ren
2004-01-01
In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then,explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obtained by means of results of the Lienard equation and proper deductions, which transform original partial differential equations into the Lienard one. These nonlinear equations include compound KdV, compound KdV-Burgers, generalized Boussinesq,generalized KP and Ginzburg-Landau equation. Some new solitary-wave solutions are found.
Smoothing and Decay Estimates for Nonlinear Diffusion Equations Equations of Porous Medium Type
Vázquez, Juan Luis
2006-01-01
This text is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of Physics, Chemistry, Biology, and Engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis.Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity ("equations of porou
Asymptotic behavior of solutions to nonlinear parabolic equation with nonlinear boundary conditions
Directory of Open Access Journals (Sweden)
Diabate Nabongo
2008-01-01
Full Text Available We show that solutions of a nonlinear parabolic equation of second order with nonlinear boundary conditions approach zero as t approaches infinity. Also, under additional assumptions, the solutions behave as a function determined here.
The Riccati Equation - An Economic Fundamental Equation which Describes Marginal Movement in Time
Directory of Open Access Journals (Sweden)
Lars P. Lystad
2006-01-01
Full Text Available The objective of this article is to demonstrate that the Riccati equation is an economic fundamental equation, which is marginally descriptive in time for the large majority of economic systems. It is also an objective to interpret the Riccati equation and the corresponding relations in terms of economics. The article shows that there is a close relationship between the marginal Hamiltonian function and the Riccati equation. The marginal Hamiltonian function will be an expression of the accounts based on optimal behaviour including both the change in result and the change in balance.
Techniques in Linear and Nonlinear Partial Differential Equations
1991-10-21
nonlinear partial differential equations , elliptic 15. NUMBER OF PAGES hyperbolic and parabolic. Variational methods. Vibration problems. Ordinary Five...NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS FINAL TECHNICAL REPORT PROFESSOR LOUIS NIRENBERG OCTOBER 21, 1991 NT)S CRA&I D FIC ,- U.S. ARMY RESEARCH OFFICE...Analysis and partial differential equations . ed. C. Sadowsky. Marcel Dekker (1990) 567-619. [7] Lin, Fanghua, Asymptotic behavior of area-minimizing
Linear and nonlinear degenerate abstract differential equations with small parameter
Shakhmurov, Veli B.
2016-01-01
The boundary value problems for linear and nonlinear regular degenerate abstract differential equations are studied. The equations have the principal variable coefficients and a small parameter. The linear problem is considered on a parameter-dependent domain (i.e., on a moving domain). The maximal regularity properties of linear problems and the optimal regularity of the nonlinear problem are obtained. In application, the well-posedness of the Cauchy problem for degenerate parabolic equation...
Travelling wave solutions for ( + 1)-dimensional nonlinear evolution equations
Indian Academy of Sciences (India)
Jonu Lee; Rathinasamy Sakthivel
2010-10-01
In this paper, we implement the exp-function method to obtain the exact travelling wave solutions of ( + 1)-dimensional nonlinear evolution equations. Four models, the ( + 1)-dimensional generalized Boussinesq equation, ( + 1)-dimensional sine-cosine-Gordon equation, ( + 1)-double sinh-Gordon equation and ( + 1)-sinh-cosinh-Gordon equation, are used as vehicles to conduct the analysis. New travelling wave solutions are derived.
Flexible alternatives to the Gompertz equation for describing growth with age in turkey hens.
Porter, T; Kebreab, E; Darmani Kuhi, H; Lopez, S; Strathe, A B; France, J
2010-02-01
A total of 49 profiles of growing turkey hens from commercial flocks were used in this study. Three flexible growth functions (von Bertalanffy, Richards, and Morgan) were evaluated with regard to their ability to describe the relationship between BW and age and were compared with the Gompertz equation with its fixed point of inflection, which might result in its overestimation. For each function, 4 ways of analysis were implemented. A basic model was fitted first, followed by implementation of a first-order autoregressive correlation structure. A model that considered only mature BW varied with year and another that considered only the rate coefficient varied with different years were applied. The results showed that the fixed point of inflection of the Gompertz equation can be a limitation and that the relationship between BW and age in turkeys was best described using flexible growth functions. However, the Richards equation failed to converge when fitted to the turkey growth data; therefore, it was not considered further. Inclusion of an autoregressive process of the first order rendered a substantially improved fit to data for the 3 growth functions. The Morgan equation provided the best fit to the data set and was used for characterizing mean growth curves for the 7 yr of production. It was estimated that the maximum growth rate occurred at 3.74, 3.65, 3.99, 4.18, 4.05, 4.01, and 3.77 kg BW for production years from 1997 to 2003, respectively. It is recommended that flexible growth functions should be considered as an alternative to the simpler functions (with a fixed point of inflection) for describing the relationship between BW and age in turkeys because they were easier to fit and very often gave a closer fit to data points because of their flexibility, and therefore a smaller residual MS value, than simpler models. It can also be recommended that studies should consider adding a first-order autoregressive process when analyzing repeated measures data with
International Conference on Differential Equations and Nonlinear Mechanics
2001-01-01
The International Conference on Differential Equations and Nonlinear Mechanics was hosted by the University of Central Florida in Orlando from March 17-19, 1999. One of the conference days was dedicated to Professor V. Lakshmikantham in th honor of his 75 birthday. 50 well established professionals (in differential equations, nonlinear analysis, numerical analysis, and nonlinear mechanics) attended the conference from 13 countries. Twelve of the attendees delivered hour long invited talks and remaining thirty-eight presented invited forty-five minute talks. In each of these talks, the focus was on the recent developments in differential equations and nonlinear mechanics and their applications. This book consists of 29 papers based on the invited lectures, and I believe that it provides a good selection of advanced topics of current interest in differential equations and nonlinear mechanics. I am indebted to the Department of Mathematics, College of Arts and Sciences, Department of Mechanical, Materials and Ae...
The effect of nonlinearity on unstable zones of Mathieu equation
Indian Academy of Sciences (India)
M GH SARYAZDI
2017-03-01
Mathieu equation is a well-known ordinary differential equation in which the excitation term appears as the non-constant coefficient. The mathematical modelling of many dynamic systems leads to Mathieu equation. The determination of the locus of unstable zone is important for the control of dynamic systems. In this paper, the stable and unstable regions of Mathieu equation are determined for three cases of linear and nonlinear equations using the homotopy perturbation method. The effect of nonlinearity is examined in the unstable zone. The results show that the transition curves of linear Mathieu equation depend on the frequency of the excitation term. However, for nonlinear equations, the curves depend also on initial conditions. In addition, increasing the amplitude of response leads to an increase in the unstable zone.
Arino, O; Kimmel, M
1989-01-01
A model of cell cycle kinetics is proposed, which includes unequal division of cells, and a nonlinear dependence of the fraction of cells re-entering proliferation on the total number of cells in the cycle. The model is described by a nonlinear functional-integral equation. It is analyzed using the operator semigroup theory combined with classical differential equations approach. A complete description of the asymptotic behavior of the model is provided for a relatively broad class of nonlinearities. The nonnegative solutions either tend to a stable steady state, or to zero. The simplicity of the model makes it an interesting step in the analysis of dynamics of nonlinear structure populations.
REDUCTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATION AND EXACT SOLUTIONS
Institute of Scientific and Technical Information of China (English)
YeCaier; PanZuliang
2003-01-01
Nonlinear partial differetial equation(NLPDE)is converted into ordinary differential equation(ODE)via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.
EXACT SOLITARY WAVE SOLUTIONS OF THETWO NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
ZhuYanjuan; ZhangChunhua
2005-01-01
The solitary wave solutions of the combined KdV-mKdV-Burgers equation and the Kolmogorov-Petrovskii-Piskunov equation are obtained by means of the direct algebra method, which can be generalized to deal with high dimensional nonlinear evolution equations.
A Family of Exact Solutions for the Nonlinear Schrodinger Equation
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
In this paper, the nonlinear Schrodinger (NLS) equation was analytically solved. Firstly, the stationary solutions of NLSequation were explicitly given by the elliptic functions. Then a family of exact solutions of NLS equation were obtained from these sta-tionary solutions by a method for finding new exact solutions from the stationary solutions of integrable evolution equations.
Boundary controllability for a nonlinear beam equation
Directory of Open Access Journals (Sweden)
Xiao-Min Cao
2015-09-01
Full Text Available This article concerns a nonlinear system modeling the bending vibrations of a nonlinear beam of length $L>0$. First, we derive the existence of long time solutions near an equilibrium. Then we prove that the nonlinear beam is locally exact controllable around the equilibrium in $H^4(0,L$ and with control functions in $H^2(0,T$. The approach we used are open mapping theorem, local controllability established by linearization, and the induction.
Energy Technology Data Exchange (ETDEWEB)
Mancas, Stefan C. [Department of Mathematics, Embry–Riddle Aeronautical University, Daytona Beach, FL 32114-3900 (United States); Rosu, Haret C., E-mail: hcr@ipicyt.edu.mx [IPICYT, Instituto Potosino de Investigacion Cientifica y Tecnologica, Apdo Postal 3-74 Tangamanga, 78231 San Luis Potosí, SLP (Mexico)
2013-09-02
We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in their first-kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers–Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second order nonlinear equations.
Analytic solutions of a class of nonlinear partial differential equations
Institute of Scientific and Technical Information of China (English)
ZHANG Hong-qing; DING Qi
2008-01-01
An approach is presented for computing the adjoint operator vector of a class of nonlinear (that is,partial-nonlinear) operator matrices by using the properties of conjugate operators to generalize a previous method proposed by the author.A unified theory is then given to solve a class of nonlinear (partial-nonlinear and including all linear)and non-homogeneous differential equations with a mathematical mechanization method.In other words,a transformation is constructed by homogenization and triangulation,which reduces the original system to a simpler diagonal system.Applications are given to solve some elasticity equations.
A modal analysis method to describe weak nonlinear effects in metamaterials
Zeng, Y; O'Hara, J; Trugman, S A
2012-01-01
We apply a rigorous eigenmode analysis to study the electromagnetic properties of linear and weakly nonlinear metamaterials. The nonlinear response can be totally described by the linear eigenmodes when weak nonlinearities are attributed to metamaterials. We use this theory to interpret intrinsic second-harmonic spectroscopy on metallic metamaterials. Our study indicates that metamaterial eigenmodes play a critical role in optimizing a nonlinear metamaterial response to the extent that a poorly optimized modal pattern overwhelms the widely recognized benefits of plasmonic resonant field enhancements.
New Optical Solitons in High-Order Dispersive Cubic-Quintic Nonlinear Schrodinger Equation
Institute of Scientific and Technical Information of China (English)
LI Hua-Mei; XU You-Shen; LIN Ji
2004-01-01
By using the generalized tanh-function method, we find bright and dark solitary wave solutions to an extended nonlinear Schrodinger equation with the third-order and fourth-order dispersion and the cubic-quintic nonlinear terms, describing the propagation of extremely short pulses. At the same time, we also obtained other types of exact solutions.
New explicit exact solutions to a nonlinear dispersive-dissipative equation
Institute of Scientific and Technical Information of China (English)
Naranmandula; Wang Ke-Xie
2004-01-01
Using the first-integral method, we obtain a series of new explicit exact solutions such as exponential function solutions, triangular function solutions, singular solitary wave solution and kink solitary wave solution of a nonlinear dispersive-dissipative equation, which describes weak nonlinear ion-acoustic waves in plasma consisting of cold ions and warm electrons.
Multiphase Weakly Nonlinear Geometric Optics for Schrödinger Equations
Carles, Rémi
2010-01-01
We describe and rigorously justify the nonlinear interaction of highly oscillatory waves in nonlinear Schrödinger equations, posed on Euclidean space or on the torus. Our scaling corresponds to a weakly nonlinear regime where the nonlinearity affects the leading order amplitude of the solution, but does not alter the rapid oscillations. We consider initial states which are superpositions of slowly modulated plane waves, and use the framework of Wiener algebras. A detailed analysis of the corresponding nonlinear wave mixing phenomena is given, including a geometric interpretation of the resonance structure for cubic nonlinearities. As an application, we recover and extend some instability results for the nonlinear Schrödinger equation on the torus in negative order Sobolev spaces. © 2010 Society for Industrial and Applied Mathematics.
control theory to systems described by partial differential equations. The intent is not to advance the theory of partial differential equations per se. Thus all considerations will be restricted to the more familiar equations of the type which often occur in mathematical physics. Specifically, the distributed parameter systems under consideration are represented by a set of field
Solutions to nonlinear Schrodinger equations for special initial data
Directory of Open Access Journals (Sweden)
Takeshi Wada
2015-11-01
Full Text Available This article concerns the solvability of the nonlinear Schrodinger equation with gauge invariant power nonlinear term in one space dimension. The well-posedness of this equation is known only for $H^s$ with $s\\ge 0$. Under some assumptions on the nonlinearity, this paper shows that this equation is uniquely solvable for special but typical initial data, namely the linear combinations of $\\delta(x$ and p.v. (1/x, which belong to $H^{-1/2-0}$. The proof in this article allows $L^2$-perturbations on the initial data.
Global existence and uniqueness of nonlinear evolutionary fluid equations
Qin, Yuming; Wang, Taige
2015-01-01
This book presents recent results on nonlinear evolutionary fluid equations such as the compressible (radiative) magnetohydrodynamics (MHD) equations, compressible viscous micropolar fluid equations, the full non-Newtonian fluid equations and non-autonomous compressible Navier-Stokes equations. These types of partial differential equations arise in many fields of mathematics, but also in other branches of science such as physics and fluid dynamics. This book will be a valuable resource for graduate students and researchers interested in partial differential equations, and will also benefit practitioners in physics and engineering.
Indian Academy of Sciences (India)
Yusuf Gurefe; Abdullah Sonmezoglu; Emine Misirli
2011-12-01
In this paper some exact solutions including soliton solutions for the KdV equation with dual power law nonlinearity and the (, ) equation with generalized evolution are obtained using the trial equation method. Also a more general trial equation method is proposed.
Nonlinear Allometric Equation for Crop Response to Soil Salinity
Directory of Open Access Journals (Sweden)
E. Misle
2015-06-01
Full Text Available Crop response to soil salinity has been extensively studied, from empirical works to modelling approach, being described by different equations, first as a piecewise linear model. The equation employed can differ with actual response, causing miscalculation in practical situations, particularly at the higher extremes of the curve. The aim of this work is to propose a new equation, which allows determining the full response to salinity of plant species and to provide a verification using different experimental data sets. A new nonlinear equation is exposed supported by the allometric approach, in which the allometric exponent is salinity-dependent and decreases with the increase in relative salinity. A conversion procedure of parameters of the threshold-slope model is presented; also, a simple procedure for estimating the maximum salinity (zero-yield point when data sets are incomplete is exposed. The equation was tested in a wide range of experimental situations, using data sets from published works, as well as new measurements on seed germination. The statistical indicators of quality (R2, absolute sum of squares and standard deviation of residuals showed that the equation accurately fits the tested empirical results. The new equation for determining crop response to soil salinity is able to follow the response curve of any crop with remarkable accuracy and flexibility. Remarkable characteristics are: a maximum at minimum salinity, a maximum salinity point can be found (zero-yield depending on the data sets, and a meaningful inflection point, as well as the two points at which the slope of the curve equals unity, can be found.
Painlevé analysis for nonlinear partial differential equations
Musette, M
1998-01-01
The Painlevé analysis introduced by Weiss, Tabor and Carnevale (WTC) in 1983 for nonlinear partial differential equations (PDE's) is an extension of the method initiated by Painlevé and Gambier at the beginning of this century for the classification of algebraic nonlinear differential equations (ODE's) without movable critical points. In these lectures we explain the WTC method in its invariant version introduced by Conte in 1989 and its application to solitonic equations in order to find algorithmically their associated so-called ``integrable'' equations but they are generically no more valid for equations modelising physical phenomema. Belonging to this second class, some equations called ``partially integrable'' sometimes keep remnants of integrability. In that case, the singularity analysis may also be useful for building closed form analytic solutions, which necessarily % Conte agree with the singularity structure of the equations. We display the privileged role played by the Riccati equation and syste...
Singular and non-topological soliton solutions for nonlinear fractional differential equations
Institute of Scientific and Technical Information of China (English)
Ozkan Guner
2015-01-01
In this article, the fractional derivatives are described in the modified Riemann–Liouville sense. We propose a new approach, namely an ansatz method, for solving fractional differential equations (FDEs) based on a fractional complex transform and apply it to solve nonlinear space–time fractional equations. As a result, the non-topological as well as the singular soliton solutions are obtained. This method can be suitable and more powerful for solving other kinds of nonlinear fractional FDEs arising in mathematical physics.
New Exact Solutions for New Model Nonlinear Partial Differential Equation
Directory of Open Access Journals (Sweden)
A. Maher
2013-01-01
Full Text Available In this paper we propose a new form of Padé-II equation, namely, a combined Padé-II and modified Padé-II equation. The mapping method is a promising method to solve nonlinear evaluation equations. Therefore, we apply it, to solve the combined Padé-II and modified Padé-II equation. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functions, and elliptic functions.
Singularity analysis of a new discrete nonlinear Schrodinger equation
Sakovich, Sergei
2001-01-01
We apply the Painleve test for integrability to a new discrete (differential-difference) nonlinear Schrodinger equation introduced by Leon and Manna. Since the singular expansions of solutions of this equation turn out to contain nondominant logarithmic terms, we conclude that the studied equation is nonintegrable. This result supports the observation of Levi and Yamilov that the Leon-Manna equation does not admit high-order generalized symmetries. As a byproduct of the singularity analysis c...
DEFF Research Database (Denmark)
Bang, O.; Juul Rasmussen, J.; Christiansen, P.L.
1994-01-01
Discretizing the continuous nonlinear Schrodinger equation with arbitrary power nonlinearity influences the time evolution of its ground state solitary solution. In the subcritical case, for grid resolutions above a certain transition value, depending on the degree of nonlinearity, the solution w...
Institute of Scientific and Technical Information of China (English)
HANG Chao; HUANG Guo-Xiang
2006-01-01
We investigate the nonlinear localized structures of optical pulses propagating in a one-dimensional photonic crystal with a quadratic nonlinearity. Using a method of multiple scales we show that the nonlinear evolution of a wave packet, formed by the superposition of short-wavelength excitations, and long-wavelength mean fields, generated by the self-interaction of the wave packet, are governed by a set of coupled high-dimensional nonlinear envelope equations, which can be reduced to Davey-Stewartson equations and thus support dromionlike high-dimensional nonlinear excitations in the system.
Bifurcation methods of dynamical systems for handling nonlinear wave equations
Indian Academy of Sciences (India)
Dahe Feng; Jibin Li
2007-05-01
By using the bifurcation theory and methods of dynamical systems to construct the exact travelling wave solutions for nonlinear wave equations, some new soliton solutions, kink (anti-kink) solutions and periodic solutions with double period are obtained.
Comparative study of homotopy continuation methods for nonlinear algebraic equations
Nor, Hafizudin Mohamad; Ismail, Ahmad Izani Md.; Majid, Ahmad Abd.
2014-07-01
We compare some recent homotopy continuation methods to see which method has greater applicability and greater accuracy. We test the methods on systems of nonlinear algebraic equations. The results obtained indicate the superior accuracy of Newton Homotopy Continuation Method (NHCM).
Lipschitz regularity results for nonlinear strictly elliptic equations and applications
Ley, Olivier; Nguyen, Vinh Duc
2017-10-01
Most of Lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic for instance). For equations with superquadratic growth power in gradient, one usually uses weak Bernstein-type arguments which require regularity and/or convex-type assumptions on the gradient nonlinearity. In this article, we obtain new Lipschitz regularity results for a large class of nonlinear strictly elliptic equations with possibly arbitrary growth power of the Hamiltonian with respect to the gradient variable using some ideas coming from Ishii-Lions' method. We use these bounds to solve an ergodic problem and to study the regularity and the large time behavior of the solution of the evolution equation.
Hamiltonian Formalism of the Derivative Nonlinear Schrodinger Equation
Institute of Scientific and Technical Information of China (English)
CAI Hao; LIU Feng-Min; HUANG Nian-Ning
2003-01-01
A particular form of poisson bracket is introduced for the derivative nonlinear Schrodinger (DNLS) equation.And its Hamiltonian formalism is developed by a linear combination method. Action-angle variables are found.
Nonlinear damped Schrodinger equation in two space dimensions
Directory of Open Access Journals (Sweden)
Tarek Saanouni
2015-04-01
Full Text Available In this article, we study the initial value problem for a semi-linear damped Schrodinger equation with exponential growth nonlinearity in two space dimensions. We show global well-posedness and exponential decay.
IDENTIFICATION OF PARAMETERS IN PARABOLIC EQUATIONS WITH NONLINEARITY
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
In this paper, we consider the identification of parameters in parabolic equations with nonlinearity. Some approximation processes for the identification problem are given. Our results improve and generalize the previous results.
Differentiability at lateral boundary for fully nonlinear parabolic equations
Ma, Feiyao; Moreira, Diego R.; Wang, Lihe
2017-09-01
For fully nonlinear uniformly parabolic equations, the first derivatives regularity of viscosity solutions at lateral boundary is studied under new Dini type conditions for the boundary, which is called Reifenberg Dini conditions and is weaker than usual Dini conditions.
Analysis of Nonlinear Fractional Nabla Difference Equations
Directory of Open Access Journals (Sweden)
Jagan Mohan Jonnalagadda
2015-01-01
Full Text Available In this paper, we establish sufficient conditions on global existence and uniqueness of solutions of nonlinear fractional nabla difference systems and investigate the dependence of solutions on initial conditions and parameters.
The theorem on existence of singular solutions to nonlinear equations
Directory of Open Access Journals (Sweden)
Prusinska А.
2005-01-01
Full Text Available The aim of this paper is to present some applications of pregularity theory to investigations of nonlinear multivalued mappings. The main result addresses to the problem of existence of solutions to nonlinear equations in the degenerate case when the linear part is singular at the considered initial point. We formulate conditions for existence of solutions of equation F(x = 0 when first p - 1 derivatives of F are singular.
MULTISCALE HOMOGENIZATION OF NONLINEAR HYPERBOLIC EQUATIONS WITH SEVERAL TIME SCALES
Institute of Scientific and Technical Information of China (English)
Jean Louis Woukeng; David Dongo
2011-01-01
We study the multiscale homogenization of a nonlinear hyperbolic equation in a periodic setting. We obtain an accurate homogenization result. We also show that as the nonlinear term depends on the microscopic time variable, the global homogenized problem thus obtained is a system consisting of two hyperbolic equations. It is also shown that in spite of the presence of several time scales, the global homogenized problem is not a reiterated one.
Ehrenfest theorem, Galilean invariance and nonlinear Schr"odinger equations
Kälbermann, G
2003-01-01
Galilean invariant Schr"odinger equations possessing nonlinear terms coupling the amplitude and the phase of the wave function can violate the Ehrenfest theorem. An example of this kind is provided. The example leads to the proof of the theorem: A Galilean invariant Schr"odinger equation derived from a lagrangian density obeys the Ehrenfest theorem. The theorem holds for any linear or nonlinear lagrangian.
Can Ising model and/or QKPZ equation properly describe reactive-wetting interface dynamics?
Efraim, Yael; Taitelbaum, Haim
2009-09-01
The reactive-wetting process, e.g. spreading of a liquid droplet on a reactive substrate is known as a complex, non-linear process with high sensitivity to minor fluctuations. The dynamics and geometry of the interface (triple line) between the materials is supposed to shed light on the main mechanisms of the process. We recently studied a room temperature reactive-wetting system of a small (˜ 150 μm) Hg droplet that spreads on a thin (˜ 4000 Å) Ag substrate. We calculated the kinetic roughening exponents (growth and roughness), as well as the persistence exponent of points on the advancing interface. In this paper we address the question whether there exists a well-defined model to describe the interface dynamics of this system, by performing two sets of numerical simulations. The first one is a simulation of an interface propagating according to the QKPZ equation, and the second one is a landscape of an Ising chain with ferromagnetic interactions in zero temperature. We show that none of these models gives a full description of the dynamics of the experimental reactivewetting system, but each one of them has certain common growth properties with it. We conjecture that this results from a microscopic behavior different from the macroscopic one. The microscopic mechanism, reflected by the persistence exponent, resembles the Ising behavior, while in the macroscopic scale, exemplified by the growth exponent, the dynamics looks more like the QKPZ dynamics.
NONLINEAR BOUNDARY STABILIZATION OF WAVE EQUATIONS WITH VARIABLE C OEFFICIENTS
Institute of Scientific and Technical Information of China (English)
冯绍继; 冯德兴
2003-01-01
The wave equation with variable coefficients with a nonlinear dissipative boundary feedbackis studied. By the Riemannian geometry method and the multiplier technique, it is shown thatthe closed loop system decays exponentially or asymptotically, and hence the relation betweenthe decay rate of the system energy and the nonlinearity behavior of the feedback function isestablished.
Applications of Elliptic Equation to Nonlinear Coupled Systems
Institute of Scientific and Technical Information of China (English)
FUZun-Tao; LIUShi-Da; LIUShi-Kuo
2003-01-01
The elliptic equation is taken as a transformation and applied to solve nonlinear coupled systems. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wave solutions, periodic wave solutions and so on, so this method can be taken as a unified method in solving nonlinear coupled systems.
Applications of Elliptic Equation to Nonlinear Coupled Systems
Institute of Scientific and Technical Information of China (English)
FU Zun-Tao; LIU Shi-Da; LIU Shi-Kuo
2003-01-01
The elliptic equation is taken as a transformation and applied to solve nonlinear coupled systems. Itis shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wavesolutions, periodic wave solutions and so on, so this method can be taken as a unified method in solving nonlinear coupled systems.
A NEW SMOOTHING EQUATIONS APPROACH TO THE NONLINEAR COMPLEMENTARITY PROBLEMS
Institute of Scientific and Technical Information of China (English)
Chang-feng Ma; Pu-yan Nie; Guo-ping Liang
2003-01-01
The nonlinear complementarity problem can be reformulated as a nonsmooth equation. In this paper we propose a new smoothing Newton algorithm for the solution of the nonlinear complementarity problem by constructing a new smoothing approximation function. Global and local superlinear convergence results of the algorithm are obtained under suitable conditions. Numerical experiments confirm the good theoretical properties of the algorithm.
Institute of Scientific and Technical Information of China (English)
YANG Yong; YAN Zhen-Ya
2002-01-01
In this letter the three-dimensional nonlinear Helmholtz equation is investigated, which describes electro-magnetic wave propagation in a nonlinear Kerr-type medium such that sixteen families of new Jacobi elliptic functionsolutions are obtained, by using our extended Jacobian elliptic function expansion method. When the modulus m → 1or0, the corresponding solitary waves including bright solitons, dark solitons and new line solitons and singly periodicsolutions can be also found.
Institute of Scientific and Technical Information of China (English)
YANGYong; YANZhen－Ya
2002-01-01
In this letter the three-dimensional nonlinear Helmholtz equation is investigated.which describes electromagnetic wave propagation in a nonlinear Kerr-type medium such that sixteen families of new Jacobi elliptic function solutions are obtained,by using our extended Jacobian elliptic function expansion method.When the modulus m-→1 or 0,the corresponding solitary waves including bright solitons,dark solitons and new line solitons and singly periodic solutions can be also found.
Application of homotopy-perturbation to non-linear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Cveticanin, L. [Faculty of Technical Sciences, 21000 Novi Sad, Trg D. Obradovica 6 (Serbia)], E-mail: cveticanin@uns.ns.ac.yu
2009-04-15
In this paper He's homotopy perturbation method has been adopted for solving non-linear partial differential equations. An approximate solution of the differential equation which describes the longitudinal vibration of a beam is obtained. The solution is compared with that found using the variational iteration method introduced by He. The difference between the two solutions is negligible.
New Doubly Periodic Solutions for the Coupled Nonlinear Klein-Gordon Equations
Institute of Scientific and Technical Information of China (English)
LIU Chun-Ping
2005-01-01
By using the general solutions of a new coupled Riccati equations, a direct algebraic method is described to construct doubly periodic solutions (Jacobi elliptic function solution) for the coupled nonlinear Klein-Gordon equations.It is shown that more doubly periodic solutions and the corresponding solitary wave solutions and trigonometric function solutions can be obtained in a unified way by this method.
Excited state nonlinear integral equations for an integrable anisotropic spin-1 chain
Energy Technology Data Exchange (ETDEWEB)
Suzuki, J [Department of Physics, Faculty of Science, Shizuoka University, Ohya 836, Shizuoka (Japan)
2004-12-17
We propose a set of nonlinear integral equations to describe the excited states of an integrable the spin-1 chain with anisotropy. The scaling dimensions, evaluated numerically in previous studies, are recovered analytically by using the equations. This result may be relevant to the study of the supersymmetric sine-Gordon model.
Elimination and nonlinear equations of Rees algebra
Busé, Laurent; Simis, Aron
2009-01-01
A new approach is established to computing the image of a rational map, whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees. In the case the map is everywhere defined this analysis provides free resolutions of graded parts of the Rees algebra of the base ideal in degrees where it does not coincide with the corresponding symmetric algebra. A surprising fact is that the torsion in those degrees only contributes to the first free module in the resolution of the symmetric algebra modulo torsion. An additional point is that this contribution -- which of course corresponds to non linear equations of the Rees algebra -- can be described in these degrees in terms of non Koszul syzygies via certain upgrading maps in the vein of the ones introduced earlier by J. Herzog, the third named author and W. Vasconcelos. As a measure of the reach of this torsion analysis we could say that, in the case of a general everywhere defined map, ...
Computer-aided Nonlinear Control System Design Using Describing Function Models
Nassirharand, Amir
2012-01-01
A systematic computer-aided approach provides a versatile setting for the control engineer to overcome the complications of controller design for highly nonlinear systems. Computer-aided Nonlinear Control System Design provides such an approach based on the use of describing functions. The text deals with a large class of nonlinear systems without restrictions on the system order, the number of inputs and/or outputs or the number, type or arrangement of nonlinear terms. The strongly software-oriented methods detailed facilitate fulfillment of tight performance requirements and help the designer to think in purely nonlinear terms, avoiding the expedient of linearization which can impose substantial and unrealistic model limitations and drive up the cost of the final product. Design procedures are presented in a step-by-step algorithmic format each step being a functional unit with outputs that drive the other steps. This procedure may be easily implemented on a digital computer with example problems from mecha...
Nonlinear Kramers equation associated with nonextensive statistical mechanics.
Mendes, G A; Ribeiro, M S; Mendes, R S; Lenzi, E K; Nobre, F D
2015-05-01
Stationary and time-dependent solutions of a nonlinear Kramers equation, as well as its associated nonlinear Fokker-Planck equations, are investigated within the context of Tsallis nonextensive statistical mechanics. Since no general analytical time-dependent solutions are found for such a nonlinear Kramers equation, an ansatz is considered and the corresponding asymptotic behavior is studied and compared with those known for the standard linear Kramers equation. The H-theorem is analyzed for this equation and its connection with Tsallis entropy is investigated. An application is discussed, namely the motion of Hydra cells in two-dimensional cellular aggregates, for which previous measurements have verified q-Gaussian distributions for velocity components and superdiffusion. The present analysis is in quantitative agreement with these experimental results.
Analytical exact solution of the non-linear Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Martins, Alisson Xavier; Rocha Filho, Tarcisio Marciano da [Universidade de Brasilia (UnB), DF (Brazil). Inst. de Fisica. Grupo de Fisica e Matematica
2011-07-01
Full text: In this work we present how to classify and obtain analytical solutions of the Schroedinger equation with a generic non-linearity in 1+1 dimensions. Our approach is based on the determination of Lie symmetry transformation mapping solutions into solutions, and non-classical symmetry transformations, mapping a given solution into itself. From these symmetries it is then possible to reduce the equation to a system of ordinary differential equations which can then be solved using standard methods. The generic non-linearity is handled by considering it as an additional unknown in the determining equations for the symmetry transformations. This results in an over-determined system of non-linear partial differential equations. Its solution can then be determined in some cases by reducing it to the so called involutive (triangular) form, and then solved. This reduction is very tedious and can only performed using a computer algebra system. Once the determining system is solved, we obtain the explicit form for the non-linearity admitting a Lie or non-classical symmetry. The analytical solutions are then derived by solving the reduced ordinary differential equations. The non-linear determining system for the non-classical symmetry transformations and Lie symmetry generators are obtaining using the computer algebra package SADE (symmetry analysis of differential equations), developed at our group. (author)
Nonlinear ordinary differential equations analytical approximation and numerical methods
Hermann, Martin
2016-01-01
The book discusses the solutions to nonlinear ordinary differential equations (ODEs) using analytical and numerical approximation methods. Recently, analytical approximation methods have been largely used in solving linear and nonlinear lower-order ODEs. It also discusses using these methods to solve some strong nonlinear ODEs. There are two chapters devoted to solving nonlinear ODEs using numerical methods, as in practice high-dimensional systems of nonlinear ODEs that cannot be solved by analytical approximate methods are common. Moreover, it studies analytical and numerical techniques for the treatment of parameter-depending ODEs. The book explains various methods for solving nonlinear-oscillator and structural-system problems, including the energy balance method, harmonic balance method, amplitude frequency formulation, variational iteration method, homotopy perturbation method, iteration perturbation method, homotopy analysis method, simple and multiple shooting method, and the nonlinear stabilized march...
Nonlinear Parabolic Equations with Singularities in Colombeau Vector Spaces
Institute of Scientific and Technical Information of China (English)
Mirjana STOJANOVI(C)
2006-01-01
We consider nonlinear parabolic equations with nonlinear non-Lipschitz's term and singular initial data like Dirac measure, its derivatives and powers. We prove existence-uniqueness theorems in Colombeau vector space gC1,w2,2([O,T),Rn),n ≤ 3. Due to high singularity in a case of parabolic equation with nonlinear conservative term we employ the regularized derivative for the conservative term, in order to obtain the global existence-uniqueness result in Colombeau vector space gC1,L2([O,T),Rn),n ≤ 3.
GHM method for obtaining rationalsolutions of nonlinear differential equations.
Vazquez-Leal, Hector; Sarmiento-Reyes, Arturo
2015-01-01
In this paper, we propose the application of the general homotopy method (GHM) to obtain rational solutions of nonlinear differential equations. It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. The obtained results show that GHM is a powerful tool, capable to generate highly accurate rational solutions. AMS subject classification 34L30.
Derivation of an Applied Nonlinear Schroedinger Equation.
Energy Technology Data Exchange (ETDEWEB)
Pitts, Todd Alan; Laine, Mark Richard; Schwarz, Jens; Rambo, Patrick K.; Karelitz, David B.
2015-01-01
We derive from first principles a mathematical physics model useful for understanding nonlinear optical propagation (including filamentation). All assumptions necessary for the development are clearly explained. We include the Kerr effect, Raman scattering, and ionization (as well as linear and nonlinear shock, diffraction and dispersion). We explain the phenomenological sub-models and each assumption required to arrive at a complete and consistent theoretical description. The development includes the relationship between shock and ionization and demonstrates why inclusion of Drude model impedance effects alters the nature of the shock operator. Unclassified Unlimited Release
Derivation of an applied nonlinear Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Pitts, Todd Alan [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Laine, Mark Richard [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Schwarz, Jens [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Rambo, Patrick K. [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States); Karelitz, David B. [Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
2015-01-01
We derive from first principles a mathematical physics model useful for understanding nonlinear optical propagation (including filamentation). All assumptions necessary for the development are clearly explained. We include the Kerr effect, Raman scattering, and ionization (as well as linear and nonlinear shock, diffraction and dispersion). We explain the phenomenological sub-models and each assumption required to arrive at a complete and consistent theoretical description. The development includes the relationship between shock and ionization and demonstrates why inclusion of Drude model impedance effects alters the nature of the shock operator. Unclassified Unlimited Release
Institute of Scientific and Technical Information of China (English)
WANG Mei-Jiao; WANG Qi
2006-01-01
In this work, by means of a new more general ansatz and the symbolic computation system Maple, we extend the Riccati equation rational expansion method [Chaos, Solitons & Fractals 25 (2005) 1019] to uniformly construct a series of stochastic nontravelling wave solutions for nonlinear stochastic evolution equation. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions including a series of rational formal nontraveling wave and coefficient functions' soliton-like solutions and trigonometric-like function solutions. The method can also be applied to solve other nonlinear stochastic evolution equation or equations.
NEW EXACT TRAVELLING WAVE SOLUTIONS TO THREE NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
Sirendaoreji
2004-01-01
Based on the computerized symbolic computation, some new exact travelling wave solutions to three nonlinear evolution equations are explicitly obtained by replacing the tanhξ in tanh-function method with the solutions of a new auxiliary ordinary differential equation.
Exact solutions for the cubic-quintic nonlinear Schroedinger equation
Energy Technology Data Exchange (ETDEWEB)
Zhu Jiamin [Department of Physics, Zhejiang Lishui University, Lishui 323000 (China)]. E-mail: zjm64@163.com; Ma Zhengyi [Department of Physics, Zhejiang Lishui University, Lishui 323000 (China); Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072 (China)
2007-08-15
In this paper, the cubic-quintic nonlinear Schroedinger equation is solved through the extended elliptic sub-equation method. As a consequence, many types of exact travelling wave solutions are obtained which including bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions.
Nonlinear partial differential equations: Integrability, geometry and related topics
Krasil'shchik, Joseph; Rubtsov, Volodya
2017-03-01
Geometry and Differential Equations became inextricably entwined during the last one hundred fifty years after S. Lie and F. Klein's fundamental insights. The two subjects go hand in hand and they mutually enrich each other, especially after the "Soliton Revolution" and the glorious streak of Symplectic and Poisson Geometry methods in the context of Integrability and Solvability problems for Non-linear Differential Equations.
An analysis of the nonlinear equation = (, ) + (, )$u^2_$ + ℎ(, ) + (, )$
Indian Academy of Sciences (India)
R M Edelstein; K S Govinder
2011-01-01
We use the method of preliminary group classiﬁcation to analyse a particular form of the nonlinear diffusion equation in which the inhomogeneity is quadratic in . The method yields an optimal system of one-dimensional subalgebras. As a result we obtain those explicit forms of the unknown functions , , ℎ and for which the equation admits additional point symmetries.
Multiple solutions to some singular nonlinear Schrodinger equations
Directory of Open Access Journals (Sweden)
Monica Lazzo
2001-01-01
Full Text Available We consider the equation $$ - h^2 Delta u + V_varepsilon(x u = |u|^{p-2} u $$ which arises in the study of standing waves of a nonlinear Schrodinger equation. We allow the potential $V_varepsilon$ to be unbounded below and prove existence and multiplicity results for positive solutions.
Several Dynamical Properties for a Nonlinear Shallow Water Equation
Directory of Open Access Journals (Sweden)
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.
Nonlinear eigenvalue approach to differential Riccati equations for contraction analysis
Kawano, Yu; Ohtsuka, Toshiyuki
2017-01-01
In this paper, we extend the eigenvalue method of the algebraic Riccati equation to the differential Riccati equation (DRE) in contraction analysis. One of the main results is showing that solutions to the DRE can be expressed as functions of nonlinear eigenvectors of the differential Hamiltonian ma
Structure of Dirac matrices and invariants for nonlinear Dirac equations
2004-01-01
We present invariants for nonlinear Dirac equations in space-time ${\\mathbb R}^{n+1}$, by which we prove that a special choice of the Cauchy data yields free solutions. Our argument works for Klein-Gordon-Dirac equations with Yukawa coupling as well. Related problems on the structure of Dirac matrices are studied.
LIMITED MEMORY BFGS METHOD FOR NONLINEAR MONOTONE EQUATIONS
Institute of Scientific and Technical Information of China (English)
Weijun Zhou; Donghui Li
2007-01-01
In this paper, we propose an algorithm for solving nonlinear monotone equations by combining the limited memory BFGS method (L-BFGS) with a projection method. We show that the method is globally convergent if the equation involves a Lipschitz continuous monotone function. We also present some preliminary numerical results.
STABILITY OF NONLINEAR NEUTRAL DIFFERENTIAL EQUATION VIA FIXED POINT
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
In this paper,a nonlinear neutral differential equation is considered.By a fixed point theory,we give some conditions to ensure that the zero solution to the equation is asymptotically stable.Some existing results are improved and generalized.
Relations between nonlinear Riccati equations and other equations in fundamental physics
Schuch, Dieter
2014-10-01
Many phenomena in the observable macroscopic world obey nonlinear evolution equations while the microscopic world is governed by quantum mechanics, a fundamental theory that is supposedly linear. In order to combine these two worlds in a common formalism, at least one of them must sacrifice one of its dogmas. Linearizing nonlinear dynamics would destroy the fundamental property of this theory, however, it can be shown that quantum mechanics can be reformulated in terms of nonlinear Riccati equations. In a first step, it will be shown that the information about the dynamics of quantum systems with analytical solutions can not only be obtainable from the time-dependent Schrödinger equation but equally-well from a complex Riccati equation. Comparison with supersymmetric quantum mechanics shows that even additional information can be obtained from the nonlinear formulation. Furthermore, the time-independent Schrödinger equation can also be rewritten as a complex Riccati equation for any potential. Extension of the Riccati formulation to include irreversible dissipative effects is straightforward. Via (real and complex) Riccati equations, other fields of physics can also be treated within the same formalism, e.g., statistical thermodynamics, nonlinear dynamical systems like those obeying a logistic equation as well as wave equations in classical optics, Bose- Einstein condensates and cosmological models. Finally, the link to abstract "quantizations" such as the Pythagorean triples and Riccati equations connected with trigonometric and hyperbolic functions will be shown.
Indian Academy of Sciences (India)
Ranjit Kumar
2012-09-01
Travelling and solitary wave solutions of certain coupled nonlinear diffusion-reaction equations have been constructed using the auxiliary equation method. These equations arise in a variety of contexts not only in biological, chemical and physical sciences but also in ecological and social sciences.
NEW ALTERNATING DIRECTION FINITE ELEMENT SCHEME FOR NONLINEAR PARABOLIC EQUATION
Institute of Scientific and Technical Information of China (English)
崔霞
2002-01-01
A new alternating direction (AD) finite element (FE) scheme for 3-dimensional nonlinear parabolic equation and parabolic integro-differential equation is studied. By using AD,the 3-dimensional problem is reduced to a family of single space variable problems, calculation work is simplified; by using FE, high accuracy is kept; by using various techniques for priori estimate for differential equations such as inductive hypothesis reasoning, the difficulty arising from the nonlinearity is treated. For both FE and ADFE schemes, the convergence properties are rigorously demonstrated, the optimal H1- and L2-norm space estimates and the O((△t)2) estimate for time variable are obtained.
Exact solutions of certain nonlinear chemotaxis diffusion reaction equations
Indian Academy of Sciences (India)
MISHRA AJAY; KAUSHAL R S; PRASAD AWADHESH
2016-05-01
Using the auxiliary equation method, we obtain exact solutions of certain nonlinear chemotaxis diffusion reaction equations in the presence of a stimulant. In particular, we account for the nonlinearities arising not only from the density-dependent source terms contributed by the particles and the stimulant but also from the coupling term of the stimulant. In addition to this, the diffusion of the stimulant and the effect of long-range interactions are also accounted for in theconstructed coupled differential equations. The results obtained here could be useful in the studies of several biological systems and processes, e.g., in bacterial infection, chemotherapy, etc.
An almost symmetric Strang splitting scheme for nonlinear evolution equations.
Einkemmer, Lukas; Ostermann, Alexander
2014-07-01
In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow cannot be computed exactly. Instead, we use a numerical approximation based on the linearization of the vector field. This is of interest in applications as it allows us to apply splitting methods to a wider class of problems from the sciences. However, in the situation described, the classic Strang splitting scheme, while still being a method of second order, is not longer symmetric. This, in turn, implies that the construction of higher order methods by composition is limited to order three only. To remedy this situation, based on previous work in the context of ordinary differential equations, we construct a class of Strang splitting schemes that are symmetric up to a desired order. We show rigorously that, under suitable assumptions on the nonlinearity, these methods are of second order and can then be used to construct higher order methods by composition. In addition, we illustrate the theoretical results by conducting numerical experiments for the Brusselator system and the KdV equation.
Generalized nonlinear Proca equation and its free-particle solutions
Energy Technology Data Exchange (ETDEWEB)
Nobre, F.D. [Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex Systems, Rio de Janeiro, RJ (Brazil); Plastino, A.R. [Universidad Nacional Buenos Aires-Noreoeste, CeBio y Secretaria de Investigacion, Junin (Argentina)
2016-06-15
We introduce a nonlinear extension of Proca's field theory for massive vector (spin 1) bosons. The associated relativistic nonlinear wave equation is related to recently advanced nonlinear extensions of the Schroedinger, Dirac, and Klein-Gordon equations inspired on the non-extensive generalized thermostatistics. This is a theoretical framework that has been applied in recent years to several problems in nuclear and particle physics, gravitational physics, and quantum field theory. The nonlinear Proca equation investigated here has a power-law nonlinearity characterized by a real parameter q (formally corresponding to the Tsallis entropic parameter) in such a way that the standard linear Proca wave equation is recovered in the limit q → 1. We derive the nonlinear Proca equation from a Lagrangian, which, besides the usual vectorial field Ψ{sup μ}(vector x,t), involves an additional field Φ{sup μ}(vector x,t). We obtain exact time-dependent soliton-like solutions for these fields having the form of a q-plane wave, and we show that both field equations lead to the relativistic energy-momentum relation E{sup 2} = p{sup 2}c{sup 2} + m{sup 2}c{sup 4} for all values of q. This suggests that the present nonlinear theory constitutes a new field theoretical representation of particle dynamics. In the limit of massless particles the present q-generalized Proca theory reduces to Maxwell electromagnetism, and the q-plane waves yield localized, transverse solutions of Maxwell equations. Physical consequences and possible applications are discussed. (orig.)
Optimal Variational Asymptotic Method for Nonlinear Fractional Partial Differential Equations.
Baranwal, Vipul K; Pandey, Ram K; Singh, Om P
2014-01-01
We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters γ 0, γ 1, γ 2,… and auxiliary functions H 0(x), H 1(x), H 2(x),… are introduced in the correction functional of the standard variational iteration method. The optimal values of these parameters are obtained by minimizing the square residual error. To test the method, we apply it to solve two important classes of nonlinear partial differential equations: (1) the fractional advection-diffusion equation with nonlinear source term and (2) the fractional Swift-Hohenberg equation. Only few iterations are required to achieve fairly accurate solutions of both the first and second problems.
Stochastic nonlinear differential equation generating 1/f noise.
Kaulakys, B; Ruseckas, J
2004-08-01
Starting from the simple point process model of 1/f noise, we derive a stochastic nonlinear differential equation for the signal exhibiting 1/f noise, in any desirably wide range of frequency. A stochastic differential equation (the general Langevin equation with a multiplicative noise) that gives 1/f noise is derived. The solution of the equation exhibits the power-law distribution. The process with 1/f noise is demonstrated by the numerical solution of the derived equation with the appropriate restriction of the diffusion of the signal in some finite interval.
Exact travelling wave solutions of nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish) Suez Canal University, AL-Arish 45111 (Egypt)]. E-mail: asoliman_99@yahoo.com; Abdou, M.A. [Theoretical Research Group, Department of Physics, Faculty of Science, Mansoura University, Mansoura 35516 (Egypt)]. E-mail: m_abdou_eg@yahoo.com
2007-04-15
An extended Fan-sub equation method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. The key idea of this method is to take full advantage of the general elliptic equation, involving five parameters, which has more new solutions and whose degeneracies can lead to special sub equation involving three parameters. As an illustration of the extended Fan method, more new solutions are obtained for three models namely, generalized KdV, Drinfeld-Sokolov system and RLW equation.
Adomian solution of a nonlinear quadratic integral equation
Directory of Open Access Journals (Sweden)
E.A.A. Ziada
2013-04-01
Full Text Available We are concerned here with a nonlinear quadratic integral equation (QIE. The existence of a unique solution will be proved. Convergence analysis of Adomian decomposition method (ADM applied to these type of equations is discussed. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of Adomian’s series solution. Two methods are used to solve these type of equations; ADM and repeated trapezoidal method. The obtained results are compared.
Iterative Solution for Systems of Nonlinear Two Binary Operator Equations
Institute of Scientific and Technical Information of China (English)
ZHANGZhi-hong; LIWen-feng
2004-01-01
Using the cone and partial ordering theory and mixed monotone operator theory, the existence and uniqueness of solutions for some classes of systems of nonlinear two binary operator equations in a Banach space with a partial ordering are discussed. And the error estimates that the iterative sequences converge to solutions are also given. Some relevant results of solvability of two binary operator equations and systems of operator equations are imnroved and generalized.
Tian, Qing; Wu, Lei; Zhang, Jie-Fang; Malomed, Boris A; Mihalache, D; Liu, W M
2011-01-01
We put forward a generic transformation which helps to find exact soliton solutions of the nonlinear Schrödinger equation with a spatiotemporal modulation of the nonlinearity and external potentials. As an example, we construct exact solitons for the defocusing nonlinearity and harmonic potential. When the soliton's eigenvalue is fixed, the number of exact solutions is determined by energy levels of the linear harmonic oscillator. In addition to the stable fundamental solitons, stable higher-order modes, describing array of dark solitons nested in a finite-width background, are constructed too. We also show how to control the instability domain of the nonstationary solitons.
Scalable nonlinear iterative methods for partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Cai, X-C
2000-10-29
We conducted a six-month investigation of the design, analysis, and software implementation of a class of singularity-insensitive, scalable, parallel nonlinear iterative methods for the numerical solution of nonlinear partial differential equations. The solutions of nonlinear PDEs are often nonsmooth and have local singularities, such as sharp fronts. Traditional nonlinear iterative methods, such as Newton-like methods, are capable of reducing the global smooth nonlinearities at a nearly quadratic convergence rate but may become very slow once the local singularities appear somewhere in the computational domain. Even with global strategies such as line search or trust region the methods often stagnate at local minima of {parallel}F{parallel}, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution u* of F(u) = 0, we solve, instead, an equivalent nonlinearly preconditioned system G(F(u*)) = 0 whose nonlinearities are more balanced. In this project, we proposed and studied a nonlinear additive Schwarz based parallel nonlinear preconditioner and showed numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, when a traditional inexact Newton method fails.
Directory of Open Access Journals (Sweden)
Veyis Turut
2013-01-01
Full Text Available Two tecHniques were implemented, the Adomian decomposition method (ADM and multivariate Padé approximation (MPA, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in Caputo sense. First, the fractional differential equation has been solved and converted to power series by Adomian decomposition method (ADM, then power series solution of fractional differential equation was put into multivariate Padé series. Finally, numerical results were compared and presented in tables and figures.
Nonlinear structures for extended Korteweg–de Vries equation in multicomponent plasma
Indian Academy of Sciences (India)
Abdelsalam U M; Allehiany F M; Moslem W M; El-Labany S K
2016-03-01
Using the fluid hydrodynamic equations of positive and negative ions, as well as$q$-nonextensive electron density distribution, an extended Korteweg–de Vries (EKdV) equation describing a small but finite amplitude dust ion-acoustic waves (DIAWs) is derived. Extended homogeneous balance method is used to obtain a new class of solutions of the EKdV equation. The effects of different physical parameters on the propagating nonlinear structures and their relevanceto particle acceleration in space plasma are reported.
Quantum theory of nonlocal nonlinear Schrodinger equation
Vyas, Vivek M
2015-01-01
Nonlocal nonlinear Schrodinger model is quantised and exactly solved using the canonical framework. It is found that the usual canonical quantisation of the model leads to a theory with pathological inner product. This problem is resolved by constructing another inner product over the vector space of the theory. The resultant theory is found to be identical to that of nonrelativistic bosons with delta function interaction potential, devoid of any nonlocality. The exact eigenstates are found using the Bethe ansatz technique.
Directory of Open Access Journals (Sweden)
Khaled A. Gepreel
2016-07-01
Full Text Available In this paper, we improve the extended trial equation method to construct the exact solutions for nonlinear coupled system of partial differential equations in mathematical physics. We use the extended trial equation method to find some different types of exact solutions such as the Jacobi elliptic function solutions, soliton solutions, trigonometric function solutions and rational, exact solutions to the nonlinear coupled Schrodinger Boussinesq equations when the balance number is a positive integer. The performance of this method is reliable, effective and powerful for solving more complicated nonlinear partial differential equations in mathematical physics. The balance number of this method is not constant as we have in other methods. This method allows us to construct many new types of exact solutions. By using the Maple software package we show that all obtained solutions satisfy the original partial differential equations.
Chaos in the fractional order nonlinear Bloch equation with delay
Baleanu, Dumitru; Magin, Richard L.; Bhalekar, Sachin; Daftardar-Gejji, Varsha
2015-08-01
The Bloch equation describes the dynamics of nuclear magnetization in the presence of static and time-varying magnetic fields. In this paper we extend a nonlinear model of the Bloch equation to include both fractional derivatives and time delays. The Caputo fractional time derivative (α) in the range from 0.85 to 1.00 is introduced on the left side of the Bloch equation in a commensurate manner in increments of 0.01 to provide an adjustable degree of system memory. Time delays for the z component of magnetization are inserted on the right side of the Bloch equation with values of 0, 10 and 100 ms to balance the fractional derivative with delay terms that also express the history of an earlier state. In the absence of delay, τ = 0 , we obtained results consistent with the previously published bifurcation diagram, with two cycles appearing at α = 0.8548 with subsequent period doubling that leads to chaos at α = 0.9436 . A periodic window is observed for the range 0.962 chaos arising again as α nears 1.00. The bifurcation diagram for the case with a 10 ms delay is similar: two cycles appear at the value α = 0.8532 , and the transition from two to four cycles at α = 0.9259 . With further increases in the fractional order, period doubling continues until at α = 0.9449 chaos ensues. In the case of a 100 millisecond delay the transitions from one cycle to two cycles and two cycles to four cycles are observed at α = 0.8441 , and α = 0.8635 , respectively. However, the system exhibits chaos at much lower values of α (α = 0.8635). A periodic window is observed in the interval 0.897 chaos again appearing for larger values of α . In general, as the value of α decreased the system showed transitions from chaos to transient chaos, and then to stability. Delays naturally appear in many NMR systems, and pulse programming allows the user control over the process. By including both the fractional derivative and time delays in the Bloch equation, we have developed a
Nonlinear Biharmonic Equations with Critical Potential
Institute of Scientific and Technical Information of China (English)
Hui XIONG; Yao Tian SHEN
2005-01-01
In this paper, we study two semilinear singular biharmonic equations: one with subcritical exponent and critical potential, another with sub-critical potential and critical exponent. By Pohozaev identity for singular solution, we prove there is no nontrivial solution for equations with critical exponent and critical potential. And by using the concentrate compactness principle and Mountain Pass theorem, respectively, we get two existence results for the two problems. Meanwhile,we have compared the changes of the critical dimensions in singular and non-singular cases, and we get an interesting result.
Energy Technology Data Exchange (ETDEWEB)
Zhang, Sheng [Department of Mathematics, Bohai University, Jinzhou 121000 (China)]. E-mail: zhshaeng@yahoo.com.cn; Xia, Tiecheng [Department of Mathematics, Bohai University, Jinzhou 121000 (China); Department of Mathematics, Shanghai University, Shanghai 200444 (China)
2007-04-09
In this Letter, a generalized new auxiliary equation method is proposed for constructing more general exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the combined KdV-mKdV equation and the (2+1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equations to illustrate the validity and advantages of the method. As a result, many new and more general exact solutions are obtained.
Institute of Scientific and Technical Information of China (English)
Zhi Hong-Yan; Zhao Xue-Qin; Zhang Hong-Qing
2005-01-01
Based on the study of tanh function method and the coupled projective Riccati equation method, we propose a new algorithm to search for explicit exact solutions of nonlinear evolution equations. We use the higher-order Schrodinger equation and mKdV equation to illustrate this algorithm. As a result, more new solutions are obtained, which include new solitary solutions, periodic solutions, and singular solutions. Some new solutions are illustrated in figures.
Backward stochastic differential equations from linear to fully nonlinear theory
Zhang, Jianfeng
2017-01-01
This book provides a systematic and accessible approach to stochastic differential equations, backward stochastic differential equations, and their connection with partial differential equations, as well as the recent development of the fully nonlinear theory, including nonlinear expectation, second order backward stochastic differential equations, and path dependent partial differential equations. Their main applications and numerical algorithms, as well as many exercises, are included. The book focuses on ideas and clarity, with most results having been solved from scratch and most theories being motivated from applications. It can be considered a starting point for junior researchers in the field, and can serve as a textbook for a two-semester graduate course in probability theory and stochastic analysis. It is also accessible for graduate students majoring in financial engineering.
Multi-diffusive nonlinear Fokker-Planck equation
Ribeiro, Mauricio S.; Casas, Gabriela A.; Nobre, Fernando D.
2017-02-01
Nonlinear Fokker-Planck equations, characterized by more than one diffusion term, have appeared recently in literature. Here, it is shown that these equations may be derived either from approximations in a master equation, or from a Langevin-type approach. An H-theorem is proven, relating these Fokker-Planck equations to an entropy composed by a sum of contributions, each of them associated with a given diffusion term. Moreover, the stationary state of the Fokker-Planck equation is shown to coincide with the equilibrium state, obtained by extremization of the entropy, in the sense that both procedures yield precisely the same equation. Due to the nonlinear character of this equation, the equilibrium probability may be obtained, in most cases, only by means of numerical approaches. Some examples are worked out, where the equilibrium probability distribution is computed for nonlinear Fokker-Planck equations presenting two diffusion terms, corresponding to an entropy characterized by a sum of two contributions. It is shown that the resulting equilibrium distribution, in general, presents a form that differs from a sum of the equilibrium distributions that maximizes each entropic contribution separately, although in some cases one may construct such a linear combination as a good approximation for the equilibrium distribution.
EXACT EXPLICIT SOLUTIONS OF THE NONLINEAR SCHR(O)DINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION
Institute of Scientific and Technical Information of China (English)
姚若侠; 李忠斌
2003-01-01
A system comprised of the nonlinear Schrodinger equation coupled to theBoussinesq equation (S-B equations) which dealing with the stationary propagation of cou-pled non-linear upper-hybrid and magnetosonic waves in magnetized plasma is proposed.To examine its solitary wave solutions, a reduced set of ordinary differential equations areconsidered by a simple traveling wave transformation. It is then shown that several newsolutions (either functional or parametrical) can be obtained systematically, in addition torederiving all known ones by means of our simple and direct algebra method with the helpof the computer algebra system Maple.
Directory of Open Access Journals (Sweden)
Khaled A. Gepreel
2012-01-01
Full Text Available We modified the rational Jacobi elliptic functions method to construct some new exact solutions for nonlinear differential difference equations in mathematical physics via the lattice equation, the discrete nonlinear Schrodinger equation with a saturable nonlinearity, the discrete nonlinear Klein-Gordon equation, and the quintic discrete nonlinear Schrodinger equation. Some new types of the Jacobi elliptic solutions are obtained for some nonlinear differential difference equations in mathematical physics. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.
Algebraic calculation of stroboscopic maps of ordinary, nonlinear differential equations
Energy Technology Data Exchange (ETDEWEB)
Wackerbauer, R. (Max-Planck-Institut fuer Extraterrestrische Physik, Garching (Germany)); Huebler, A. (Illinois Univ., Urbana, IL (United States). Center for Complex Systems Research); Mayer-Kress, G. (Los Alamos National Lab., NM (United States) California Univ., Santa Cruz, CA (United States). Dept. of Mathematics)
1991-07-25
The relation between the parameters of a differential equation and corresponding discrete maps are becoming increasingly important in the study of nonlinear dynamical systems. Maps are well adopted for numerical computation and several universal properties of them are known. Therefore some perturbation methods have been proposed to deduce them for physical systems, which can be modeled by an ordinary differential equation (ODE) with a small nonlinearity. A new iterative, rigorous algebraic method for the calculation of the coefficients of a Taylor expansion of a stroboscopic map from ODE's with not necessarily small nonlinearities is presented. It is shown analytically that most of the coefficients are small for a small integration time and grow slowly in the course of time if the flow vector field of the ODE is polynomial and if the ODE has fixed point in the origin. Approximations of different orders respectively of the rest term are investigated for several nonlinear systems. 31 refs., 16 figs.
Equations to describe brain size across the continuum of human lifespan.
Borzage, Matthew; Blüml, Stefan; Seri, Istvan
2014-01-01
Equations fitting the normative values for gender-specific brain size changes are available. However, these equations do not fit for all age ranges across the human lifespan and particularly have failed to examine the fit across the continuum of prenatal and postnatal human life. We sought to develop a parametric equation that best describes the changes in gender-specific brain size as a function of age across the continuum of prenatal and postnatal human life. Brain weight and brain volume data retrieved from the literature were combined to perform a meta-analysis. Additions to previously published findings included collecting a dataset that spanned the continuum of human lifespan, logarithmic transformation of the data and utilization of the Birch equation. We used Akaike's Information Criterion (AIC) for quantitative evaluation of the new equations. A total of 2,011 brain weight data points spanning from 10 weeks of fetal gestation to over 90 years of age were retrieved. Using our approach, we developed equations with improved fits and lower or similar AIC values compared to the published equations. The new equations are modifications of the basic Birch model. These equations are the first to describe the gender-specific brain weight changes through the continuum of both prenatal and postnatal human life while achieving a level of accuracy similar to or better than the previous, more age-restricted models. The new equations are improved compared to previously used equations and may be useful to those who study brain development, particularly researchers interested in prenatal and postnatal brain size.
In a book "Tsunami and Nonlinear Waves": Numerical Verification of the Hasselmann equation
Korotkevich, A O; Resio, D; Zakharov, V E; Korotkevich, Alexander O.; Pushkarev, Andrei N.; Resio, Don; Zakharov, Vladimir E.
2007-01-01
The purpose of this article is numerical verification of the thory of weak turbulence. We performed numerical simulation of an ensemble of nonlinearly interacting free gravity waves (swell) by two different methods: solution of primordial dynamical equations describing potential flow of the ideal fluid with a free surface and, solution of the kinetic Hasselmann equation, describing the wave ensemble in the framework of the theory of weak turbulence. Comparison of the results demonstrates pretty good applicability of the weak turbulent approach.
Institute of Scientific and Technical Information of China (English)
2008-01-01
In this article, we consider the existence of local and global solution to the Cauchy problem of a doubly nonlinear equation. By introducing the norms |||f|||h and
Stochasticity in numerical solutions of the nonlinear Schroedinger equation
Shen, Mei-Mei; Nicholson, D. R.
1987-01-01
The cubically nonlinear Schroedinger equation is an important model of nonlinear phenomena in fluids and plasmas. Numerical solutions in a spatially periodic system commonly involve truncation to a finite number of Fourier modes. These solutions are found to be stochastic in the sense that the largest Liapunov exponent is positive. As the number of modes is increased, the size of this exponent appears to converge to zero, in agreement with the recent demonstration of the integrability of the spatially periodic case.
Energy Technology Data Exchange (ETDEWEB)
Belmonte-Beitia, Juan [Departamento de Matematicas, E. T. S. de Ingenieros Industriales and Instituto de Matematica Aplicada a la Ciencia y la IngenierIa (IMACI), E. T. S. I. Industriales, Avda. Camilo Jose Cela, s/n Universidad de Castilla-La Mancha 13071 Ciudad Real (Spain)
2009-01-23
We introduce a model of a Bose-Einstein condensate based on the one-dimensional nonlinear Schroedinger equation, in which the nonlinear term depends on the domain. The nonlinear term changes a cubic term into a quintic term, according to the domain considered. We study the existence, stability and bifurcation of solutions, and use the qualitative theory of dynamical systems to study certain properties of such solutions.
Indian Academy of Sciences (India)
R S Kaushal; Ranjit Kumar; Awadhesh Prasad
2006-08-01
Attempts have been made to look for the soliton content in the solutions of the recently studied nonlinear diffusion-reaction equations [R S Kaushal, J. Phys. 38, 3897 (2005)] involving quadratic or cubic nonlinearities in addition to the convective flux term which renders the system nonconservative and the corresponding Hamiltonian non-Hermitian.
Energy Technology Data Exchange (ETDEWEB)
Belmonte-Beitia, Juan [Departamento de Matematicas, E.T.S. de Ingenieros Industriales and Instituto de Matematica Aplicada a la Ciencia y la Ingenieria (IMACI), Avda. Camilo Jose Cela 3, Universidad de Castilla-La Mancha, 13071 Ciudad Real (Spain)], E-mail: juan.belmonte@uclm.es; Calvo, Gabriel F. [Departamento de Matematicas, E.T.S. de Ingenieros Industriales and Instituto de Matematica Aplicada a la Ciencia y la Ingenieria (IMACI), Avda. Camilo Jose Cela 3, Universidad de Castilla-La Mancha, 13071 Ciudad Real (Spain)], E-mail: gabriel.fernandez@uclm.es
2009-01-19
In this Letter, by means of similarity transformations, we construct explicit solutions to the quintic nonlinear Schroedinger equation with potentials and nonlinearities depending both on time and on the spatial coordinates. We present the general approach and use it to study some examples and find nontrivial explicit solutions such as periodic (breathers), quasiperiodic and bright and dark soliton solutions.
AN INTEGRAL EQUATION DESCRIBING RIDING WAVES IN SHALLOW WATER OF FINITE DEPTH
Institute of Scientific and Technical Information of China (English)
An Shu-ping; Le Jia-chun; Dai Shi-qiang
2003-01-01
An integral equation describing riding waves, i.e., small-scale perturbation waves superposed on unperturbed surface waves, in shallow water of finite depth was studied via explicit Hamiltonian formulation, and the water was regarded as ideal incompressible fluid of uniform density. The kinetic energy, density of the perturbed fluid motion was formulated with Hamiltonian canonical variables[1], elevation of the free surface and the velocity potential at the free surface. Then the variables were expanded to the first order at the free surface of unperturbed waves. An integal equation for velocity potential of perturbed waves on the unperturbed free surface was derived by conformal mapping and the Fourier transformation. The integral equation could replace the Hamiltonian canonical equations which are difficult to solve. An explicit expression of Lagrangian density function could be obtained by solving the integral equation. The method used in this paper provides a new path to study the Hamiltonian formulation of riding waves and wave interaction problems.
Ultrashort light bullets described by the two-dimensional sine-Gordon equation
Leblond, Hervé; 10.1103/PHYSREVA.81.063815
2011-01-01
By using a reductive perturbation technique applied to a two-level model, this study puts forward a generic two-dimensional sine-Gordon evolution equation governing the propagation of femtosecond spatiotemporal optical solitons in Kerr media beyond the slowly varying envelope approximation. Direct numerical simulations show that, in contrast to the long-wave approximation, no collapse occurs, and that robust (2+1)-dimensional ultrashort light bullets may form from adequately chosen few-cycle input spatiotemporal wave forms. In contrast to the case of quadratic nonlinearity, the light bullets oscillate in both space and time and are therefore not steady-state lumps.
Analytic treatment of nonlinear evolution equations using ﬁrst integral method
Indian Academy of Sciences (India)
Ahmet Bekir; Ömer Ünsal
2012-07-01
In this paper, we show the applicability of the ﬁrst integral method to combined KdV-mKdV equation, Pochhammer–Chree equation and coupled nonlinear evolution equations. The power of this manageable method is conﬁrmed by applying it for three selected nonlinear evolution equations. This approach can also be applied to other nonlinear differential equations.
AD GALERKIN ANALYSIS FOR NONLINEAR PSEUDO-HYPERBOLIC EQUATIONS
Institute of Scientific and Technical Information of China (English)
Xia Cui
2003-01-01
AD (Alternating direction) Galerkin schemes for d-dimensional nonlinear pseudo-hyperbolic equations are studied. By using patch approximation technique, AD procedure is realized,and calculation work is simplified. By using Galerkin approach, highly computational accuracy is kept. By using various priori estimate techniques for differential equations,difficulty coming from non-linearity is treated, and optimal H1 and L2 convergence properties are demonstrated. Moreover, although all the existed AD Galerkin schemes using patch approximation are limited to have only one order accuracy in time increment, yet the schemes formulated in this paper have second order accuracy in it. This implies an essential advancement in AD Galerkin analysis.
New traveling wave solutions for nonlinear evolution equations
Energy Technology Data Exchange (ETDEWEB)
El-Wakil, S.A. [Theoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura (Egypt); Madkour, M.A. [Theoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura (Egypt); Abdou, M.A. [Theoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura (Egypt)]. E-mail: m_abdou_eg@yahoo.com
2007-06-11
The generalized Jacobi elliptic function expansion method is used with a computerized symbolic computation for constructing the new exact traveling wave solutions. The validity and reliability of the method is tested by its applications on a class of nonlinear evolution equations of special interest in mathematical physics. As a result, many exact traveling wave solutions are obtained which include the kink-shaped solutions, bell-shaped solutions, singular solutions and periodic solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.
THE MORTAR ELEMENT METHOD FOR A NONLINEAR BIHARMONIC EQUATION
Institute of Scientific and Technical Information of China (English)
Zhong-ci Shi; Xue-jun Xu
2005-01-01
The mortar element method is a new domain decomposition method(DDM) with nonoverlapping subdomains. It can handle the situation where the mesh on different subdomains need not align across interfaces, and the matching of discretizations on adjacent subdomains is only enforced weakly. But until now there has been very little work for nonlinear PDEs. In this paper, we will present a mortar-type Morley element method for a nonlinear biharmonic equation which is related to the well-known Navier-Stokes equation. Optimal energy and H1-norm estimates are obtained under a reasonable elliptic regularity assumption.
Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms
Carrillo, José A.
2016-09-22
In this paper we present a numerical scheme for nonlinear continuity equations, which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance. It can be applied to a large class of nonlinear continuity equations, whose dynamics are driven by internal energies, given external potentials and/or interaction energies. The solver is based on its variational formulation as a gradient flow with respect to the Wasserstein distance. Positivity of solutions as well as energy decrease of the semi-discrete scheme are guaranteed by its construction. We illustrate this property with various examples in spatial dimension one and two.
Solving Nonlinear Partial Differential Equations with Maple and Mathematica
Shingareva, Inna K
2011-01-01
The emphasis of the book is given in how to construct different types of solutions (exact, approximate analytical, numerical, graphical) of numerous nonlinear PDEs correctly, easily, and quickly. The reader can learn a wide variety of techniques and solve numerous nonlinear PDEs included and many other differential equations, simplifying and transforming the equations and solutions, arbitrary functions and parameters, presented in the book). Numerous comparisons and relationships between various types of solutions, different methods and approaches are provided, the results obtained in Maple an
Critical Exponents for Fast Diffusion Equations with Nonlinear Boundary Sources
Institute of Scientific and Technical Information of China (English)
WANG LU-SHENG; WANG ZE-JIA
2011-01-01
In this paper, we study the large time behavior of solutions to a class of fast diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball. We are interested in the critical global exponent q0 and the critical Fujita exponent qc for the problem considered, and show that q0 ＝ qc for the multidimensional Non-Newtonian polytropic filtration equation with nonlinear boundary sources, which is quite different from the known results that q0 ＜ qc for the onedimensional case; moreover, the value is different from the slow case.
Properties of some nonlinear Schroedinger equations motivated through information theory
Energy Technology Data Exchange (ETDEWEB)
Yuan, Liew Ding; Parwani, Rajesh R, E-mail: parwani@nus.edu.s [Department of Physics, National University of Singapore, Kent Ridge (Singapore)
2009-06-01
We update our understanding of nonlinear Schroedinger equations motivated through information theory. In particular we show that a q-deformation of the basic nonlinear equation leads to a perturbative increase in the energy of a system, thus favouring the simplest q = 1 case. Furthermore the energy minimisation criterion is shown to be equivalent, at leading order, to an uncertainty maximisation argument. The special value eta = 1/4 for the interpolation parameter, where leading order energy shifts vanish, implies the preservation of existing supersymmetry in nonlinearised supersymmetric quantum mechanics. Physically, eta might be encoding relativistic effects.
Direct Perturbation Method for Derivative Nonlinear Schrodinger Equation
Institute of Scientific and Technical Information of China (English)
CHENG Xue-Ping; LIN Ji; HAN Ping
2008-01-01
We extend Lou's direct perturbation method for solving the nonlinear SchrSdinger equation to the case of the derivative nonlinear Schrodinger equation (DNLSE). By applying this method, different types of perturbation solutions axe obtained. Based on these approximate solutions, the analytical forms of soliton parameters, such as the velocity, the width and the initial position, are carried out and the effects of perturbation on solitons are analyzed at the same time. A numerical simulation of perturbed DNLSE finally verifies the results of the perturbation method.
A nonlinear viscoelastic constitutive equation - Yield predictions in multiaxial deformations
Shay, R. M., Jr.; Caruthers, J. M.
1987-01-01
Yield stress predictions of a nonlinear viscoelastic constitutive equation for amorphous polymer solids have been obtained and are compared with the phenomenological von Mises yield criterion. Linear viscoelasticity theory has been extended to include finite strains and a material timescale that depends on the instantaneous temperature, volume, and pressure. Results are presented for yield and the correct temperature and strain-rate dependence in a variety of multiaxial deformations. The present nonlinear viscoelastic constitutive equation can be formulated in terms of either a Cauchy or second Piola-Kirchhoff stress tensor, and in terms of either atmospheric or hydrostatic pressure.
BIHARMONIC EQUATIONS WITH ASYMPTOTICALLY LINEAR NONLINEARITIES
Institute of Scientific and Technical Information of China (English)
Liu Yue; Wang Zhengping
2007-01-01
This article considers the equation △2u = f(x, u)with boundary conditions either u|(a)Ω = (a)u/(a)n|(a)Ω = 0 or u|(a)Ω = △u|(a)Ω = 0, where f(x,t) is asymptotically linear with respect to t at infinity, and Ω is a smooth bounded domain in RN, N ＞ 4. By a variant version of Mountain Pass Theorem, it is proved that the above problems have a nontrivial solution under suitable assumptions of f(x, t).
Leblond, Hervé; Mihalache, Dumitru; 10.1103/PHYSREVA.81.033824
2011-01-01
By using a reductive perturbation method, we derive from Maxwell-Bloch equations a cubic generalized Kadomtsev-Petviashvili equation for ultrashort spatiotemporal optical pulse propagation in cubic (Kerr-like) media without the use of the slowly varying envelope approximation. We calculate the collapse threshold for the propagation of few-cycle spatiotemporal pulses described by the generic cubic generalized Kadomtsev-Petviashvili equation by a direct numerical method and compare it to analytic results based on a rigorous virial theorem. Besides, typical evolution of the spectrum (integrated over the transverse spatial coordinate) is given and a strongly asymmetric spectral broadening of ultrashort spatiotemporal pulses during collapse is evidenced.
Weakly nonlinear Schr\\"odinger equation with random initial data
Lukkarinen, Jani
2009-01-01
There is wide interest in weakly nonlinear wave equations with random initial data. A common approach is the approximation through a kinetic transport equation, which clearly poses the issue of understanding its validity in the kinetic limit. While for the general case a proof of the kinetic limit remains open, we report here on first progress. As wave equation we consider the nonlinear Schrodinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to a Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution psi_t(x) of the nonlinear Schrodinger equation yields then a stochastic process stationary in x in Z^d and t in R. If lambda denotes the strength of the nonlinearity, we prove that the space-time covariance of psi_t(x) has a limit as lambda -> 0 for t=lambda^{-2} tau, with tau fixed and |tau| sufficiently small. The limit agrees with the prediction from ...
Nonlinear electrostatic wave equations for magnetized plasmas - II
DEFF Research Database (Denmark)
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....
Multisymplectic five-point scheme for the nonlinear wave equation
Institute of Scientific and Technical Information of China (English)
WANG Yushun; WANG Bin; YANG Hongwei; WANG Yunfeng
2003-01-01
In this paper, we introduce the multisymplectic structure of the nonlinear wave equation, and prove that the classical five-point scheme for the equation is multisymplectic. Numerical simulations of this multisymplectic scheme on highly oscillatory waves of the nonlinear Klein-Gordon equation and the collisions between kink and anti-kink solitons of the sine-Gordon equation are also provided. The multisymplectic schemes do not need to discrete PDEs in the space first as the symplectic schemes do and preserve not only the geometric structure of the PDEs accurately, but also their first integrals approximately such as the energy, the momentum and so on. Thus the multisymplectic schemes have better numerical stability and long-time numerical behavior than the energy-conserving scheme and the symplectic scheme.
Navier-Stokes equation describes the movement of a special superfluid medium
Sbitnev, Valeriy I
2015-01-01
The Navier-Stokes equation contains two terms which have been subjected to slight modification: (a) the viscosity term depends of time (the viscosity in average on time is zero, but its variance is nonzero); (b) the pressure gradient contains an added term describing the entropy gradient multiplied by the pressure. Owing to these modifications, the Navier-Stokes equation can be reduced to the Schr\\"odinger equation describing behavior of a particle into the vacuum, where the vacuum is a superfluid medium. Vortex structures arising in this medium show infinitely long life owing to zero average viscosity. The nonzero variance describes exchange of the vortex energy with zero-point energy of the vacuum. Radius of the vortex trembles around some average value. This observation sheds the light to the Zitterbewegung phenomenon. The vortex has a non-zero core where the vortex velocity vanishes. Its organization is discussed with the point of view of the Calabi-Yau manifold.
Inverse Coefficient Problems for Nonlinear Parabolic Differential Equations
Institute of Scientific and Technical Information of China (English)
Yun Hua OU; Alemdar HASANOV; Zhen Hai LIU
2008-01-01
This paper is devoted to a class of inverse problems for a nonlinear parabolic differential equation.The unknown coefficient of the equation depends on the gradient of the solution and belongs to a set of admissible coefficients.It is proved that the convergence of solutions for the corresponding direct problems continuously depends on the coefficient convergence.Based on this result the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.
Intermittency and solitons in the driven dissipative nonlinear Schroedinger equation
Moon, H. T.; Goldman, M. V.
1984-01-01
The cubic nonlinear Schroedinger equation, in the presence of driving and Landau damping, is studied numerically. As the pump intensity is increased, the system exhibits a transition from intermittency to a two-torus to chaos. The laminar phase of the intermittency is also a two-torus motion which corresponds in physical space to two identical solitons of amplitude determined by a power-balance equation.
Conservation laws of inviscid Burgers equation with nonlinear damping
Abdulwahhab, Muhammad Alim
2014-06-01
In this paper, the new conservation theorem presented in Ibragimov (2007) [14] is used to find conservation laws of the inviscid Burgers equation with nonlinear damping ut+g(u)ux+λh(u)=0. We show that this equation is both quasi self-adjoint and self-adjoint, and use these concepts to simplify conserved quantities for various choices of g(u) and h(u).
Oscillation criteria for nonlinear fractional differential equation with damping term
Directory of Open Access Journals (Sweden)
Bayram Mustafa
2016-01-01
Full Text Available In this paper, we study the oscillation of solutions to a non-linear fractional differential equation with damping term. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. By using a variable transformation, a generalized Riccati transformation, inequalities, and integration average techniquewe establish new oscillation criteria for the fractional differential equation. Several illustrative examples are also given.
Stochastic Cahn-Hilliard equation with singular nonlinearity and reflection
Goudenège, Ludovic
2008-01-01
International audience; We consider a stochastic partial differential equation with logarithmic (or negative power) nonlinearity, with one reflection at 0 and with a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of the maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being ...
SINGULAR AND RAREFACTIVE SOLUTIONS TO A NONLINEAR VARIATIONAL WAVE EQUATION
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Following a recent paper of the authors in Communications in Partial Differential Equations, this paper establishes the global existence of weak solutions to a nonlinear variational wave equation under relaxed conditions on the initial data so that the solutions can contain singularities (blow-up). Propagation of local oscillations along one family of characteristics remains under control despite singularity formation in the other family of characteristics.
Asymptotic Behavior of Solutions for Nonlinear Volterra Discrete Equations
Directory of Open Access Journals (Sweden)
E. Messina
2008-01-01
Full Text Available We consider nonlinear difference equations of unbounded order of the form xi=bi−∑j=0iai,jfi−j(xj, i=0,1,2,…, where fj(x (j=0,…,i are suitable functions. We establish sufficient conditions for the boundedness and the convergence of xi as i→+∞. Some of these conditions are interesting mainly for studying stability of numerical methods for Volterra integral equations.
Institute of Scientific and Technical Information of China (English)
杨灵娥
2003-01-01
In this paper, we prove that in the inviscid limit the solution of the gen eralized derivative Ginzburg-Landau equations converges to the solution of derivative nonlinear Schrodinger equation, we also give the convergence rates for the difference of the solution.
The chaotic effects in a nonlinear QCD evolution equation
Zhu, Wei; Shen, Zhenqi; Ruan, Jianhong
2016-10-01
The corrections of gluon fusion to the DGLAP and BFKL equations are discussed in a united partonic framework. The resulting nonlinear evolution equations are the well-known GLR-MQ-ZRS equation and a new evolution equation. Using the available saturation models as input, we find that the new evolution equation has the chaos solution with positive Lyapunov exponents in the perturbative range. We predict a new kind of shadowing caused by chaos, which blocks the QCD evolution in a critical small x range. The blocking effect in the evolution equation may explain the Abelian gluon assumption and even influence our expectations to the projected Large Hadron Electron Collider (LHeC), Very Large Hadron Collider (VLHC) and the upgrade (CppC) in a circular e+e- collider (SppC).
The modified simple equation method for solving some fractional-order nonlinear equations
Indian Academy of Sciences (India)
KAPLAN MELIKE; BEKIR AHMET
2016-07-01
Nonlinear fractional differential equations are encountered in various fields of mathematics, physics, chemistry, biology, engineering and in numerous other applications. Exact solutions of these equations play a crucial role in the proper understanding of the qualitative features of many phenomena and processes in various areas of natural science. Thus, many effective and powerful methods have been established and improved. In this study, we establish exact solutions of the time fractional biological population model equation and nonlinearfractional Klein–Gordon equation by using the modified simple equation method.
SOME DISCRETE NONLINEAR INEQUALITIES AND APPLICATIONS TO DIFFERENCE EQUATIONS
Institute of Scientific and Technical Information of China (English)
Cheung Wing-Sum; Ma Qing-Hua; Josip Pe(c)ari(c)
2008-01-01
In this article, the authors establish some new nonlinear difference inequalities in two independent variables, which generalize some existing results and can be used as handy tools in the study of qualitative as well as quantitative properties of solutions of certain classes of difference equations.
BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR NONLINEAR EVOLUTION EQUATIONS
Institute of Scientific and Technical Information of China (English)
彭艳
2014-01-01
In this paper, we consider an initial-boundary value problem for some nonlinear evolution equations with damping and diffusion. The main purpose is to investigate the boundary layer effect and the convergence rates as the diffusion parameterαgoes to zero.
The Peridic Wave Solutions for Two Nonlinear Evolution Equations
Institute of Scientific and Technical Information of China (English)
ZHANG Jin-Liang; WANG Ming-Liang; CHENG Dong-Ming; FANG Zong-De
2003-01-01
By using the F-expansion method proposed recently, the periodic wave solutions expressed by Jacobielliptic functions for two nonlinear evolution equations are derived. In the limit cases, the solitary wave solutions andthe other type of traveling wave solutions for the system are obtained.
The Local Stability of Solutions for a Nonlinear Equation
Directory of Open Access Journals (Sweden)
Haibo Yan
2014-01-01
Full Text Available The approach of Kruzkov’s device of doubling the variables is applied to establish the local stability of strong solutions for a nonlinear partial differential equation in the space L1(R by assuming that the initial value only lies in the space L1(R∩L∞(R.
Further studies of a simple gyrotron equation: nonlinear theory
Energy Technology Data Exchange (ETDEWEB)
Shi Meixuan, E-mail: meixuan@cims.nyu.ed [Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012-1185 (United States)
2010-11-05
A nonlinear version of a standard system of gyrotron model equations is studied using asymptotic analysis and variational methods. The condition for obtaining a high-amplitude wave is achieved in the study. A simple method for obtaining the patterns and amplitude of the wave based on the given free-space wave-number pattern is shown.
Oscillation criteria for first-order forced nonlinear difference equations
Grace Said R; Agarwal Ravi P.; Smith Tim
2006-01-01
Some new criteria for the oscillation of first-order forced nonlinear difference equations of the form Δx(n)+q1(n)xμ(n+1) = q2(n)xλ(n+1)+e(n), where λ, μ are the ratios of positive odd integers 0 <μ < 1 and λ > 1, are established.
Oscillation Theorems for Nonlinear Second Order Elliptic Equations
Institute of Scientific and Technical Information of China (English)
2007-01-01
Some oscillation theorems are given for the nonlinear second order elliptic equation N ∑i,j=1 Di[aij(x)Ψ(y)||(△)y||p-2Djy]+c(x)f(y)=0. The results are extensions of modified Riccati techniques and include recent results of Usami.
Exact controllability for a nonlinear stochastic wave equation
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available The exact controllability for a semilinear stochastic wave equation with a boundary control is established. The target and initial spaces are L 2 ( G × H −1 ( G with G being a bounded open subset of R 3 and the nonlinear terms having at most a linear growth.
Exact periodic solution in coupled nonlinear Schrodinger equations
Institute of Scientific and Technical Information of China (English)
Li Qi-Liang; Chen Jun-Lang; Sun Li-Li; Yu Shu-Yi; Qian Sheng
2007-01-01
The coupled nonlinear Schrodinger equations (CNLSEs) of two symmetrical optical fibres are nonintegrable, however the transformed CNLSEs have integrability. Integrability of the transformed CNLSEs is proved by the Hamilton dynamics theory and Galilei transform. Making use of a transform for CNLSEs and using the ansatz with Jacobi elliptic function form, this paper obtains the exact optical pulse solutions.
EXISTENCE OF SOLUTIONS OF NONLINEAR FRACTIONAL PANTOGRAPH EQUATIONS
Institute of Scientific and Technical Information of China (English)
K. BALACHANDRAN; S. KIRUTHIKA; J.J. TRUJILLO
2013-01-01
This article deals with the existence of solutions of nonlinear fractional pantograph equations.Such model can be considered suitable to be applied when the corresponding process occurs through strongly anomalous media.The results are obtained using fractional calculus and fixed point theorems.An example is provided to illustrate the main result obtained in this article.
Nonlocal Cauchy problem for nonlinear mixed integrodifferential equations
Directory of Open Access Journals (Sweden)
H.L. Tidke
2010-12-01
Full Text Available The present paper investigates the existence and uniqueness of mild and strong solutions of a nonlinear mixed Volterra-Fredholm integrodifferential equation with nonlocal condition. The results obtained by using Schauder fixed point theorem and the theory of semigroups.
An Orthogonal Residual Procedure for Nonlinear Finite Element Equations
DEFF Research Database (Denmark)
Krenk, S.
A general and robust solution procedure for nonlinear finite element equations with limit points is developed. At each equilibrium iteration the magnitude of the load is adjusted such that the residual force is orthogonal to the current displacement increment from the last equilibrium state...
Maximum Likelihood Estimation of Nonlinear Structural Equation Models.
Lee, Sik-Yum; Zhu, Hong-Tu
2002-01-01
Developed an EM type algorithm for maximum likelihood estimation of a general nonlinear structural equation model in which the E-step is completed by a Metropolis-Hastings algorithm. Illustrated the methodology with results from a simulation study and two real examples using data from previous studies. (SLD)
Case-Deletion Diagnostics for Nonlinear Structural Equation Models
Lee, Sik-Yum; Lu, Bin
2003-01-01
In this article, a case-deletion procedure is proposed to detect influential observations in a nonlinear structural equation model. The key idea is to develop the diagnostic measures based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm. An one-step pseudo approximation is proposed to reduce the…
Local Influence Analysis of Nonlinear Structural Equation Models
Lee, Sik-Yum; Tang, Nian-Sheng
2004-01-01
By regarding the latent random vectors as hypothetical missing data and based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm, we investigate assessment of local influence of various perturbation schemes in a nonlinear structural equation model. The basic building blocks of local influence analysis…
Probabilistic methods for discrete nonlinear Schr\\"odinger equations
Chatterjee, Sourav
2010-01-01
Using techniques from probability theory, we show that the thermodynamics of the focusing cubic discrete nonlinear Schrodinger equation (NLS) are exactly solvable in dimensions three and higher. A number of explicit formulas are derived. The probabilistic results, combined with dynamical information, prove the existence and typicality of solutions to the discrete NLS with highly stable localized modes that are sometimes called discrete breathers.
An inhomogeneous wave equation and non-linear Diophantine approximation
DEFF Research Database (Denmark)
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...... is studied. Both the Lebesgue and Hausdorff measures of this set are obtained....
Tensor methods for large sparse systems of nonlinear equations
Energy Technology Data Exchange (ETDEWEB)
Bouaricha, A. [Argonne National Lab., IL (United States). Mathematics and Computer Science Div.; Schnabel, R.B. [Colorado Univ., Boulder, CO (United States). Dept. of Computer Science
1996-12-31
This paper introduces censor methods for solving, large sparse systems of nonlinear equations. Tensor methods for nonlinear equations were developed in the context of solving small to medium- sized dense problems. They base each iteration on a quadratic model of the nonlinear equations. where the second-order term is selected so that the model requires no more derivative or function information per iteration than standard linear model-based methods, and hardly more storage or arithmetic operations per iteration. Computational experiments on small to medium-sized problems have shown censor methods to be considerably more efficient than standard Newton-based methods, with a particularly large advantage on singular problems. This paper considers the extension of this approach to solve large sparse problems. The key issue that must be considered is how to make efficient use of sparsity in forming and solving the censor model problem at each iteration. Accomplishing this turns out to require an entirely new way of solving the tensor model that successfully exploits the sparsity of the Jacobian, whether the Jacobian is nonsingular or singular. We develop such an approach and, based upon it, an efficient tensor method for solving large sparse systems of nonlinear equations. Test results indicate that this tensor method is significantly more efficient and robust than an efficient sparse Newton-based method. in terms of iterations, function evaluations. and execution time.
Large Time Asymptotics for Solutions of Nonlinear Partial Differential Equations
Sachdev, PL
2010-01-01
A large number of physical phenomena are modeled by nonlinear partial differential equations, subject to appropriate initial/boundary conditions. This title presents the constructive mathematical techniques. It deals with the asymptotic methods which include self-similarity, balancing argument, and matched asymptotic expansions
A Dual Orthogonality Procedure for Nonlinear Finite Element Equations
DEFF Research Database (Denmark)
Krenk, S.; Hededal, O.
In the orthogonal residual procedure for solution of nonlinear finite element equations the load is adjusted in each equilibrium iteration to satisfy an orthogonality condition to the current displacement increment. It is here shown that the quasi-newton formulation of the orthogonal residual...
ON THE NONLINEAR TIMOSHENKO-KIRCHHOFF BEAM EQUATION
Institute of Scientific and Technical Information of China (English)
A.AROSIO
1999-01-01
When an elastic string with fixed ends is subjected to transverse vibrations, its length vaxiewith the time: this introduces chvages of the tension in the string. Thls induced Kirchoffto propose a nonlinear correction of the classical D'Alembert equation. Later on, Wolnowsky-
Numerical treatments for solving nonlinear mixed integral equation
Directory of Open Access Journals (Sweden)
M.A. Abdou
2016-12-01
Full Text Available We consider a mixed type of nonlinear integral equation (MNLIE of the second kind in the space C[0,T]×L2(Ω,T<1. The Volterra integral terms (VITs are considered in time with continuous kernels, while the Fredholm integral term (FIT is considered in position with singular general kernel. Using the quadratic method and separation of variables method, we obtain a nonlinear system of Fredholm integral equations (NLSFIEs with singular kernel. A Toeplitz matrix method, in each case, is then used to obtain a nonlinear algebraic system. Numerical results are calculated when the kernels take a logarithmic form or Carleman function. Moreover, the error estimates, in each case, are then computed.
Mapping deformation method and its application to nonlinear equations
Institute of Scientific and Technical Information of China (English)
李画眉
2002-01-01
An extended mapping deformation method is proposed for finding new exact travelling wave solutions of nonlinearpartial differential equations (PDEs). The key idea of this method is to take full advantage of the simple algebraicmapping relation between the solutions of the PDEs and those of the cubic nonlinear Klein-Gordon equation. This isapplied to solve a system of variant Boussinesq equations. As a result, many explicit and exact solutions are obtained,including solitary wave solutions, periodic wave solutions, Jacobian elliptic function solutions and other exact solutions.
A procedure to construct exact solutions of nonlinear evolution equations
Indian Academy of Sciences (India)
Adem Cengiz Çevikel; Ahmet Bekir; Mutlu Akar; Sait San
2012-09-01
In this paper, we implemented the functional variable method for the exact solutions of the Zakharov-Kuznetsov-modified equal-width (ZK-MEW), the modified Benjamin-Bona-Mohany (mBBM) and the modified kdV-Kadomtsev-Petviashvili (kdV-KP) equation. By using this scheme, we found some exact solutions of the above-mentioned equation. The obtained solutions include solitary wave solutions, periodic wave solutions and combined formal solutions. The functional variable method presents a wider-applicability for handling nonlinear wave equations.
An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations
Directory of Open Access Journals (Sweden)
Moh’d Khier Al-Srihin
2017-01-01
Full Text Available In this paper, we introduce an efficient series solution for a class of nonlinear multiterm fractional differential equations of Caputo type. The approach is a generalization to our recent work for single fractional differential equations. We extend the idea of the Taylor series expansion method to multiterm fractional differential equations, where we overcome the difficulty of computing iterated fractional derivatives, which are difficult to be computed in general. The terms of the series are obtained sequentially using a closed formula, where only integer derivatives have to be computed. Several examples are presented to illustrate the efficiency of the new approach and comparison with the Adomian decomposition method is performed.
Numerical solution of nonlinear fractional-order Volterra integro-differential equations by SCW
Zhu, Li; Fan, Qibin
2013-05-01
Fractional calculus is an extension of derivatives and integrals to non-integer orders and has been widely used to model scientific and engineering problems. In this paper, we describe the fractional derivative in the Caputo sense and give the second kind Chebyshev wavelet (SCW) operational matrix of fractional integration. Then based on above results we propose the SCW operational matrix method to solve a kind of nonlinear fractional-order Volterra integro-differential equations. The main characteristic of this approach is that it reduces the integro-differential equations into a nonlinear system of algebraic equations. Thus, it can simplify the problem of fractional order equation solving. The obtained numerical results indicate that the proposed method is efficient and accurate for this kind equations.
Time-dependent exact solutions of the nonlinear Kompaneets equation
Energy Technology Data Exchange (ETDEWEB)
Ibragimov, N H, E-mail: nib@bth.s [Department of Mathematics and Science, Blekinge Institute of Technology, 371 79 Karlskrona (Sweden)
2010-12-17
Time-dependent exact solutions of the Kompaneets photon diffusion equation are obtained for several approximations of this equation. One of the approximations describes the case when the induced scattering is dominant. In this case, the Kompaneets equation has an additional symmetry which is used for constructing some exact solutions as group invariant solutions. (fast track communication)
On the sharp front-type solution of the Nagumo equation with nonlinear diffusion and convection
Indian Academy of Sciences (India)
M B A Mansour
2013-03-01
This paper is concerned with the Nagumo equation with nonlinear degenerate diffusion and convection which arises in several problems of population dynamics, chemical reactions and others. A sharp front-type solution with a minimum speed to this model equation is analysed using different methods. One of the methods is to solve the travelling wave equations and compute an exact solution which describes the sharp travelling wavefront. The second method is to solve numerically an initial-moving boundary-value problem for the partial differential equation and obtain an approximation for this sharp front-type solution.
A simple equation for describing the temperature dependent growth of free-floating macrophytes
Heide, van Tj.; Roijackers, R.M.M.; Nes, van E.H.; Peeters, E.T.H.M.
2006-01-01
Temperature is one of the most important factors determining growth rates of free-floating macrophytes in the field. To analyse and predict temperature dependent growth rates of these pleustophytes, modelling may play an important role. Several equations have been published for describing temperatur
A granular computing method for nonlinear convection-diffusion equation
Directory of Open Access Journals (Sweden)
Tian Ya Lan
2016-01-01
Full Text Available This paper introduces a method of solving nonlinear convection-diffusion equation (NCDE, based on the combination of granular computing (GrC and characteristics finite element method (CFEM. The key idea of the proposed method (denoted as GrC-CFEM is to reconstruct the solution from coarse-grained layer to fine-grained layer. It first gets the nonlinear solution on the coarse-grained layer, and then the function (Taylor expansion is applied to linearize the NCDE on the fine-grained layer. Switch to the fine-grained layer, the linear solution is directly derived from the nonlinear solution. The full nonlinear problem is solved only on the coarse-grained layer. Numerical experiments show that the GrC-CFEM can accelerate the convergence and improve the computational efficiency without sacrificing the accuracy.
The First Integral Method to the Nonlinear Schrodinger Equations in Higher Dimensions
Directory of Open Access Journals (Sweden)
Shoukry Ibrahim Atia El-Ganaini
2013-01-01
Full Text Available The first integral method introduced by Feng is adopted for solving some important nonlinear partial differential equations, including the (2 + 1-dimensional hyperbolic nonlinear Schrodinger (HNLS equation, the generalized nonlinear Schrodinger (GNLS equation with a source, and the higher-order nonlinear Schrodinger equation in nonlinear optical fibers. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner.
A mixed finite element method for nonlinear diffusion equations
Burger, Martin
2010-01-01
We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model. © American Institute of Mathematical Sciences.
Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations
Indekeu, Joseph O.; Smets, Ruben
2017-08-01
Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.
Modelling of nonlinear shoaling based on stochastic evolution equations
DEFF Research Database (Denmark)
Kofoed-Hansen, Henrik; Rasmussen, Jørgen Hvenekær
1998-01-01
A one-dimensional stochastic model is derived to simulate the transformation of wave spectra in shallow water including generation of bound sub- and super-harmonics, near-resonant triad wave interaction and wave breaking. Boussinesq type equations with improved linear dispersion characteristics...... are recast into evolution equations for the complex amplitudes, and serve as the underlying deterministic model. Next, a set of evolution equations for the cumulants is derived. By formally introducing the well-known Gaussian closure hypothesis, nonlinear evolution equations for the power spectrum...... and bispectrum are derived. A simple description of depth-induced wave breaking is incorporated in the model equations, assuming that the total rate of dissipation may be distributed in proportion to the spectral energy density on each discrete frequency. The proposed phase-averaged model is compared...
Exact Solutions to Extended Nonlinear Schr(o)dinger Equation in Monomode Optical Fiber
Institute of Scientific and Technical Information of China (English)
BAI Cheng-Lin; ZHAO Hong; Wang Wei-Tao
2006-01-01
By using the generally projective Riccati equation method, more new exact travelling wave solutions to extended nonlinear Schr(o)dinger equation (NLSE), which describes the femtosecond pulse propagation in monomode optical fiber, are found, which include bright soliton solution, dark soliton solution, new solitary waves, periodic solutions, and rational solutions. The finding of abundant solution structures for extended NLSE helps to study the movement rule of femtosecond pulse propagation in monomode optical fiber.
Levchenko, E. A.; Trifonov, A. Yu.; Shapovalov, A. V.
2017-06-01
The one-dimensional Fokker-Planck-Kolmogorov equation with a special type of nonlocal quadratic nonlinearity is represented as a consistent system of differential equations, including a dynamical system describing the evolution of the moments of the unknown function. Lie symmetries are found for the consistent system using methods of classical group analysis. An example of an invariant-group solution obtained with an additional integral constraint imposed on the system is considered.
Modeling Solution of Nonlinear Dispersive Partial Differential Equations using the Marker Method
Energy Technology Data Exchange (ETDEWEB)
Jerome L.V. Lewandowski
2005-01-25
A new method for the solution of nonlinear dispersive partial differential equations is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details.
Relativistic superfluidity and vorticity from the nonlinear Klein-Gordon equation
Xiong, Chi; Guo, Yulong; Liu, Xiaopei; Huang, Kerson
2014-01-01
We investigate superfluidity, and the mechanism for creation of quantized vortices, in the relativistic regime. The general framework is a nonlinear Klein-Gordon equation in curved spacetime for a complex scalar field, whose phase dynamics gives rise to superfluidity. The mechanisms discussed are local inertial forces (Coriolis and centrifugal), and current-current interaction with an external source. The primary application is to cosmology, but we also discuss the reduction to the non-relativistic nonlinear Schr\\"{o}dinger equation, which is widely used in describing superfluidity and vorticity in liquid helium and cold-trapped atomic gases.
A geometric characterization of the nonlinear Schrodinger equation and its applications
Institute of Scientific and Technical Information of China (English)
DING; Qing(丁青); ZHU; Zuonong(朱佐农)
2002-01-01
We prove that the nonlinear Schrodinger equation of attractive type (NLS+ describes just spher-ical surfaces (SS) and the nonlinear Schrodinger equation of repulsive type (NLS-) determines only pseudo-spherical surfaces (PSS). This implies that, though we show that given two differential PSS (resp. SS) equationsthere exists a local gauge transformation (despite of changing the independent variables or not) which trans-forms a solution of one into any solution of the other, it is impossible to have such a gauge transformationbetween the NLS+ and the NLS-.
Discrete Nonlinear Schrodinger Equation, Solitons and Organizing Principles for Protein Folding
Molkenthin, Nora; Niemi, Antti J
2010-01-01
We introduce a novel generalization of the discrete nonlinear Schr\\"odinger equation. It supports solitons that describe how proteins fold. As an example we scrutinize the villin headpiece HP35, an archetypal protein for testing both experimental and theoretical approaches to protein folding. Using explicit soliton profiles we construct its carbon backbone with an unprecedented accuracy.
A new differential equations-based model for nonlinear history-dependent magnetic behaviour
Aktaa, J
2000-01-01
The paper presents a new kind of numerical model describing nonlinear magnetic behaviour. The model is formulated as a set of differential equations taking into account history dependence phenomena like the magnetisation hysteresis as well as saturation effects. The capability of the model is demonstrated carrying out comparisons between measurements and calculations.
Mazur, A. K.; Dorofeev, V. E.; Abagyan, R. A.
1991-02-01
General Lagrange's equations of motion for a system of polymeric molecules are obtained in an explicit form. They can be used for simulating molecular dynamics of large molecules. The molecular conformations are described by internal coordinates, i.e., bond lengths, valence angles, and torsion angles. The equations derived permit any internal degrees of freedom to be frozen. The method is applied to an oligopeptide in an α-helical conformation. Three models of the molecule with different degrees of fixation are compared. It is shown that the method permits one to increase significantly the time step in molecular dynamics calculations.
Stability of planar diffusion wave for nonlinear evolution equation
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
It is known that the one-dimensional nonlinear heat equation ut = f(u)x1x1,f'(u) 0,u(±∞,t) = u±,u+ = u_ has a unique self-similar solution u(x1/1+t).In multi-dimensional space,u(x1/1+t) is called a planar diffusion wave.In the first part of the present paper,it is shown that under some smallness conditions,such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation:ut-△f(u) = 0,x ∈ Rn.The optimal time decay rate is obtained.In the second part of this paper,it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping:utt + utt+ △f(u) = 0,x ∈ Rn.The time decay rate is also obtained.The proofs are given by an elementary energy method.
Maĭmistov, A. I.
2003-02-01
We discuss propagation of an ultimately short (single-cycle) pulse of an electromagnetic field in a medium whose dispersion and nonlinear properties can be described by the cubic-quintic Duffing model, i.e., by an oscillator with third-and fifth-order anharmonicity. A system of equations governing the evolution of a unidirectional electromagnetic wave is analyzed without using the approximation of slowly varying envelopes. Three types of solutions of this system describing stationary propagation of a pulse in such a medium are found. When the signs of the anharmonicity constants are different, then the amplitude of a steady-state pulse is limited, but its energy may grow on account of an increase in its duration. The characteristics of such a pulse, referred to as an electromagnetic domain, are discussed.
On nonlinear equation of Schrödinger type
Soltanov, Kamal N.
2012-11-01
In this paper we study a mixed problem for the nonlinear Schrödinger equation that have a nonlinear adding, in which the coefficient is a generalized function. Here is proved a solvability theorem of the considered problem with use of the general solvability theorem of the article [28]. Furthermore here is investigated also the behaviour of the solution of the studied problem. aipproc class produce a paper with the correct layout for AIP Conference Proceedings 8.5in × 11in double column.
Flow Equation Approach to the Statistics of Nonlinear Dynamical Systems
Marston, J. B.; Hastings, M. B.
2005-03-01
The probability distribution function of non-linear dynamical systems is governed by a linear framework that resembles quantum many-body theory, in which stochastic forcing and/or averaging over initial conditions play the role of non-zero . Besides the well-known Fokker-Planck approach, there is a related Hopf functional methodootnotetextUriel Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, 1995) chapter 9.5.; in both formalisms, zero modes of linear operators describe the stationary non-equilibrium statistics. To access the statistics, we investigate the method of continuous unitary transformationsootnotetextS. D. Glazek and K. G. Wilson, Phys. Rev. D 48, 5863 (1993); Phys. Rev. D 49, 4214 (1994). (also known as the flow equation approachootnotetextF. Wegner, Ann. Phys. 3, 77 (1994).), suitably generalized to the diagonalization of non-Hermitian matrices. Comparison to the more traditional cumulant expansion method is illustrated with low-dimensional attractors. The treatment of high-dimensional dynamical systems is also discussed.
Comparing non-linear mathematical models to describe growth of different animals
Directory of Open Access Journals (Sweden)
Jhony Tiago Teleken
2017-02-01
Full Text Available The main objective of this study was to compare the goodness of fit of five non-linear growth models, i.e. Brody, Gompertz, Logistic, Richards and von Bertalanffy in different animals. It also aimed to evaluate the influence of the shape parameter on the growth curve. To accomplish this task, published growth data of 14 different groups of animals were used and four goodness of fit statistics were adopted: coefficient of determination (R2, root mean square error (RMSE, Akaike information criterion (AIC and Bayesian information criterion (BIC. In general, the Richards growth equation provided better fits to experimental data than the other models. However, for some animals, different models exhibited better performance. It was obtained a possible interpretation for the shape parameter, in such a way that can provide useful insights to predict animal growth behavior.
ERROR ESTIMATES FOR THE TIME DISCRETIZATION FOR NONLINEAR MAXWELL'S EQUATIONS
Institute of Scientific and Technical Information of China (English)
Marián Slodi(c)ka; Ján Bu(s)a Jr.
2008-01-01
This paper is devoted to the study of a nonlinear evolution eddy current model of the type (б)tB(H) +▽×(▽×H) = 0 subject to homogeneous Dirichlet boundary conditions H×v = 0 and a given initial datum. Here, the magnetic properties of a soft ferromagnet are linked by a nonlinear material law described by B(H). We apply the backward Euler method for the time discretization and we derive the error estimates in suitable function spaces. The results depend on the nonlinearity of B(H).
Quasi-periodic solutions of nonlinear beam equations with quintic quasi-periodic nonlinearities
Directory of Open Access Journals (Sweden)
Qiuju Tuo
2015-01-01
Full Text Available In this article, we consider the one-dimensional nonlinear beam equations with quasi-periodic quintic nonlinearities $$ u_{tt}+u_{xxxx}+(B+ \\varepsilon\\phi(tu^5=0 $$ under periodic boundary conditions, where B is a positive constant, $\\varepsilon$ is a small positive parameter, $\\phi(t$ is a real analytic quasi-periodic function in t with frequency vector $\\omega=(\\omega_1,\\omega_2,\\dots,\\omega_m$. It is proved that the above equation admits many quasi-periodic solutions by KAM theory and partial Birkhoff normal form.
Charged anisotropic matter with linear or nonlinear equation of state
Varela, Victor; Ray, Saibal; Chakraborty, Kaushik; Kalam, Mehedi
2010-01-01
Ivanov pointed out substantial analytical difficulties associated with self-gravitating, static, isotropic fluid spheres when pressure explicitly depends on matter density. Simplification achieved with the introduction of electric charge were noticed as well. We deal with self-gravitating, charged, anisotropic fluids and get even more flexibility in solving the Einstein-Maxwell equations. In order to discuss analytical solutions we extend Krori and Barua's method to include pressure anisotropy and linear or non-linear equations of state. The field equations are reduced to a system of three algebraic equations for the anisotropic pressures as well as matter and electrostatic energy densities. Attention is paid to compact sources characterized by positive matter density and positive radial pressure. Arising solutions satisfy the energy conditions of general relativity. Spheres with vanishing net charge contain fluid elements with unbounded proper charge density located at the fluid-vacuum interface. Notably the...
Nonlinear electromagnetic gyrokinetic equation for plasmas with large mean flows
Energy Technology Data Exchange (ETDEWEB)
Sugama, H. [National Inst. for Fusion Science, Toki, Gifu (Japan); Horton, W.
1998-02-01
A new nonlinear electromagnetic gyrokinetic equation is derived for plasmas with large flow velocities on the order of the ion thermal speed. The gyrokinetic equation derived here is given in the form which is valid for general magnetic geometries including the slab, cylindrical and toroidal configurations. The source term for the anomalous viscosity arising through the Reynolds stress is identified in the gyrokinetic equation. For the toroidally rotating plasma, particle, energy and momentum balance equations as well as the detailed definitions of the anomalous transport fluxes and the anomalous entropy production are shown. The quasilinear anomalous transport matrix connecting the conjugate pairs of the anomalous fluxes and the forces satisfies the Onsager symmetry. (author)
Generalized creation and annihilation operators via complex nonlinear Riccati equations
Schuch, Dieter; Castaños, Octavio; Rosas-Ortiz, Oscar
2013-06-01
Based on Gaussian wave packet solutions of the time-dependent Schrödinger equation, a generalization of the conventional creation and annihilation operators and the corresponding coherent states can be obtained. This generalization includes systems where also the width of the coherent states is time-dependent as they occur for harmonic oscillators with time-dependent frequency or systems in contact with a dissipative environment. The key point is the replacement of the frequency ω0 that occurs in the usual definition of the creation/annihilation operator by a complex time-dependent function that fulfils a nonlinear Riccati equation. This equation and its solutions depend on the system under consideration and on the (complex) initial conditions. Formal similarities also exist with supersymmetric quantum mechanics. The generalized creation and annihilation operators also allow to construct exact analytic solutions of the free motion Schrödinger equation in terms of Hermite polynomials with time-dependent variable.
Multi-soliton rational solutions for some nonlinear evolution equations
Directory of Open Access Journals (Sweden)
Osman Mohamed S.
2016-01-01
Full Text Available The Korteweg-de Vries equation (KdV and the (2+ 1-dimensional Nizhnik-Novikov-Veselov system (NNV are presented. Multi-soliton rational solutions of these equations are obtained via the generalized unified method. The analysis emphasizes the power of this method and its capability of handling completely (or partially integrable equations. Compared with Hirota’s method and the inverse scattering method, the proposed method gives more general exact multi-wave solutions without much additional effort. The results show that, by virtue of symbolic computation, the generalized unified method may provide us with a straightforward and effective mathematical tool for seeking multi-soliton rational solutions for solving many nonlinear evolution equations arising in different branches of sciences.
Numerical solution of control problems governed by nonlinear differential equations
Energy Technology Data Exchange (ETDEWEB)
Heinkenschloss, M. [Virginia Polytechnic Institute and State Univ., Blacksburg, VA (United States)
1994-12-31
In this presentation the author investigates an iterative method for the solution of optimal control problems. These problems are formulated as constrained optimization problems with constraints arising from the state equation and in the form of bound constraints on the control. The method for the solution of these problems uses the special structure of the problem arising from the bound constraint and the state equation. It is derived from SQP methods and projected Newton methods and combines the advantages of both methods. The bound constraint is satisfied by all iterates using a projection, the nonlinear state equation is satisfied in the limit. Only a linearized state equation has to be solved in every iteration. The solution of the linearized problems are done using multilevel methods and GMRES.
Interconnection between static regimes in the LJJs described by the double sine-Gordon equation
Atanasova, P. Kh; Zemlyanaya, E. V.; Shukrinov, Yu M.
2012-11-01
The second harmonic contribution to the current-phase relation changes the properties of the static magnetic flux distributions in the long Josephson junction (LJJ) and inspires new homogenious and fluxon static states. We study stability properties and bifurcations of these static regimes within the frame of a model described by the double sine-Gordon equation. The critical curves behavior and the interconnection between different types of magnetic flux distributions are analyzed.
Uniqueness of Inversion Problems Described by First-Kind Integral Equations
Institute of Scientific and Technical Information of China (English)
徐铁峰
2002-01-01
We propose a general method to prove the uniqueness of the inversion problems described by first-kind integral equations. The method depends on the analytical properties of the Fourier transform of the integral kernel and the finiteness of the total states (or probability, if normalized), the integration of the "local" density of states, which is a rather moderate condition and satisfied by many inversion problems arising from physics and engineering.
Radial selfsimilar solutions of a nonlinear Ornstein-Uhlenbeck equation
Directory of Open Access Journals (Sweden)
Arij Bouzelmate
2007-05-01
Full Text Available This paper concerns the existence, uniqueness and asymptotic properties (as $r=|x|oinfty$ of radial self-similar solutions to the nonlinear Ornstein-Uhlenbeck equation [ v_t=Delta_p v+xcdot abla (|v|^{q-1}v ] in $mathbb{R}^Nimes (0, +infty$. Here $q>p-1>1$, $Ngeq 1$, and $Delta_p$ denotes the $p$-Laplacian operator. These solutions are of the form [ v(x,t=t^{-gamma} U(cxt^{-sigma}, ] where $gamma$ and $sigma$ are fixed powers given by the invariance properties of differential equation, while $U$ is a radial function, $U(y=u(r$, $r=|y|$. With the choice $c=(q-1^{-1/p}$, the radial profile $u$ satisfies the nonlinear ordinary differential equation $$ (|u'|^{p-2}u''+frac{N-1}r |u'|^{p-2}u'+frac{q+1-p}{p} r u'+(q-1 r(|u|^{q-1}u'+u=0 $$in $mathbb{R}_+$. We carry out a careful analysis of this equation anddeduce the corresponding consequences for the Ornstein-Uhlenbeck equation.
On the Amplitude Equations for Weakly Nonlinear Surface Waves
Benzoni-Gavage, Sylvie; Coulombel, Jean-François
2012-09-01
Nonlocal generalizations of Burgers' equation were derived in earlier work by Hunter (Contemp Math, vol 100, pp 185-202. AMS, 1989), and more recently by Benzoni-Gavage and Rosini (Comput Math Appl 57(3-4):1463-1484, 2009), as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage (Differ Integr Equ 22(3-4):303-320, 2009) under an appropriate stability condition originally pointed out by Hunter. The same stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov (Benzoni-Gavage et al. in Adv Math 227(6):2220-2240, 2011). In this article, we show how the verification of Hunter's stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. We also show that the resulting amplitude equation has a Hamiltonian structure when the original boundary value problem has a variational origin. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity.
Singular solutions of fully nonlinear elliptic equations and applications
Armstrong, Scott N; Smart, Charles K
2011-01-01
We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\\mathbb{R}^n$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragm\\'en-Lindel\\"of result as well as a principle of positive singularities in certain Lipschitz domains.
Critical exponent for damped wave equations with nonlinear memory
Fino, Ahmad
2010-01-01
We consider the Cauchy problem in $\\mathbb{R}^n,$ $n\\geq 1,$ for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as $t\\to\\infty$ of small data solutions have been established in the case when $1\\leq n\\leq3.$ Moreover, we derive a blow-up result under some positive data for in any dimensional space. It turns out that the critical exponent indeed coincides with the one to the corresponding semilinear heat equation.
Nonzero solutions of nonlinear integral equations modeling infectious disease
Energy Technology Data Exchange (ETDEWEB)
Williams, L.R. (Indiana Univ., South Bend); Leggett, R.W.
1982-01-01
Sufficient conditions to insure the existence of periodic solutions to the nonlinear integral equation, x(t) = ..integral../sup t//sub t-tau/f(s,x(s))ds, are given in terms of simple product and product integral inequalities. The equation can be interpreted as a model for the spread of infectious diseases (e.g., gonorrhea or any of the rhinovirus viruses) if x(t) is the proportion of infectives at time t and f(t,x(t)) is the proportion of new infectives per unit time.
Various Newton-type iterative methods for solving nonlinear equations
Directory of Open Access Journals (Sweden)
Manoj Kumar
2013-10-01
Full Text Available The aim of the present paper is to introduce and investigate new ninth and seventh order convergent Newton-type iterative methods for solving nonlinear equations. The ninth order convergent Newton-type iterative method is made derivative free to obtain seventh-order convergent Newton-type iterative method. These new with and without derivative methods have efficiency indices 1.5518 and 1.6266, respectively. The error equations are used to establish the order of convergence of these proposed iterative methods. Finally, various numerical comparisons are implemented by MATLAB to demonstrate the performance of the developed methods.
Comparison Criteria for Nonlinear Functional Dynamic Equations of Higher Order
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Taher S. Hassan
2016-01-01
Full Text Available We will consider the higher order functional dynamic equations with mixed nonlinearities of the form xnt+∑j=0Npjtϕγjxφjt=0, on an above-unbounded time scale T, where n≥2, xi(t≔ri(tϕαixi-1Δ(t, i=1,…,n-1, with x0=x, ϕβ(u≔uβsgnu, and α[i,j]≔αi⋯αj. The function φi:T→T is a rd-continuous function such that limt→∞φi(t=∞ for j=0,1,…,N. The results extend and improve some known results in the literature on higher order nonlinear dynamic equations.
Quasi-periodic Solutions of the General Nonlinear Beam Equations
Institute of Scientific and Technical Information of China (English)
GAO YI-XIAN
2012-01-01
In this paper,one-dimensional (1D) nonlinear beam equations of the form utt - uxx + uxxxx + mu = f(u)with Dirichlet boundary conditions are considered,where the nonlinearity f is an analytic,odd function and f(u) = O(u3).It is proved that for all m ∈ (0,M*] (∈) R(M* is a fixed large number),but a set of small Lebesgue measure,the above equations admit small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system.The proof is based on an infinite dimensional KAM theory and a partial Birkhoff normal form technique.
An Explicit High Resolution Scheme for Nonlinear Shallow Water Equations
Institute of Scientific and Technical Information of China (English)
FANG Ke-zhao; ZOU Zhi-li; WANG Yan
2005-01-01
The present study develops a numerical model of the two-dimensional fully nonlinear shallow water equations (NSWE) for the wave run-up on a beach. The finite volume method (FVM) is used to solve the equations, and a second-order explicit scheme is developed to improve the computation efficiency. The numerical fluxes are obtained by the two dimensional Roe's flux function to overcome the errors caused by the use of one dimensional fluxes in dimension splitting methods. The high-resolution Godunov-type TVD upwind scheme is employed and a second-order accuracy is achieved based on monotonic upstream schemes for conservation laws (MUSCL) variable extrapolation; a nonlinear limiter is applied to prevent unwanted spurious oscillation. A simple but efficient technique is adopted to deal with the moving shoreline boundary. The verification of the solution technique is carried out by comparing the model output with documented results and it shows that the solution technique is robust.
RESTRICTED NONLINEAR APPROXIMATION AND SINGULAR SOLUTIONS OF BOUNDARY INTEGRAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
Reinhard Hochmuth
2002-01-01
This paper studies several problems, which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1 ] are chosen as a starting point for characterizations of functions in Besov spaces B , (0,1) with 0＜σ＜∞ and (1+σ)-1＜τ＜∞. Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered. Besides characterization results Jackson type estimates for various tree-type and tresholding algorithms are investigated. Finally known approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results.
Nonlinear partial differential equations for scientists and engineers
Debnath, Lokenath
1997-01-01
"An exceptionally complete overview. There are numerous examples and the emphasis is on applications to almost all areas of science and engineering. There is truly something for everyone here. This reviewer feels that it is a very hard act to follow, and recommends it strongly. [This book] is a jewel." ---Applied Mechanics Review (Review of First Edition) This expanded and revised second edition is a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied applications. Building upon the successful material of the first book, this edition contains updated modern examples and applications from areas of fluid dynamics, gas dynamics, plasma physics, nonlinear dynamics, quantum mechanics, nonlinear optics, acoustics, and wave propagation. Methods and properties of solutions are presented, along with their physical significance, making the book more useful for a diverse readership. Topics and key features: * Thorough coverage of derivation and methods of soluti...
Till, Bernie C.; Driessen, Peter F.
2014-01-01
Starting from first principles, we derive the telegraph equation to describe the propagation of sound waves in rigid tubes by using a simple approach that yields a lossy transmission line model with frequency-independent parameters. The approach is novel in the sense that it has not been found in the literature or textbooks. To derive the lossy acoustic telegraph equation from the lossless wave equation, we need only to relax the assumption that the dynamical variables are constant over the entire cross-sectional area of the tube. In this paper, we do this by introducing a relatively narrow boundary layer at the wall of the tube, over which the dynamical variables decrease linearly from the constant value to zero. This allows us to make very simple corrections to the lossless case, and to express them in terms of two parameters, namely the viscous diffusion time constant and the thermal diffusion time constant. The coefficients of the resulting telegraph equation are frequency-independent. A comparison with the telegraph equation for the electrical transmission line establishes precise relationships between the electrical circuit elements and the physical properties of the fluid. These relationships are thus proven a posteriori rather than asserted a priori. In this way, we arrive at an instructive and useful derivation of the acoustic telegraph equation, which takes viscous damping and thermal dissipation into account, and is accessible to students at the undergraduate level. This derivation does not resort to the combined heavy machinery of fluid dynamics and thermodynamics, does not assume that the waveforms are sinusoidal, and does not assume any particular cross-sectional shape of the tube. Surprisingly, we have been unable to find a comparable treatment in the standard introductory physics and acoustics texts, or in the literature.
Walcott, Sam
2014-10-01
Molecular motors, by turning chemical energy into mechanical work, are responsible for active cellular processes. Often groups of these motors work together to perform their biological role. Motors in an ensemble are coupled and exhibit complex emergent behavior. Although large motor ensembles can be modeled with partial differential equations (PDEs) by assuming that molecules function independently of their neighbors, this assumption is violated when motors are coupled locally. It is therefore unclear how to describe the ensemble behavior of the locally coupled motors responsible for biological processes such as calcium-dependent skeletal muscle activation. Here we develop a theory to describe locally coupled motor ensembles and apply the theory to skeletal muscle activation. The central idea is that a muscle filament can be divided into two phases: an active and an inactive phase. Dynamic changes in the relative size of these phases are described by a set of linear ordinary differential equations (ODEs). As the dynamics of the active phase are described by PDEs, muscle activation is governed by a set of coupled ODEs and PDEs, building on previous PDE models. With comparison to Monte Carlo simulations, we demonstrate that the theory captures the behavior of locally coupled ensembles. The theory also plausibly describes and predicts muscle experiments from molecular to whole muscle scales, suggesting that a micro- to macroscale muscle model is within reach.
Nonlinear disintegration of sine wave in the framework of the Gardner equation
Kurkina, Oxana; Rouvinskaya, Ekaterina; Talipova, Tatiana; Kurkin, Andrey; Pelinovsky, Efim
2016-10-01
Internal tidal wave entering shallow waters transforms into an undular bore and this process can be described in the framework of the Gardner equation (extended version of the Korteweg-de Vries equation with both quadratic and cubic nonlinear terms). Our numerical computations demonstrate the features of undular bore developing for different signs of the cubic nonlinear term. If cubic nonlinear term is negative, and initial wave amplitude is large enough, two undular bores are generated from the two breaking points formed on both crest slopes (within dispersionless Gardner equation). Undular bore consists of one table-top soliton and a group of small soliton-like waves passing through the table-top soliton. If the cubic nonlinear term is positive and again the wave amplitude is large enough, the breaking points appear on crest and trough generating groups of positive and negative soliton-like pulses. This is the main difference with respect to the classic Korteweg-de Vries equation, where the breaking point is single. It is shown also that nonlinear interaction of waves happens similarly to one of scenarios of two-soliton interaction of "exchange" or "overtake" types with a phase shift. If small-amplitude pulses interact with large-amplitude soliton-like pulses, their speed in average is negative in the case when "free" velocity is positive. Nonlinear interaction leads to the generation of higher harmonics and spectrum width increases with amplitude increase independently of the sign of cubic nonlinear term. The breaking asymptotic k 4 / 3 predicted within the dispersionless Gardner equation emerges during the process of undular bore development. The formation of soliton-like perturbations leads to appearance of several spectral peaks which are downshifting with time.
Nonlinear Self-Adjoint Classification of a Burgers-KdV Family of Equations
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Júlio Cesar Santos Sampaio
2014-01-01
Full Text Available The concepts of strictly, quasi, weak, and nonlinearly self-adjoint differential equations are revisited. A nonlinear self-adjoint classification of a class of equations with second and third order is carried out.
Periodic Wave Solutions of Generalized Derivative Nonlinear Schr(o)dinger Equation
Institute of Scientific and Technical Information of China (English)
ZHA Qi-Lao; LI Zhi-Bin
2008-01-01
A Darboux transformation of the generalized derivative nonlinear Schr(o)dinger equation is derived. As an application, some new periodic wave solutions of the generalized derivative nonlinear Schr(o)dinger equation are explicitly given.
A Hierarchy of New Nonlinear Evolution Equations Associated with a 3 × 3 Matrix Spectral Problem
Institute of Scientific and Technical Information of China (English)
GENG Xian-Guo; LI Fang
2009-01-01
A 3 × 3 matrix spectral problem with three potentials and the corresponding hierarchy of new nonlinear evolution equations are proposed. Generalized Hamiltonian structures for the hierarchy of nonlinear evolution equations are derived with the aid of trace identity.
Local H\\"older continuity for doubly nonlinear parabolic equations
Kuusi, Tuomo; Urbano, José Miguel
2010-01-01
We give a proof of the H\\"older continuity of weak solutions of certain degenerate doubly nonlinear parabolic equations in measure spaces. We only assume the measure to be a doubling non-trivial Borel measure which supports a Poincar\\'e inequality. The proof discriminates between large scales, for which a Harnack inequality is used, and small scales, that require intrinsic scaling methods.
Solitary wave solutions to nonlinear evolution equations in mathematical physics
Indian Academy of Sciences (India)
Anwar Ja’afar Mohamad Jawad; M Mirzazadeh; Anjan Biswas
2014-10-01
This paper obtains solitons as well as other solutions to a few nonlinear evolution equations that appear in various areas of mathematical physics. The two analytical integrators that are applied to extract solutions are tan–cot method and functional variable approaches. The soliton solutions can be used in the further study of shallow water waves in (1+1) as well as (2+1) dimensions.
Parallel Evolutionary Modeling for Nonlinear Ordinary Differential Equations
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
We introduce a new parallel evolutionary algorithm in modeling dynamic systems by nonlinear higher-order ordinary differential equations (NHODEs). The NHODEs models are much more universal than the traditional linear models. In order to accelerate the modeling process, we propose and realize a parallel evolutionary algorithm using distributed CORBA object on the heterogeneous networking. Some numerical experiments show that the new algorithm is feasible and efficient.
Symmetries and Similarity Reductions of Nonlinear Diffusion Equation
Institute of Scientific and Technical Information of China (English)
LI Hui-Jun; RUAN Hang-Yu
2004-01-01
The inverse recursion operator, three new sets of symmetries, and infinite-dimensional Lie algebras for the nonlinear diffusion equation are given. Some nonlocal symmetries related to eigenvectors of the recursion operator Ф with the eigenvalue λi are also obtained with the help of the recursion operator Фi = Ф - λi. Using a part of these symmetries we get twelve types of nontrivial new similarity reduction.
Symmetries and Similarity Reductions of Nonlinear Diffusion Equation
Institute of Scientific and Technical Information of China (English)
LIHui-Jun; RUANHang-Yu
2004-01-01
The inverse recursion operator, three new sets of symmetries, and infinite-dimensional Lie algebras for the nonlinear diffusion equation are given. Some nonlocal symmetries related to eigenvectors of the recursion operator with the eigenvalue λi are also obtained with the help of the recursion operator φi=φ-λi. Using a part of these symmetries we get twelve types of nontrivial new similarity reduction.
ANALYTIC INVARIANT CURVES OF A NONLINEAR SECOND ORDER DIFFERENCE EQUATION
Institute of Scientific and Technical Information of China (English)
Wang Wusheng
2009-01-01
This article studies the existence of analytic invariant curves for a nonlinear second order difference equation which was modeled from macroeconomics of the business cycle. The author not only discusses the case of the eigenvalue off the unit circle S1 and the case on S1 with the Diophantine condition but also considers the case of the eigenvalue at a root of the unity, which obviously violates the Diophantine condition.
Symposium on Nonlinear Semigroups, Partial Differential Equations and Attractors
Zachary, Woodford
1987-01-01
The original idea of the organizers of the Washington Symposium was to span a fairly narrow range of topics on some recent techniques developed for the investigation of nonlinear partial differential equations and discuss these in a forum of experts. It soon became clear, however, that the dynamical systems approach interfaced significantly with many important branches of applied mathematics. As a consequence, the scope of this resulting proceedings volume is an enlarged one with coverage of a wider range of research topics.
New Efficient Fourth Order Method for Solving Nonlinear Equations
Directory of Open Access Journals (Sweden)
Farooq Ahmad
2013-12-01
Full Text Available In a paper [Appl. Math. Comput., 188 (2 (2007 1587--1591], authors have suggested and analyzed a method for solving nonlinear equations. In the present work, we modified this method by using the finite difference scheme, which has a quintic convergence. We have compared this modified Halley method with some other iterative of fifth-orders convergence methods, which shows that this new method having convergence of fourth order, is efficient.
Lu, Bin
2012-06-01
In this Letter, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Bäcklund transformation of fractional Riccati equation are employed for constructing the exact solutions of nonlinear fractional partial differential equations. The power of this manageable method is presented by applying it to several examples. This approach can also be applied to other nonlinear fractional differential equations.
Solution behaviors in coupled Schrödinger equations with full-modulated nonlinearities
Pınar, Zehra; Deliktaş, Ekin
2017-02-01
The nonlinear partial differential equations have an important role in real life problems. To obtain the exact solutions of the nonlinear partial differential equations, a number of approximate methods are known in the literature. In this work, a time- space modulated nonlinearities of coupled Schrödinger equations are considered. We provide a large class of Jacobi-elliptic solutions via the auxiliary equation method with sixth order nonlinearity and the Chebyshev approximation.
Directory of Open Access Journals (Sweden)
Ping Liu
2015-08-01
Full Text Available The symmetry reduction equations, similarity solutions, sub-groups and exact solutions of the (3+1-dimensional nonlinear incompressible non-hydrostatic Boussinesq equations with viscosity (INHBV equations, which describe the atmospheric gravity waves, are researched in this paper. Calculation on symmetry shows that the equations are invariant under the Galilean transformations, scaling transformations, rotational transformations and space-time translations. Three types of symmetry reduction equations and similar solutions for the (3+1-dimensional INHBV equations are proposed. Traveling wave solutions of the INHBV equations are demonstrated by means of symmetry method. The evolutions on the wind velocities and temperature perturbation are demonstrated by figures.
Liu, Ping; Zeng, Bao-Qing; Deng, Bo-Bo; Yang, Jian-Rong
2015-08-01
The symmetry reduction equations, similarity solutions, sub-groups and exact solutions of the (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq equations with viscosity (INHBV equations), which describe the atmospheric gravity waves, are researched in this paper. Calculation on symmetry shows that the equations are invariant under the Galilean transformations, scaling transformations, rotational transformations and space-time translations. Three types of symmetry reduction equations and similar solutions for the (3+1)-dimensional INHBV equations are proposed. Traveling wave solutions of the INHBV equations are demonstrated by means of symmetry method. The evolutions on the wind velocities and temperature perturbation are demonstrated by figures.
Periodic and Chaotic Breathers in the Nonlinear Schr(o)dinger Equation
Institute of Scientific and Technical Information of China (English)
LIU Xue-Shen; QI Yue-Ying; DING Pei-Zhu
2004-01-01
@@ The breathers in the cubic nonlinear Schrodinger equation are investigated numerically by using the symplectic method. We show that the solitonlike wave, the periodic, quasiperiodic and chaotic breathers can be observed with the increase of cubic nonlinear perturbation. Finally, we discuss the breathers in the cubic-quintic nonlinear Schrodinger equation with the increase of quintic nonlinear perturbation.
Multi-Order Exact Solutions for a generalized shallow water wave equation and other nonlinear PDEs
Bagchi, Bijan; Ganguly, Asish
2011-01-01
We seek multi-order exact solutions of a generalized shallow water wave equation along with those corresponding to a class of nonlinear systems described by the KdV, modified KdV, Boussinesq, Klein-Gordon and modified Benjamin-Bona-Mahony equation. We employ a modified version of a generalized Lame equation and subject it to a perturbative treatment identifying the solutions order by order in terms of Jacobi elliptic functions. Our solutions are new and hold the key feature that they are expressible in terms of an auxiliary function f in a generic way. For appropriate choices of f we recover the previous results reported in the literature.
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
In this paper, we extend the mapping transformation method through introducing variable coefficients.By means of the extended mapping transformation method, many explicit and exact general solutions with arbitrary functions for some nonlinear partial differential equations, which contain solitary wave solutions, trigonometric function solutions, and rational solutions, are obtained.
Describing Growth Pattern of Bali Cows Using Non-linear Regression Models
Directory of Open Access Journals (Sweden)
Mohd. Hafiz A.W
2016-12-01
Full Text Available The objective of this study was to evaluate the best fit non-linear regression model to describe the growth pattern of Bali cows. Estimates of asymptotic mature weight, rate of maturing and constant of integration were derived from Brody, von Bertalanffy, Gompertz and Logistic models which were fitted to cross-sectional data of body weight taken from 74 Bali cows raised in MARDI Research Station Muadzam Shah Pahang. Coefficient of determination (R2 and residual mean squares (MSE were used to determine the best fit model in describing the growth pattern of Bali cows. Von Bertalanffy model was the best model among the four growth functions evaluated to determine the mature weight of Bali cattle as shown by the highest R2 and lowest MSE values (0.973 and 601.9, respectively, followed by Gompertz (0.972 and 621.2, respectively, Logistic (0.971 and 648.4, respectively and Brody (0.932 and 660.5, respectively models. The correlation between rate of maturing and mature weight was found to be negative in the range of -0.170 to -0.929 for all models, indicating that animals of heavier mature weight had lower rate of maturing. The use of non-linear model could summarize the weight-age relationship into several biologically interpreted parameters compared to the entire lifespan weight-age data points that are difficult and time consuming to interpret.
Extended Elliptic Mild Slope Equation Incorporating the Nonlinear Shoaling Effect
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Xiao Qian-lu
2016-10-01
Full Text Available The transformation during wave propagation is significantly important for the calculations of hydraulic and coastal engineering, as well as the sediment transport. The exact wave height deformation calculation on the coasts is essential to near-shore hydrodynamics research and the structure design of coastal engineering. According to the wave shoaling results gained from the elliptical cosine wave theory, the nonlinear wave dispersion relation is adopted to develop the expression of the corresponding nonlinear wave shoaling coefficient. Based on the extended elliptic mild slope equation, an efficient wave numerical model is presented in this paper for predicting wave deformation across the complex topography and the surf zone, incorporating the nonlinear wave dispersion relation, the nonlinear wave shoaling coefficient and other energy dissipation factors. Especially, the phenomenon of wave recovery and second breaking could be shown by the present model. The classical Berkhoff single elliptic topography wave tests, the sinusoidal varying topography experiment, and complex composite slopes wave flume experiments are applied to verify the accuracy of the calculation of wave heights. Compared with experimental data, good agreements are found upon single elliptical topography and one-dimensional beach profiles, including uniform slope and step-type profiles. The results indicate that the newly-developed nonlinear wave shoaling coefficient improves the calculated accuracy of wave transformation in the surf zone efficiently, and the wave breaking is the key factor affecting the wave characteristics and need to be considered in the nearshore wave simulations.
Focusing of Spherical Nonlinear Pulses for Nonlinear Wave Equations Ⅲ. Subcritical Case
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
This paper studied spherical pulses of solutions of the system of semilinear wave equations with the pulses focusing at a point in three space variables. It is shown that there is no nonlinear effect at leading terms of pulses, when the initial data is subcritical.
Algebraic dynamics solution to and algebraic dynamics algorithm for nonlinear advection equation
Institute of Scientific and Technical Information of China (English)
2008-01-01
Algebraic dynamics approach and algebraic dynamics algorithm for the solution of nonlinear partial differential equations are applied to the nonlinear advection equa-tion. The results show that the approach is effective for the exact analytical solu-tion and the algorithm has higher precision than other existing algorithms in nu-merical computation for the nonlinear advection equation.
On localization in the discrete nonlinear Schrödinger equation
DEFF Research Database (Denmark)
Bang, O.; Juul Rasmussen, J.; Christiansen, P.L.
1993-01-01
For some values of the grid resolution, depending on the nonlinearity, the discrete nonlinear Schrodinger equation with arbitrary power nonlinearity can be approximated by the corresponding continuum version of the equation. When the discretization becomes too coarse, the discrete equation exhibits...
An efficient algorithm for solving nonlinear system of differential equations and applications
Directory of Open Access Journals (Sweden)
Mustafa GÜLSU
2015-10-01
Full Text Available In this article, we apply Chebyshev collocation method to obtain the numerical solutions of nonlinear systems of differential equations. This method transforms the nonlinear systems of differential equation to nonlinear systems of algebraic equations. The convergence of the numerical method are given and their applicability is illustrated with some examples.
Nonlinear Schrodinger equations on the half-line with nonlinear boundary conditions
Directory of Open Access Journals (Sweden)
Ahmet Batal
2016-08-01
Full Text Available In this article, we study the initial boundary value problem for nonlinear Schrodinger equations on the half-line with nonlinear boundary conditions $$ u_x(0,t+\\lambda|u(0,t|^ru(0,t=0,\\quad \\lambda\\in\\mathbb{R}-\\{0\\},\\; r> 0. $$ We discuss the local well-posedness when the initial data $u_0=u(x,0$ belongs to an $L^2$-based inhomogeneous Sobolev space $H^s(\\mathbb{R}_+$ with $s\\in (\\frac{1}{2},\\frac{7}{2}-\\{\\frac{3}{2}\\}$. We deal with the nonlinear boundary condition by first studying the linear Schrodinger equation with a time-dependent inhomogeneous Neumann boundary condition $u_x(0,t=h(t$ where $h\\in H^{\\frac{2s-1}{4}}(0,T$.
Describing Adequacy of cure with maximum hardness ratios and non-linear regression.
Bouschlicher, Murray; Berning, Kristen; Qian, Fang
2008-01-01
Knoop Hardness (KH) ratios (HR) > or = 80% are commonly used as criteria for the adequate cure of a composite. These per-specimen HRs can be misleading, as both numerator and denominator may increase concurrently, prior to reaching an asymptotic, top-surface maximum hardness value (H(MAX)). Extended cure times were used to establish H(MAX) and descriptive statistics, and non-linear regression analysis were used to describe the relationship between exposure duration and HR and predict the time required for HR-H(MAX) = 80%. Composite samples 2.00 x 5.00 mm diameter (n = 5/grp) were cured for 10 seconds, 20 seconds, 40 seconds, 60 seconds, 90 seconds, 120 seconds, 180 seconds and 240 seconds in a 2-composite x 2-light curing unit design. A microhybrid (Point 4, P4) or microfill resin (Heliomolar, HM) composite was cured with a QTH or LED light curing unit and then stored in the dark for 24 hours prior to KH testing. Non-linear regression was calculated with: H = (H(MAX)-c)(1-e(-kt)) +c, H(MAX) = maximum hardness (a theoretical asymptotic value), c = constant (t = 0), k = rate constant and t = exposure duration describes the relationship between radiant exposure (irradiance x time) and HRs. Exposure durations for HR-H(MAX) = 80% were calculated. Two-sample t-tests for pairwise comparisons evaluated relative performance of the light curing units for similar surface x composite x exposure (10-90s). A good measure of goodness-of-fit of the non-linear regression, r2, ranged from 0.68-0.95. (mean = 0.82). Microhybrid (P4) exposure to achieve HR-H(MAX = 80% was 21 seconds for QTH and 34 seconds for the LED light curing unit. Corresponding values for microfill (HM) were 71 and 74 seconds, respectively. P4 HR-H(MAX) of LED vs QTH was statistically similar for 10 to 40 seconds, while HM HR-H(MAX) of LED was significantly lower than QTH for 10 to 40 seconds. It was concluded that redefined hardness ratios based on maximum hardness used in conjunction with non-linear regression
Dynamic behavior of a nonlinear rational difference equation and generalization
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Shi Qihong
2011-01-01
Full Text Available Abstract This paper is concerned about the dynamic behavior for the following high order nonlinear difference equation x n = (x n-k + x n-m + x n-l /(x n-k x n-m + x n-m x n-l +1 with the initial data { x - l , x - l + 1 , … , x - 1 } ∈ ℝ + l and 1 ≤ k ≤ m ≤ l. The convergence of solution to this equation is investigated by introducing a new sequence, which extends and includes corresponding results obtained in the references (Li in J Math Anal Appl 312:103-111, 2005; Berenhaut et al. Appl. Math. Lett. 20:54-58, 2007; Papaschinopoulos and Schinas J Math Anal Appl 294:614-620, 2004 to a large extent. In addition, some propositions for generalized equations are reported.
Barker, Blake; Noble, Pascal; Rodrigues, L Miguel; Zumbrun, Kevin
2012-01-01
In this paper we consider the spectral and nonlinear stability of periodic traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In particular, we resolve the long-standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized (integrable) perturbations. Our analysis is based upon detailed estimates of the linearized solution operator, which are complicated by the fact that the (necessarily essential) spectrum of the associated linearization intersects the imaginary axis at the origin. We carry out a numerical Evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves, in close agreement with previous numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy, Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also compare predictions of the associated Whitham modulation equations, which formally describe the dynamics of weak large s...
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Potemki, Valeri G. [Moscow State Engineering Physics Institute (Technical University), Moscow (Russian Federation). Dept. of Automatics and Electronics; Borisevich, Valentine D.; Yupatov, Sergei V. [Moscow State Enineering Physics Institute (Technical University), Moscow (Russian Federation). Dept. of Technical Physics
1996-12-31
This paper describes the the next evolution step in development of the direct method for solving systems of Nonlinear Algebraic Equations (SNAE). These equations arise from the finite difference approximation of original nonlinear partial differential equations (PDE). This method has been extended on the SNAE with three variables. The solving SNAE bases on Reiterating General Singular Value Decomposition of rectangular matrix pencils (RGSVD-algorithm). In contrast to the computer algebra algorithm in integer arithmetic based on the reduction to the Groebner`s basis that algorithm is working in floating point arithmetic and realizes the reduction to the Kronecker`s form. The possibilities of the method are illustrated on the example of solving the one-dimensional diffusion equation for 3-component model isotope mixture in a ga centrifuge. The implicit scheme for the finite difference equations without simplifying the nonlinear properties of the original equations is realized. The technique offered provides convergence to the solution for the single run. The Toolbox SNAE is developed in the framework of the high performance numeric computation and visualization software MATLAB. It includes more than 30 modules in MATLAB language for solving SNAE with two and three variables. (author) 7 refs., 10 figs.
Indian Academy of Sciences (India)
Wang Zhi-Yun; Chen Pei-Jie
2016-06-01
A generalized Langevin equation driven by fractional Brownian motion is used to describe the vertical oscillations of general relativistic disks. By means of numerical calculation method, the displacements, velocities and luminosities of oscillating disks are explicitly obtained for different Hurst exponent $H$. The results show that as $H$ increases, the energies and luminosities of oscillating disk are enhanced, and the spectral slope at high frequencies of the power spectrum density of disk luminosity is also increased. This could explain the observational features related to the Intra Day Variability of the BL Lac objects.
Accuracy Assessment for Cad Modeling of Freeform Surface Described by Equation
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Golba Grzegorz
2015-06-01
Full Text Available This paper presents the results of comparative analysis of modeling accuracy the freeform surface constructed by using a variety of algorithms for surface modeling. Also determined the accuracy of mapping the theoretical freeform surface described by mathematical equation. To model surface objects used: SolidWorks 2012, CATIA v5 and Geomagic Studio 12. During the design process of CAD models were used: profile curves, fitting parametric surface and polygonal mesh. To assess the accuracy of the CAD models used Geomagic Qualify 12. On the basis of analyse defined the scope of application of each modeling techniques depending on the nature of the constructed object.
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Elsayed M.E. Zayed
2016-02-01
Full Text Available In this article, the modified extended tanh-function method is employed to solve fractional partial differential equations in the sense of the modified Riemann–Liouville derivative. Based on a nonlinear fractional complex transformation, certain fractional partial differential equations can be turned into nonlinear ordinary differential equations of integer orders. For illustrating the validity of this method, we apply it to four nonlinear equations namely, the space–time fractional generalized nonlinear Hirota–Satsuma coupled KdV equations, the space–time fractional nonlinear Whitham–Broer–Kaup equations, the space–time fractional nonlinear coupled Burgers equations and the space–time fractional nonlinear coupled mKdV equations.
Zia, Haider
2017-06-01
This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a notable example being the generalized derivative non-linear Schrödinger equation that arises in non-linear optics with self-steepening terms. These differential equations feature terms that were previously inaccessible to model accurately with low computational resources. The new method maintains a 3rd order error even with these additional terms and models the equation in all three spatial dimensions and time. The class of non-linear differential equations that this method applies to is shown. The method is fully derived and implementation of the method in the split-step architecture is shown. This paper lays the mathematical ground work for an upcoming paper employing this method in white-light generation simulations in bulk material.
Accelerating Inexact Newton Schemes for Large Systems of Nonlinear Equations
Fokkema, D.R.; Sleijpen, G.L.G.; Vorst, H.A. van der
1998-01-01
Classical iteration methods for linear systems, such as Jacobi iteration, can be accelerated considerably by Krylov subspace methods like GMRES. In this paper, we describe how inexact Newton methods for nonlinear problems can be accelerated in a similar way and how this leads to a general framework
Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics
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Khaled A. Gepreel
2012-01-01
Full Text Available We put a direct new method to construct the rational Jacobi elliptic solutions for nonlinear differential difference equations which may be called the rational Jacobi elliptic functions method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential difference equations in mathematical physics via the lattice equation and the discrete nonlinear Schrodinger equation with a saturable nonlinearity. The proposed method is more effective and powerful to obtain the exact solutions for nonlinear differential difference equations.
Evaluation of model fit in nonlinear multilevel structural equation modeling
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Karin eSchermelleh-Engel
2014-03-01
Full Text Available Evaluating model fit in nonlinear multilevel structural equation models (MSEM presents a challenge as no adequate test statistic is available. Nevertheless, using a product indicator approach a likelihood ratio test for linear models is provided which may also be useful for nonlinear MSEM. The main problem with nonlinear models is that product variables are nonnormally distributed. Although robust test statistics have been developed for linear SEM to ensure valid results under the condition of nonnormality, they were not yet investigated for nonlinear MSEM. In a Monte Carlo study, the performance of the robust likelihood ratio test was investigated for models with single-level latent interaction effects using the unconstrained product indicator approach. As overall model fit evaluation has a potential limitation in detecting the lack of fit at a single level even for linear models, level-specific model fit evaluation was also investigated using partially saturated models. Four population models were considered: a model with interaction effects at both levels, an interaction effect at the within-group level, an interaction effect at the between-group level, and a model with no interaction effects at both levels. For these models the number of groups, predictor correlation, and model misspecification was varied. The results indicate that the robust test statistic performed sufficiently well. Advantages of level-specific model fit evaluation for the detection of model misfit are demonstrated.
Evaluation of model fit in nonlinear multilevel structural equation modeling.
Schermelleh-Engel, Karin; Kerwer, Martin; Klein, Andreas G
2014-01-01
Evaluating model fit in nonlinear multilevel structural equation models (MSEM) presents a challenge as no adequate test statistic is available. Nevertheless, using a product indicator approach a likelihood ratio test for linear models is provided which may also be useful for nonlinear MSEM. The main problem with nonlinear models is that product variables are non-normally distributed. Although robust test statistics have been developed for linear SEM to ensure valid results under the condition of non-normality, they have not yet been investigated for nonlinear MSEM. In a Monte Carlo study, the performance of the robust likelihood ratio test was investigated for models with single-level latent interaction effects using the unconstrained product indicator approach. As overall model fit evaluation has a potential limitation in detecting the lack of fit at a single level even for linear models, level-specific model fit evaluation was also investigated using partially saturated models. Four population models were considered: a model with interaction effects at both levels, an interaction effect at the within-group level, an interaction effect at the between-group level, and a model with no interaction effects at both levels. For these models the number of groups, predictor correlation, and model misspecification was varied. The results indicate that the robust test statistic performed sufficiently well. Advantages of level-specific model fit evaluation for the detection of model misfit are demonstrated.
Domain decomposition solvers for nonlinear multiharmonic finite element equations
Copeland, D. M.
2010-01-01
In many practical applications, for instance, in computational electromagnetics, the excitation is time-harmonic. Switching from the time domain to the frequency domain allows us to replace the expensive time-integration procedure by the solution of a simple elliptic equation for the amplitude. This is true for linear problems, but not for nonlinear problems. However, due to the periodicity of the solution, we can expand the solution in a Fourier series. Truncating this Fourier series and approximating the Fourier coefficients by finite elements, we arrive at a large-scale coupled nonlinear system for determining the finite element approximation to the Fourier coefficients. The construction of fast solvers for such systems is very crucial for the efficiency of this multiharmonic approach. In this paper we look at nonlinear, time-harmonic potential problems as simple model problems. We construct and analyze almost optimal solvers for the Jacobi systems arising from the Newton linearization of the large-scale coupled nonlinear system that one has to solve instead of performing the expensive time-integration procedure. © 2010 de Gruyter.
Directory of Open Access Journals (Sweden)
S. S. Motsa
2014-01-01
Full Text Available This paper presents a new application of the homotopy analysis method (HAM for solving evolution equations described in terms of nonlinear partial differential equations (PDEs. The new approach, termed bivariate spectral homotopy analysis method (BISHAM, is based on the use of bivariate Lagrange interpolation in the so-called rule of solution expression of the HAM algorithm. The applicability of the new approach has been demonstrated by application on several examples of nonlinear evolution PDEs, namely, Fisher’s, Burgers-Fisher’s, Burger-Huxley’s, and Fitzhugh-Nagumo’s equations. Comparison with known exact results from literature has been used to confirm accuracy and effectiveness of the proposed method.
Nonlinear Whitham-Broer-Kaup Wave Equation in an Analytical Solution
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S. A. Zahedi
2008-01-01
Full Text Available This study presented a new approach for the analysis of a nonlinear Whitham-Broer-Kaup equation dealing with propagation of shallow water waves with different dispersion relations. The analysis was based on a kind of analytical method, called Variational Iteration Method (VIM. To illustrate the capability of the approach, some numerical examples were given and the propagation and the error of solutions were shown in comparison to those of exact solution. In clear conclusion, the approach was efficient and capable to obtain the analytical approximate solution of this set of wave equations while these solutions could straightforwardly show some facts of the described process deeply such as the propagation. This method can be easily extended to other nonlinear wave equations and so can be found widely applicable in this field of science.
Nonlinear wave structures as exact solutions of Vlasov-Maxwell equations.
Dasgupta, B.; Tsurutani, B. T.; Janaki, M. S.; Sharma, A. S.
2001-12-01
Many recent observations by POLAR and Geotail spacecraft of the low-latitudes magnetopause boundary layer (LLBL) and the polar cap boundary layer (PCBL) have detected nonlinear wave structures [Tsurutani et al, Geophys. Res. Lett., 25, 4117, 1998]. These nonlinear waves have electromagnetic signatures that are identified with Alfven and Whistler modes. Also solitary waves with mono- and bi-polar features were observed. In general such electromagnetic structures are described by the full Vlasov-Maxwell equations for waves propagating at an angle to the ambient magnetic field, but it has been a diffficult task obtaining the solutions because of the inherent nonlinearity. We have obtained an exact nonlinear solution of the full Vlasov-Maxwell equations in the presence of an electromagnetic wave propagating at an arbitrary direction with an ambient magnetic field. This is accomplished by finding the constants of motion of the charged particles in the electromagnetic field of the wave and then constructing a realistic distribution function as a function of these constants of motion. The corresponding trapping conditions for such waves are obtained, yielding the self-consistent description for the particles in the presence of the nonlinear waves. The interpretation of the observed nonlinear structures in terms of these general solutions will be presented.
A Master Equation for Multi-Dimensional Non-Linear Field Theories
Park, Q H
1992-01-01
A master equation ( $n$ dimensional non--Abelian current conservation law with mutually commuting current components ) is introduced for multi-dimensional non-linear field theories. It is shown that the master equation provides a systematic way to understand 2-d integrable non-linear equations as well as 4-d self-dual equations and, more importantly, their generalizations to higher dimensions.
Nonlinear evolution operators and semigroups applications to partial differential equations
Pavel, Nicolae H
1987-01-01
This research monograph deals with nonlinear evolution operators and semigroups generated by dissipative (accretive), possibly multivalued operators, as well as with the application of this theory to partial differential equations. It shows that a large class of PDE's can be studied via the semigroup approach. This theory is not available otherwise in the self-contained form provided by these Notes and moreover a considerable part of the results, proofs and methods are not to be found in other books. The exponential formula of Crandall and Liggett, some simple estimates due to Kobayashi and others, the characterization of compact semigroups due to Brézis, the proof of a fundamental property due to Ursescu and the author and some applications to PDE are of particular interest. Assuming only basic knowledge of functional analysis, the book will be of interest to researchers and graduate students in nonlinear analysis and PDE, and to mathematical physicists.
Institute of Scientific and Technical Information of China (English)
WANG Shunjin; ZHANG Hua
2006-01-01
The problem of preserving fidelity in numerical computation of nonlinear ordinary differential equations is studied in terms of preserving local differential structure and approximating global integration structure of the dynamical system.The ordinary differential equations are lifted to the corresponding partial differential equations in the framework of algebraic dynamics,and a new algorithm-algebraic dynamics algorithm is proposed based on the exact analytical solutions of the ordinary differential equations by the algebraic dynamics method.In the new algorithm,the time evolution of the ordinary differential system is described locally by the time translation operator and globally by the time evolution operator.The exact analytical piece-like solution of the ordinary differential equations is expressd in terms of Taylor series with a local convergent radius,and its finite order truncation leads to the new numerical algorithm with a controllable precision better than Runge Kutta Algorithm and Symplectic Geometric Algorithm.
Tkachova, P.; Krot, A.; Minervina, H.
It is well known that there is chaos in convective process in atmosphere and ocean. In particular,dynamic model of Lorenz [1] describes the Rayleigh-Benard convection phenomenon. Phase trajectories of Lorenz equation system are characterized by strange alternative properties: on the one hand, they diverge (because of positive Lyapunov exponents), on the second hand, they attract to the limited domain of phase space called an attractor [1]. The Lorenz attractor has specific geometrical structure and can be characterized by means of fractal dimension. In this connection the aim of this work is development of analysis of Lorenz attractor based on the proposed nonlinear decomposition into matrix series [2]. This analysis permits to estimate the values of characteristic parameters (including control one) of Lorenz attractors and predict their evolution in time. Using results of matrix decomposition [2], it is not difficult to see that the change of vector function (describing the Lorenz attractor) can be approximated by only linear and quadratic terms [3]. Because values of the first and second order derivatives can be calculated by means of numerical methods we can estimate the change of the vector function from computational experiment. In result, the values of parameters of the Lorenz's attractor can be estimated. This permits us to solve the identification task of the current dynamical state of a convective aerodynamic flows. Moreover, using the results of matrix decomposition we can estimate the minimal embedding dimension [4] for the Lorenz attractor based on experimental data. References: [1] P.Berge,Y.Pomeau and C.Vidal. L'ordre dans le chaos: Vers une approche deterministe de la turbulence. Hermann:Paris,1988. [2] A.M.Krot, "Matrix decompositions of vector functions and shift operators on the trajectories of a nonlinear dynamical system", Nonlinear Phenomena in Complex Systems,vol.4, N2, pp.106- 115, 2001. [3] A.M.Krot and P
Caffarelli, Luis; Nirenberg, Louis
2011-01-01
The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.
Picone-type inequalities for nonlinear elliptic equations and their applications
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Takaŝi Kusano
2001-01-01
Full Text Available Picone-type inequalities are derived for nonlinear elliptic equations, and Sturmian comparison theorems are established as applications. Oscillation theorems for forced super-linear elliptic equations and superlinear-sublinear elliptic equations are also obtained.
Method of the Logistic Function for Finding Analytical Solutions of Nonlinear Differential Equations
Kudryashov, N. A.
2015-01-01
The method of the logistic function is presented for finding exact solutions of nonlinear differential equations. The application of the method is illustrated by using the nonlinear ordinary differential equation of the fourth order. Analytical solutions obtained by this method are presented. These solutions are expressed via exponential functions.logistic function, nonlinear wave, nonlinear ordinary differential equation, Painlev´e test, exact solution
Directory of Open Access Journals (Sweden)
Xia Liu
2017-02-01
Full Text Available The discrete nonlinear Schrodinger equation is a nonlinear lattice system that appears in many areas of physics such as nonlinear optics, biomolecular chains and Bose-Einstein condensates. In this article, we consider a class of discrete nonlinear Schrodinger equations with unbounded potentials. We obtain some new sufficient conditions on the multiplicity results of ground state solutions for the equations by using the symmetric mountain pass lemma. Recent results in the literature are greatly improved.
Modified extended tanh-function method for solving nonlinear partial differential equations
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El-Wakil, S.A. [Department of Physics, Faculty of Science, Theoretical Research Group, Mansoura University, 35516 Mansoura (Egypt); Abdou, M.A. [Department of Physics, Faculty of Science, Theoretical Research Group, Mansoura University, 35516 Mansoura (Egypt)]. E-mail: m_abdou_eg@yahoo.com
2007-03-15
Based on computerized symbolic computation, modified extended tanh-method for constructing multiple travelling wave solutions of nonlinear evolution equations is presented and implemented in a computer algebraic system. Applying this method, with the aid of Maple, we consider some nonlinear evolution equations in mathematical physics such as the nonlinear partial differential equation, nonlinear Fisher-type equation, ZK-BBM equation, generalized Burgers-Fisher equation and Drinfeld-Sokolov system. As a result, we can successfully recover the previously known solitary wave solutions that had been found by the extended tanh-function method and other more sophisticated methods.
Superposition of elliptic functions as solutions for a large number of nonlinear equations
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Khare, Avinash [Raja Ramanna Fellow, Indian Institute of Science Education and Research (IISER), Pune 411021 (India); Saxena, Avadh [Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)
2014-03-15
For a large number of nonlinear equations, both discrete and continuum, we demonstrate a kind of linear superposition. We show that whenever a nonlinear equation admits solutions in terms of both Jacobi elliptic functions cn(x, m) and dn(x, m) with modulus m, then it also admits solutions in terms of their sum as well as difference. We have checked this in the case of several nonlinear equations such as the nonlinear Schrödinger equation, MKdV, a mixed KdV-MKdV system, a mixed quadratic-cubic nonlinear Schrödinger equation, the Ablowitz-Ladik equation, the saturable nonlinear Schrödinger equation, λϕ{sup 4}, the discrete MKdV as well as for several coupled field equations. Further, for a large number of nonlinear equations, we show that whenever a nonlinear equation admits a periodic solution in terms of dn{sup 2}(x, m), it also admits solutions in terms of dn {sup 2}(x,m)±√(m) cn (x,m) dn (x,m), even though cn(x, m)dn(x, m) is not a solution of these nonlinear equations. Finally, we also obtain superposed solutions of various forms for several coupled nonlinear equations.
Heteroclinic structure of parametric resonance in the nonlinear Schr\\"odinger equation
Conforti, M; Kudlinski, A; Rota-Nodari, S; Dujardin, G; De Bievre, S; Armaroli, A; Trillo, S
2016-01-01
We show that the nonlinear stage of modulational instability induced by parametric driving in the {\\em defocusing} nonlinear Schr\\"odinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearised Floquet analysis.
An analytical solution to the equation of motion for the damped nonlinear pendulum
DEFF Research Database (Denmark)
Johannessen, Kim
2014-01-01
An analytical approximation of the solution to the differential equation describing the oscillations of the damped nonlinear pendulum at large angles is presented. The solution is expressed in terms of the Jacobi elliptic functions by including a parameter-dependent elliptic modulus. The analytical...... of the damped nonlinear pendulum is presented, and it is shown that the period of oscillation is dependent on time. It is established that, in general, the period is longer than that of a linearized model, asymptotically approaching the period of oscillation of a damped linear pendulum....
Static solution of the general relativistic nonlinear $\\sigma$model equation
Lee, C H; Lee, H K; Lee, Chul H; Kim, Joon Ha; Lee, Hyun Kyu
1994-01-01
The nonlinear \\sigma-model is considered to be useful in describing hadrons (Skyrmions) in low energy hadron physics and the approximate behavior of the global texture. Here we investigate the properties of the static solution of the nonlinear \\sigma-model equation coupled with gravity. As in the case where gravity is ignored, there is still no scale parameter that determines the size of the static solution and the winding number of the solution is 1/2. The geometry of the spatial hyperspace in the asymptotic region of large r is explicitly shown to be that of a flat space with some missing solid angle.
Computational uncertainty principle in nonlinear ordinary differential equations
Institute of Scientific and Technical Information of China (English)
LI; Jianping
2001-01-01
［1］Li Jianping, Zeng Qingcun, Chou Jifan, Computational Uncertainty Principle in Nonlinear Ordinary Differential Equations I. Numerical Results, Science in China, Ser. E, 2000, 43(5): 449［2］Henrici, P., Discrete Variable Methods in Ordinary Differential Equations, New York: John Wiley, 1962, 1; 187.［3］Henrici, P., Error Propagation for Difference Methods, New York: John Whiley, 1963.［4］Gear, C. W., Numerical Initial Value Problems in Ordinary Differential Equations, Englewood Cliffs, NJ: Prentice-Hall, 1971, 1; 72.［5］Hairer, E., Nrsett, S. P., Wanner, G., Solving Ordinary Differential Equations I. Nonstiff Problems, 2nd ed., Berlin-Heidelberg-New York: Springer-Verlag, 1993, 130.［6］Stoer, J., Bulirsch, R., Introduction to Numerical Analysis, 2nd ed., Vol. 1, Berlin-Heidelberg-New York: Springer-Verlag (reprinted in China by Beijing Wold Publishing Corporation), 1998, 428.［7］Li Qingyang, Numerical Methods in Ordinary Differential Equations (Stiff Problems and Boundary Value Problems), in Chinese Beijing: Higher Education Press, 1991, 1.［8］Li Ronghua, Weng Guochen, Numerical Methods in Differential Equations (in Chinese), 3rd ed., Beijing: Higher Education Press, 1996, 1.［9］Dahlquist, G., Convergence and stability in the numerical integration of ordinary differential equations, Math. Scandinavica, 1956, 4: 33.［10］Dahlquist, G., 33 years of numerical instability, Part I, BIT, 1985, 25: 188.［11］Heisenberg, W., The Physical Principles of Quantum Theory, Chicago: University of Chicago Press, 1930.［12］McMurry, S. M., Quantum Mechanics, London: Addison-Wesley Longman Ltd (reprined in China by Beijing World Publishing Corporation), 1998.
Some existence results on nonlinear fractional differential equations.
Baleanu, Dumitru; Rezapour, Shahram; Mohammadi, Hakimeh
2013-05-13
In this paper, by using fixed-point methods, we study the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem D(α)u(t)=f(t,u(t)) with a Riemann-Liouville fractional derivative via the different boundary-value problems u(0)=u(T), and the three-point boundary condition u(0)=β(1)u(η) and u(T)=β(2)u(η), where T>0, t∈I=[0,T], 0<α<1, 0<η
On a Nonlinear Partial Integro-Differential Equation
Abergel, Frederic
2009-01-01
Consistently fitting vanilla option surfaces is an important issue when it comes to modelling in finance. Local volatility models introduced by Dupire in 1994 are widely used to price and manage the risks of structured products. However, the inconsistencies observed between the dynamics of the smile in those models and in real markets motivate researches for stochastic volatility modelling. Combining both those ideas to form Local and Stochastic Volatility models is of interest for practitioners. In this paper, we study the calibration of the vanillas in those models. This problem can be written as a nonlinear and nonlocal partial differential equation, for which we prove short-time existence of solutions.
Fourth order wave equations with nonlinear strain and source terms
Liu, Yacheng; Xu, Runzhang
2007-07-01
In this paper we study the initial boundary value problem for fourth order wave equations with nonlinear strain and source terms. First we introduce a family of potential wells and prove the invariance of some sets and vacuum isolating of solutions. Then we obtain a threshold result of global existence and nonexistence. Finally we discuss the global existence of solutions for the problem with critical initial condition I(u0)[greater-or-equal, slanted]0, E(0)=d. So the Esquivel-Avila's results are generalized and improved.
TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations.By using the bifurcation theory of dynamical systems to do qualitative analysis,all possible phase portraits in the parametric space for the traveling wave systems are obtained.It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied.The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.
Energy Technology Data Exchange (ETDEWEB)
Feng, Wenqiang, E-mail: wfeng1@vols.utk.edu [Department of Mathematics, The University of Tennessee, Knoxville, TN 37996 (United States); Salgado, Abner J., E-mail: asalgad1@utk.edu [Department of Mathematics, The University of Tennessee, Knoxville, TN 37996 (United States); Wang, Cheng, E-mail: cwang1@umassd.edu [Department of Mathematics, The University of Massachusetts, North Dartmouth, MA 02747 (United States); Wise, Steven M., E-mail: swise1@utk.edu [Department of Mathematics, The University of Tennessee, Knoxville, TN 37996 (United States)
2017-04-01
We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems – including thin film epitaxy with slope selection and the square phase field crystal model – are carried out to verify the efficiency of the scheme.
Feng, Wenqiang; Salgado, Abner J.; Wang, Cheng; Wise, Steven M.
2017-04-01
We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of the equations are constant-coefficient, positive linear operators, which suggests a natural preconditioning strategy. Such nonlinear elliptic equations often arise from time discretization of parabolic equations that model various biological and physical phenomena, in particular, liquid crystals, thin film epitaxial growth and phase transformations. The analyses of the schemes involve the characterization of the strictly convex energies associated with the equations. We first give a general framework for PSD in Hilbert spaces. Based on certain reasonable assumptions of the linear pre-conditioner, a geometric convergence rate is shown for the nonlinear PSD iteration. We then apply the general theory to the fourth and sixth-order problems of interest, making use of Sobolev embedding and regularity results to confirm the appropriateness of our pre-conditioners for the regularized p-Lapacian problems. Our results include a sharper theoretical convergence result for p-Laplacian systems compared to what may be found in existing works. We demonstrate rigorously how to apply the theory in the finite dimensional setting using finite difference discretization methods. Numerical simulations for some important physical application problems - including thin film epitaxy with slope selection and the square phase field crystal model - are carried out to verify the efficiency of the scheme.
LOCAL DISCONTINUOUS GALERKIN METHODS FOR THREE CLASSES OF NONLINEAR WAVE EQUATIONS
Institute of Scientific and Technical Information of China (English)
Yan Xu; Chi-wang Shu
2004-01-01
In this paper, we further develop the local discontinuous Galerkin method to solve three classes of nonlinear wave equations formulated by the general KdV-Burgers type equations, the general fifth-order KdV type equations and the fully nonlinear K(n, n, n)equations, and prove their stability for these general classes of nonlinear equations. The schemes we present extend the previous work of Yan and Shu [30, 31] and of Levy, Shu and Yan [24] on local discontinuous Galerkin method solving partial differential equations with higher spatial derivatives. Numerical examples for nonlinear problems are shown to illustrate the accuracy and capability of the methods. The numerical experiments include stationary solitons, soliton interactions and oscillatory solitary wave solutions.The numerical experiments also include the compacton solutions of a generalized fifthorder KdV equation in which the highest order derivative term is nonlinear and the fully nonlinear K(n, n, n) equations.
Multiple scales analysis and travelling wave solutions for KdV type nonlinear evolution equations
Ayhan, Burcu; Ozer, M. Naci; Bekir, Ahmet
2017-01-01
Nonlinear evolution equations are the mathematical models of problems that arise in many field of science. These equations has become an important field of study in applied mathematics in recent years. We apply exact solution methods and multiple scale method which is known as a perturbation method to nonlinear evolution equations. Using exact solution methods we get travelling wave solutions expressed by hyperbolic functions, trigonometric functions and rational functions. Also we derive Nonlinear Schrödinger (NLS) type equations from Korteweg-de Vries (KdV) type nonlinear evolution equations and we get approximate solutions for KdV type equations using multiple scale method. The proposed methods are direct and effective and can be used for many nonlinear evolution equations. It is shown that these methods provide a powerful mathematical tool to solve nonlinear evolution equations in mathematical physics.
A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
BAI Cheng-Lin
2003-01-01
For the Noyes-Fields equations, two-dimensional hyperbolic equations of conversation laws, and theBurgers-KdV equation, a class of traveling wave solutions has been obtained by constructing appropriate functiontransformations. The main idea of solving the equations is that nonlinear partial differential equations are changed intosolving algebraic equations. This method has a wide-rangingpracticability.
A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations
Institute of Scientific and Technical Information of China (English)
BAICheng-Lin
2003-01-01
For the Noyes-Fields equations, two-dimenslonal hyperbolic equations of conversation laww and the Burgers-KdV equation, a class of travellng wave solutions has been obtained by constructhag appropriate function transformations. The main idea of solving the equations is that nonlinear partial differential equations are changed into solving algebraic equations. This method has a wide-ranging practicability.
A procedure to construct exact solutions of nonlinear fractional differential equations.
Güner, Özkan; Cevikel, Adem C
2014-01-01
We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.
On invariant analysis of some time fractional nonlinear systems of partial differential equations. I
Singla, Komal; Gupta, R. K.
2016-10-01
An investigation of Lie point symmetries for systems of time fractional partial differential equations including Ito system, coupled Burgers equations, coupled Korteweg de Vries equations, Hirota-Satsuma coupled KdV equations, and coupled nonlinear Hirota equations has been done. Using the obtained symmetries, each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving Erdélyi-Kober fractional differential operator depending on a parameter α.
Institute of Scientific and Technical Information of China (English)
HUANGYi; ZHANGYin-ke
2003-01-01
The nonlinear constitutive equations and field equations of unsaturated soils were constructed on the basis of mixture theory.The soils were treated as the mixture composed of three constituents.First ,from the researches of soil mechanics,some basic assumptions about the unsaturated soil mixture were mode,and the entropy inequality unsaturated soil mixture was derived.Then,with the common method usually used to deal with the constitutive problems in mixture theory,the nonlinear constitutive equations were obtained.Finally,putting the constiutive equtions of constituents into the balance equations of momentum,the nonlinear field equations of constitutents into the balance equations of momentum,the nonliear field equations of constitutents were set up.The balance equation of energy of unsaturated soil was also given,and thus the complete equations for solving the thermodynamic process of unsaturated soil was formed.
Generalized multiscale finite element methods. nonlinear elliptic equations
Efendiev, Yalchin R.
2013-01-01
In this paper we use the Generalized Multiscale Finite Element Method (GMsFEM) framework, introduced in [26], in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing the equation so that coarse-grid quantities of previous solution iterates can be regarded as auxiliary parameters within the problem formulation. With this convention, we systematically construct respective coarse solution spaces that lend themselves to either continuous Galerkin (CG) or discontinuous Galerkin (DG) global formulations. Here, we use Symmetric Interior Penalty Discontinuous Galerkin approach. Both methods yield a predictable error decline that depends on the respective coarse space dimension, and we illustrate the effectiveness of the CG and DG formulations by offering a variety of numerical examples. © 2014 Global-Science Press.
New variable separation solutions for the generalized nonlinear diffusion equations
Fei-Yu, Ji; Shun-Li, Zhang
2016-03-01
The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A(u,ux)uxx + B(u,ux) is studied by using the conditional Lie-Bäcklund symmetry method. The variant forms of the considered equations, which admit the corresponding conditional Lie-Bäcklund symmetries, are characterized. To construct functionally generalized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided. Project supported by the National Natural Science Foundation of China (Grant Nos. 11371293, 11401458, and 11501438), the National Natural Science Foundation of China, Tian Yuan Special Foundation (Grant No. 11426169), and the Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2015JQ1014).
Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations
Institute of Scientific and Technical Information of China (English)
Junaid Ali Khan; Muhammad Asif Zahoor Raja; Ijaz Mansoor Qureshi
2011-01-01
@@ We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs).The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error.The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique.The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations.We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods.The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy.With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.%We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs). The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error. The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique. The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations. We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods. The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed.
DEFF Research Database (Denmark)
Miansari, Mo; Miansari, Me; Barari, Amin
2009-01-01
In this article, He’s variational iteration method (VIM), is implemented to solve the linear Helmholtz partial differential equation and some nonlinear fifth-order Korteweg-de Vries (FKdV) partial differential equations with specified initial conditions. The initial approximations can be freely...... and nonlinear problems. It is predicted that VIM can be widely applied in engineering....
Sahadevan, R.; Prakash, P.
2017-01-01
We show how invariant subspace method can be extended to time fractional coupled nonlinear partial differential equations and construct their exact solutions. Effectiveness of the method has been illustrated through time fractional Hunter-Saxton equation, time fractional coupled nonlinear diffusion system, time fractional coupled Boussinesq equation and time fractional Whitman-Broer-Kaup system. Also we explain how maximal dimension of the time fractional coupled nonlinear partial differential equations can be estimated.
Institute of Scientific and Technical Information of China (English)
Chang Jing; Gao Yi-xian; Cai Hua
2014-01-01
In this paper, the generalized extended tanh-function method is used for constructing the traveling wave solutions of nonlinear evolution equations. We choose Fisher’s equation, the nonlinear schr¨odinger equation to illustrate the validity and ad-vantages of the method. Many new and more general traveling wave solutions are obtained. Furthermore, this method can also be applied to other nonlinear equations in physics.
A combined modification of Newton`s method for systems of nonlinear equations
Energy Technology Data Exchange (ETDEWEB)
Monteiro, M.T.; Fernandes, E.M.G.P. [Universidade do Minho, Braga (Portugal)
1996-12-31
To improve the performance of Newton`s method for the solution of systems of nonlinear equations a modification to the Newton iteration is implemented. The modified step is taken as a linear combination of Newton step and steepest descent directions. In the paper we describe how the coefficients of the combination can be generated to make effective use of the two component steps. Numerical results that show the usefulness of the combined modification are presented.
Homotopy perturbation method for nonlinear partial differential equations of fractional order
Energy Technology Data Exchange (ETDEWEB)
Momani, Shaher [Department of Mathematics and Physics, Qatar University (Qatar)]. E-mail: shahermm@yahoo.com; Odibat, Zaid [Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa' Applied University, Salt (Jordan)]. E-mail: odibat@bau.edu.jo
2007-06-11
The aim of this Letter is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for nonlinear partial differential equations with fractional time derivative. The fractional derivative is described in the Caputo sense. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. The obtained results are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, efficient and easy to implement.
Non-linear equation: energy conservation and impact parameter dependence
Kormilitzin, Andrey
2010-01-01
In this paper we address two questions: how energy conservation affects the solution to the non-linear equation, and how impact parameter dependence influences the inclusive production. Answering the first question we solve the modified BK equation which takes into account energy conservation. In spite of the fact that we used the simplified kernel, we believe that the main result of the paper: the small ($\\leq 40%$) suppression of the inclusive productiondue to energy conservation, reflects a general feature. This result leads us to believe that the small value of the nuclear modification factor is of a non-perturbative nature. In the solution a new scale appears $Q_{fr} = Q_s \\exp(-1/(2 \\bas))$ and the production of dipoles with the size larger than $2/Q_{fr}$ is suppressed. Therefore, we can expect that the typical temperature for hadron production is about $Q_{fr}$ ($ T \\approx Q_{fr}$). The simplified equation allows us to obtain a solution to Balitsky-Kovchegov equation taking into account the impact pa...
Energy Technology Data Exchange (ETDEWEB)
Huang Dingjiang [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)]. E-mail: hdj8116@163.com; Zhang Hongqing [Department of Applied Mathematics, Dalian University of Technology, Dalian 116024 (China)
2006-08-15
Many travelling wave solutions of nonlinear evolution equations can be written as a polynomial in several elementary or special functions which satisfy a first order nonlinear ordinary differential equation with a sixth-degree nonlinear term. From that property, we deduce an algebraic method for constructing those solutions by determining only a finite number of coefficients. Being concise and straightforward, the method is applied to three nonlinear evolution equations. As a result, many exact travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions.
DEFF Research Database (Denmark)
Gaididei, Yu. B.; Christiansen, Peter Leth
2008-01-01
We study a parametrically driven Ginzburg-Landau equation model with nonlinear management. The system is made of laterally coupled long active waveguides placed along a circumference. Stationary solutions of three kinds are found: periodic Ising states and two types of Bloch states, staggered...... and unstaggered. The stability of these states is investigated analytically and numerically. The nonlinear dynamics of the Bloch states are described by a complex Ginzburg-Landau equation with linear and nonlinear parametric driving. The switching between the staggered and unstaggered Bloch states under...
Energy Technology Data Exchange (ETDEWEB)
Alvarez-Estrada, R.F.
1979-08-01
A comprehensive review of the inverse scattering solution of certain non-linear evolution equations of physical interest in one space dimension is presented. We explain in some detail the interrelated techniques which allow to linearize exactly the following equations: (1) the Korteweg and de Vries equation; (2) the non-linear Schrodinger equation; (3) the modified Korteweg and de Vries equation; (4) the Sine-Gordon equation. We concentrate in discussing the pairs of linear operators which accomplish such an exact linearization and the solution of the associated initial value problem. The application of the method to other non-linear evolution equations is reviewed very briefly.
Asymptotic behavior for a quadratic nonlinear Schrodinger equation
Directory of Open Access Journals (Sweden)
Pavel I. Naumkin
2008-02-01
Full Text Available We study the initial-value problem for the quadratic nonlinear Schrodinger equation $$displaylines{ iu_{t}+frac{1}{2}u_{xx}=partial _{x}overline{u}^{2},quad xin mathbb{R},; t>1, cr u(1,x=u_{1}(x,quad xin mathbb{R}. }$$ For small initial data $u_{1}in mathbf{H}^{2,2}$ we prove that there exists a unique global solution $uin mathbf{C}([1,infty ;mathbf{H}^{2,2}$ of this Cauchy problem. Moreover we show that the large time asymptotic behavior of the solution is defined in the region $|x|leq Csqrt{t}$ by the self-similar solution $frac{1}{sqrt{t}}MS(frac{x}{sqrt{t}}$ such that the total mass $$ frac{1}{sqrt{t}}int_{mathbb{R}}MS(frac{x}{sqrt{t}} dx=int_{mathbb{R}}u_{1}(xdx, $$ and in the far region $|x|>sqrt{t}$ the asymptotic behavior of solutions has rapidly oscillating structure similar to that of the cubic nonlinear Schrodinger equations.
Nonlinear subelliptic Schrodinger equations with external magnetic field
Directory of Open Access Journals (Sweden)
Kyril Tintarev
2004-10-01
Full Text Available To account for an external magnetic field in a Hamiltonian of a quantum system on a manifold (modelled here by a subelliptic Dirichlet form, one replaces the the momentum operator $frac 1i d$ in the subelliptic symbol by $frac 1i d-alpha$, where $alphain TM^*$ is called a magnetic potential for the magnetic field $eta=dalpha$. We prove existence of ground state solutions (Sobolev minimizers for nonlinear Schrodinger equation associated with such Hamiltonian on a generally, non-compact Riemannian manifold, generalizing the existence result of Esteban-Lions [5] for the nonlinear Schrödinger equation with a constant magnetic field on $mathbb{R}^N$ and the existence result of [6] for a similar problem on manifolds without a magnetic field. The counterpart of a constant magnetic field is the magnetic field, invariant with respect to a subgroup of isometries. As an example to the general statement we calculate the invariant magnetic fields in the Hamiltonians associated with the Kohn Laplacian and for the Laplace-Beltrami operator on the Heisenberg group.
Nonlinear differential equations with exact solutions expressed via the Weierstrass function
Kudryashov, NA
2004-01-01
A new problem is studied, that is to find nonlinear differential equations with special solutions expressed via the Weierstrass function. A method is discussed to construct nonlinear ordinary differential equations with exact solutions. The main step of our method is the assumption that nonlinear di
Nonlinear Fokker-Planck Equation in the Model of Asset Returns
Directory of Open Access Journals (Sweden)
Alexander Shapovalov
2008-04-01
Full Text Available The Fokker-Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker-Planck equation we describe a construction of exact solution of the Cauchy problem. In the general case, we construct the leading term of the Cauchy problem solution asymptotic in a formal small parameter in semiclassical approximation following the complex WKB-Maslov method in the class of trajectory concentrated functions.
Directory of Open Access Journals (Sweden)
C. Ünlü
2013-01-01
Full Text Available A modification of the variational iteration method (VIM for solving systems of nonlinear fractional-order differential equations is proposed. The fractional derivatives are described in the Caputo sense. The solutions of fractional differential equations (FDE obtained using the traditional variational iteration method give good approximations in the neighborhood of the initial position. The main advantage of the present method is that it can accelerate the convergence of the iterative approximate solutions relative to the approximate solutions obtained using the traditional variational iteration method. Illustrative examples are presented to show the validity of this modification.
Institute of Scientific and Technical Information of China (English)
黄义; 张引科
2003-01-01
The nonlinear constitutive equations and field equations of unsaturated soils were cons tructed on the basis of mixture theory. The soils were treated as the mixture composed of three constituents. First, from the researches of soil mechanics, some basic assumptions about the unsaturated soil mixture were made, and the entropy inequality of unsaturated soil mixture was derived. Then, with the common method usually used to deal with the constitutive problems in mixture theory, the nonlinear constitutive equations were obtained. Finally, putting the constitutive equations of constituents into the balance equations of momentum, the nonlinear field equations of constituents were set up. The balance equation of energy of unsaturated soil was also given, and thus the complete equations for solving the thermodynamic process of unsaturated soil was formed.
Numerical solution of the nonlinear Schrödinger equation with wave operator on unbounded domains.
Li, Hongwei; Wu, Xiaonan; Zhang, Jiwei
2014-09-01
In this paper, we generalize the unified approach proposed in Zhang et al. [J. Zhang, Z. Xu, and X. Wu, Phys. Rev. E 78, 026709 (2008)] to design the nonlinear local absorbing boundary conditions (LABCs) for the nonlinear Schrödinger equation with wave operator on unbounded domains. In fact, based on the methodology underlying the unified approach, we first split the original equation into two parts-the linear equation and the nonlinear equation-then achieve a one-way operator to approximate the linear equation to make the wave outgoing, and finally combine the one-way operator with the nonlinear equation to achieve the nonlinear LABCs. The stability of the equation with the nonlinear LABCs is also analyzed by introducing some auxiliary variables, and some numerical examples are presented to verify the accuracy and effectiveness of our proposed method.
An effective analytic approach for solving nonlinear fractional partial differential equations
Ma, Junchi; Zhang, Xiaolong; Liang, Songxin
2016-08-01
Nonlinear fractional differential equations are widely used for modelling problems in applied mathematics. A new analytic approach with two parameters c1 and c2 is first proposed for solving nonlinear fractional partial differential equations. These parameters are used to improve the accuracy of the resulting series approximations. It turns out that much more accurate series approximations are obtained by choosing proper values of c1 and c2. To demonstrate the applicability and effectiveness of the new method, two typical fractional partial differential equations, the nonlinear gas dynamics equation and the nonlinear KdV-Burgers equation, are solved.
Nonpoint Symmetry and Reduction of Nonlinear Evolution and Wave Type Equations
Directory of Open Access Journals (Sweden)
Ivan Tsyfra
2015-01-01
Full Text Available We study the symmetry reduction of nonlinear partial differential equations with two independent variables. We propose new ansätze reducing nonlinear evolution equations to system of ordinary differential equations. The ansätze are constructed by using operators of nonpoint classical and conditional symmetry. Then we find solution to nonlinear heat equation which cannot be obtained in the framework of the classical Lie approach. By using operators of Lie-Bäcklund symmetries we construct the solutions of nonlinear hyperbolic equations depending on arbitrary smooth function of one variable too.
Chavanis, Pierre-Henri
2008-01-01
We consider a generalized class of Keller-Segel models describing the chemotaxis of biological populations (bacteria, amoebae, endothelial cells, social insects,...). We show the analogy with nonlinear mean field Fokker-Planck equations and generalized thermodynamics. As an illustration, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). We also discuss the analogy between biological populations described by the Keller-Segel model and self-gravitating Brownian particles described by the Smoluchowski-Poisson system.
Energy Technology Data Exchange (ETDEWEB)
Belmonte-Beitia, J [Departamento de Matematicas, E T S de Ingenieros Industriales and Instituto de Matematica Aplicada a la Ciencia y la IngenierIa (IMACI), Avda Camilo Jose Cela, 3 Universidad de Castilla-La Mancha 13071 Ciudad Real (Spain); Cuevas, J [Grupo de Fisica No Lineal, Departamento de Fisica Aplicada I, Escuela Universitaria Politecnica, C/Virgen de Africa, 7, 41011 Sevilla (Spain)], E-mail: juan.belmonte@uclm.es, E-mail: jcuevas@us.es
2009-04-24
In this paper, we construct, by means of similarity transformations, explicit solutions to the cubic-quintic nonlinear Schroedinger equation with potentials and nonlinearities depending on both time and spatial coordinates. We present the general approach and use it to calculate bright and dark soliton solutions for nonlinearities and potentials of physical interest in applications to Bose-Einstein condensates and nonlinear optics.
OSCILLATION FOR NONLINEAR SECOND-ORDER DYNAMIC EQUATIONS ON TIME SCALES
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
Through the use of generalized Riccati transformation techniques, we establish some oscillation criteria for one type of nonlinear dynamic equation on time scales. Several examples, including a semilinear dynamic equation and a nonlinear Emden-Fowler dynamic equation, are also given to illustrate these criteria and to improve the results obtained in some references.
Energy Technology Data Exchange (ETDEWEB)
Zhao Xiqiang [Department of Mathematics, Ocean University of China, Qingdao Shandong 266071 (China)] e-mail: zhaodss@yahoo.com.cn; Wang Limin [Shandong University of Technology, Zibo Shandong 255049 (China); Sun Weijun [Shandong University of Technology, Zibo Shandong 255049 (China)
2006-04-01
In this letter, a new method, called the repeated homogeneous balance method, is proposed for seeking the traveling wave solutions of nonlinear partial differential equations. The Burgers-KdV equation is chosen to illustrate our method. It has been confirmed that more traveling wave solutions of nonlinear partial differential equations can be effectively obtained by using the repeated homogeneous balance method.
Institute of Scientific and Technical Information of China (English)
Wan-sheng WANG; Shou-fu LI; Run-sheng YANG
2012-01-01
A series of contractivity and exponential stability results for the solutions to nonlinear neutral functional differential equations (NFDEs) in Banach spaces are obtained,which provide unified theoretical foundation for the contractivity analysis of solutions to nonlinear problems in functional differential equations (FDEs),neutral delay differential equations (NDDEs) and NFDEs of other types which appear in practice.
Institute of Scientific and Technical Information of China (English)
Chuan Qiang CHEN; Bo Wen HU
2013-01-01
We study microscopic spacetime convexity properties of fully nonlinear parabolic partial differential equations.Under certain general structure condition,we establish a constant rank theorem for the spacetime convex solutions of fully nonlinear parabolic equations.At last,we consider the parabolic convexity of solutions to parabolic equations and the convexity of the spacetime second fundamental form of geometric flows.
Deterministic and stochastic evolution equations for fully dispersive and weakly nonlinear waves
DEFF Research Database (Denmark)
Eldeberky, Y.; Madsen, Per A.
1999-01-01
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 c...
Kengne, Emmanuel; Saydé, Michel; Ben Hamouda, Fathi; Lakhssassi, Ahmed
2013-11-01
Analytical entire traveling wave solutions to the 1+1 density-dependent nonlinear reaction-diffusion equation via the extended generalized Riccati equation mapping method are presented in this paper. This equation can be regarded as an extension case of the Fisher-Kolmogoroff equation, which is used for studying insect and animal dispersal with growth dynamics. The analytical solutions are then used to investigate the effect of equation parameters on the population distribution.
Nonlinear unified equations for water waves propagating over uneven bottoms in the nearshore region
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Considering the continuous characteristics for water waves propagating over complex topography in the nearshore region, the unified nonlinear equations, based on the hypothesis for a typical uneven bottom, are presented by employing the Hamiltonian variational principle for water waves. It is verified that the equations include the following special cases: the extension of Airy's nonlinear shallow-water equations, the generalized mild-slope equation, the dispersion relation for the second-order Stokes waves and the higher order Boussinesq-type equations.
Exponential Polynomials as Solutions of Certain Nonlinear Difference Equations
Institute of Scientific and Technical Information of China (English)
Zhi Tao WEN; Janne HEITTOKANGAS; Ilpo LAINE
2012-01-01
Recently,C.-C.Yang and I.Laine have investigated finite order entire solutions f of nonlinear differential-difference equations of the form fn+L(z,f)=h,where n ≥ 2 is an integer.In particular,it is known that the equation f(z)2+q(z)f(z+1) =p(z),where p(z),q(z) are polynomials,has no transcendental entire solutions of finite order.Assuming that Q(z) is also a polynomial and c ∈ C,equations of the form f(z)n + q(z)eQ(z)f(z + c) =p(z) do posses finite order entire solutions.A classification of these solutions in terms of growth and zero distribution will be given.In particular,it is shown that any exponential polynomial solution must reduce to a rather specific form.This reasoning relies on an earlier paper due to N.Steinmetz.
A new method for parameter estimation in nonlinear dynamical equations
Wang, Liu; He, Wen-Ping; Liao, Le-Jian; Wan, Shi-Quan; He, Tao
2015-01-01
Parameter estimation is an important scientific problem in various fields such as chaos control, chaos synchronization and other mathematical models. In this paper, a new method for parameter estimation in nonlinear dynamical equations is proposed based on evolutionary modelling (EM). This will be achieved by utilizing the following characteristics of EM which includes self-organizing, adaptive and self-learning features which are inspired by biological natural selection, and mutation and genetic inheritance. The performance of the new method is demonstrated by using various numerical tests on the classic chaos model—Lorenz equation (Lorenz 1963). The results indicate that the new method can be used for fast and effective parameter estimation irrespective of whether partial parameters or all parameters are unknown in the Lorenz equation. Moreover, the new method has a good convergence rate. Noises are inevitable in observational data. The influence of observational noises on the performance of the presented method has been investigated. The results indicate that the strong noises, such as signal noise ratio (SNR) of 10 dB, have a larger influence on parameter estimation than the relatively weak noises. However, it is found that the precision of the parameter estimation remains acceptable for the relatively weak noises, e.g. SNR is 20 or 30 dB. It indicates that the presented method also has some anti-noise performance.
Small-amplitude excitations in a deformable discrete nonlinear Schrödinger equation
Konotop, V V
1996-01-01
A detailed analysis of the small-amplitude solutions of a deformed discrete nonlinear Schrödinger equation is performed. For generic deformations the system possesses "singular" points which split the infinite chain in a number of independent segments. We show that small-amplitude dark solitons in the vicinity of the singular points are described by the Toda-lattice equation while away from the singular points are described by the Korteweg-de Vries equation. Depending on the value of the deformation parameter and of the background level several kinds of solutions are possible. In particular we delimit the regions in the parameter space in which dark solitons are stable in contrast with regions in which bright pulses on nonzero background are possible. On the boundaries of these regions we find that shock waves and rapidly spreading solutions may exist.
Hafez, M. G.; Talukder, M. R.; Hossain Ali, M.
2017-04-01
The Burgers equation is obtained to study the characteristics of nonlinear propagation of ionacoustic shock, singular kink, and periodic waves in weakly relativistic plasmas containing relativistic thermal ions, nonextensive distributed electrons, Boltzmann distributed positrons, and kinematic viscosity of ions using the well-known reductive perturbation technique. This equation is solved by employing the ( G'/ G)-expansion method taking unperturbed positron-to-electron concentration ratio, electron-to-positron temperature ratio, strength of electrons nonextensivity, ion kinematic viscosity, and weakly relativistic streaming factor. The influences of plasma parameters on nonlinear propagation of ion-acoustic shock, periodic, and singular kink waves are displayed graphically and the relevant physical explanations are described. It is found that these parameters extensively modify the shock structures excitation. The obtained results may be useful in understanding the features of small but finite amplitude localized relativistic ion-acoustic shock waves in an unmagnetized plasma system for some astrophysical compact objects and space plasmas.
(3+1)-dimensional nonlinear propagation equation for ultrashort pulsed beam in left-handed material
Institute of Scientific and Technical Information of China (English)
Hu Yong-Hua; Fu Xi-Quan; Wen Shuang-Chun; Su Wen-Hua; Fan Dian-Yuan
2006-01-01
In this paper a comprehensive framework for treating the nonlinear propagation of ultrashort pulse in metamaterial with dispersive dielectric susceptibility and magnetic permeability is presented. Under the slowly-evolving-wave approximation, a generalized (3+1)-dimensional wave equation first order in the propagation coordinate and suitable for both right-handed material (RHM) and left-handed material (LHM) is derived. By the commonly used Drude dispersive model for LHM, a (3+1)-dimensional nonlinear Schr(o)dinger equation describing ultrashort pulsed beam propagation in LHM is obtained, and its difference from that for conventional RHM is discussed. Particularly, the self-steeping effect of ultrashort pulse is found to be anomalous in LHM.
Kierkels, A. H. M.; Velázquez, J. J. L.
2016-06-01
We construct a family of self-similar solutions with fat tails to a quadratic kinetic equation. This equation describes the long time behaviour of weak solutions with finite mass to the weak turbulence equation associated to the nonlinear Schrödinger equation. The solutions that we construct have finite mass, but infinite energy. In Kierkels and Velázquez (J Stat Phys 159:668-712, 2015) self-similar solutions with finite mass and energy were constructed. Here we prove upper and lower exponential bounds on the tails of these solutions.
On global attraction to stationary states for wave equations with concentrated nonlinearities
Kopylova, E.
2016-01-01
The global attraction to stationary states is established for solutions to 3D wave equations with concentrated nonlinearities: each finite energy solution converges as $t\\to\\pm\\infty$ to stationary states. The attraction is caused by nonlinear energy radiation.
Solitons and periodic solutions to a couple of fractional nonlinear evolution equations
Indian Academy of Sciences (India)
M Mirzazadeh; M Eslami; Anjan Biswas
2014-03-01
This paper studies a couple of fractional nonlinear evolution equations using first integral method. These evolution equations are foam drainage equation and Klein–Gordon equation (KGE), the latter of which is considered in (2 + 1) dimensions. For the fractional evolution, the Jumarie’s modified Riemann–Liouville derivative is considered. Exact solutions to these equations are obtained.
Kvrekidis, Panayotis G
2009-01-01
This book constitutes the first effort to summarize a large volume of results obtained over the past 20 years in the context of the Discrete Nonlinear Schrödinger equation and the physical settings that it describes. It contains an introduction to the model, its systematic derivation and its connection to applications, a subsequent analysis of the existence and the stability of fundamental nonlinear structures in 1, 2 and even 3 spatial lattice dimensions. It also covers the case of defocusing nonlinearities, the modulational instabilities of plane wave solutions, and the extension to multi-component lattices. In addition, it features a final chapter on special topics written by a wide array of experts in the field, addressing through short reviews, areas of particular recent interest.
Kolli, Avinash; Nazir, Ahsan; Olaya-Castro, Alexandra
2011-10-21
We derive a many-site version of the non-Markovian time-convolutionless polaron master equation [Jang et al., J. Chem Phys. 129, 101104 (2008)] to describe electronic excitation dynamics in multichromophoric systems. By treating electronic and vibrational degrees of freedom in a combined frame (polaron frame), this theory is capable of interpolating between weak and strong exciton-phonon coupling and is able to account for initial non-equilibrium bath states and spatially correlated environments. Besides outlining a general expression for the expected value of any electronic system observable in the original frame, we also discuss implications of the Markovian and Secular approximations highlighting that they need not hold in the untransformed frame despite being strictly satisfied in the polaron frame. The key features of the theory are illustrated using as an example a four-site subsystem of the Fenna-Mathews-Olson light-harvesting complex. For a spectral density including a localised mode, we show that oscillations of site populations may only be observed when non-equilibrium bath effects are taken into account. Furthermore, we illustrate how this formalism allows us to identify the electronic and vibrational components of the oscillatory dynamics.
Note on Nonlinear Schr\\"odinger Equation, KdV Equation and 2D Topological Yang-Mills-Higgs Theory
Nian, Jun
2016-01-01
In this paper we discuss the relation between the (1+1)D nonlinear Schr\\"odinger equation and the KdV equation. By applying the boson/vortex duality, we can map the classical nonlinear Schr\\"odinger equation into the classical KdV equation in the small coupling limit, which corresponds to the UV regime of the theory. At quantum level, the two theories satisfy the Bethe Ansatz equations of the spin-$\\frac{1}{2}$ XXX chain and the XXZ chain in the continuous limit respectively. Combining these relations with the dualities discussed previously in the literature, we propose a duality web in the UV regime among the nonlinear Schr\\"odinger equation, the KdV equation and the 2D $\\mathcal{N}=(2,2)^*$ topological Yang-Mills-Higgs theory.
Energy Technology Data Exchange (ETDEWEB)
Lu, Bin, E-mail: lubinhb@163.com [School of Mathematical Sciences, Anhui University, Hefei 230601 (China)
2012-06-04
In this Letter, the fractional derivatives in the sense of modified Riemann–Liouville derivative and the Bäcklund transformation of fractional Riccati equation are employed for constructing the exact solutions of nonlinear fractional partial differential equations. The power of this manageable method is presented by applying it to several examples. This approach can also be applied to other nonlinear fractional differential equations. -- Highlights: ► Backlund transformation of fractional Riccati equation is presented. ► A new method for solving nonlinear fractional differential equations is proposed. ► Three important fractional differential equations are solved successfully. ► Some new exact solutions of the fractional differential equations are obtained.
Origin of heavy tail statistics in equations of the Nonlinear Schr\\"odinger type: an exact result
Onorato, M; El, G; Randoux, S; Suret, P
2016-01-01
We study the formation of extreme events in incoherent systems described by envelope equations, such as the Nonliner Schr\\"odinger equation. We derive an identity that relates the evolution of the kurtosis (a measure of the relevance of the tails in a probability density function) of the wave amplitude to the rate of change of the width of the Fourier spectrum of the wave field. The result is exact for all dispersive systems characterized by a nonlinear term of the form of the one contained in the Nonlinear Schr\\"odinger equation. Numerical simulations are also performed to confirm our findings. Our work sheds some light on the origin of rogue waves in incoherent dispersive nonlinear media ruled by local cubic nonlinearity.
Islam, Md Shafiqul; Khan, Kamruzzaman; Akbar, M Ali; Mastroberardino, Antonio
2014-10-01
The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin-Bona-Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.
Dynamic properties of the cubic nonlinear Schr(o)dinger equation by symplectic method
Institute of Scientific and Technical Information of China (English)
Liu Xue-Shen; Wei Jia-Yu; Ding Pei-Zhu
2005-01-01
The dynamic properties of a cubic nonlinear Schrodinger equation are investigated numerically by using the symplectic method with different space approximations. The behaviours of the cubic nonlinear Schrodinger equation are discussed with different cubic nonlinear parameters in the harmonically modulated initial condition. We show that the conserved quantities will be preserved for long-time computation but the system will exhibit different dynamic behaviours in space difference approximation for the strong cubic nonlinearity.
Nonlinear equations on controlling interface patterns during solidification of a dilute binary alloy
Institute of Scientific and Technical Information of China (English)
王自东; 周永利; 常国威; 胡汉起
1999-01-01
In nonequilibrium nonlinear region, by assuming that there is local equilibrium at the solid/liquid interface, and considering that curvature, temperature and composition at the solid/liquid interface which are related to perturbation amplitude are nonlinear, nonlinear equations of the time dependence of the perturbation amplitude of the solid/liquid interface during solidification of a dilute binary alloy are established. Crystal growth from nonsteady state to steady state can be controlled by these nonlinear equations.
Soliton solution for nonlinear partial differential equations by cosine-function method
Energy Technology Data Exchange (ETDEWEB)
Ali, A.H.A. [Mathematics Department, Faculty of Science, Menoufia University, Shebein El-Koom (Egypt); Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish), Suez Canal University, AL-Arish 45111 (Egypt)], E-mail: asoliman_99@yahoo.com; Raslan, K.R. [Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo (Egypt)
2007-08-20
In this Letter, we established a traveling wave solution by using Cosine-function algorithm for nonlinear partial differential equations. The method is used to obtain the exact solutions for five different types of nonlinear partial differential equations such as, general equal width wave equation (GEWE), general regularized long wave equation (GRLW), general Korteweg-de Vries equation (GKdV), general improved Korteweg-de Vries equation (GIKdV), and Coupled equal width wave equations (CEWE), which are the important soliton equations.
Frank, T. D.
2008-02-01
We discuss two central claims made in the study by Bassler et al. [K.E. Bassler, G.H. Gunaratne, J.L. McCauley, Physica A 369 (2006) 343]. Bassler et al. claimed that Green functions and Langevin equations cannot be defined for nonlinear diffusion equations. In addition, they claimed that nonlinear diffusion equations are linear partial differential equations disguised as nonlinear ones. We review bottom-up and top-down approaches that have been used in the literature to derive Green functions for nonlinear diffusion equations and, in doing so, show that the first claim needs to be revised. We show that the second claim as well needs to be revised. To this end, we point out similarities and differences between non-autonomous linear Fokker-Planck equations and autonomous nonlinear Fokker-Planck equations. In this context, we raise the question whether Bassler et al.’s approach to financial markets is physically plausible because it necessitates the introduction of external traders and causes. Such external entities can easily be eliminated when taking self-organization principles and concepts of nonextensive thermostatistics into account and modeling financial processes by means of nonlinear Fokker-Planck equations.
Institute of Scientific and Technical Information of China (English)
Jeffery C. DiFranco; Peter D. Miller; Benson K. Muite
2011-01-01
The purpose of this paper is to present a comparison between the modified nonlinear Schr(o)dinger (MNLS) equation and the focusing and defocusing variants of the (unmodified) nonlinear Schr(o)dinger (NLS) equation in the semiclassical limit.We describe aspects of the limiting dynamics and discuss how the nature of the dynamics is evident theoretically through inverse-scattering and noncommutative steepest descent methods.The main message is that,depending on initial data,the MNLS equation can behave either like the defocusing NLS equation,like the focusing NLS equation (in both cases the analogy is asymptotically accurate in the semiclassical limit when the NLS equation is posed with appropriately modified initial data),or like an interesting mixture of the two.In the latter case,we identify a feature of the dynamics analogous to a sonic line in gas dynamics,a free boundary separating subsonic flow from supersonic flow.
Canonical equations of Hamilton for the nonlinear Schr\\"{o}dinger equation
Liang, Guo; Ren, Zhanmei
2013-01-01
We define two different systems of mathematical physics: the second-order differential system (SODS) and the first-order differential system (FODS). The Newton's second law of motion and the nonlinear Schr\\"{o}dinger equation (NLSE) are the exemplary SODS and FODS, respectively. We obtain a new kind of canonical equations of Hamilton (CEH), which are of some kind of symmetry in form and are formally different with the conventional CEH without symmetry [H. Goldstein, C. Poole, J. Safko, Classical Mechanics, third ed., Addison-Wesley, 2001]. We also prove that the number of the CEHs is equal to the number of the generalized coordinates for the FODS, but twice the number of the generalized coordinates for the SODS. We show that the FODS can only be expressed by the new CEH, but do not by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs. As an example, we prove that the nonlinear Schr\\"{o}dinger equation can be expressed with the new CEH in a consistent way.
Canonical equations of Hamilton for the nonlinear Schrödinger equation
Liang, Guo; Guo, Qi; Ren, Zhanmei
2015-09-01
We define two different systems of mathematical physics: the second order differential system (SODS) and the first order differential system (FODS). The Newton's second law of motion and the nonlinear Schrödinger equation (NLSE) are the exemplary SODS and FODS, respectively. We obtain a new kind of canonical equations of Hamilton (CEH), which exhibit some kind of symmetry in form and are formally different from the conventional CEH without symmetry [H. Goldstein, C. Poole, J. Safko, Classical Mechanics, third ed., Addison- Wesley, 2001]. We also prove that the number of the CEHs is equal to the number of the generalized coordinates for the FODS, but twice the number of the generalized coordinates for the SODS. We show that the FODS can only be expressed by the new CEH, but not introduced by the conventional CEH, while the SODS can be done by both the new and the conventional CEHs. As an example, we prove that the nonlinear Schrödinger equation can be expressed with the new CEH in a consistent way.
On the so called rogue waves in nonlinear Schrodinger equations
Directory of Open Access Journals (Sweden)
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.
THREE POINT BOUNDARY VALUE PROBLEMS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
Mujeeb ur Rehman; Rahmat Ali Khan; Naseer Ahmad Asif
2011-01-01
In this paper,we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type cDδ0+u(t) =f(t,u(t),cDσ0+u(t)),t ∈[0,T],u(0) =αu(η),u(T) =βu(η),where1 ＜δ＜2,0＜σ＜ 1,α,β∈R,η∈(0,T),αη(1-β)+(1-α)(T-βη) ≠0 and cDoδ+,cDσ0+ are the Caputo fractional derivatives.We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results.Examples are also included to show the applicability of our results.
A Unified and Explicit Construction of N-Soliton Solutions for the Nonlinear Schrfdinger Equation
Institute of Scientific and Technical Information of China (English)
FAN En-Gui
2001-01-01
An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrodinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrfdinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrfdinger equation is given.``
Approach in Theory of Nonlinear Evolution Equations: The Vakhnenko-Parkes Equation
Directory of Open Access Journals (Sweden)
V. O. Vakhnenko
2016-01-01
Full Text Available A variety of methods for examining the properties and solutions of nonlinear evolution equations are explored by using the Vakhnenko equation (VE as an example. The VE, which arises in modelling the propagation of high-frequency waves in a relaxing medium, has periodic and solitary traveling wave solutions some of which are loop-like in nature. The VE can be written in an alternative form, known as the Vakhnenko-Parkes equation (VPE, by a change of independent variables. The VPE has an N-soliton solution which is discussed in detail. Individual solitons are hump-like in nature whereas the corresponding solution to the VE comprises N-loop-like solitons. Aspects of the inverse scattering transform (IST method, as applied originally to the KdV equation, are used to find one- and two-soliton solutions to the VPE even though the VPE’s spectral equation is third-order and not second-order. A Bäcklund transformation for the VPE is used to construct conservation laws. The standard IST method for third-order spectral problems is used to investigate solutions corresponding to bound states of the spectrum and to a continuous spectrum. This leads to N-soliton solutions and M-mode periodic solutions, respectively. Interactions between these types of solutions are investigated.
DEFF Research Database (Denmark)
Christiansen, Peter Leth; Gaididei, Yuri Borisovich; Rasmussen, Kim
1996-01-01
We study the effect of adding noise and nonlinear damping in the two-dimensional nonlinear Schrodinger equation (NLS). Using a collective approach, we find that for initial conditions where total collapse occurs in the unperturbed NLS, the presence of the damping term will instead in an exponenti......We study the effect of adding noise and nonlinear damping in the two-dimensional nonlinear Schrodinger equation (NLS). Using a collective approach, we find that for initial conditions where total collapse occurs in the unperturbed NLS, the presence of the damping term will instead...
Energy Technology Data Exchange (ETDEWEB)
Aslan, İsmail, E-mail: ismailaslan@iyte.edu.tr [Department of Mathematics, Izmir Institute of Technology, Urla, İzmir 35430 (Turkey)
2011-11-14
We analyze the discrete nonlinear Schrödinger equation with a saturable nonlinearity through the (G{sup ′}/G)-expansion method to present some improved results. Three types of analytic solutions with arbitrary parameters are constructed; hyperbolic, trigonometric, and rational which have not been explicitly computed before. -- Highlights: ► Discrete nonlinear Schrödinger equation with a saturable nonlinearity. ► We confirm that the model supports three types of solutions with arbitrary parameters. ► A new application of the (G{sup ′}/G)-expansion method presented.
Directory of Open Access Journals (Sweden)
Juan Belmonte-Beitia
2008-01-01
Full Text Available We give a proof of the existence of stationary bright soliton solutions of the cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearity. By using bifurcation theory, we prove that the norm of the positive solution goes to zero as the parameter λ, called chemical potential in the Bose-Einstein condensates' literature, tends to zero. Moreover, we solve the time-dependent cubic-quintic nonlinear Schrödinger equation with inhomogeneous nonlinearities by using a numerical method.
Chaos control in the nonlinear Schrödinger equation with Kerr law nonlinearity
Yin, Jiu-Li; Zhao, Liu-Wei; Tian, Li-Xin
2014-02-01
The nonlinear Schrödinger equation with Kerr law nonlinearity in the two-frequency interference is studied by the numerical method. Chaos occurs easily due to the absence of damping. This phenomenon will cause the distortion in the process of information transmission. We find that fiber-optic transmit signals still present chaotic phenomena if the control intensity is smaller. With the increase of intensity, the fiber-optic signal can stay in a stable state in some regions. When the strength is suppressed to a certain value, an unstable phenomenon of the fiber-optic signal occurs. Moreover we discuss the sensitivities of the parameters to be controlled. The results show that the linear term coefficient and the environment of two quite different frequences have less effects on the fiber-optic transmission. Meanwhile the phenomena of vibration, attenuation and escape occur in some regions.
Directory of Open Access Journals (Sweden)
K. R. McCall
1996-01-01
Full Text Available The velocity of sound in rock is a strong function of pressure, indicating that wave propagation in rocks is very nonlinear. The quasistatic elastic properties of rocks axe hysteretic, possessing discrete memory. In this paper a new theory is developed, placing all of these properties (nonlinearity, hysteresis, and memory on equal footing. The starting point of the new theory is closer to a microscopic description of a rock than the starting point of the traditional five-constant theory of nonlinear elasticity. However, this starting point (the number density Ï? of generic mechanical elements in an abstract space is deliberately independent of a specific microscopic model. No prejudice is imposed as to the mechanism causing nonlinear response in the microscopic mechanical elements. The new theory (1 relates suitable stress-strain measurements to the number density Ï? and (2 uses the number density Ï? to find the behaviour of nonlinear elastic waves. Thus the new theory provides for the synthesis of the full spectrum of elastic behaviours of a rock. Early development of the new theory is sketched in this contribution.
Lanzafame, Giuseppe
2012-01-01
Physical damping, regarding the nonlinear Navier-Stokes viscous flow dynamics, refers to a tensorial turbulent dissipation term, attributed to adjacent moving macroscopic flow components. Mutual dissipation among these parts of fluid is described by a braking term in the momentum equation together with a heating term in the energy equation, both responsible of the damping of the momentum variation and of the viscous conversion of mechanical energy into heat. A macroscopic mixing scale length is currently the only characteristic length needed in the nonlinear modelling of viscous fluid dynamics describing the nonlinear eddy viscosity through the kinematic viscosity coefficient in the viscous stress tensor, without any reference to the chemical composition and to the atomic dimensions. Therefore, in this paper, we write a new formulation for the kinematic viscosity coefficient to the turbulent viscous physical dissipation in the Navier-Stokes equations, where molecular parameters are also included. Results of 2...
Self-similar solutions for some nonlinear evolution equations: KdV, mKdV and Burgers equations
Directory of Open Access Journals (Sweden)
S.A. El-Wakil
2016-02-01
Full Text Available A method for solving three types of nonlinear evolution equations namely KdV, modified KdV and Burgers equations, with self-similar solutions is presented. The method employs ideas from symmetry reduction to space and time variables and similarity reductions for nonlinear evolution equations are performed. The obtained self-similar solutions of KdV and mKdV equations are related to Bessel and Airy functions whereas those of Burgers equation are related to the error and Hermite functions. These solutions appear as new types of solitary, shock and periodic waves. Also, the method can be applied to other nonlinear evolution equations in mathematical physics.
Modeling extreme wave heights from laboratory experiments with the nonlinear Schrödinger equation
Zhang, H. D.; Guedes Soares, C.; Cherneva, Z.; Onorato, M.
2014-04-01
Spatial variation of nonlinear wave groups with different initial envelope shapes is theoretically studied first, confirming that the simplest nonlinear theoretical model is capable of describing the evolution of propagating wave packets in deep water. Moreover, three groups of laboratory experiments run in the wave basin of CEHIPAR (Canal de Experiencias Hidrodinámicas de El Pardo, known also as El Pardo Model Basin) was founded in 1928 by the Spanish Navy. are systematically compared with the numerical simulations of the nonlinear Schrödinger equation. Although a little overestimation is detected, especially in the set of experiments characterized by higher initial wave steepness, the numerical simulation still displays a high degree of agreement with the laboratory experiments. Therefore, the nonlinear Schrödinger equation catches the essential characteristics of the extreme waves and provides an important physical insight into their generation. The modulation instability, resulting from the quasi-resonant four-wave interaction in a unidirectional sea state, can be indicated by the coefficient of kurtosis, which shows an appreciable correlation with the extreme wave height and hence is used in the modified Edgeworth-Rayleigh distribution. Finally, some statistical properties on the maximum wave heights in different sea states have been related with the initial Benjamin-Feir index.
The nonlinear Schrödinger equation singular solutions and optical collapse
Fibich, Gadi
2015-01-01
This book is an interdisciplinary introduction to optical collapse of laser beams, which is modelled by singular (blow-up) solutions of the nonlinear Schrödinger equation. With great care and detail, it develops the subject including the mathematical and physical background and the history of the subject. It combines rigorous analysis, asymptotic analysis, informal arguments, numerical simulations, physical modelling, and physical experiments. It repeatedly emphasizes the relations between these approaches, and the intuition behind the results. The Nonlinear Schrödinger Equation will be useful to graduate students and researchers in applied mathematics who are interested in singular solutions of partial differential equations, nonlinear optics and nonlinear waves, and to graduate students and researchers in physics and engineering who are interested in nonlinear optics and Bose-Einstein condensates. It can be used for courses on partial differential equations, nonlinear waves, and nonlinear optics. Gadi Fib...