Irreversible energy flow in forced Vlasov dynamics

KAUST Repository

Plunk, Gabriel G.; Parker, Joseph T.

2014-01-01

© EDP Sciences, Società Italiana di Fisica, Springer-Verlag. The recent paper of Plunk [G.G. Plunk, Phys. Plasmas 20, 032304 (2013)] considered the forced linear Vlasov equation as a model for the quasi-steady state of a single stable plasma wavenumber interacting with a bath of turbulent fluctuations. This approach gives some insight into possible energy flows without solving for nonlinear dynamics. The central result of the present work is that the forced linear Vlasov equation exhibits asymptotically zero (irreversible) dissipation to all orders under a detuning of the forcing frequency and the characteristic frequency associated with particle streaming. We first prove this by direct calculation, tracking energy flow in terms of certain exact conservation laws of the linear (collisionless) Vlasov equation. Then we analyze the steady-state solutions in detail using a weakly collisional Hermite-moment formulation, and compare with numerical solution. This leads to a detailed description of the Hermite energy spectrum, and a proof of no dissipation at all orders, complementing the collisionless Vlasov result.

Irreversible energy flow in forced Vlasov dynamics

KAUST Repository

Plunk, Gabriel G.

2014-10-01

© EDP Sciences, Società Italiana di Fisica, Springer-Verlag. The recent paper of Plunk [G.G. Plunk, Phys. Plasmas 20, 032304 (2013)] considered the forced linear Vlasov equation as a model for the quasi-steady state of a single stable plasma wavenumber interacting with a bath of turbulent fluctuations. This approach gives some insight into possible energy flows without solving for nonlinear dynamics. The central result of the present work is that the forced linear Vlasov equation exhibits asymptotically zero (irreversible) dissipation to all orders under a detuning of the forcing frequency and the characteristic frequency associated with particle streaming. We first prove this by direct calculation, tracking energy flow in terms of certain exact conservation laws of the linear (collisionless) Vlasov equation. Then we analyze the steady-state solutions in detail using a weakly collisional Hermite-moment formulation, and compare with numerical solution. This leads to a detailed description of the Hermite energy spectrum, and a proof of no dissipation at all orders, complementing the collisionless Vlasov result.

Nonadiabatic quantum Vlasov equation for Schwinger pair production

International Nuclear Information System (INIS)

Kim, Sang Pyo; Schubert, Christian

2011-01-01

Using Lewis-Riesenfeld theory, we derive an exact nonadiabatic master equation describing the time evolution of the QED Schwinger pair-production rate for a general time-varying electric field. This equation can be written equivalently as a first-order matrix equation, as a Vlasov-type integral equation, or as a third-order differential equation. In the last version it relates to the Korteweg-de Vries equation, which allows us to construct an exact solution using the well-known one-soliton solution to that equation. The case of timelike delta function pulse fields is also briefly considered.

Non-linear free streaming in Vlasov plasma

Czech Academy of Sciences Publication Activity Database

Sedláček, Zdeněk

2004-01-01

Roč. 54, suppl.C (2004), C82-C88 ISSN 0011-4626. [Symposium on Plasma Physics and Technology/21th./. Prague, 14.06.2004-17.06.2004] Institutional research plan: CEZ:AV0Z2043910 Keywords : plasma oscillations * Vlasov equation Subject RIV: BL - Plasma and Gas Discharge Physics Impact factor: 0.292, year: 2004

Mathematical and numerical methods for Vlasov-Maxwell equations: the contributions of data mining

International Nuclear Information System (INIS)

Assous, F.; Chaskalovic, J.

2014-01-01

There exist a lot of formulations that can model plasma physics or particle accelerators problems as the Vlasov- Maxwell equations. This paper deals with the applications of data mining techniques in the evaluation of numerical solutions of Vlasov-Maxwell models. This is part of the topic of characterizing the model and approximation errors via learning techniques. We give two examples of application. The first one aims at comparing two Vlasov-Maxwell approximate models. In the second one, a scheme based on data mining techniques is proposed to characterize the errors between a P1 and a P2 finite element Particle-In-Cell approach. Beyond these examples, this original approach should operate in all cases where intricate numerical simulations like for the Vlasov-Maxwell equations take a central part. (authors)

Stability of Nonlinear Wave Patterns to the Bipolar Vlasov-Poisson-Boltzmann System

Science.gov (United States)

Li, Hailiang; Wang, Yi; Yang, Tong; Zhong, Mingying

2018-04-01

The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction waves for the bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, motivated by the micro-macro decomposition to the Boltzmann equation in Liu and Yu (Commun Math Phys 246:133-179, 2004) and Liu et al. (Physica D 188:178-192, 2004), we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave patterns, viscous shock waves and rarefaction waves are proved for the 1D bipolar VPB system. More precisely, it is first proved that the linear superposition of two Boltzmann shock profiles in the first and third characteristic fields is nonlinearly stable to the 1D bipolar VPB system up to some suitable shifts without the zero macroscopic mass conditions on the initial perturbations. Then the time-asymptotic stability of the rarefaction wave fan to compressible Euler equations is proved for the 1D bipolar VPB system. These two results are concerned with the nonlinear stability of wave patterns for Boltzmann equation coupled with additional (electric) forces, which together with spectral analysis made in Li et al. (Indiana Univ Math J 65(2):665-725, 2016) sheds light on understanding the complicated dynamic behaviors around the wave patterns in the transportation of charged particles under the binary collisions, mutual interactions, and the effect of the electrostatic potential forces.

Relativistic simulation of the Vlasov equation for plasma expansion into vacuum

Directory of Open Access Journals (Sweden)

H Abbasi

2012-12-01

Full Text Available   In this study, relativistic Vlasov simulation of plasma for expansion of collisionless plasma for into vacuum is presented. The model is based on 1+1 dimensional phase space and electrostatic approximation. For this purpose, the electron dynamics is studied by the relativistic Vlasov equation. Regardless of the ions temperature, fluid equations are used for their dynamics. The initial electrons distribution function is the relativistic Maxwellian. The results show that due to the electrons relativistic temperature, the process of the plasma expansion takes place faster, the resulting electric field is stronger and the ions are accelerated to higher velocities, in comparison to the non-relativistic case.

Reduced Vlasov-Maxwell simulations

International Nuclear Information System (INIS)

Helluy, P.; Navoret, L.; Pham, N.; Crestetto, A.

2014-01-01

The Maxwell-Vlasov system is a fundamental model in physics. It can be applied to plasma simulations, charged particles beam, astrophysics, etc. The unknowns are the electromagnetic field, solution to the Maxwell equations and the distribution function, solution to the Vlasov equation. In this paper we review two different numerical methods for Vlasov-Maxwell simulations. The first method is based on a coupling between a Discontinuous Galerkin (DG) Maxwell solver and a Particle-In-Cell (PIC) Vlasov solver. The second method only uses a DG approach for the Vlasov and Maxwell equations. The Vlasov equation is first reduced to a space-only hyperbolic system thanks to the finite-element method. The two numerical methods are implemented using OpenCL in order to achieve high performance on recent Graphic Processing Units (GPU). We obtained interesting speedups, but we also observe that the PIC method is the most expensive part of the computation. Therefore we propose another fully Eulerian approach. Thanks to a decomposition of the distribution function on velocity basis functions, we obtain a reduced Vlasov model, which appears to be a hyperbolic system of conservation laws written only in the (x,t) space. We can thus adapt very easily our DG solver to the reduced model

Instability of the filtering method for Vlasov's equation

International Nuclear Information System (INIS)

Figua, H.; Bouchut, F.; Fijalkow, E.

1999-01-01

Klimas has introduced a smoothed Fourier-Fourier method. This method consists in convolving the original distribution function with a Gaussian distribution function, and, next, in solving the new system with a transformed splitting algorithm. Unfortunately, a second-order term appears in the new equation. In this work, it is studied how this term affects the numerical equation. In particular it is proven that instability occurs in the linear version of the Vlasov equation obtained by considering only free non-interacting particles. It is proved that the use of Fourier-Fourier transform is a fundamental requirement to solve this new equation. An important property is pointed out concerning the filtered distribution function in the transformed space. (K.A.)

Appearance of eigen modes for the linearized Vlasov-Poisson equation

International Nuclear Information System (INIS)

Degond, P.

1983-01-01

In order to determine the asymptotic behaviour, when the time goes to infinity, of the solution of the linearized Vlasov-Poisson equation, we use eigen modes, associated to continuous linear functionals on a Banach space of analytic functions [fr

On the relativistic Vlasov equation in guiding-center coordinates

International Nuclear Information System (INIS)

Salimullah, M.; Chaudhry, M.B.; Hassan, M.H.A.

1989-11-01

The relativistic Vlasov equation has been expressed in terms of the guiding-center coordinates in a hot magnetized plasma. It is noted that the relativistic effect reduces the cyclotron resonance frequency for electrostatic and electromagnetic waves propagating transverse to the direction of the static magnetic field in the plasma. (author). 4 refs

On invariant measures for the Vlasov equation with a regular potential

International Nuclear Information System (INIS)

Zhidkov, P.E.

2003-01-01

We consider a Vlasov equation with a smooth bounded potential of interaction between particles in a class of measure-valued solutions and construct a measure which is invariant for this problem in a sense

Nonlinear wave evolution in VLASOV plasma: a lie-transform analysis

International Nuclear Information System (INIS)

Cary, J.R.

1979-08-01

Nonlinear wave evolution in Vlasov plasma is analyzed using the Lie transform, a powerful mathematical tool which is applicable to Hamiltonian systems. The first part of this thesis is an exposition of the Lie transform. Dewar's general Lie transform theory is explained and is used to construct Deprit's Lie transform perturbation technique. The basic theory is illustrated by simple examples

Nonlinear behavior of a monochromatic wave in a one-dimensional Vlasov plasma

International Nuclear Information System (INIS)

Shoucri, M.M.; Gagne, R.R.J.

1978-01-01

The nonlinear evolution of a monochromatic wave in a one-dimensional Vlasov plasma is studied numerically. The numerical results are carried out far enough in time for phase mixing to dominate the asymptotic state of the system. A qualitative comparison with previously reported simulations is given

Fully Electromagnetic Nonlinear Gyrokinetic Equations for Tokamak Edge Turbulence

International Nuclear Information System (INIS)

Hahm, T.S.; Wang, Lu; Madsen, J.

2008-01-01

An energy conserving set of the fully electromagnetic nonlinear gyrokinetic Vlasov equation and Maxwell's equations, which is applicable to both L-mode turbulence with large amplitude and H-mode turbulence in the presence of high E x B shear has been derived. The phase-space action variational Lie perturbation method ensures the preservation of the conservation laws of the underlying Vlasov-Maxwell system. Our generalized ordering takes ρ i θi ∼ L E ∼ L p i is the thermal ion Larmor radius and ρ θi = B/B θ ρ i ), as typically observed in the tokamak H-mode edge, with L E and L p being the radial electric field and pressure gradient lengths. We take k # perpendicular# ρ i ∼ 1 for generality, and keep the relative fluctuation amplitudes e(delta)φ/T i ∼ (delta)B/B up to the second order. Extending the electrostatic theory in the presence of high E x B shear [Hahm, Phys. Plasmas 3, 4658 (1996)], contributions of electromagnetic fluctuations to the particle charge density and current are explicitly evaluated via pull-back transformation from the gyrocenter distribution function in the gyrokinetic Maxwell's equation

Local WKB dispersion relation for the Vlasov-Maxwell equations

International Nuclear Information System (INIS)

Berk, H.L.; Dominguez, R.R.

1982-10-01

A formalism for analyzing systems of integral equations, based on the Wentzel-Kramers-Brillouin (WKB) approximation, is applied to the Vlasov-Maxwell integral equations in an arbitrary-β, spatially inhomogenous plasma model. It is shown that when treating frequencies comparable with and larger than the cyclotron frequency, relevant new terms must be accounted for to treat waves that depend upon local spatial gradients. For a specific model, the response for very short wavelength and high frequency is shown to reduce to the straight-line orbit approximation when the WKB rules are correctly followed

Single particle dynamics of many-body systems described by Vlasov-Fokker-Planck equations

International Nuclear Information System (INIS)

Frank, T.D.

2003-01-01

Using Langevin equations we describe the random walk of single particles that belong to particle systems satisfying Vlasov-Fokker-Planck equations. In doing so, we show that Haissinski distributions of bunched particles in electron storage rings can be derived from a particle dynamics model

Fully nonlinear phenomenology of the Berk-Breizman augmentation of the Vlasov-Maxwell system

International Nuclear Information System (INIS)

Vann, R.G.L.; Dendy, R.O.; Rowlands, G.; Arber, T.D.; D'Ambrumenil, N.

2003-01-01

The Berk-Breizman augmentation of the Vlasov-Maxwell system is widely used to model self-consistent resonant excitation and damping of wave fields by evolving energetic particle populations in magnetic fusion plasmas. The key model parameters are the particle annihilation rate ν a , which drives bump-on-tail structure, and the linear wave damping rate γ d . A code, based on the piecewise parabolic method, is used to integrate the fully nonlinear Berk-Breizman system of equations across the whole (ν a ,γ d ) parameter space. The results of this code show that the system's behavior can be classified into one of four types, each of which occurs in a well-defined region of parameter space: chaotic, periodic, steady state, and damped. The corresponding evolution in (x,v) phase space is also examined

Numerical Integration of the Vlasov Equation of Two Colliding Beams

CERN Document Server

Zorzano-Mier, M P

2000-01-01

In a circular collider the motion of particles of one beam is strongly perturbed at the interaction points by the electro-magnetic field associated with the counter-rotating beam. For any two arbitrary initial particle distributions the time evolution of the two beams can be known by solving the coupled system of two Vlasov equations. This collective description is mandatory when the two beams have similar strengths, as in the case of LEP or LHC. The coherent modes excited by this beam-beam interaction can be a strong limitation for the operation of LHC. In this work, the coupled Vlasov equations of two colliding flat beams are solved numerically using a finite difference scheme. The results suggest that, for the collision of beams with equal tunes, the tune shift between the $\\sigma$- and $\\pi$- coherent dipole mode depends on the unperturbed tune $q$ because of the deformation that the so-called dynamic beta effect induces on the beam distribution. Only when the unperturbed tune $q\\rightarrow 0.25$ this tun...

Numerical study of a Vlasov equation for systems with interacting particles

Energy Technology Data Exchange (ETDEWEB)

Herrera, Dianela; Curilef, Sergio [Departamento de Física, Universidad Católica del Norte, Avenida Angamos 0610, Antofagasta (Chile)

2015-03-10

We solve numerically the Vlasov equation for the self-gravitating sheet model. We used the method introduced by Cheng and Knorr [Comput Phys 22, 330-351 (1976)]. We discuss the quasi-stationary state for some thermodynamical observables, specifically the kinetic energy, whose trend is depicted for early evolution.

Non-Linear Excitation of Ion Acoustic Waves

DEFF Research Database (Denmark)

Michelsen, Poul; Hirsfield, J. L.

1974-01-01

The excitation of ion acoustic waves by nonlinear coupling of two transverse magnetic waves generated in a microwave cavity was investigated. Measurements of the wave amplitude showed good agreement with calculations based on the Vlasov equation.......The excitation of ion acoustic waves by nonlinear coupling of two transverse magnetic waves generated in a microwave cavity was investigated. Measurements of the wave amplitude showed good agreement with calculations based on the Vlasov equation....

VLASOVIA 2013 - International workshop on the theory and applications of Vlasov equation - Slides of the presentations

International Nuclear Information System (INIS)

Aunai, N.; Belmont, G.; Smets, R.; Chandre, C.; Tassi, E.; Morrison, P.J.; Back, A.; Guillebon, L. de; Qin, H.; Squire, J.; Tang, W.M.; Garbet, X.; Esteve, D.; Sarazin, Y.; Abiteboul, J.; Bourdelle, C.; Dif-Pradalier, G.; Ghendrih, P.; Grandgirard, V.; Latu, G.; Smolyakov, A.; Hervieux, P.A.; Manfredi, G.; Jasiak, R.; Kraus, M.; Mora, P.; Morel, P.; Dreydemy Ghiro, F.; Berionni, V.; Gurcan, O.D.; Morrison, P.J.; Negulescu, C.; Pegoraro, F.; Bulanov, S.V.; Califano, F.; Fedeli, L.; Grassi, A.; Macchi, A.; Petri, J.; Pezzi, O.; Valentini, F.; Perrone, D.; Veltri, P.; Taccogna, F.; Minelli, P.; Thide, B.; Tamburini, F.; Throumoulopoulos, G.; Tasso, H.

2014-01-01

The Vlasov equation is used for the modelling of a wide range of phenomena occurring in natural and man-made plasmas, as well as in other many-particle systems displaying a collective behaviour. The purpose of this workshop is to bring together scientists to discuss the latest results on Vlasov theory and related applications. The topics discussed include: space plasmas, inertial confinement plasmas, magnetic confinement plasmas, quantum effects in collisionless plasmas, gravitational systems, Hamiltonian Vlasov dynamics, and computational and numerical approaches. This document gathers the slides of the presentations.

Action principles for the Vlasov equation: Four old, one new

International Nuclear Information System (INIS)

Ye, Huanchun; Morrison, P.J.

1991-01-01

Action principles for the Vlasov equation are presented. Four previously known action principles, which differ by the choice of dynamical variables, are described and the interrelationship between them discussed. A new action principle called the leaf action, which manifestly preserves the Casimir invariants and possess a single function as the dynamical variable, is presented. The relationship to the noncanonical Hamiltonian formalism is also explored. 21 refs

Quantum kinetic field theory in curved spacetime: Covariant Wigner function and Liouville-Vlasov equations

International Nuclear Information System (INIS)

Calzetta, E.; Habib, S.; Hu, B.L.

1988-01-01

We consider quantum fields in an external potential and show how, by using the Fourier transform on propagators, one can obtain the mass-shell constraint conditions and the Liouville-Vlasov equation for the Wigner distribution function. We then consider the Hadamard function G 1 (x 1 ,x 2 ) of a real, free, scalar field in curved space. We postulate a form for the Fourier transform F/sup (//sup Q//sup )/(X,k) of the propagator with respect to the difference variable x = x 1 -x 2 on a Riemann normal coordinate centered at Q. We show that F/sup (//sup Q//sup )/ is the result of applying a certain Q-dependent operator on a covariant Wigner function F. We derive from the wave equations for G 1 a covariant equation for the distribution function and show its consistency. We seek solutions to the set of Liouville-Vlasov equations for the vacuum and nonvacuum cases up to the third adiabatic order. Finally we apply this method to calculate the Hadamard function in the Einstein universe. We show that the covariant Wigner function can incorporate certain relevant global properties of the background spacetime. Covariant Wigner functions and Liouville-Vlasov equations are also derived for free fermions in curved spacetime. The method presented here can serve as a basis for constructing quantum kinetic theories in curved spacetime or for near-uniform systems under quasiequilibrium conditions. It can also be useful to the development of a transport theory of quantum fields for the investigation of grand unification and post-Planckian quantum processes in the early Universe

Theoretical and numerical study of the equations of Vlasov-Maxwell in the covariant formalism

International Nuclear Information System (INIS)

Back, A.

2011-11-01

A new point of view is proposed for the simulation of plasmas using the kinetic model which links the equations of Vlasov for the distribution of particles and the equations of Maxwell for the electromagnetic contribution of fields. We use the following principle: the equations of Physics are mathematical objects which put in relation geometrical objects. To preserve the geometrical properties of the various objects in an equation, we use, for the theoretical and numerical study, the differential geometry. All the equations of Physics can be written with differential forms and this point of view is not dependent on the choice of coordinates. We propose then a discretization of the differential forms by using B-Splines. To be coherent with the theory, we also propose a discretization of the various operations of the differential geometry. We test our scheme, first on the equations of Maxwell with several boundary conditions and since it does not depend on the system of coordinates, we also test it when we change coordinates. Finally, we apply the same method to the equations of Vlasov-Poisson in one-dimension and we propose several numerical schemes. (author)

Vlasov treatment of coherent synchrotron radiation from arbitrary planar orbits

International Nuclear Information System (INIS)

Warnock, R.; Bassi, G.; Ellison, J.A.

2006-01-01

We study the influence of coherent synchrotron radiation (CSR) on particle bunches traveling on arbitrary planar orbits between parallel conducting plates which represent the vacuum chamber. Our goal is to follow the time evolution of the phase space distribution by solving the Vlasov-Maxwell equations in the time domain. This should provide simulations with lower numerical noise than the macro-particle method, and allow one to study such issues as emittance degradation and microbunching due to CSR in bunch compressors. The fields excited by the bunch are computed in the laboratory frame from a new formula that leads to much simpler computations than usual methods. The nonlinear Vlasov equation, formulated in the interaction picture, is integrated in the beam frame by approximating the Perron-Frobenius operator. For application to a chicane bunch compressor we take steps to deal with energy chirp

Stability analysis of cylindrical Vlasov equilibria

International Nuclear Information System (INIS)

Short, R.W.

1979-01-01

A general method of stability analysis is described which may be applied to a large class of such problems, namely those which are described dynamically by the Vlasov equation, and geometrically by cylindrical symmetry. The method is presented for the simple case of the Vlasov-Poisson (electrostatic) equations, and the results are applied to a calculation of the lower-hybrid-drift instability in a plasma with a rigid rotor distribution function. The method is extended to the full Vlasov-Maxwell (electromagnetic) equations. These results are applied to a calculation of the instability of the extraordinary electromagnetic mode in a relativistic E-layer interacting with a background plasma

Stability analysis of cylindrical Vlasov equilibria

International Nuclear Information System (INIS)

Short, R.W.

1979-01-01

A general method of stability analysis is described which may be applied to a large class of such problems, namely those which are described dynamically by the Vlasov equation, and geometrically by clindrical symmetry. The method is presented for the simple case of the Vlasov-Poisson (electrostatic) equations, and the results are applied to a calculation of the lower-hybrid-drift instability in a plasma with a rigid rotor distribution function. The method is extended to the full Vlasov-Maxwell (electromagnetic) equations. These results are applied to a calculation of the instability of the extraordinary electromagnetic mode in a relativistic E-layer interacting with a background plasma

Integral propagator solvers for Vlasov-Fokker-Planck equations

International Nuclear Information System (INIS)

Donoso, J M; Rio, E del

2007-01-01

We briefly discuss the use of short-time integral propagators on solving the so-called Vlasov-Fokker-Planck equation for the dynamics of a distribution function. For this equation, the diffusion tensor is singular and the usual Gaussian representation of the short-time propagator is no longer valid. However, we prove that the path-integral approach on solving the equation is, in fact, reliable by means of our generalized propagator, which is obtained through the construction of an auxiliary solvable Fokker-Planck equation. The new representation of the grid-free advancing scheme describes the inherent cross- and self-diffusion processes, in both velocity and configuration spaces, in a natural manner, although these processes are not explicitly depicted in the differential equation. We also show that some splitting methods, as well as some finite-difference schemes, could fail in describing the aforementioned diffusion processes, governed in the whole phase space only by the velocity diffusion tensor. The short-time transition probability offers a stable and robust numerical algorithm that preserves the distribution positiveness and its norm, ensuring the smoothness of the evolving solution at any time step. (fast track communication)

On kinetic Boltzmann equations and related hydrodynamic flows with dry viscosity

Directory of Open Access Journals (Sweden)

Nikolai N. Bogoliubov (Jr.

2007-01-01

Full Text Available A two-component particle model of Boltzmann-Vlasov type kinetic equations in the form of special nonlinear integro-differential hydrodynamic systems on an infinite-dimensional functional manifold is discussed. We show that such systems are naturally connected with the nonlinear kinetic Boltzmann-Vlasov equations for some one-dimensional particle flows with pointwise interaction potential between particles. A new type of hydrodynamic two-component Benney equations is constructed and their Hamiltonian structure is analyzed.

Vlasov equation for photons and quasi-particles in a plasma

International Nuclear Information System (INIS)

Mendonca, J.T.

2014-01-01

We show that, in quite general conditions, a Vlasov equation can be derived for photons in a medium. The same is true for other quasi-particles, such as plasmons, phonons or driftons, associated with other wave modes in a plasma. The range of validity of this equation is discussed. We also discuss the Landau resonance, and its relation with photon acceleration. Exact and approximate expressions for photon and quasi-particle Landau damping are stated. Photon and quasi-particle acceleration and trapping is also discussed. Specific applications to laser-plasma interaction, and to magnetic fusion turbulence, are considered as illustrations of the general approach. (author)

Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent Collisions to Diffusive Limit

Science.gov (United States)

Herda, Maxime; Rodrigues, L. Miguel

2018-03-01

The present contribution investigates the dynamics generated by the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. We provide global in time estimates that are uniform with respect to initial data taken in a bounded set of a weighted L^2 space, and where dependencies on the mean-free path τ and the Debye length δ are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions τ → ∞ to the strongly collisional regime τ → 0. As a counterpart, the largeness of the Debye length, that enforces a weakly nonlinear regime, is used to close our nonlinear estimates. Accordingly we pay a special attention to relax as much as possible the τ -dependent constraint on δ ensuring exponential decay with explicit τ -dependent rates towards the stationary solution. In the strongly collisional limit τ → 0, we also examine all possible asymptotic regimes selected by a choice of observation time scale. Here also, our emphasis is on strong convergence, uniformity with respect to time and to initial data in bounded sets of a L^2 space. Our proofs rely on a detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive/hypoelliptic estimates.

Asymptotic solution of the Vlasov and Poisson equations for an inhomogeneous plasma

International Nuclear Information System (INIS)

Croci, R.

1991-01-01

The asymptotic solutions to a class of inhomogeneous integral equations that reduce to algebraic equations when a parameter η goes to zero (the kernel becoming proportional to a Dirac δ function) are derived. This class includes the integral equations obtained from the system of Vlasov and Poisson equations for the Fourier transform in space and the Laplace transform in time of the electrostatic potential, when the equilibrium magnetic field is uniform and the equilibrium plasma density depends on ηx, with the co-ordinate z being the direction of the magnetic field. In this case the inhomogeneous term is given by the initial conditions and possibly by sources, and the Laplace-transform variable ω is the eigenvalue parameter. (Author)

Hamiltonian field description of the one-dimensional Poisson-Vlasov equations

International Nuclear Information System (INIS)

Morrison, P.J.

1981-07-01

The one-dimensional Poisson-Vlasov equations are cast into Hamiltonian form. A Poisson Bracket in terms of the phase space density, as sole dynamical variable, is presented. This Poisson bracket is not of the usual form, but possesses the commutator properties of antisymmetry, bilinearity, and nonassociativity by virtue of the Jacobi requirement. Clebsch potentials are seen to yield a conventional (canonical) formulation. This formulation is discretized by expansion in terms of an arbitrary complete set of basis functions. In particular, a wave field representation is obtained

On classical solutions of the relativistic Vlasov-Klein-Gordon system

Directory of Open Access Journals (Sweden)

Michael Kunzinger

2005-01-01

Full Text Available We consider a collisionless ensemble of classical particles coupled with a Klein-Gordon field. For the resulting nonlinear system of partial differential equations, the relativistic Vlasov-Klein-Gordon system, we prove local-in-time existence of classical solutions and a continuation criterion which says that a solution can blow up only if the particle momenta become large. We also show that classical solutions are global in time in the one-dimensional case.

Continuum Vlasov Simulation in Four Phase-space Dimensions

Science.gov (United States)

Cohen, B. I.; Banks, J. W.; Berger, R. L.; Hittinger, J. A.; Brunner, S.

2010-11-01

In the VALHALLA project, we are developing scalable algorithms for the continuum solution of the Vlasov-Maxwell equations in two spatial and two velocity dimensions. We use fourth-order temporal and spatial discretizations of the conservative form of the equations and a finite-volume representation to enable adaptive mesh refinement and nonlinear oscillation control [1]. The code has been implemented with and without adaptive mesh refinement, and with electromagnetic and electrostatic field solvers. A goal is to study the efficacy of continuum Vlasov simulations in four phase-space dimensions for laser-plasma interactions. We have verified the code in examples such as the two-stream instability, the weak beam-plasma instability, Landau damping, electron plasma waves with electron trapping and nonlinear frequency shifts [2]^ extended from 1D to 2D propagation, and light wave propagation.^ We will report progress on code development, computational methods, and physics applications. This work was performed under the auspices of the U.S. DOE by LLNL under contract no. DE-AC52-07NA27344. This work was funded by the Lab. Dir. Res. and Dev. Prog. at LLNL under project tracking code 08-ERD-031. [1] J.W. Banks and J.A.F. Hittinger, to appear in IEEE Trans. Plas. Sci. (Sept., 2010). [2] G.J. Morales and T.M. O'Neil, Phys. Rev. Lett. 28,417 (1972); R. L. Dewar, Phys. Fluids 15,712 (1972).

Equations of motion of test particles for solving the spin-dependent Boltzmann–Vlasov equation

Energy Technology Data Exchange (ETDEWEB)

Xia, Yin [Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800 (China); University of Chinese Academy of Science, Beijing 100049 (China); Xu, Jun, E-mail: xujun@sinap.ac.cn [Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800 (China); Li, Bao-An [Department of Physics and Astronomy, Texas A& M University-Commerce, Commerce, TX 75429-3011 (United States); Department of Applied Physics, Xi' an Jiao Tong University, Xi' an 710049 (China); Shen, Wen-Qing [Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800 (China)

2016-08-10

A consistent derivation of the equations of motion (EOMs) of test particles for solving the spin-dependent Boltzmann–Vlasov equation is presented. The resulting EOMs in phase space are similar to the canonical equations in Hamiltonian dynamics, and the EOM of spin is the same as that in the Heisenburg picture of quantum mechanics. Considering further the quantum nature of spin and choosing the direction of total angular momentum in heavy-ion reactions as a reference of measuring nucleon spin, the EOMs of spin-up and spin-down nucleons are given separately. The key elements affecting the spin dynamics in heavy-ion collisions are identified. The resulting EOMs provide a solid foundation for using the test-particle approach in studying spin dynamics in heavy-ion collisions at intermediate energies. Future comparisons of model simulations with experimental data will help to constrain the poorly known in-medium nucleon spin–orbit coupling relevant for understanding properties of rare isotopes and their astrophysical impacts.

The Einstein-Vlasov System/Kinetic Theory

Directory of Open Access Journals (Sweden)

Håkan Andréasson

2002-12-01

Full Text Available The main purpose of this article is to provide a guide to theorems on global properties of solutions to the Einstein-Vlasov system. This system couples Einstein's equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades in which the main focus has been on nonrelativistic and special relativistic physics, i.e., to model the dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems. In 1990, Rendall and Rein initiated a mathematical study of the Einstein-Vlasov system. Since then many theorems on global properties of solutions to this system have been established. The Vlasov equation describes matter phenomenologically, and it should be stressed that most of the theorems presented in this article are not presently known for other such matter models (i.e., fluid models. This paper gives introductions to kinetic theory in non-curved spacetimes and then the Einstein-Vlasov system is introduced. We believe that a good understanding of kinetic theory in non-curved spacetimes is fundamental to good comprehension of kinetic theory in general relativity.

An adaptive, implicit, conservative, 1D-2V multi-species Vlasov-Fokker-Planck multi-scale solver in planar geometry

Science.gov (United States)

Taitano, W. T.; Chacón, L.; Simakov, A. N.

2018-07-01

We consider a 1D-2V Vlasov-Fokker-Planck multi-species ionic description coupled to fluid electrons. We address temporal stiffness with implicit time stepping, suitably preconditioned. To address temperature disparity in time and space, we extend the conservative adaptive velocity-space discretization scheme proposed in [Taitano et al., J. Comput. Phys., 318, 391-420, (2016)] to a spatially inhomogeneous system. In this approach, we normalize the velocity-space coordinate to a temporally and spatially varying local characteristic speed per species. We explicitly consider the resulting inertial terms in the Vlasov equation, and derive a discrete formulation that conserves mass, momentum, and energy up to a prescribed nonlinear tolerance upon convergence. Our conservation strategy employs nonlinear constraints to enforce these properties discretely for both the Vlasov operator and the Fokker-Planck collision operator. Numerical examples of varying degrees of complexity, including shock-wave propagation, demonstrate the favorable efficiency and accuracy properties of the scheme.

KEEN Wave Simulations: Comparing various PIC to various fixed grid Vlasov to Phase-Space Adaptive Sparse Tiling & Effective Lagrangian (PASTEL) Techniques

Science.gov (United States)

Afeyan, Bedros; Larson, David; Shadwick, Bradley; Sydora, Richard

2017-10-01

We compare various ways of solving the Vlasov-Poisson and Vlasov-Maxwell equations on rather demanding nonlinear kinetic phenomena associated with KEEN and KEEPN waves. KEEN stands for Kinetic, Electrostatic, Electron Nonlinear, and KEEPN, for electron-positron or pair plasmas analogs. Because these self-organized phase space structures are not steady-state, or single mode, or fluid or low order moment equation limited, typical techniques with low resolution or too much noise will distort the answer too much, too soon, and fail. This will be shown via Penrose criteria triggers for instability at the formation stage as well as particle orbit statistics in fully formed KEEN waves and KEEN-KEEN and KEEN-EPW interacting states. We will argue that PASTEL is a viable alternative to traditional methods with reasonable chances of success in higher dimensions. Work supported by a Grant from AFOSR PEEP.

Dielectric energy versus plasma energy, and Hamiltonian action-angle variables for the Vlasov equation

International Nuclear Information System (INIS)

Morrison, P.J.

1992-04-01

Expressions for the energy content of one-dimensional electrostatic perturbations about homogeneous equilibria are revisited. The well-known dielectric energy, var-epsilon D , is compared with the exact plasma free energy expression, δ 2 F, that is conserved by the Vlasov-Poisson system. The former is an expression in terms of the perturbed electric field amplitude, while the latter is determined by a generating function, which describes perturbations of the distribution function that respect the important constraint of dynamical accessibility of the system. Thus the comparison requires solving the Vlasov equation for such a perturbations of the distribution function in terms of the electric field. This is done for neutral modes of oscillation that occur for equilibria with stationary inflection points, and it is seen that for these special modes δ 2 F = var-epsilon D . In the case of unstable and corresponding damped modes it is seen that δ 2 F ≠ var-epsilon D ; in fact δ 2 F ≡ 0. This failure of the dielectric energy expression persists even for arbitrarily small growth and damping rates since var-epsilon D is nonzero in this limit, whereas δ 2 F remains zero. The connection between the new exact energy expression and the at-best approximate var-epsilon D is described. The new expression motivates natural definitions of Hamiltonian action variables and signature. A general linear integral transform is introduced that maps the linear version of the noncanonical Hamiltonian structure, which describes the Vlasov equation, to action-angle (diagonal) form

Numerical solution of the Maxwell-Vlasov equations in the periodic regime. Application to the study of isotope separation by ion cyclotron resonance; Resolution numerique des equations de Maxwell-Vlasov en regime periodique. Application a l'etude de la separation isotopique par resonance cyclotron ionique

Energy Technology Data Exchange (ETDEWEB)

Omnes, P

1999-01-25

This work is dedicated to the study of the behaviour of a magnetic confined plasma that is excited by a purely sinusoidal electric current delivered by an antenna. The response of the electrons to the electromagnetic field is considered as linear,whereas the ions of the plasma are represented by a non-relativistic Vlasov equation. In order to avoid transients, the coupled Maxwell-Vlasov equations are solved in a periodic mode and in a bounded domain. An equivalent electric conductivity tensor has been defined, this tensor is a linear operator that links the electric current generated by the movement of the particles to the electromagnetic field. Theoretical considerations can assure the existence and uniqueness of a periodical solution to Vlasov equations and of a solution to Maxwell equations in harmonic mode. The system of equations is periodical and has been solved by using an iterative method. The application of this method to the simulation of a isotopic separation device based on ionic cyclotron resonance has shown that the convergence is reached in a few iterations and that the solution is valid. Furthermore a method based on a finite-volume formulation of Maxwell equations in the time domain is presented. 2 new variables are defined in order to better take into account the Gauss' law and the conservation of the magnetic flux, the new system is still hyperbolic. The parallelization of the process has been successfully realized. (A.C.)

Multi-dimensional, fully-implicit, spectral method for the Vlasov-Maxwell equations with exact conservation laws in discrete form

Science.gov (United States)

Delzanno, G. L.

2015-11-01

A spectral method for the numerical solution of the multi-dimensional Vlasov-Maxwell equations is presented. The plasma distribution function is expanded in Fourier (for the spatial part) and Hermite (for the velocity part) basis functions, leading to a truncated system of ordinary differential equations for the expansion coefficients (moments) that is discretized with an implicit, second order accurate Crank-Nicolson time discretization. The discrete non-linear system is solved with a preconditioned Jacobian-Free Newton-Krylov method. It is shown analytically that the Fourier-Hermite method features exact conservation laws for total mass, momentum and energy in discrete form. Standard tests involving plasma waves and the whistler instability confirm the validity of the conservation laws numerically. The whistler instability test also shows that we can step over the fastest time scale in the system without incurring in numerical instabilities. Some preconditioning strategies are presented, showing that the number of linear iterations of the Krylov solver can be drastically reduced and a significant gain in performance can be obtained.

Comparison of two forms of Vlasov-type relativistic kinetic equations in hadrodynamics

International Nuclear Information System (INIS)

Mashnik, S.G.; Maino, G.

1996-01-01

A comparison of two methods in the relativistic kinetic theory of the Fermi systems is carried out assuming, as an example, the simplest σω-version of quantum hadrodynamics with allowance for strong mean meson fields. It is shown that the Vlasov-type relativistic kinetic equation (VRKE) obtained by means of the procedure of squaring at an intermediate step is responsible for unphysical features. A direct method of derivation of kinetic equations is proposed. This method does not contain such drawback and gives rise to VRKE in hydrodynamics of a non-contradictory form in which both spin degrees of freedom and states with positive and negative energies are taken into account. 17 refs

Nonlinear evolution equations

CERN Document Server

Uraltseva, N N

1995-01-01

This collection focuses on nonlinear problems in partial differential equations. Most of the papers are based on lectures presented at the seminar on partial differential equations and mathematical physics at St. Petersburg University. Among the topics explored are the existence and properties of solutions of various classes of nonlinear evolution equations, nonlinear imbedding theorems, bifurcations of solutions, and equations of mathematical physics (Navier-Stokes type equations and the nonlinear Schrödinger equation). The book will be useful to researchers and graduate students working in p

Nonlinear Dirac Equations

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.

Vlasov Treatment of Coherent Synchrotron Radiation from Arbitrary Planar Orbits

International Nuclear Information System (INIS)

Warnock, R

2004-01-01

We study the influence of coherent synchrotron radiation (CSR) on particle bunches traveling on arbitrary planar orbits between parallel conducting plates. The plates represent shielding due to the vacuum chamber. The vertical distribution of charge is an arbitrary fixed function. Our goal is to follow the time evolution of the phase space distribution by solving the Vlasov-Maxwell equations in the time domain. This provides simulations with lower numerical noise than the macroparticle method, and allows one to study such issues as emittance degradation and microbunching due to CSR in bunch compressors. The fields excited by the bunch are computed in the laboratory frame from a new formula that leads to much simpler computations than the usual retarded potentials or Lienard-Wiechert potentials. The nonlinear Vlasov equation, formulated in the interaction picture, is integrated in the beam frame by approximating the Perron-Frobenius operator. The distribution function is represented by B-splines, in a scheme preserving positivity and normalization of the distribution. For application to a chicane bunch compressor we take steps to deal with energy chirp, an initial near-perfect correlation of energy with position in the bunch

The Vlasov equation with strong magnetic field and oscillating electric field as a model for isotop resonant separation

Directory of Open Access Journals (Sweden)

Emmanuel Frenod

2002-01-01

Full Text Available We study the qualitative behavior of solutions to the Vlasov equation with strong external magnetic field and oscillating electric field. This model is relevant to the understanding of isotop resonant separation. We show that the effective equation is a kinetic equation with a memory term. This memory term involves a pseudo-differential operator whose kernel is characterized by an integral equation involving Bessel functions. The kernel is explicitly given in some particular cases.

Hamiltonian dynamics of spatially-homogeneous Vlasov-Einstein systems

International Nuclear Information System (INIS)

Okabe, Takahide; Morrison, P. J.; Friedrichsen, J. E. III; Shepley, L. C.

2011-01-01

We introduce a new matter action principle, with a wide range of applicability, for the Vlasov equation in terms of a conjugate pair of functions. Here we apply this action principle to the study of matter in Bianchi cosmological models in general relativity. The Bianchi models are spatially-homogeneous solutions to the Einstein field equations, classified by the three-dimensional Lie algebra that describes the symmetry group of the model. The Einstein equations for these models reduce to a set of coupled ordinary differential equations. The class A Bianchi models admit a Hamiltonian formulation in which the components of the metric tensor and their time derivatives yield the canonical coordinates. The evolution of anisotropy in the vacuum Bianchi models is determined by a potential due to the curvature of the model, according to its symmetry. For illustrative purposes, we examine the evolution of anisotropy in models with Vlasov matter. The Vlasov content is further simplified by the assumption of cold, counter-streaming matter, a kind of matter that is far from thermal equilibrium and is not describable by an ordinary fluid model nor other more simplistic matter models. Qualitative differences and similarities are found in the dynamics of certain vacuum class A Bianchi models and Bianchi type I models with cold, counter-streaming Vlasov-matter potentials analogous to the curvature potentials of corresponding vacuum models.

Nonlinear Delta-f Particle Simulations of Collective Effects in High-Intensity Bunched Beams

CERN Document Server

Qin, Hong; Hudson, Stuart R; Startsev, Edward

2005-01-01

The collective effects in high-intensity 3D bunched beams are described self-consistently by the nonlinear Vlasov-Maxwell equations.* The nonlinear delta-f method,** a particle simulation method for solving the nonlinear Vlasov-Maxwell equations, is being used to study the collective effects in high-intensity 3D bunched beams. The delta-f method, as a nonlinear perturbative scheme, splits the distribution function into equilibrium and perturbed parts. The perturbed distribution function is represented as a weighted summation over discrete particles, where the particle orbits are advanced by equations of motion in the focusing field and self-consistent fields, and the particle weights are advanced by the coupling between the perturbed fields and the zero-order distribution function. The nonlinear delta-f method exhibits minimal noise and accuracy problems in comparison with standard particle-in-cell simulations. A self-consistent 3D kinetic equilibrium is first established for high intensity bunched beams. The...

Deterministic methods for the relativistic Vlasov-Maxwell equations and the Van Allen belts dynamics

International Nuclear Information System (INIS)

Le Bourdiec, S.

2007-03-01

Artificial satellites operate in an hostile radiation environment, the Van Allen radiation belts, which partly condition their reliability and their lifespan. In order to protect them, it is necessary to characterize the dynamics of the energetic electrons trapped in these radiation belts. This dynamics is essentially determined by the interactions between the energetic electrons and the existing electromagnetic waves. This work consisted in designing a numerical scheme to solve the equations modelling these interactions: the relativistic Vlasov-Maxwell system of equations. Our choice was directed towards methods of direct integration. We propose three new spectral methods for the momentum discretization: a Galerkin method and two collocation methods. All of them are based on scaled Hermite functions. The scaling factor is chosen in order to obtain the proper velocity resolution. We present in this thesis the discretization of the one-dimensional Vlasov-Poisson system and the numerical results obtained. Then we study the possible extensions of the methods to the complete relativistic problem. In order to reduce the computing time, parallelization and optimization of the algorithms were carried out. Finally, we present 1Dx-3Dv (mono-dimensional for x and three-dimensional for velocity) computations of Weibel and whistler instabilities with one or two electrons species. (author)

Wigner transformation in curved space-time and the curvature correction of the Vlasov equation for semiclassical gravitating systems

International Nuclear Information System (INIS)

Winter, J.

1985-01-01

A covariant generalization of the Wigner transformation of quantum equations is proposed for gravitating many-particle systems, which modifies the Einstein-Liouville equations for the coupled gravity-matter problem by inclusion of quantum effects of the matter moving in its self-consistent classical gravitational field, in order to extend their realm of validity to higher particle densities. The corrections of the Vlasov equation (Liouville equation in one-particle phase space) are exhibited as combined effects of quantum mechanics and the curvature of space-time arranged in a semiclassical expansion in powers of h 2 , the first-order term of which is explicitly calculated. It is linear in the Riemann tensor and in its gradient; the Riemann tensor occurs in a similar position as the tensor of the Yang-Mills field strength in a corresponding Vlasov equation for systems with local gauge invariance in the purely classical limit. The performance of the Wigner transformation is based on expressing the equation of motion for the two-point function of the Klein-Gordon field, in particular the Beltrami operator, in terms of a midpoint and a distance vector covariantly defined for the two points. This implies the calculation of deviations of the geodesic between these points, the standard concept of which has to be refined to include infinitesimal variations of the second order. A differential equation for the second-order deviation is established

Numerical solution of the Maxwell-Vlasov equations in the periodic regime. Application to the study of isotope separation by ion cyclotron resonance

International Nuclear Information System (INIS)

Omnes, P.

1999-01-01

This work is dedicated to the study of the behaviour of a magnetic confined plasma that is excited by a purely sinusoidal electric current delivered by an antenna. The response of the electrons to the electromagnetic field is considered as linear, whereas the ions of the plasma are represented by a non-relativistic Vlasov equation. In order to avoid transients, the coupled Maxwell-Vlasov equations are solved in a periodic mode and in a bounded domain. An equivalent electric conductivity tensor has been defined, this tensor is a linear operator that links the electric current generated by the movement of the particles to the electromagnetic field. Theoretical considerations can assure the existence and uniqueness of a periodical solution to Vlasov equations and of a solution to Maxwell equations in harmonic mode. The system of equations is periodical and has been solved by using an iterative method. The application of this method to the simulation of a isotopic separation device based on ionic cyclotron resonance has shown that the convergence is reached in a few iterations and that the solution is valid. Furthermore a method based on a finite-volume formulation of Maxwell equations in the time domain is presented. 2 new variables are defined in order to better take into account the Gauss' law and the conservation of the magnetic flux, the new system is still hyperbolic. The parallelization of the process has been successfully realized. (A.C.)

Nonlinear turbulence theory and simulation of Buneman instability

International Nuclear Information System (INIS)

Yoon, P. H.; Umeda, T.

2010-01-01

In the present paper, the weak turbulence theory for reactive instabilities, formulated in a companion paper [P. H. Yoon, Phys. Plasmas 17, 112316 (2010)], is applied to the strong electron-ion two-stream (or Buneman) instability. The self-consistent theory involves quasilinear velocity space diffusion equation for the particles and nonlinear wave kinetic equation that includes quasilinear (or induced emission) term as well as nonlinear wave-particle interaction term (or a term that represents an induced scattering off ions). We have also performed one-dimensional electrostatic Vlasov simulation in order to benchmark the theoretical analysis. Under the assumption of self-similar drifting Gaussian distribution function for the electrons it is shown that the current reduction and the accompanying electron heating as well as electric field turbulence generation can be discussed in a self-consistent manner. Upon comparison with the Vlasov simulation result it is found that quasilinear wave kinetic equation alone is insufficient to account for the final saturation amplitude. Upon including the nonlinear scattering term in the wave kinetic equation, however, we find that a qualitative agreement with the simulation is recovered. From this, we conclude that the combined quasilinear particle diffusion plus induced emission and scattering (off ions) processes adequately account for the nonlinear development of the Buneman instability.

Antisymmetrized molecular dynamics of wave packets with stochastic incorporation of Vlasov equation

International Nuclear Information System (INIS)

Ono, Akira; Horiuchi, Hisashi.

1996-01-01

The first purpose of this report is to present an extended AMD model which can generally describe such minor branching processes by removing the restriction on the one-body distribution function. This is done not by generalizing the wave packets to arbitrary single-particle wave functions but by representing the diffused and/or deformed wave packet as an ensemble of Gaussian wave packets. In other words, stochastic displacements are given to the wave packets in phase space so that the ensemble-average of the time evolution of the one-body distribution function is essentially equivalent to the solution of Vlasov equation which does not have any restriction on the shape of wave packets. This new model is called AMD-V. Although AMD-V is equivalent to Vlasov equation in the instantaneous time evolution of the one-body distribution function for an AMD wave function, AMD-V describes the branching into channels and the fluctuation of the mean field which are caused by the spreading or the splitting of the single-particle wave function. The second purpose of this report is to show the drastic effect of this new stochastic process of wave packet splitting on the dynamics of heavy ion collisions, especially in the fragmentation mechanism. We take the 40 Ca + 40 Ca system at the incident energy 35 MeV/nucleon. It will be shown that the reproduction of data by the AMD-V calculation is surprisingly good. We will see that the effect of the wave packet diffusion is crucially important to remove the spurious binary feature of the AMD calculation and to enable the multi-fragment final state. (J.P.N.)

Electromagnetic nonlinear gyrokinetics with polarization drift

International Nuclear Information System (INIS)

Duthoit, F.-X.; Hahm, T. S.; Wang, Lu

2014-01-01

A set of new nonlinear electromagnetic gyrokinetic Vlasov equation with polarization drift and gyrokinetic Maxwell equations is systematically derived by using the Lie-transform perturbation method in toroidal geometry. For the first time, we recover the drift-kinetic expression for parallel acceleration [R. M. Kulsrud, in Basic Plasma Physics, edited by A. A. Galeev and R. N. Sudan (North-Holland, Amsterdam, 1983)] from the nonlinear gyrokinetic equations, thereby bridging a gap between the two formulations. This formalism should be useful in addressing nonlinear ion Compton scattering of intermediate-mode-number toroidal Alfvén eigenmodes for which the polarization current nonlinearity [T. S. Hahm and L. Chen, Phys. Rev. Lett. 74, 266 (1995)] and the usual finite Larmor radius effects should compete

Deterministic methods for the relativistic Vlasov-Maxwell equations and the Van Allen belts dynamics; Methodes deterministes de resolution des equations de Vlasov-Maxwell relativistes en vue du calcul de la dynamique des ceintures de Van Allen

Energy Technology Data Exchange (ETDEWEB)

Le Bourdiec, S

2007-03-15

Artificial satellites operate in an hostile radiation environment, the Van Allen radiation belts, which partly condition their reliability and their lifespan. In order to protect them, it is necessary to characterize the dynamics of the energetic electrons trapped in these radiation belts. This dynamics is essentially determined by the interactions between the energetic electrons and the existing electromagnetic waves. This work consisted in designing a numerical scheme to solve the equations modelling these interactions: the relativistic Vlasov-Maxwell system of equations. Our choice was directed towards methods of direct integration. We propose three new spectral methods for the momentum discretization: a Galerkin method and two collocation methods. All of them are based on scaled Hermite functions. The scaling factor is chosen in order to obtain the proper velocity resolution. We present in this thesis the discretization of the one-dimensional Vlasov-Poisson system and the numerical results obtained. Then we study the possible extensions of the methods to the complete relativistic problem. In order to reduce the computing time, parallelization and optimization of the algorithms were carried out. Finally, we present 1Dx-3Dv (mono-dimensional for x and three-dimensional for velocity) computations of Weibel and whistler instabilities with one or two electrons species. (author)

Resolution of the Vlasov-Maxwell system by PIC discontinuous Galerkin method on GPU with OpenCL

Directory of Open Access Journals (Sweden)

Crestetto Anaïs

2013-01-01

Full Text Available We present an implementation of a Vlasov-Maxwell solver for multicore processors. The Vlasov equation describes the evolution of charged particles in an electromagnetic field, solution of the Maxwell equations. The Vlasov equation is solved by a Particle-In-Cell method (PIC, while the Maxwell system is computed by a Discontinuous Galerkin method. We use the OpenCL framework, which allows our code to run on multicore processors or recent Graphic Processing Units (GPU. We present several numerical applications to two-dimensional test cases.

The Einstein-Vlasov System/Kinetic Theory.

Science.gov (United States)

Andréasson, Håkan

2011-01-01

The main purpose of this article is to provide a guide to theorems on global properties of solutions to the Einstein-Vlasov system. This system couples Einstein's equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades in which the main focus has been on non-relativistic and special relativistic physics, i.e., to model the dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems. In 1990, Rendall and Rein initiated a mathematical study of the Einstein-Vlasov system. Since then many theorems on global properties of solutions to this system have been established. This paper gives introductions to kinetic theory in non-curved spacetimes and then the Einstein-Vlasov system is introduced. We believe that a good understanding of kinetic theory in non-curved spacetimes is fundamental to a good comprehension of kinetic theory in general relativity.

Degenerate nonlinear diffusion equations

CERN Document Server

Favini, Angelo

2012-01-01

The aim of these notes is to include in a uniform presentation style several topics related to the theory of degenerate nonlinear diffusion equations, treated in the mathematical framework of evolution equations with multivalued m-accretive operators in Hilbert spaces. The problems concern nonlinear parabolic equations involving two cases of degeneracy. More precisely, one case is due to the vanishing of the time derivative coefficient and the other is provided by the vanishing of the diffusion coefficient on subsets of positive measure of the domain. From the mathematical point of view the results presented in these notes can be considered as general results in the theory of degenerate nonlinear diffusion equations. However, this work does not seek to present an exhaustive study of degenerate diffusion equations, but rather to emphasize some rigorous and efficient techniques for approaching various problems involving degenerate nonlinear diffusion equations, such as well-posedness, periodic solutions, asympt...

The forced nonlinear Schroedinger equation

International Nuclear Information System (INIS)

Kaup, D.J.; Hansen, P.J.

1985-01-01

The nonlinear Schroedinger equation describes the behaviour of a radio frequency wave in the ionosphere near the reflexion point where nonlinear processes are important. A simple model of this phenomenon leads to the forced nonlinear Schroedinger equation in terms of a nonlinear boundary value problem. A WKB analysis of the time evolution equations for the nonlinear Schroedinger equation in the inverse scattering transform formalism gives a crude order of magnitude estimation of the qualitative behaviour of the solutions. This estimation is compared with the numerical solutions. (D.Gy.)

The free energy of Maxwell-Vlasov equilibria

International Nuclear Information System (INIS)

Morrison, P.J.; Pfirsch, D.

1989-10-01

A previously derived expression for the energy of arbitrary perturbations about arbitrary Vlasov-Maxwell equilibria is transformed into a very compact form. The new form is also obtained by a canonical transformation method for solving Vlasov's equation, which is based on Lie group theory. This method is simpler than the one used before and provides better physical insight. Finally a procedure is presented for determining the existence of negative-energy modes. In this context the question of why there is an accessibility constraint for the particles, but not for the fields, is discussed. 16 refs

Vlasov dynamics of periodically driven systems

Science.gov (United States)

Banerjee, Soumyadip; Shah, Kushal

2018-04-01

Analytical solutions of the Vlasov equation for periodically driven systems are of importance in several areas of plasma physics and dynamical systems and are usually approximated using ponderomotive theory. In this paper, we derive the plasma distribution function predicted by ponderomotive theory using Hamiltonian averaging theory and compare it with solutions obtained by the method of characteristics. Our results show that though ponderomotive theory is relatively much easier to use, its predictions are very restrictive and are likely to be very different from the actual distribution function of the system. We also analyse all possible initial conditions which lead to periodic solutions of the Vlasov equation for periodically driven systems and conjecture that the irreducible polynomial corresponding to the initial condition must only have squares of the spatial and momentum coordinate. The resulting distribution function for other initial conditions is aperiodic and can lead to complex relaxation processes within the plasma.

Introduction to nonlinear dispersive equations

CERN Document Server

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...

Cosmology in one dimension: Vlasov dynamics.

Science.gov (United States)

Manfredi, Giovanni; Rouet, Jean-Louis; Miller, Bruce; Shiozawa, Yui

2016-04-01

Numerical simulations of self-gravitating systems are generally based on N-body codes, which solve the equations of motion of a large number of interacting particles. This approach suffers from poor statistical sampling in regions of low density. In contrast, Vlasov codes, by meshing the entire phase space, can reach higher accuracy irrespective of the density. Here, we perform one-dimensional Vlasov simulations of a long-standing cosmological problem, namely, the fractal properties of an expanding Einstein-de Sitter universe in Newtonian gravity. The N-body results are confirmed for high-density regions and extended to regions of low matter density, where the N-body approach usually fails.

Turbulence in unmagnetized Vlasov plasmas

International Nuclear Information System (INIS)

Kuo, S.P.

1985-01-01

The classical technique of transformation and characteristics is employed to analyze the problem of strong turbulence in unmagnetized plasmas. The effect of resonance broadening and perturbation expansion are treated simultaneously, without time secularities. The renormalization procedure of Dupree and Tetreault is used in the transformed Vlasov equation to analyze the turbulence and to derive explicitly a diffusion equation. Analyses are extended to inhomogeneous plasmas and the relationship between the transformation and ponderomotive force is obtained. (author)

Exact Solutions to Nonlinear Schroedinger Equation and Higher-Order Nonlinear Schroedinger Equation

International Nuclear Information System (INIS)

Ren Ji; Ruan Hangyu

2008-01-01

We study solutions of the nonlinear Schroedinger equation (NLSE) and higher-order nonlinear Schroedinger equation (HONLSE) with variable coefficients. By considering all the higher-order effect of HONLSE as a new dependent variable, the NLSE and HONLSE can be changed into one equation. Using the generalized Lie group reduction method (GLGRM), the abundant solutions of NLSE and HONLSE are obtained

Connection between hydrodynamic, water bag and Vlasov models

International Nuclear Information System (INIS)

Gros, M.; Bertrand, P.; Feix, M.R.

1978-01-01

The connection between hydrodynamic, water bag and Vlasov models is still under consideration with numerical experiments. For long wavelength, slightly non linear excitations and initial preparations such as the usual adiabatic invariant Pn -3 is space independent, the hydrodynamic model is equivalent to the water bag, and for long wavelengths a nice agreement is found with the full numerical solution of the Vlasov equation. For other initial conditions when the water bag cannot be defined, the hydrodynamic approach does not represent the correct behaviour. (author)

Discrete nonlinear Schrodinger equations with arbitrarily high-order nonlinearities

DEFF Research Database (Denmark)

Khare, A.; Rasmussen, Kim Ø; Salerno, M.

2006-01-01

-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.......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 Ablowitz...

Comparison of free-streaming ELM formulae to a Vlasov simulation

Energy Technology Data Exchange (ETDEWEB)

Moulton, D., E-mail: david.moulton@cea.fr [CEA, IRFM, F-13108 Saint-Paul Lez Durance (France); Fundamenski, W. [Imperial College of Science, Technology and Medicine, London (United Kingdom); Manfredi, G. [Institut de Physique et Chimie des Matériaux, CNRS and Université de Strasbourg, BP 43, F-67034 Strasbourg (France); Hirstoaga, S. [INRIA Nancy Grand-Est and Institut de Recherche en Mathématiques Avancées, 7 rue René Descartes, F-67084 Strasbourg (France); Tskhakaya, D. [Association EURATOM-ÖAW, University of Innsbruck, A-6020 Innsbruck (Austria)

2013-07-15

The main drawbacks of the original free-streaming equations for edge localised mode transport in the scrape-off layer [W. Fundamenski, R.A. Pitts, Plasma Phys. Control Fusion 48 (2006) 109] are that the plasma potential is not accounted for and that only solutions for ion quantities are considered. In this work, the equations are modified and augmented in order to address these two issues. The new equations are benchmarked against (and justified by) a numerical simulation which solves the Vlasov equation in 1d1v. When the source function due to an edge localised mode is instantaneous, the modified free-streaming ‘impulse response’ equations agree closely with the Vlasov simulation results. When the source has a finite duration in time, the agreement worsens. However, in all cases the match is encouragingly good, thus justifying the applicability of the free-streaming approach.

Modeling Microbunching from Shot Noise Using Vlasov Solvers

International Nuclear Information System (INIS)

Venturini, Marco; Venturini, Marco; Zholents, Alexander

2008-01-01

Unlike macroparticle simulations, which are sensitive to unphysical statistical fluctuations when the number of macroparticles is smaller than the bunch population, direct methods for solving the Vlasov equation are free from sampling noise and are ideally suited for studying microbunching instabilities evolving from shot noise. We review a 2D (longitudinal dynamics) Vlasov solver we have recently developed to study the microbunching instability in the beam delivery systems for x-ray FELs and present an application to FERMI(at)Elettra. We discuss, in particular, the impact of the spreader design on microbunching

Electron-acoustic Instability Simulated By Modified Zakharov Equations

Science.gov (United States)

Jásenský, V.; Fiala, V.; Vána, O.; Trávnícek, P.; Hellinger, P.

We present non-linear equations describing processes in plasma when electron - acoustic waves are excited. These waves are present for instance in the vicinity of Earth's bow shock and in the polar ionosphere. Frequently they are excited by an elec- tron beam in a plasma with two electron populations, a cold and hot one. We derive modified Zakharov equations from kinetic theory for such a case together with numer- ical method for solving of this type of equations. Bispectral analysis is used to show which non-linear wave processes are of importance in course of the instability. Finally, we compare these results with similar simulations using Vlasov approach.

Nonlinear theory of surface-wave--particle interactions in a cylindrical plasma

International Nuclear Information System (INIS)

Dengra, A.; Palop, J.I.F.

1994-01-01

This work is an application of the specular reflection hypothesis to the study of the nonlinear surface-wave--particle interactions in a cylindrical plasma. The model is based on nonlinear resolution of the Vlasov equation by the method of characteristics. The expression obtained for the rate of increase of kinetic energy per electron has permitted us to investigate the temporal behavior of nonlinear collisionless damping for different situations as a function of the critical parameters

Nonlinear wave equations

CERN Document Server

Li, Tatsien

2017-01-01

This book focuses on nonlinear wave equations, which are of considerable significance from both physical and theoretical perspectives. It also presents complete results on the lower bound estimates of lifespan (including the global existence), which are established for classical solutions to the Cauchy problem of nonlinear wave equations with small initial data in all possible space dimensions and with all possible integer powers of nonlinear terms. Further, the book proposes the global iteration method, which offers a unified and straightforward approach for treating these kinds of problems. Purely based on the properties of solut ions to the corresponding linear problems, the method simply applies the contraction mapping principle.

Nonlinear Alfvén Waves in a Vlasov Plasma

DEFF Research Database (Denmark)

Bell, T.F.

1965-01-01

Stationary solutions to the nonlinear Vlasov—Boltzmann equations are considered which represent one-dimensional electromagnetic waves in a hot magnetoplasma. These solutions appear in arbitrary reference frames as circularly polarized, sinusoidal waves of unlimited amplitude, i.e., as nonlinear...... Alfvén waves. Solutions are found implicitly by deriving a set of integral dispersion relations which link the wave characteristics with the particle distribution functions. A physical discussion is given of the way in which the Alfvén waves can trap particles, and it is shown that the presence...

Instability of the time splitting scheme for the one-dimensional and relativistic Vlasov-Maxwell system

CERN Document Server

Huot, F; Bertrand, P; Sonnendrücker, E; Coulaud, O

2003-01-01

The Time Splitting Scheme (TSS) has been examined within the context of the one-dimensional (1D) relativistic Vlasov-Maxwell model. In the strongly relativistic regime of the laser-plasma interaction, the TSS cannot be applied to solve the Vlasov equation. We propose a new semi-Lagrangian scheme based on a full 2D advection and study its advantages over the classical Splitting procedure. Details of the underlying integration of the Vlasov equation appear to be important in achieving accurate plasma simulations. Examples are given which are related to the relativistic modulational instability and the self-induced transparency of an ultra-intense electromagnetic pulse in the relativistic regime.

Flows of non-smooth vector fields and degenerate elliptic equations with applications to the Vlasov-Poisson and semigeostrophic systems

CERN Document Server

Colombo, Maria

2017-01-01

The first part of the book is devoted to the transport equation for a given vector field, exploiting the lagrangian structure of solutions. It also treats the regularity of solutions of some degenerate elliptic equations, which appear in the eulerian counterpart of some transport models with congestion. The second part of the book deals with the lagrangian structure of solutions of the Vlasov-Poisson system, which describes the evolution of a system of particles under the self-induced gravitational/electrostatic field, and the existence of solutions of the semigeostrophic system, used in meteorology to describe the motion of large-scale oceanic/atmospheric flows.

Nonlinear differential equations

Energy Technology Data Exchange (ETDEWEB)

Dresner, L.

1988-01-01

This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.

Nonlinear differential equations

International Nuclear Information System (INIS)

Dresner, L.

1988-01-01

This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics

A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations

International Nuclear Information System (INIS)

Yomba, Emmanuel

2008-01-01

With the aid of symbolic computation, a generalized auxiliary equation method is proposed to construct more general exact solutions to two types of NLPDEs. First, we present new family of solutions to a nonlinear Klein-Gordon equation, by using this auxiliary equation method including a new first-order nonlinear ODE with six-degree nonlinear term proposed by Sirendaoreji. Then, we apply an indirect F-function method very close to the F-expansion method to solve the generalized Camassa-Holm equation with fully nonlinear dispersion and fully nonlinear convection C(l,n,p). Taking advantage of the new first-order nonlinear ODE with six degree nonlinear term, this indirect F-function method is used to map the solutions of C(l,n,p) equations to those of that nonlinear ODE. As a result, we can successfully obtain in a unified way, many exact solutions

L2-stability of the Vlasov-Maxwell-Boltzmann system near global Maxwellians

International Nuclear Information System (INIS)

Ha, Seung-Yeal; Xiao, Qinghua; Xiong, Linjie; Zhao, Huijiang

2013-01-01

We present a L 2 -stability theory of the Vlasov-Maxwell-Boltzmann system for the two-species collisional plasma. We show that in a perturbative regime of a global Maxwellian, the L 2 -distance between two strong solutions can be controlled by that between initial data in a Lipschitz manner. Our stability result extends earlier results [Ha, S.-Y. and Xiao, Q.-H., “A revisiting to the L 2 -stability theory of the Boltzmann equation near global Maxwellians,” (submitted) and Ha, S.-Y., Yang, X.-F., and Yun, S.-B., “L 2 stability theory of the Boltzmann equation near a global Maxwellian,” Arch. Ration. Mech. Anal. 197, 657–688 (2010)] on the L 2 -stability of the Boltzmann equation to the Boltzmann equation coupled with self-consistent external forces. As a direct application of our stability result, we show that classical solutions in Duan et al. [“Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,” Commun. Pure Appl. Math. 24, 1497–1546 (2011)] and Guo [“The Vlasov-Maxwell-Boltzmann system near Maxwellians,” Invent. Math. 153(3), 593–630 (2003)] satisfy a uniform L 2 -stability estimate. This is the first result on the L 2 -stability of the Boltzmann equation coupled with self-consistent field equations in three dimensions

Solving Nonlinear Coupled Differential Equations

Science.gov (United States)

Mitchell, L.; David, J.

1986-01-01

Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.

The energy of perturbations for Vlasov plasmas

International Nuclear Information System (INIS)

Morrison, P.J.

1994-02-01

The energy content of electrostatic perturbations about homogeneous equilibria is discussed. The calculation leading to the well-known dielectric (or as it is sometimes called the wave) energy is revisited and interpreted in light of Vlasov theory. It is argued that this quantity is deficient because resonant particles are not correctly handled. A linear integral transform is presented that solves the linear Vlasov-Poisson equation. This solution together with the Kruskal-Oberman energy [Phys. Fluids 1, 275 (1958)] is used to obtain an energy expression in terms of the electric field [Phys. Fluids B 4, 3038 (1992)]. It is described how the integral transform amounts to a change to normal coordinates in an infinite dimensional Hamiltonian system

Nonlinear Poisson equation for heterogeneous media.

Science.gov (United States)

Hu, Langhua; Wei, Guo-Wei

2012-08-22

The Poisson equation is a widely accepted model for electrostatic analysis. However, the Poisson equation is derived based on electric polarizations in a linear, isotropic, and homogeneous dielectric medium. This article introduces a nonlinear Poisson equation to take into consideration of hyperpolarization effects due to intensive charges and possible nonlinear, anisotropic, and heterogeneous media. Variational principle is utilized to derive the nonlinear Poisson model from an electrostatic energy functional. To apply the proposed nonlinear Poisson equation for the solvation analysis, we also construct a nonpolar solvation energy functional based on the nonlinear Poisson equation by using the geometric measure theory. At a fixed temperature, the proposed nonlinear Poisson theory is extensively validated by the electrostatic analysis of the Kirkwood model and a set of 20 proteins, and the solvation analysis of a set of 17 small molecules whose experimental measurements are also available for a comparison. Moreover, the nonlinear Poisson equation is further applied to the solvation analysis of 21 compounds at different temperatures. Numerical results are compared to theoretical prediction, experimental measurements, and those obtained from other theoretical methods in the literature. A good agreement between our results and experimental data as well as theoretical results suggests that the proposed nonlinear Poisson model is a potentially useful model for electrostatic analysis involving hyperpolarization effects. Copyright © 2012 Biophysical Society. Published by Elsevier Inc. All rights reserved.

Auxiliary equation method for solving nonlinear partial differential equations

International Nuclear Information System (INIS)

Sirendaoreji,; Jiong, Sun

2003-01-01

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

The multi-order envelope periodic solutions to the nonlinear Schrodinger equation and cubic nonlinear Schrodinger equation

International Nuclear Information System (INIS)

Xiao Yafeng; Xue Haili; Zhang Hongqing

2011-01-01

Based on Jacobi elliptic function and the Lame equation, the perturbation method is applied to get the multi-order envelope periodic solutions of the nonlinear Schrodinger equation and cubic nonlinear Schrodinger equation. These multi-order envelope periodic solutions can degenerate into the different envelope solitary solutions. (authors)

A reliable treatment for nonlinear Schroedinger equations

International Nuclear Information System (INIS)

Khani, F.; Hamedi-Nezhad, S.; Molabahrami, A.

2007-01-01

Exp-function method is used to find a unified solution of nonlinear wave equation. Nonlinear Schroedinger equations with cubic and power law nonlinearity are selected to illustrate the effectiveness and simplicity of the method. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving nonlinear equation

A study of the disintegration of highly excited nuclei with the Vlasov-Uehling-Uhlenbeck equation

International Nuclear Information System (INIS)

Vinet, L.; Gregoire, C.; Schuck, P.; Remaud, B.; Sebille, F.

1987-01-01

The disintegration of hot and/or compressed nuclei is studied using (i) the Vlasov equation (VE) with imposed spherical symmetry, (ii) the VE in three dimensions (3D) and (iii) the VE in three dimensions supplemented by the Uehling-Uhlenbeck collision term (VUU). We find that case (ii) is slightly more unstable with respect to disintegration compared to case (i) whereas (iii) tends to make nuclei more stable. In all cases the thermal energies (15-20 MeV per nucleon) needed to totally disintegrate a nucleus seem to be higher than those found in static and hydrodynamic calculation. On the contrary, compressional energy very much helps disintegration. Some comments on the introduction of fluctuations and corresponding fragmentation are added. (orig.)

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.

Numerical simulation of Vlasov equation with parallel tools

International Nuclear Information System (INIS)

Peyroux, J.

2005-11-01

This project aims to make even more powerful the resolution of Vlasov codes through the various parallelization tools (MPI, OpenMP...). A simplified test case served as a base for constructing the parallel codes for obtaining a data-processing skeleton which, thereafter, could be re-used for increasingly complex models (more than four variables of phase space). This will thus make it possible to treat more realistic situations linked, for example, to the injection of ultra short and ultra intense impulses in inertial fusion plasmas, or the study of the instability of trapped ions now taken as being responsible for the generation of turbulence in tokamak plasmas. (author)

Relations between nonlinear Riccati equations and other equations in fundamental physics

International Nuclear Information System (INIS)

Schuch, Dieter

2014-01-01

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

Numerical simulation of Vlasov equation with parallel tools; Simulations numeriques de l'equation de Vlasov a l'aide d'outils paralleles

Energy Technology Data Exchange (ETDEWEB)

Peyroux, J

2005-11-15

This project aims to make even more powerful the resolution of Vlasov codes through the various parallelization tools (MPI, OpenMP...). A simplified test case served as a base for constructing the parallel codes for obtaining a data-processing skeleton which, thereafter, could be re-used for increasingly complex models (more than four variables of phase space). This will thus make it possible to treat more realistic situations linked, for example, to the injection of ultra short and ultra intense impulses in inertial fusion plasmas, or the study of the instability of trapped ions now taken as being responsible for the generation of turbulence in tokamak plasmas. (author)

Solution of the linearised Vlasov equation for collisionless plasmas evolving in external fields of arbitrary spatial and time dependence: Pt. 2

International Nuclear Information System (INIS)

Skarka, V.; Coveney, P.V.

1990-01-01

We solve perturbatively the linearised Vlasov equation describing inhomogeneous collisionless plasmas evolving in time-dependent external fields. The method employs an explicitly time-dependent formalism and is facilitated by the used of diagrammatic techniques. It leads to a straightforward algorithm for computing the contribution to the solution, order by order in the external field. In the previous paper we provided the solution to first order; higher orders are described in the present paper. (author)

The Arrow of Time in the Collapse of Collisionless Self-gravitating Systems: Non-validity of the Vlasov-Poisson Equation during Violent Relaxation

Science.gov (United States)

Beraldo e Silva, Leandro; de Siqueira Pedra, Walter; Sodré, Laerte; Perico, Eder L. D.; Lima, Marcos

2017-09-01

The collapse of a collisionless self-gravitating system, with the fast achievement of a quasi-stationary state, is driven by violent relaxation, with a typical particle interacting with the time-changing collective potential. It is traditionally assumed that this evolution is governed by the Vlasov-Poisson equation, in which case entropy must be conserved. We run N-body simulations of isolated self-gravitating systems, using three simulation codes, NBODY-6 (direct summation without softening), NBODY-2 (direct summation with softening), and GADGET-2 (tree code with softening), for different numbers of particles and initial conditions. At each snapshot, we estimate the Shannon entropy of the distribution function with three different techniques: Kernel, Nearest Neighbor, and EnBiD. For all simulation codes and estimators, the entropy evolution converges to the same limit as N increases. During violent relaxation, the entropy has a fast increase followed by damping oscillations, indicating that violent relaxation must be described by a kinetic equation other than the Vlasov-Poisson equation, even for N as large as that of astronomical structures. This indicates that violent relaxation cannot be described by a time-reversible equation, shedding some light on the so-called “fundamental paradox of stellar dynamics.” The long-term evolution is well-described by the orbit-averaged Fokker-Planck model, with Coulomb logarithm values in the expected range 10{--}12. By means of NBODY-2, we also study the dependence of the two-body relaxation timescale on the softening length. The approach presented in the current work can potentially provide a general method for testing any kinetic equation intended to describe the macroscopic evolution of N-body systems.

Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation

Science.gov (United States)

Vitanov, Nikolay K.; Dimitrova, Zlatinka I.

2018-03-01

We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.

Symmetry and exact solutions of nonlinear spinor equations

International Nuclear Information System (INIS)

Fushchich, W.I.; Zhdanov, R.Z.

1989-01-01

This review is devoted to the application of algebraic-theoretical methods to the problem of constructing exact solutions of the many-dimensional nonlinear systems of partial differential equations for spinor, vector and scalar fields widely used in quantum field theory. Large classes of nonlinear spinor equations invariant under the Poincare group P(1, 3), Weyl group (i.e. Poincare group supplemented by a group of scale transformations), and the conformal group C(1, 3) are described. Ansaetze invariant under the Poincare and the Weyl groups are constructed. Using these we reduce the Poincare-invariant nonlinear Dirac equations to systems of ordinary differential equations and construct large families of exact solutions of the nonlinear Dirac-Heisenberg equation depending on arbitrary parameters and functions. In a similar way we have obtained new families of exact solutions of the nonlinear Maxwell-Dirac and Klein-Gordon-Dirac equations. The obtained solutions can be used for quantization of nonlinear equations. (orig.)

On a new series of integrable nonlinear evolution equations

International Nuclear Information System (INIS)

Ichikawa, Y.H.; Wadati, Miki; Konno, Kimiaki; Shimizu, Tohru.

1980-10-01

Recent results of our research are surveyed in this report. The derivative nonlinear Schroedinger equation for the circular polarized Alfven wave admits the spiky soliton solutions for the plane wave boundary condition. The nonlinear equation for complex amplitude associated with the carrier wave is shown to be a generalized nonlinear Schroedinger equation, having the ordinary cubic nonlinear term and the derivative of cubic nonlinear term. A generalized scheme of the inverse scattering transformation has confirmed that superposition of the A-K-N-S scheme and the K-N scheme for the component equations valids for the generalized nonlinear Schroedinger equation. Then, two types of new integrable nonlinear evolution equation have been derived from our scheme of the inverse scattering transformation. One is the type of nonlinear Schroedinger equation, while the other is the type of Korteweg-de Vries equation. Brief discussions are presented for physical phenomena, which could be accounted by the second type of the new integrable nonlinear evolution equation. Lastly, the stationary solitary wave solutions have been constructed for the integrable nonlinear evolution equation of the second type. These solutions have peculiar structure that they are singular and discrete. It is a new challenge to construct singular potentials by the inverse scattering transformation. (author)

Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation

Directory of Open Access Journals (Sweden)

Vitanov Nikolay K.

2018-03-01

Full Text Available We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.

Fractional analysis for nonlinear electrical transmission line and nonlinear Schroedinger equations with incomplete sub-equation

Science.gov (United States)

Fendzi-Donfack, Emmanuel; Nguenang, Jean Pierre; Nana, Laurent

2018-02-01

We use the fractional complex transform with the modified Riemann-Liouville derivative operator to establish the exact and generalized solutions of two fractional partial differential equations. We determine the solutions of fractional nonlinear electrical transmission lines (NETL) and the perturbed nonlinear Schroedinger (NLS) equation with the Kerr law nonlinearity term. The solutions are obtained for the parameters in the range (0<α≤1) of the derivative operator and we found the traditional solutions for the limiting case of α =1. We show that according to the modified Riemann-Liouville derivative, the solutions found can describe physical systems with memory effect, transient effects in electrical systems and nonlinear transmission lines, and other systems such as optical fiber.

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....

Relativistic extension of a charge-conservative finite element solver for time-dependent Maxwell-Vlasov equations

Science.gov (United States)

Na, D.-Y.; Moon, H.; Omelchenko, Y. A.; Teixeira, F. L.

2018-01-01

Accurate modeling of relativistic particle motion is essential for physical predictions in many problems involving vacuum electronic devices, particle accelerators, and relativistic plasmas. A local, explicit, and charge-conserving finite-element time-domain (FETD) particle-in-cell (PIC) algorithm for time-dependent (non-relativistic) Maxwell-Vlasov equations on irregular (unstructured) meshes was recently developed by Moon et al. [Comput. Phys. Commun. 194, 43 (2015); IEEE Trans. Plasma Sci. 44, 1353 (2016)]. Here, we extend this FETD-PIC algorithm to the relativistic regime by implementing and comparing three relativistic particle-pushers: (relativistic) Boris, Vay, and Higuera-Cary. We illustrate the application of the proposed relativistic FETD-PIC algorithm for the analysis of particle cyclotron motion at relativistic speeds, harmonic particle oscillation in the Lorentz-boosted frame, and relativistic Bernstein modes in magnetized charge-neutral (pair) plasmas.

Nonlinear diffusion equations

CERN Document Server

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

Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations

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.

Longitudinal traveling waves bifurcating from Vlasov plasma equilibria

International Nuclear Information System (INIS)

Holloway, J.P.

1989-01-01

The kinetic equations governing longitudinal motion along a straight magnetic field in a multi-species collisionless plasma are investigated. A necessary condition for the existence of small amplitude spatially periodic equilibria and traveling waves near a given spatially uniform background equilibrium is derived, and the wavelengths which such solutions must approach as their amplitude decreases to zero are discussed. A sufficient condition for the existence of these small amplitude waves is also established. This is accomplished by studying the nonlinear ODE for the potential which arises when the distribution functions are represented in a BGK form; the arbitrary functions of energy that describe the BGK representation are tested as an infinite dimensional set of parameters in a bifurcation theory for the ODE. The positivity and zero current condition in the wave frame of the BGK distribution functions are maintained. The undamped small amplitude nonlinear waves so constructed can be made to satisfy the Vlasov dispersion relation exactly, but in general they need only satisfy it approximately. Numerical calculations reveal that even a thermal equilibrium electron-proton plasma with equal ion and electron temperatures will support undamped traveling waves with phase speeds greater than 1.3 times the electron velocity; the dispersion relation for this case exhibits both Langmuir and ion-acoustic branches as long wavelength limits, and shows how these branches are in fact connected by short wavelength waves of intermediate frequency. In apparent contradiction to the linear theory of Landau, these exact solutions of the kinetic equations do not damp; this contradiction is explained by observing that the linear theory is, in general, fundamentally incapable of describing undamped traveling waves

Linear superposition solutions to nonlinear wave equations

International Nuclear Information System (INIS)

Liu Yu

2012-01-01

The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic, triangle, and exponential functions, and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics. The linear superposition solutions to the generalized KdV equation K(2,2,1), the Oliver water wave equation, and the k(n, n) equation are given. The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed, and the reason why the solutions with the forms of hyperbolic, triangle, and exponential functions can form the linear superposition solutions is also discussed

Decomposition of a hierarchy of nonlinear evolution equations

International Nuclear Information System (INIS)

Geng Xianguo

2003-01-01

The generalized Hamiltonian structures for a hierarchy of nonlinear evolution equations are established with the aid of the trace identity. Using the nonlinearization approach, the hierarchy of nonlinear evolution equations is decomposed into a class of new finite-dimensional Hamiltonian systems. The generating function of integrals and their generator are presented, based on which the finite-dimensional Hamiltonian systems are proved to be completely integrable in the Liouville sense. As an application, solutions for the hierarchy of nonlinear evolution equations are reduced to solving the compatible Hamiltonian systems of ordinary differential equations

A new auxiliary equation and exact travelling wave solutions of nonlinear equations

International Nuclear Information System (INIS)

Sirendaoreji

2006-01-01

A new auxiliary ordinary differential equation and its solutions are used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the auxiliary equation which has more new exact solutions. More new exact travelling wave solutions are obtained for the quadratic nonlinear Klein-Gordon equation, the combined KdV and mKdV equation, the sine-Gordon equation and the Whitham-Broer-Kaup equations

Quasi-exact solutions of nonlinear differential equations

OpenAIRE

Kudryashov, Nikolay A.; Kochanov, Mark B.

2014-01-01

The concept of quasi-exact solutions of nonlinear differential equations is introduced. Quasi-exact solution expands the idea of exact solution for additional values of parameters of differential equation. These solutions are approximate solutions of nonlinear differential equations but they are close to exact solutions. Quasi-exact solutions of the the Kuramoto--Sivashinsky, the Korteweg--de Vries--Burgers and the Kawahara equations are founded.

One-Dimensional Vlasov-Maxwell Equilibrium for the Force-Free Harris Sheet

International Nuclear Information System (INIS)

Harrison, Michael G.; Neukirch, Thomas

2009-01-01

In this Letter, the first nonlinear force-free Vlasov-Maxwell equilibrium is presented. One component of the equilibrium magnetic field has the same spatial structure as the Harris sheet, but whereas the Harris sheet is kept in force balance by pressure gradients, in the force-free solution presented here force balance is maintained by magnetic shear. Magnetic pressure, plasma pressure and plasma density are constant. The method used to find the equilibrium is based on the analogy of the one-dimensional Vlasov-Maxwell equilibrium problem to the motion of a pseudoparticle in a two-dimensional conservative potential. The force-free solution can be generalized to a complete family of equilibria that describe the transition between the purely pressure-balanced Harris sheet to the force-free Harris sheet

Generalized solutions of nonlinear partial differential equations

CERN Document Server

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

Hamiltonian structures of some non-linear evolution equations

International Nuclear Information System (INIS)

Tu, G.Z.

1983-06-01

The Hamiltonian structure of the O(2,1) non-linear sigma model, generalized AKNS equations, are discussed. By reducing the O(2,1) non-linear sigma model to its Hamiltonian form some new conservation laws are derived. A new hierarchy of non-linear evolution equations is proposed and shown to be generalized Hamiltonian equations with an infinite number of conservation laws. (author)

Nonlinear von Neumann equations for quantum dissipative systems

International Nuclear Information System (INIS)

Messer, J.; Baumgartner, B.

1978-01-01

For pure states nonlinear Schroedinger equations, the so-called Schroedinger-Langevin equations are well-known to model quantum dissipative systems of the Langevin type. For mixtures it is shown that these wave equations do not extend to master equations, but to corresponding nonlinear von Neumann equations. Solutions for the damped harmonic oscillator are discussed. (Auth.)

Nonlinear von Neumann equations for quantum dissipative systems

International Nuclear Information System (INIS)

Messer, J.; Baumgartner, B.

For pure states nonlinear Schroedinger equations, the so-called Schroedinger-Langevin equations are well-known to model quantum dissipative systems of the Langevin type. For mixtures it is shown that these wave equations do not extend to master equations, but to corresponding nonlinear von Neumann equations. Solutions for the damped harmonic oscillator are discussed. (Author)

Multi-diffusive nonlinear Fokker–Planck equation

International Nuclear Information System (INIS)

Ribeiro, Mauricio S; Casas, Gabriela A; Nobre, Fernando D

2017-01-01

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. (paper)

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.

Exact solutions of a nonpolynomially nonlinear Schrodinger equation

International Nuclear Information System (INIS)

Parwani, R.; Tan, H.S.

2007-01-01

A nonlinear generalisation of Schrodinger's equation had previously been obtained using information-theoretic arguments. The nonlinearities in that equation were of a nonpolynomial form, equivalent to the occurrence of higher-derivative nonlinear terms at all orders. Here we construct some exact solutions to that equation in 1+1 dimensions. On the half-line, the solutions resemble (exponentially damped) Bloch waves even though no external periodic potential is included. The solutions are nonperturbative as they do not reduce to solutions of the linear theory in the limit that the nonlinearity parameter vanishes. An intriguing feature of the solutions is their infinite degeneracy: for a given energy, there exists a very large arbitrariness in the normalisable wavefunctions. We also consider solutions to a q-deformed version of the nonlinear equation and discuss a natural discretisation implied by the nonpolynomiality. Finally, we contrast the properties of our solutions with other solutions of nonlinear Schrodinger equations in the literature and suggest some possible applications of our results in the domains of low-energy and high-energy physics

Evolution of the phase-space density and the Jeans scale for dark matter derived from the Vlasov-Einstein equation

International Nuclear Information System (INIS)

Piattella, O.F.; Rodrigues, D.C.; Fabris, J.C.; Pacheco, J.A. de Freitas

2013-01-01

We discuss solutions of Vlasov-Einstein equation for collisionless dark matter particles in the context of a flat Friedmann universe. We show that, after decoupling from the primordial plasma, the dark matter phase-space density indicator Q = ρ/(σ 1D 2 ) 3/2 remains constant during the expansion of the universe, prior to structure formation. This well known result is valid for non-relativistic particles and is not ''observer dependent'' as in solutions derived from the Vlasov-Poisson system. In the linear regime, the inclusion of velocity dispersion effects permits to define a physical Jeans length for collisionless matter as function of the primordial phase-space density indicator: λ J = (5π/G) 1/2 Q −1/3 ρ dm −1/6 . The comoving Jeans wavenumber at matter-radiation equality is smaller by a factor of 2-3 than the comoving wavenumber due to free-streaming, contributing to the cut-off of the density fluctuation power spectrum at the lowest scales. We discuss the physical differences between these two scales. For dark matter particles of mass equal to 200 GeV, the derived Jeans mass is 4.3 × 10 −6 M ⊙

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 interes...... nonlinear Schrodinger equation. In this way eve are able to construct coherent solitonlike structures of profile determined by the map parameters.......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 hydrodynamic equations for superfluid helium in aerogel

International Nuclear Information System (INIS)

Brusov, Peter N.; Brusov, Paul P.

2003-01-01

Aerogel in superfluids is studied very intensively during last decade. The importance of these systems is connected to the fact that this allows to investigate the influence of impurities on superfluidity. We have derived for the first time nonlinear hydrodynamic equations for superfluid helium in aerogel. These equations are generalization of McKenna et al. equations for nonlinear hydrodynamics case and could be used to study sound propagation phenomena in aerogel-superfluid system, in particular--to study sound conversion phenomena. We have obtained two alternative sets of equations, one of which is a generalization of a traditional set of nonlinear hydrodynamics equations for the case of an aerogel-superfluid system and, the other one represents a la Putterman equations (equation for v→ s is replaced by equation for A→=((ρ n )/(ρσ))w→, where w→=v→ n -v→ s )

Nonlinear wave equation in frequency domain: accurate modeling of ultrafast interaction in anisotropic nonlinear media

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...... nonlinearities, delayed Raman effects, and anisotropic nonlinearities. The full potential of this wave equation is demonstrated by investigating simulations of solitons generated in the process of ultrafast cascaded second-harmonic generation. We show that a balance in the soliton delay can be achieved due...

Linear differential equations to solve nonlinear mechanical problems: A novel approach

OpenAIRE

Nair, C. Radhakrishnan

2004-01-01

Often a non-linear mechanical problem is formulated as a non-linear differential equation. A new method is introduced to find out new solutions of non-linear differential equations if one of the solutions of a given non-linear differential equation is known. Using the known solution of the non-linear differential equation, linear differential equations are set up. The solutions of these linear differential equations are found using standard techniques. Then the solutions of the linear differe...

A new sine-Gordon equation expansion algorithm to investigate some special nonlinear differential equations

International Nuclear Information System (INIS)

Yan Zhenya

2005-01-01

A new transformation method is developed using the general sine-Gordon travelling wave reduction equation and a generalized transformation. With the aid of symbolic computation, this method can be used to seek more types of solutions of nonlinear differential equations, which include not only the known solutions derived by some known methods but new solutions. Here we choose the double sine-Gordon equation, the Magma equation and the generalized Pochhammer-Chree (PC) equation to illustrate the method. As a result, many types of new doubly periodic solutions are obtained. Moreover when using the method to these special nonlinear differential equations, some transformations are firstly needed. The method can be also extended to other nonlinear differential equations

Nonlinear Fokker-Planck Equations Fundamentals and Applications

CERN Document Server

Frank, Till Daniel

2005-01-01

Providing an introduction to the theory of nonlinear Fokker-Planck equations, this book discusses fundamental properties of transient and stationary solutions, emphasizing the stability analysis of stationary solutions by means of self-consistency equations, linear stability analysis, and Lyapunov's direct method. Also treated are Langevin equations and correlation functions. Nonlinear Fokker-Planck Equations addresses various phenomena such as phase transitions, multistability of systems, synchronization, anomalous diffusion, cut-off solutions, travelling-wave solutions and the emergence of power law solutions. A nonlinear Fokker-Planck perspective to quantum statistics, generalized thermodynamics, and linear nonequilibrium thermodynamics is given. Theoretical concepts are illustrated where possible by simple examples. The book also reviews several applications in the fields of condensed matter physics, the physics of porous media and liquid crystals, accelerator physics, neurophysics, social sciences, popul...

Discrete Time McKean–Vlasov Control Problem: A Dynamic Programming Approach

Energy Technology Data Exchange (ETDEWEB)

Pham, Huyên, E-mail: pham@math.univ-paris-diderot.fr; Wei, Xiaoli, E-mail: tyswxl@gmail.com [Laboratoire de Probabilités et Modèles Aléatoires, CNRS, UMR 7599, Université Paris Diderot (France)

2016-12-15

We consider the stochastic optimal control problem of nonlinear mean-field systems in discrete time. We reformulate the problem into a deterministic control problem with marginal distribution as controlled state variable, and prove that dynamic programming principle holds in its general form. We apply our method for solving explicitly the mean-variance portfolio selection and the multivariate linear-quadratic McKean–Vlasov control problem.

Discrete Time McKean–Vlasov Control Problem: A Dynamic Programming Approach

International Nuclear Information System (INIS)

Pham, Huyên; Wei, Xiaoli

2016-01-01

We consider the stochastic optimal control problem of nonlinear mean-field systems in discrete time. We reformulate the problem into a deterministic control problem with marginal distribution as controlled state variable, and prove that dynamic programming principle holds in its general form. We apply our method for solving explicitly the mean-variance portfolio selection and the multivariate linear-quadratic McKean–Vlasov control problem.

The Cauchy problem for non-linear Klein-Gordon equations

International Nuclear Information System (INIS)

Simon, J.C.H.; Taflin, E.

1993-01-01

We consider in R n+1 , n≥2, the non-linear Klein-Gordon equation. We prove for such an equation that there is neighbourhood of zero in a Hilbert space of initial conditions for which the Cauchy problem has global solutions and on which there is asymptotic completeness. The inverse of the wave operator linearizes the non-linear equation. If, moreover, the equation is manifestly Poincare covariant then the non-linear representation of the Poincare-Lie algebra, associated with the non-linear Klein-Gordon equation is integrated to a non-linear representation of the Poincare group on an invariant neighbourhood of zero in the Hilbert space. This representation is linearized by the inverse of the wave operator. The Hilbert space is, in both cases, the closure of the space of the differentiable vectors for the linear representation of the Poincare group, associated with the Klein-Gordon equation, with respect to a norm defined by the representation of the enveloping algebra. (orig.)

International Conference on Differential Equations and Nonlinear Mechanics

CERN Document Server

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...

Numerical simulation of Vlasov equation with parallel tools; Simulations numeriques de l'equation de Vlasov a l'aide d'outils paralleles

Energy Technology Data Exchange (ETDEWEB)

Peyroux, J

2005-11-15

This project aims to make even more powerful the resolution of Vlasov codes through the various parallelization tools (MPI, OpenMP...). A simplified test case served as a base for constructing the parallel codes for obtaining a data-processing skeleton which, thereafter, could be re-used for increasingly complex models (more than four variables of phase space). This will thus make it possible to treat more realistic situations linked, for example, to the injection of ultra short and ultra intense impulses in inertial fusion plasmas, or the study of the instability of trapped ions now taken as being responsible for the generation of turbulence in tokamak plasmas. (author)

Nonlinear elliptic equations of the second order

CERN Document Server

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...

Multilevel and Multi-index Monte Carlo methods for the McKean–Vlasov equation

KAUST Repository

Haji-Ali, Abdul-Lateef

2017-09-12

We address the approximation of functionals depending on a system of particles, described by stochastic differential equations (SDEs), in the mean-field limit when the number of particles approaches infinity. This problem is equivalent to estimating the weak solution of the limiting McKean–Vlasov SDE. To that end, our approach uses systems with finite numbers of particles and a time-stepping scheme. In this case, there are two discretization parameters: the number of time steps and the number of particles. Based on these two parameters, we consider different variants of the Monte Carlo and Multilevel Monte Carlo (MLMC) methods and show that, in the best case, the optimal work complexity of MLMC, to estimate the functional in one typical setting with an error tolerance of $$\\\\mathrm {TOL}$$TOL, is when using the partitioning estimator and the Milstein time-stepping scheme. We also consider a method that uses the recent Multi-index Monte Carlo method and show an improved work complexity in the same typical setting of . Our numerical experiments are carried out on the so-called Kuramoto model, a system of coupled oscillators.

Fuchs indices and the first integrals of nonlinear differential equations

International Nuclear Information System (INIS)

Kudryashov, Nikolai A.

2005-01-01

New method of finding the first integrals of nonlinear differential equations in polynomial form is presented. Basic idea of our approach is to use the scaling of solution of nonlinear differential equation and to find the dimensions of arbitrary constants in the Laurent expansion of the general solution. These dimensions allows us to obtain the scalings of members for the first integrals of nonlinear differential equations. Taking the polynomials with unknown coefficients into account we present the algorithm of finding the first integrals of nonlinear differential equations in the polynomial form. Our method is applied to look for the first integrals of eight nonlinear ordinary differential equations of the fourth order. The general solution of one of the fourth order ordinary differential equations is given

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...

Exact solutions for the higher-order nonlinear Schoerdinger equation in nonlinear optical fibres

International Nuclear Information System (INIS)

Liu Chunping

2005-01-01

First, by using the generally projective Riccati equation method, many kinds of exact solutions for the higher-order nonlinear Schoerdinger equation in nonlinear optical fibres are obtained in a unified way. Then, some relations among these solutions are revealed

Explosions in Landau Vlasov dynamics

International Nuclear Information System (INIS)

Suraud, E.; Cussol, D.; Gregoire, C.; Boilley, D.; Pi, M.; Schuck, P.; Remaud, B.; Sebille, F.

1988-01-01

A microscopic study of the quasi-fusion/explosion transition is presented in the framework of Landau-Vlasov simulations of intermediate energy heavy-ion collisions (bombarding energies between 10 and 100 MeV/A). A detailed analysis in terms of the Equation of State of the system is performed. In agreement with schematic models we find that the composite nuclear system formed in the collision does explode when it stays long enough in the mechanically unstable region (spinodal region). Quantitative estimates of the explosion threshold are given for central symmetric reactions (Ca+Ca and Ar+Ti). The effect of the nuclear matter compressibility modulus is discussed

Nonlinear H-infinity control, Hamiltonian systems and Hamilton-Jacobi equations

CERN Document Server

Aliyu, MDS

2011-01-01

A comprehensive overview of nonlinear Haeu control theory for both continuous-time and discrete-time systems, Nonlinear Haeu-Control, Hamiltonian Systems and Hamilton-Jacobi Equations covers topics as diverse as singular nonlinear Haeu-control, nonlinear Haeu -filtering, mixed H2/ Haeu-nonlinear control and filtering, nonlinear Haeu-almost-disturbance-decoupling, and algorithms for solving the ubiquitous Hamilton-Jacobi-Isaacs equations. The link between the subject and analytical mechanics as well as the theory of partial differential equations is also elegantly summarized in a single chapter

OSCILLATION OF NONLINEAR DELAY DIFFERENCE EQUATIONS

Institute of Scientific and Technical Information of China (English)

无

2006-01-01

This paper deals with the oscillatory properties of a class of nonlinear difference equations with several delays. Sufficient criteria in the form of infinite sum for the equations to be oscillatory are obtained.

Algorithms For Integrating Nonlinear Differential Equations

Science.gov (United States)

Freed, A. D.; Walker, K. P.

1994-01-01

Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.

Jacobi Elliptic Solutions for Nonlinear Differential Difference Equations in Mathematical Physics

Directory of Open Access Journals (Sweden)

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.

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....

Advances in iterative methods for nonlinear equations

CERN Document Server

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...

Exact solutions of nonlinear generalizations of the Klein Gordon and Schrodinger equations

International Nuclear Information System (INIS)

Burt, P.B.

1978-01-01

Exact solutions of sine Gordon and multiple sine Gordon equations are constructed in terms of solutions of a linear base equation, the Klein Gordon equation and also in terms of nonlinear base equations where the nonlinearity is polynomial in the dependent variable. Further, exact solutions of nonlinear generalizations of the Schrodinger equation and of additional nonlinear generalizations of the Klein Gordon equation are constructed in terms of solutions of linear base equations. Finally, solutions with spherical symmetry, of nonlinear Klein Gordon equations are given. 14 references

Hybrid (Vlasov-Fluid) simulation of ion-acoustic soliton chain formation and validity of Korteweg de-Vries model

Energy Technology Data Exchange (ETDEWEB)

Aminmansoor, F.; Abbasi, H., E-mail: abbasi@aut.ac.ir [Faculty of Energy Engineering and Physics, Amirkabir University of Technology, P.O. Box 15875-4413, Tehran (Iran, Islamic Republic of)

2015-08-15

The present paper is devoted to simulation of nonlinear disintegration of a localized perturbation into ion-acoustic solitons train in a plasma with hot electrons and cold ions. A Gaussian initial perturbation is used to model the localized perturbation. For this purpose, first, we reduce fluid system of equations to a Korteweg de-Vries equation by the following well-known assumptions. (i) On the ion-acoustic evolution time-scale, the electron velocity distribution function (EVDF) is assumed to be stationary. (ii) The calculation is restricted to small amplitude cases. Next, in order to generalize the model to finite amplitudes cases, the evolution of EVDF is included. To this end, a hybrid code is designed to simulate the case, in which electrons dynamics is governed by Vlasov equation, while cold ions dynamics is, like before, studied by the fluid equations. A comparison between the two models shows that although the fluid model is capable of demonstrating the general features of the process, to have a better insight into the relevant physics resulting from the evolution of EVDF, the use of kinetic treatment is of great importance.

The modified simplest equation method to look for exact solutions of nonlinear partial differential equations

OpenAIRE

Efimova, Olga Yu.

2010-01-01

The modification of simplest equation method to look for exact solutions of nonlinear partial differential equations is presented. Using this method we obtain exact solutions of generalized Korteweg-de Vries equation with cubic source and exact solutions of third-order Kudryashov-Sinelshchikov equation describing nonlinear waves in liquids with gas bubbles.

Contributions to mathematical analysis and to numerical approximation in plasma physics

International Nuclear Information System (INIS)

Besse, N.

2009-01-01

The author's scientific works deal with numerical analysis and the simulation of the partial differential equations that intervene in the transport of charged particles and in plasma physics. In the chapters 2 and 3, a reduction of the Vlasov equation is presented, this method is based on the Liouville geometric invariants and it leads to a mathematical model named water-bag model that can be coupled with various equations of the electromagnetic field: the Poisson equation, the quasi-neutral equation or Maxwell equations. In the chapter 3 this reduction method is applied to the Vlasov gyro-kinetic equation to form the gyro-water-bag model. The mathematical analysis of this model produces interesting analytical results such as: threshold instabilities, instability growth rate, transport coefficient and non-linear turbulence mechanisms. Simulations have been performed to study turbulence in magnetized plasmas. In these plasmas occurred numerous instabilities due to the presence of high density and temperature gradients. These instabilities generate turbulence that deteriorates plasma confinement conditions required for thermonuclear fusion. The numerical calculation of turbulent thermal diffusivities is important since confinement time is determined by these transport coefficients. The chapter 4 gathers mathematical analysis issues like convergence or prior knowledge of errors concerning several high-order numerical methods used to solve Vlasov-Poisson or Vlasov-Einstein equation systems as well as the induction equation of an idealistic MHD system. The chapter 5 presents original numerical methods to solve several non-linear Vlasov equations such as Vlasov-Poisswell, Vlasov-Darwin, Vlasov-Maxwell and Vlasov-gyrokinetic that are involved either in inertial fusion or in magnetic confinement fusion

On the Stochastic Wave Equation with Nonlinear Damping

International Nuclear Information System (INIS)

Kim, Jong Uhn

2008-01-01

We discuss an initial boundary value problem for the stochastic wave equation with nonlinear damping. We establish the existence and uniqueness of a solution. Our method for the existence of pathwise solutions consists of regularization of the equation and data, the Galerkin approximation and an elementary measure-theoretic argument. We also prove the existence of an invariant measure when the equation has pure nonlinear damping

Green functions and Langevin equations for nonlinear diffusion equations: A comment on ‘Markov processes, Hurst exponents, and nonlinear diffusion equations’ by Bassler et al.

Science.gov (United States)

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.

Stability analysis of a Vlasov-Wave system describing particles interacting with their environment

Science.gov (United States)

De Bièvre, Stephan; Goudon, Thierry; Vavasseur, Arthur

2018-06-01

We study a kinetic equation of the Vlasov-Wave type, which arises in the description of the behavior of a large number of particles interacting weakly with an environment, composed of an infinite collection of local vibrational degrees of freedom, modeled by wave equations. We use variational techniques to establish the existence of large families of stationary states for this system, and analyze their stability.

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.)

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....

The matrix nonlinear Schrodinger equation in dimension 2

DEFF Research Database (Denmark)

Zuhan, L; Pedersen, Michael

2001-01-01

In this paper we study the existence of global solutions to the Cauchy problem for the matrix nonlinear Schrodinger equation (MNLS) in 2 space dimensions. A sharp condition for the global existence is obtained for this equation. This condition is in terms of an exact stationary solution...... of a semilinear elliptic equation. In the scalar case, the MNLS reduces to the well-known cubic nonlinear Schrodinger equation for which existence of solutions has been studied by many authors. (C) 2001 Academic Press....

Global solutions of nonlinear Schrödinger equations

CERN Document Server

Bourgain, J

1999-01-01

This volume presents recent progress in the theory of nonlinear dispersive equations, primarily the nonlinear Schrödinger (NLS) equation. The Cauchy problem for defocusing NLS with critical nonlinearity is discussed. New techniques and results are described on global existence and properties of solutions with large Cauchy data. Current research in harmonic analysis around Strichartz's inequalities and its relevance to nonlinear PDE is presented. Several topics in NLS theory on bounded domains are reviewed. Using the NLS as an example, the book offers comprehensive insight on current research r

Prolongation Structure of Semi-discrete Nonlinear Evolution Equations

International Nuclear Information System (INIS)

Bai Yongqiang; Wu Ke; Zhao Weizhong; Guo Hanying

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 Schroedinger equation is discussed in terms of this theory and the corresponding Lax pairs are also given.

Blow-Up Time for Nonlinear Heat Equations with Transcendental Nonlinearity

Directory of Open Access Journals (Sweden)

Hee Chul Pak

2012-01-01

Full Text Available A blow-up time for nonlinear heat equations with transcendental nonlinearity is investigated. An upper bound of the first blow-up time is presented. It is pointed out that the upper bound of the first blow-up time depends on the support of the initial datum.

Handbook of Nonlinear Partial Differential Equations

CERN Document Server

Polyanin, Andrei D

2011-01-01

New to the Second Edition More than 1,000 pages with over 1,500 new first-, second-, third-, fourth-, and higher-order nonlinear equations with solutions Parabolic, hyperbolic, elliptic, and other systems of equations with solutions Some exact methods and transformations Symbolic and numerical methods for solving nonlinear PDEs with Maple(t), Mathematica(R), and MATLAB(R) Many new illustrative examples and tables A large list of references consisting of over 1,300 sources To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology. They

Exact solutions to two higher order nonlinear Schroedinger equations

International Nuclear Information System (INIS)

Xu Liping; Zhang Jinliang

2007-01-01

Using the homogeneous balance principle and F-expansion method, the exact solutions to two higher order nonlinear Schroedinger equations which describe the propagation of femtosecond pulses in nonlinear fibres are obtained with the aid of a set of subsidiary higher order ordinary differential equations (sub-equations for short)

A nonlinear bounce kinetic equation for trapped electrons

International Nuclear Information System (INIS)

Gang, F.Y.

1990-03-01

A nonlinear bounce averaged drift kinetic equation for trapped electrons is derived. This equation enables one to compute the nonlinear response of the trapped electron distribution function in terms of the field-line projection of a potential fluctuation left-angle e -inqθ φ n right-angle b . It is useful for both analytical and computational studies of the nonlinear evolution of short wavelength (n much-gt 1) trapped electron mode-driven turbulence. 7 refs

Integrable discretization s of derivative nonlinear Schroedinger equations

International Nuclear Information System (INIS)

Tsuchida, Takayuki

2002-01-01

We propose integrable discretizations of derivative nonlinear Schroedinger (DNLS) equations such as the Kaup-Newell equation, the Chen-Lee-Liu equation and the Gerdjikov-Ivanov equation by constructing Lax pairs. The discrete DNLS systems admit the reduction of complex conjugation between two dependent variables and possess bi-Hamiltonian structure. Through transformations of variables and reductions, we obtain novel integrable discretizations of the nonlinear Schroedinger (NLS), modified KdV (mKdV), mixed NLS, matrix NLS, matrix KdV, matrix mKdV, coupled NLS, coupled Hirota, coupled Sasa-Satsuma and Burgers equations. We also discuss integrable discretizations of the sine-Gordon equation, the massive Thirring model and their generalizations. (author)

Evans functions and bifurcations of nonlinear waves of some nonlinear reaction diffusion equations

Science.gov (United States)

Zhang, Linghai

2017-10-01

The main purposes of this paper are to accomplish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear system of reaction diffusion equations ut =uxx + α [ βH (u - θ) - u ] - w, wt = ε (u - γw) and to establish the existence, stability, instability and bifurcation of the nonlinear waves of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ], under different conditions on the model constants. To establish the bifurcation for the system, we will study the existence and instability of a standing pulse solution if 0 1; the existence and instability of two standing wave fronts if 2 (1 + αγ) θ = αβγ and 0 traveling wave front as well as the existence and instability of a standing pulse solution if 0 traveling wave front as well as the existence and instability of an upside down standing pulse solution if 0 traveling wave back of the nonlinear scalar reaction diffusion equation ut =uxx + α [ βH (u - θ) - u ] -w0, where w0 = α (β - 2 θ) > 0 is a positive constant, if 0 motivation to study the existence, stability, instability and bifurcations of the nonlinear waves is to study the existence and stability/instability of infinitely many fast/slow multiple traveling pulse solutions of the nonlinear system of reaction diffusion equations. The existence and stability of infinitely many fast multiple traveling pulse solutions are of great interests in mathematical neuroscience.

Superexponentially damped Vlasov plasma oscillations in the Fourier transformed velocity space

International Nuclear Information System (INIS)

Sedlacek, Z.; Nocera, L.

2002-01-01

The Landau (exponentially) damped solutions of the Vlasov-Poisson equation Fourier transformed with respect to velocity are genuine eigenmodes corresponding to complex eigenvalues. In addition there exist solutions decaying faster than exponentially which exhibit no oscillatory behaviour. A new characterization is given of the initial conditions that give rise to these solutions together with a numerical demonstration

Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations

International Nuclear Information System (INIS)

Lu, Bin

2012-01-01

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. -- 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.

Spurious Solutions Of Nonlinear Differential Equations

Science.gov (United States)

Yee, H. C.; Sweby, P. K.; Griffiths, D. F.

1992-01-01

Report utilizes nonlinear-dynamics approach to investigate possible sources of errors and slow convergence and non-convergence of steady-state numerical solutions when using time-dependent approach for problems containing nonlinear source terms. Emphasizes implications for development of algorithms in CFD and computational sciences in general. Main fundamental conclusion of study is that qualitative features of nonlinear differential equations cannot be adequately represented by finite-difference method and vice versa.

Fourth-Order Conservative Vlasov-Maxwell Solver for Cartesian and Cylindrical Phase Space Coordinates

Science.gov (United States)

Vogman, Genia

Plasmas are made up of charged particles whose short-range and long-range interactions give rise to complex behavior that can be difficult to fully characterize experimentally. One of the most complete theoretical descriptions of a plasma is that of kinetic theory, which treats each particle species as a probability distribution function in a six-dimensional position-velocity phase space. Drawing on statistical mechanics, these distribution functions mathematically represent a system of interacting particles without tracking individual ions and electrons. The evolution of the distribution function(s) is governed by the Boltzmann equation coupled to Maxwell's equations, which together describe the dynamics of the plasma and the associated electromagnetic fields. When collisions can be neglected, the Boltzmann equation is reduced to the Vlasov equation. High-fidelity simulation of the rich physics in even a subset of the full six-dimensional phase space calls for low-noise high-accuracy numerical methods. To that end, this dissertation investigates a fourth-order finite-volume discretization of the Vlasov-Maxwell equation system, and addresses some of the fundamental challenges associated with applying these types of computationally intensive enhanced-accuracy numerical methods to phase space simulations. The governing equations of kinetic theory are described in detail, and their conservation-law weak form is derived for Cartesian and cylindrical phase space coordinates. This formulation is well known when it comes to Cartesian geometries, as it is used in finite-volume and finite-element discretizations to guarantee local conservation for numerical solutions. By contrast, the conservation-law weak form of the Vlasov equation in cylindrical phase space coordinates is largely unexplored, and to the author's knowledge has never previously been solved numerically. Thereby the methods described in this dissertation for simulating plasmas in cylindrical phase space

Yang-Mills-Vlasov system in the temporal gauge

International Nuclear Information System (INIS)

Choquet-Bruhat, Y.; Noutchegueme, N.

1991-01-01

We prove a local in time existence theorem of a solution of the Cauchy problem for the Yang-Mills-Vlasov integrodifferential system. Such equations govern the evolution of plasmas, for instance of quarks and gluons (quagmas), where non abelian gauge fields and Yang-Mills charges replace the usual electromagnetic field and electric charge. We work with the temporal gauge and use functional spaces with appropriate weight on the momenta, but no fall off is required in the space direction [fr

MINPACK-1, Subroutine Library for Nonlinear Equation System

International Nuclear Information System (INIS)

Garbow, Burton S.

1984-01-01

1 - Description of problem or function: MINPACK1 is a package of FORTRAN subprograms for the numerical solution of systems of non- linear equations and nonlinear least-squares problems. The individual programs are: Identification/Description: - CHKDER: Check gradients for consistency with functions, - DOGLEG: Determine combination of Gauss-Newton and gradient directions, - DPMPAR: Provide double precision machine parameters, - ENORM: Calculate Euclidean norm of vector, - FDJAC1: Calculate difference approximation to Jacobian (nonlinear equations), - FDJAC2: Calculate difference approximation to Jacobian (least squares), - HYBRD: Solve system of nonlinear equations (approximate Jacobian), - HYBRD1: Easy-to-use driver for HYBRD, - HYBRJ: Solve system of nonlinear equations (analytic Jacobian), - HYBRJ1: Easy-to-use driver for HYBRJ, - LMDER: Solve nonlinear least squares problem (analytic Jacobian), - LMDER1: Easy-to-use driver for LMDER, - LMDIF: Solve nonlinear least squares problem (approximate Jacobian), - LMDIF1: Easy-to-use driver for LMDIF, - LMPAR: Determine Levenberg-Marquardt parameter - LMSTR: Solve nonlinear least squares problem (analytic Jacobian, storage conserving), - LMSTR1: Easy-to-use driver for LMSTR, - QFORM: Accumulate orthogonal matrix from QR factorization QRFAC Compute QR factorization of rectangular matrix, - QRSOLV: Complete solution of least squares problem, - RWUPDT: Update QR factorization after row addition, - R1MPYQ: Apply orthogonal transformations from QR factorization, - R1UPDT: Update QR factorization after rank-1 addition, - SPMPAR: Provide single precision machine parameters. 4. Method of solution - MINPACK1 uses the modified Powell hybrid method and the Levenberg-Marquardt algorithm

Nonlinear elliptic equations and nonassociative algebras

CERN Document Server

Nadirashvili, Nikolai; Vlăduţ, Serge

2014-01-01

This book presents applications of noncommutative and nonassociative algebras to constructing unusual (nonclassical and singular) solutions to fully nonlinear elliptic partial differential equations of second order. The methods described in the book are used to solve a longstanding problem of the existence of truly weak, nonsmooth viscosity solutions. Moreover, the authors provide an almost complete description of homogeneous solutions to fully nonlinear elliptic equations. It is shown that even in the very restricted setting of "Hessian equations", depending only on the eigenvalues of the Hessian, these equations admit homogeneous solutions of all orders compatible with known regularity for viscosity solutions provided the space dimension is five or larger. To the contrary, in dimension four or less the situation is completely different, and our results suggest strongly that there are no nonclassical homogeneous solutions at all in dimensions three and four. Thus this book gives a complete list of dimensions...

Discrete coupled derivative nonlinear Schroedinger equations and their quasi-periodic solutions

International Nuclear Information System (INIS)

Geng Xianguo; Su Ting

2007-01-01

A hierarchy of nonlinear differential-difference equations associated with a discrete isospectral problem is proposed, in which a typical differential-difference equation is a discrete coupled derivative nonlinear Schroedinger equation. With the help of the nonlinearization of the Lax pairs, the hierarchy of nonlinear differential-difference equations is decomposed into a new integrable symplectic map and a class of finite-dimensional integrable Hamiltonian systems. Based on the theory of algebraic curve, the Abel-Jacobi coordinates are introduced to straighten out the corresponding flows, from which quasi-periodic solutions for these differential-difference equations are obtained resorting to the Riemann-theta functions. Moreover, a (2+1)-dimensional discrete coupled derivative nonlinear Schroedinger equation is proposed and its quasi-periodic solutions are derived

Nonlinear quantum fluid equations for a finite temperature Fermi plasma

International Nuclear Information System (INIS)

Eliasson, Bengt; Shukla, Padma K

2008-01-01

Nonlinear quantum electron fluid equations are derived, taking into account the moments of the Wigner equation and by using the Fermi-Dirac equilibrium distribution for electrons with an arbitrary temperature. A simplified formalism with the assumptions of incompressibility of the distribution function is used to close the moments in velocity space. The nonlinear quantum diffraction effects into the fluid equations are incorporated. In the high-temperature limit, we retain the nonlinear fluid equations for a dense hot plasma and in the low-temperature limit, we retain the correct fluid equations for a fully degenerate plasma

Positive Solutions for Coupled Nonlinear Fractional Differential Equations

Directory of Open Access Journals (Sweden)

Wenning Liu

2014-01-01

Full Text Available We consider the existence of positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary values. Assume the nonlinear term is superlinear in one equation and sublinear in the other equation. By constructing two cones K1, K2 and computing the fixed point index in product cone K1×K2, we obtain that the system has a pair of positive solutions. It is remarkable that it is established on the Cartesian product of two cones, in which the feature of two equations can be opposite.

Superposition of elliptic functions as solutions for a large number of nonlinear equations

International Nuclear Information System (INIS)

Khare, Avinash; Saxena, Avadh

2014-01-01

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 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 2 (x, m), it also admits solutions in terms of dn 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

Exact analytical solutions for nonlinear reaction-diffusion equations

International Nuclear Information System (INIS)

Liu Chunping

2003-01-01

By using a direct method via the computer algebraic system of Mathematica, some exact analytical solutions to a class of nonlinear reaction-diffusion equations are presented in closed form. Subsequently, the hyperbolic function solutions and the triangular function solutions of the coupled nonlinear reaction-diffusion equations are obtained in a unified way

Induced scattering due to nonlinear Landau and cyclotron damping of electromagnetic and electrostatic waves in a magnetized plasma

International Nuclear Information System (INIS)

Sugaya, Reiji

1989-01-01

General expressions of the matrix elements for nonlinear wave-particle scattering (nonlinear Landau and cyclotron damping) of electromagnetic and electrostatic waves in a homogeneous magnetized plasma are derived from the Vlasov-Maxwell equations. The kinetic wave equations obtained for electromagnetic waves are expressed by four-order tensors in the rotating and cartesian coordinates. No restrictions are imposed on the propagation angle to a uniform magnetic field, the Larmor radius, the frequencies, or the wave numbers. By electrostatic approximation of the dielectric tensor and the matrix elements the kinetic wave equations can be applied to the case in which two scattering waves are electrostatic or they are partially electrostatic. Further, the matrix elements in the limit of parallel or perpendicular propagation to the magnetic field are given. (author)

Analytical exact solution of the non-linear Schroedinger equation

International Nuclear Information System (INIS)

Martins, Alisson Xavier; Rocha Filho, Tarcisio Marciano da

2011-01-01

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)

Progress on a Vlasov Treatment of Coherent Synchrotron Radiation from Arbitrary Planar Orbits

CERN Document Server

Bassi, Gabriele; Warnock, Robert L

2005-01-01

We study the influence of coherent synchrotron radiation (CSR) on particle bunches traveling on arbitrary planar orbits between parallel conducting plates (shielding). The time evolution of the phase space distribution is determined by solving the Vlasov-Maxwell equations in the time domain. This provides lower numerical noise than the macroparticle method, and allows the study of emittance degradation and microbunching in bunch compressors. We calculate the fields excited by the bunch in the lab frame using a formula simpler than that based on retarded potentials.* We have developed an algorithm for solving the Vlasov equation in the beam frame using arc length as the independent variable and our method of local characteristics (discretized Perron-Frobenius operator).We integrate in the interaction picture in the hope that we can adopt a fixed grid. The distribution function will be represented by B-splines, in a scheme preserving positivity and normalization of the distribution. The transformation between l...

Nontrivial Periodic Solutions for Nonlinear Second-Order Difference Equations

Directory of Open Access Journals (Sweden)

Tieshan He

2011-01-01

Full Text Available This paper is concerned with the existence of nontrivial periodic solutions and positive periodic solutions to a nonlinear second-order difference equation. Under some conditions concerning the first positive eigenvalue of the linear equation corresponding to the nonlinear second-order equation, we establish the existence results by using the topological degree and fixed point index theories.

Nonlinear streak computation using boundary region equations

Energy Technology Data Exchange (ETDEWEB)

Martin, J A; Martel, C, E-mail: juanangel.martin@upm.es, E-mail: carlos.martel@upm.es [Depto. de Fundamentos Matematicos, E.T.S.I Aeronauticos, Universidad Politecnica de Madrid, Plaza Cardenal Cisneros 3, 28040 Madrid (Spain)

2012-08-01

The boundary region equations (BREs) are applied for the simulation of the nonlinear evolution of a spanwise periodic array of streaks in a flat plate boundary layer. The well-known BRE formulation is obtained from the complete Navier-Stokes equations in the high Reynolds number limit, and provides the correct asymptotic description of three-dimensional boundary layer streaks. In this paper, a fast and robust streamwise marching scheme is introduced to perform their numerical integration. Typical streak computations present in the literature correspond to linear streaks or to small-amplitude nonlinear streaks computed using direct numerical simulation (DNS) or the nonlinear parabolized stability equations (PSEs). We use the BREs to numerically compute high-amplitude streaks, a method which requires much lower computational effort than DNS and does not have the consistency and convergence problems of the PSE. It is found that the flow configuration changes substantially as the amplitude of the streaks grows and the nonlinear effects come into play. The transversal motion (in the wall normal-streamwise plane) becomes more important and strongly distorts the streamwise velocity profiles, which end up being quite different from those of the linear case. We analyze in detail the resulting flow patterns for the nonlinearly saturated streaks and compare them with available experimental results. (paper)

Spline Collocation Method for Nonlinear Multi-Term Fractional Differential Equation

OpenAIRE

Choe, Hui-Chol; Kang, Yong-Suk

2013-01-01

We study an approximation method to solve nonlinear multi-term fractional differential equations with initial conditions or boundary conditions. First, we transform the nonlinear multi-term fractional differential equations with initial conditions and boundary conditions to nonlinear fractional integral equations and consider the relations between them. We present a Spline Collocation Method and prove the existence, uniqueness and convergence of approximate solution as well as error estimatio...

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.

Relativistic electron beam acceleration by cascading nonlinear Landau damping of electromagnetic waves in a plasma

International Nuclear Information System (INIS)

Sugaya, R.; Ue, A.; Maehara, T.; Sugawa, M.

1996-01-01

Acceleration and heating of a relativistic electron beam by cascading nonlinear Landau damping involving three or four intense electromagnetic waves in a plasma are studied theoretically based on kinetic wave equations and transport equations derived from relativistic Vlasov endash Maxwell equations. Three or four electromagnetic waves excite successively two or three nonresonant beat-wave-driven relativistic electron plasma waves with a phase velocity near the speed of light [v p =c(1-γ -2 p ) 1/2 , γ p =ω/ω pe ]. Three beat waves interact nonlinearly with the electron beam and accelerate it to a highly relativistic energy γ p m e c 2 more effectively than by the usual nonlinear Landau damping of two electromagnetic waves. It is proved that the electron beam can be accelerated to more highly relativistic energy in the plasma whose electron density decreases temporally with an appropriate rate because of the temporal increase of γ p . copyright 1996 American Institute of Physics

Numerical Simulations of Self-Focused Pulses Using the Nonlinear Maxwell Equations

Science.gov (United States)

Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)

1994-01-01

This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that are currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations. Abstract of a proposed paper for presentation at the meeting NONLINEAR OPTICS: Materials, Fundamentals, and Applications, Hyatt Regency Waikaloa, Waikaloa, Hawaii, July 24-29, 1994, Cosponsored by IEEE/Lasers and Electro-Optics Society and Optical Society of America

Generalized Kapchinskij-Vladimirskij Distribution and Envelope Equation for High-intensity Beams in a Coupled Transverse Focusing Lattice

International Nuclear Information System (INIS)

Qin, Hong; Chung, Moses; Davidson, Ronald C.

2009-01-01

In an uncoupled lattice, the Kapchinskij-Vladimirskij (KV) distribution function first analyzed in 1959 is the only known exact solution of the nonlinear Vlasov-Maxwell equations for high- intensity beams including self-fields in a self-consistent manner. The KV solution is generalized here to high-intensity beams in a coupled transverse lattice using the recently developed generalized Courant-Snyder invariant for coupled transverse dynamics. This solution projects to a rotating, pulsating elliptical beam in transverse configuration space, determined by the generalized matrix envelope equation.

Universality in an information-theoretic motivated nonlinear Schrodinger equation

International Nuclear Information System (INIS)

Parwani, R; Tabia, G

2007-01-01

Using perturbative methods, we analyse a nonlinear generalization of Schrodinger's equation that had previously been obtained through information-theoretic arguments. We obtain analytical expressions for the leading correction, in terms of the nonlinearity scale, to the energy eigenvalues of the linear Schrodinger equation in the presence of an external potential and observe some generic features. In one space dimension these are (i) for nodeless ground states, the energy shifts are subleading in the nonlinearity parameter compared to the shifts for the excited states; (ii) the shifts for the excited states are due predominantly to contribution from the nodes of the unperturbed wavefunctions, and (iii) the energy shifts for excited states are positive for small values of a regulating parameter and negative at large values, vanishing at a universal critical value that is not manifest in the equation. Some of these features hold true for higher dimensional problems. We also study two exactly solved nonlinear Schrodinger equations so as to contrast our observations. Finally, we comment on the possible significance of our results if the nonlinearity is physically realized

Lectures on nonlinear evolution equations initial value problems

CERN Document Server

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...

Integrability in Dynamical Systems: Florida Workshop in Nonlinear Astronomy, 3rd, University of Florida, Gainesville, Oct. 1, 2, 1987, Proceedings

International Nuclear Information System (INIS)

Buchler, J.R.; Ipser, J.R.; Williams, C.A.

1988-01-01

Recent advances in theoretical celestial mechanics are examined in reviews and reports. Topics addressed include resonant integrable models of galaxies, new integrable systems, Painleve expansions for integrable and nonintegrable ordinary differential equations, and particle-simulation solutions of the Vlasov equation in general relativity. Consideration is given to repulsive and attractive double-bubble space-times, the integrability of magnetic-confinement systems, Hannay's angle and Berry's phase in the classical adiabatic motion of charged particles, the integrability of the nonlinear wave equations, normalization in the face of integrability, and simplifications toward the integrability of perturbed Keplerian systems

Polynomial solutions of nonlinear integral equations

International Nuclear Information System (INIS)

Dominici, Diego

2009-01-01

We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials

Polynomial solutions of nonlinear integral equations

Energy Technology Data Exchange (ETDEWEB)

Dominici, Diego [Department of Mathematics, State University of New York at New Paltz, 1 Hawk Dr. Suite 9, New Paltz, NY 12561-2443 (United States)], E-mail: dominicd@newpaltz.edu

2009-05-22

We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials.

Vlasov simulation of the relativistic effect on the breaking of large amplitude plasma waves

International Nuclear Information System (INIS)

Xu Hui; Sheng Zhengming; Zhang Jie

2007-01-01

The influence of relativistic and thermal effects on plasma wave breaking has been studied by solving the coupled Vlasov-Poisson equations. When the relativistic effect is not considered, the wave breaking will not occur, provided the initial perturbation is less than certain value as predicted previously, and the largest amplitude of the plasma wave will decrease with the increase of the initial temperature. When the relativistic effect is considered, wave breaking always occurs during the time evolution, irrespective of the initial perturbation amplitude. Yet the smaller the initial perturbation amplitude is, the longer is the time for wave breaking to occur. With large initial perturbations, wave breaking can always occur with the without the relativistic effect. However, the results are significantly different in the two cases. The thermal effects of electrons decrease the threshold value to initial amplitude for wave breaking and large phase velocity makes the nonlinear phenomenon occur more easily. (authors)

The energy-momentum tensor for the linearized Maxwell-Vlasov and kinetic guiding center theories

International Nuclear Information System (INIS)

Pfirsch, D.; Morrison, P.J.; Texas Univ., Austin

1990-02-01

A modified Hamilton-Jacobi formalism is introduced as a tool to obtain the energy-momentum and angular-momentum tensors for any kind of nonlinear or linearized Maxwell-collisionless kinetic theories. The emphasis is on linearized theories, for which these tensors are derived for the first time. The kinetic theories treated - which need not be the same for all particle species in a plasma - are the Vlasov and kinetic guiding center theories. The Hamiltonian for the guiding center motion is taken in the form resulting from Dirac's constraint theory for non-standard Lagrangian systems. As an example of the Maxwell-kinetic guiding center theory, the second-order energy for a perturbed homogeneous magnetized plasma is calculated with initially vanishing field perturbations. The expression obtained is compared with the corresponding one of Maxwell-Vlasov theory. (orig.)

The energy-momentum tensor for the linearized Maxwell-Vlasov and kinetic guiding center theories

International Nuclear Information System (INIS)

Pfirsch, D.; Morrison, P.J.

1990-02-01

A modified Hamilton-Jacobi formalism is introduced as a tool to obtain the energy-momentum and angular-momentum tensors for any king of nonlinear or linearized Maxwell-collisionless kinetic theories. The emphasis is on linearized theories, for which these tensors are derived for the first time. The kinetic theories treated --- which need not be the same for all particle species in a plasma --- are the Vlasov and kinetic guiding center theories. The Hamiltonian for the guiding center motion is taken in the form resulting from Dirac's constraint theory for non-standard Lagrangian systems. As an example of the Maxwell-kinetic guiding center theory, the second-order energy for a perturbed homogeneous magnetized plasma is calculated with initially vanishing field perturbations. The expression obtained is compared with the corresponding one of Maxwell-Vlasov theory. 11 refs

Entropy and convexity for nonlinear partial differential equations.

Science.gov (United States)

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.

Analysis of an Nth-order nonlinear differential-delay equation

Science.gov (United States)

Vallée, Réal; Marriott, Christopher

1989-01-01

The problem of a nonlinear dynamical system with delay and an overall response time which is distributed among N individual components is analyzed. Such a system can generally be modeled by an Nth-order nonlinear differential delay equation. A linear-stability analysis as well as a numerical simulation of that equation are performed and a comparison is made with the experimental results. Finally, a parallel is established between the first-order differential equation with delay and the Nth-order differential equation without delay.

The nonlinear Schrödinger equation singular solutions and optical collapse

CERN Document Server

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...

Separation of variables for the nonlinear wave equation in polar coordinates

International Nuclear Information System (INIS)

Shermenev, Alexander

2004-01-01

Some classical types of nonlinear wave motion in polar coordinates are studied within quadratic approximation. When the nonlinear quadratic terms in the wave equation are arbitrary, the usual perturbation techniques used in polar coordinates leads to overdetermined systems of linear algebraic equations for the unknown coefficients. However, we show that these overdetermined systems are compatible with the special case of the nonlinear shallow water equation and express explicitly the coefficients of the first two harmonics as polynomials of the Bessel functions of radius and of the trigonometric functions of angle. It gives a series of solutions to the nonlinear shallow water equation that are periodic in time and found with the same accuracy as the equation is derived

Nonlinear scalar field equations. Pt. 1

International Nuclear Information System (INIS)

Berestycki, H.; Lions, P.L.

1983-01-01

This paper as well as a subsequent one is concerned with the existence of nontrivial solutions for some semi-linear elliptic equations in Rsup(N). Such problems are motivated in particular by the search for certain kinds of solitary waves (stationary states) in nonlinear equations of the Klein-Gordon or Schroedinger type. (orig./HSI)

Solitary wave for a nonintegrable discrete nonlinear Schrödinger equation in nonlinear optical waveguide arrays

Science.gov (United States)

Ma, Li-Yuan; Ji, Jia-Liang; Xu, Zong-Wei; Zhu, Zuo-Nong

2018-03-01

We study a nonintegrable discrete nonlinear Schrödinger (dNLS) equation with the term of nonlinear nearest-neighbor interaction occurred in nonlinear optical waveguide arrays. By using discrete Fourier transformation, we obtain numerical approximations of stationary and travelling solitary wave solutions of the nonintegrable dNLS equation. The analysis of stability of stationary solitary waves is performed. It is shown that the nonlinear nearest-neighbor interaction term has great influence on the form of solitary wave. The shape of solitary wave is important in the electric field propagating. If we neglect the nonlinear nearest-neighbor interaction term, much important information in the electric field propagating may be missed. Our numerical simulation also demonstrates the difference of chaos phenomenon between the nonintegrable dNLS equation with nonlinear nearest-neighbor interaction and another nonintegrable dNLS equation without the term. Project supported by the National Natural Science Foundation of China (Grant Nos. 11671255 and 11701510), the Ministry of Economy and Competitiveness of Spain (Grant No. MTM2016-80276-P (AEI/FEDER, EU)), and the China Postdoctoral Science Foundation (Grant No. 2017M621964).

On the invariant measure for the nonlinear Schroedinger equation

International Nuclear Information System (INIS)

Zhidkov, P.R.

1991-01-01

The invariant measure for the nonlinear Schroedinger equation is constructed. In fact, it is assumed that the nonlinearity in the equation is semilinear. The main aim of the paper is the explanation of the Fermi - Past - Ulam phenomenon. Poincare theorem gives the answer to this question. 17 refs

Non-linear wave equations:Mathematical techniques

International Nuclear Information System (INIS)

1978-01-01

An account of certain well-established mathematical methods, which prove useful to deal with non-linear partial differential equations is presented. Within the strict framework of Functional Analysis, it describes Semigroup Techniques in Banach Spaces as well as variational approaches towards critical points. Detailed proofs are given of the existence of local and global solutions of the Cauchy problem and of the stability of stationary solutions. The formal approach based upon invariance under Lie transformations deserves attention due to its wide range of applicability, even if the explicit solutions thus obtained do not allow for a deep analysis of the equations. A compre ensive introduction to the inverse scattering approach and to the solution concept for certain non-linear equations of physical interest are also presented. A detailed discussion is made about certain convergence and stability problems which arise in importance need not be emphasized. (author) [es

Regarding on the exact solutions for the nonlinear fractional differential equations

Directory of Open Access Journals (Sweden)

Kaplan Melike

2016-01-01

Full Text Available In this work, we have considered the modified simple equation (MSE method for obtaining exact solutions of nonlinear fractional-order differential equations. The space-time fractional equal width (EW and the modified equal width (mEW equation are considered for illustrating the effectiveness of the algorithm. It has been observed that all exact solutions obtained in this paper verify the nonlinear ordinary differential equations which was obtained from nonlinear fractional-order differential equations under the terms of wave transformation relationship. The obtained results are shown graphically.

Inverse Boundary Value Problem for Non-linear Hyperbolic Partial Differential Equations

OpenAIRE

Nakamura, Gen; Vashisth, Manmohan

2017-01-01

In this article we are concerned with an inverse boundary value problem for a non-linear wave equation of divergence form with space dimension $n\\geq 3$. This non-linear wave equation has a trivial solution, i.e. zero solution. By linearizing this equation at the trivial solution, we have the usual linear isotropic wave equation with the speed $\\sqrt{\\gamma(x)}$ at each point $x$ in a given spacial domain. For any small solution $u=u(t,x)$ of this non-linear equation, we have the linear isotr...

Quasi-periodic solutions of nonlinear beam equations with quintic quasi-periodic nonlinearities

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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.

On the solution of the nonlinear Schrodinger equation

International Nuclear Information System (INIS)

Zayed, E.M.E.; Zedan, Hassan A.

2003-01-01

In this paper we study the nonlinear Schrodinger equation with respect to the unknown function S(x,t). New dimensional reduction and exact solution for a nonlinear Schrodinger equation are presented and a complete group classification is given with respect to the function S(x,t). Moreover, specializing the potential function S(x,t), new classes of invariant solution and group classification are obtained in the cases of physical interest

A new sub-equation method applied to obtain exact travelling wave solutions of some complex nonlinear equations

International Nuclear Information System (INIS)

Zhang Huiqun

2009-01-01

By using a new coupled Riccati equations, a direct algebraic method, which was applied to obtain exact travelling wave solutions of some complex nonlinear equations, is improved. And the exact travelling wave solutions of the complex KdV equation, Boussinesq equation and Klein-Gordon equation are investigated using the improved method. The method presented in this paper can also be applied to construct exact travelling wave solutions for other nonlinear complex equations.

Sensitivity theory for general non-linear algebraic equations with constraints

International Nuclear Information System (INIS)

Oblow, E.M.

1977-04-01

Sensitivity theory has been developed to a high state of sophistication for applications involving solutions of the linear Boltzmann equation or approximations to it. The success of this theory in the field of radiation transport has prompted study of possible extensions of the method to more general systems of non-linear equations. Initial work in the U.S. and in Europe on the reactor fuel cycle shows that the sensitivity methodology works equally well for those non-linear problems studied to date. The general non-linear theory for algebraic equations is summarized and applied to a class of problems whose solutions are characterized by constrained extrema. Such equations form the basis of much work on energy systems modelling and the econometrics of power production and distribution. It is valuable to have a sensitivity theory available for these problem areas since it is difficult to repeatedly solve complex non-linear equations to find out the effects of alternative input assumptions or the uncertainties associated with predictions of system behavior. The sensitivity theory for a linear system of algebraic equations with constraints which can be solved using linear programming techniques is discussed. The role of the constraints in simplifying the problem so that sensitivity methodology can be applied is highlighted. The general non-linear method is summarized and applied to a non-linear programming problem in particular. Conclusions are drawn in about the applicability of the method for practical problems

Soliton solution for nonlinear partial differential equations by cosine-function method

International Nuclear Information System (INIS)

Ali, A.H.A.; Soliman, A.A.; Raslan, K.R.

2007-01-01

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

Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations

International Nuclear Information System (INIS)

Hong Jialin; Li Chun

2006-01-01

In this paper, we consider the multi-symplectic Runge-Kutta (MSRK) methods applied to the nonlinear Dirac equation in relativistic quantum physics, based on a discovery of the multi-symplecticity of the equation. In particular, the conservation of energy, momentum and charge under MSRK discretizations is investigated by means of numerical experiments and numerical comparisons with non-MSRK methods. Numerical experiments presented reveal that MSRK methods applied to the nonlinear Dirac equation preserve exactly conservation laws of charge and momentum, and conserve the energy conservation in the corresponding numerical accuracy to the method utilized. It is verified numerically that MSRK methods are stable and convergent with respect to the conservation laws of energy, momentum and charge, and MSRK methods preserve not only the inner geometric structure of the equation, but also some crucial conservative properties in quantum physics. A remarkable advantage of MSRK methods applied to the nonlinear Dirac equation is the precise preservation of charge conservation law

Renormalized perturbation theory: Vlasov-Poisson System, weak turbulence limit and gyrokinetics

International Nuclear Information System (INIS)

Zhang, Y.Z.; Mahajan, S.M.

1987-10-01

The Self-consistency of the renormalized perturbation theory is demonstrated by applying it to the Vlasov-Poisson System and showing that the theory has the correct weak turbulence limit. Energy conservation is proved to arbitrary high order for the electrostatic drift waves. The theory is applied to derive renormalized equations for a low-β gyrokinetic system. Comparison of our theory with other current theories is presented. 22 refs

Nonlinear differential equations with exact solutions expressed via the Weierstrass function

NARCIS (Netherlands)

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

The numerical dynamic for highly nonlinear partial differential equations

Science.gov (United States)

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.

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....

Exact solutions for nonlinear evolution equations using Exp-function method

International Nuclear Information System (INIS)

Bekir, Ahmet; Boz, Ahmet

2008-01-01

In this Letter, the Exp-function method is used to construct solitary and soliton solutions of nonlinear evolution equations. The Klein-Gordon, Burger-Fisher and Sharma-Tasso-Olver equations are chosen to illustrate the effectiveness of the method. The method is straightforward and concise, and its applications are promising. The Exp-function method presents a wider applicability for handling nonlinear wave equations

GHM method for obtaining rationalsolutions of nonlinear differential equations.

Science.gov (United States)

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.

Explicit Solutions for Generalized (2+1)-Dimensional Nonlinear Zakharov-Kuznetsov Equation

International Nuclear Information System (INIS)

Sun Yuhuai; Ma Zhimin; Li Yan

2010-01-01

The exact solutions of the generalized (2+1)-dimensional nonlinear Zakharov-Kuznetsov (Z-K) equation are explored by the method of the improved generalized auxiliary differential equation. Many explicit analytic solutions of the Z-K equation are obtained. The methods used to solve the Z-K equation can be employed in further work to establish new solutions for other nonlinear partial differential equations. (general)

Exact solutions for a system of nonlinear plasma fluid equations

International Nuclear Information System (INIS)

Prahovic, M.G.; Hazeltine, R.D.; Morrison, P.J.

1991-04-01

A method is presented for constructing exact solutions to a system of nonlinear plasma fluid equations that combines the physics of reduced magnetohydrodynamics and the electrostatic drift-wave description of the Charney-Hasegawa-Mima equation. The system has nonlinearities that take the form of Poisson brackets involving the fluid field variables. The method relies on modifying a class of simple equilibrium solutions, but no approximations are made. A distinguishing feature is that the original nonlinear problem is reduced to the solution of two linear partial differential equations, one fourth-order and the other first-order. The first-order equation has Hamiltonian characteristics and is easily integrated, supplying information about the general structure of solutions. 6 refs

Weierstrass Elliptic Function Solutions to Nonlinear Evolution Equations

International Nuclear Information System (INIS)

Yu Jianping; Sun Yongli

2008-01-01

This paper is based on the relations between projection Riccati equations and Weierstrass elliptic equation, combined with the Groebner bases in the symbolic computation. Then the novel method for constructing the Weierstrass elliptic solutions to the nonlinear evolution equations is given by using the above relations

Multiphase Weakly Nonlinear Geometric Optics for Schrödinger Equations

KAUST Repository

Carles, Ré mi; Dumas, Eric; Sparber, Christof

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.

Quantum hydrodynamics and nonlinear differential equations for degenerate Fermi gas

International Nuclear Information System (INIS)

Bettelheim, Eldad; Abanov, Alexander G; Wiegmann, Paul B

2008-01-01

We present new nonlinear differential equations for spacetime correlation functions of Fermi gas in one spatial dimension. The correlation functions we consider describe non-stationary processes out of equilibrium. The equations we obtain are integrable equations. They generalize known nonlinear differential equations for correlation functions at equilibrium [1-4] and provide vital tools for studying non-equilibrium dynamics of electronic systems. The method we developed is based only on Wick's theorem and the hydrodynamic description of the Fermi gas. Differential equations appear directly in bilinear form. (fast track communication)

From nonlinear Schroedinger hierarchy to some (2+1)-dimensional nonlinear pseudodifferential equations

International Nuclear Information System (INIS)

Yang Xiao; Du Dianlou

2010-01-01

The Poisson structure on C N xR N is introduced to give the Hamiltonian system associated with a spectral problem which yields the nonlinear Schroedinger (NLS) hierarchy. The Hamiltonian system is proven to be Liouville integrable. Some (2+1)-dimensional equations including NLS equation, Kadomtesev-Petviashvili I (KPI) equation, coupled KPI equation, and modified Kadomtesev-Petviashvili (mKP) equation, are decomposed into Hamilton flows via the NLS hierarchy. The algebraic curve, Abel-Jacobi coordinates, and Riemann-Jacobi inversion are used to obtain the algebrogeometric solutions of these equations.

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....

Nonlinear localized dust acoustic waves in a charge varying dusty plasma with nonthermal ions

International Nuclear Information System (INIS)

Tribeche, Mouloud; Amour, Rabia

2007-01-01

A numerical investigation is presented to show the existence, formation, and possible realization of large-amplitude dust acoustic (DA) solitary waves in a charge varying dusty plasma with nonthermal ions. These nonlinear localized structures are self-consistent solutions of the collisionless Vlasov equation with a population of fast particles. The spatial patterns of the variable charge DA solitary wave are significantly modified by the nonthermal effects. The results complement and provide new insights into previously published results on this problem

A procedure to construct exact solutions of nonlinear fractional differential equations.

Science.gov (United States)

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.

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.

Maintaining the stability of nonlinear differential equations by the enhancement of HPM

International Nuclear Information System (INIS)

Hosein Nia, S.H.; Ranjbar, A.N.; Ganji, D.D.; Soltani, H.; Ghasemi, J.

2008-01-01

Homotopy perturbation method is an effective method to find a solution of a nonlinear differential equation. In this method, a nonlinear complex differential equation is transformed to a series of linear and nonlinear parts, almost simpler differential equations. These sets of equations are then solved iteratively. Finally, a linear series of the solutions completes the answer if the convergence is maintained. In this Letter, the need for stability verification is shown through some examples. Consequently, HPM is enhanced by a preliminary assumption. The idea is to keep the inherent stability of nonlinear dynamic, even the selected linear part is not

Large Time Behavior of the Vlasov-Poisson-Boltzmann System

Directory of Open Access Journals (Sweden)

Li Li

2013-01-01

Full Text Available The motion of dilute charged particles can be modeled by Vlasov-Poisson-Boltzmann system. We study the large time stability of the VPB system. To be precise, we prove that when time goes to infinity, the solution of VPB system tends to global Maxwellian state in a rate Ot−∞, by using a method developed for Boltzmann equation without force in the work of Desvillettes and Villani (2005. The improvement of the present paper is the removal of condition on parameter λ as in the work of Li (2008.

Lattice Boltzmann model for high-order nonlinear partial differential equations.

Science.gov (United States)

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂_{t}ϕ+∑_{k=1}^{m}α_{k}∂_{x}^{k}Π_{k}(ϕ)=0 (1≤k≤m≤6), α_{k} are constant coefficients, Π_{k}(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009)1672-179910.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009)PHYADX0378-437110.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

Lattice Boltzmann model for high-order nonlinear partial differential equations

Science.gov (United States)

Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang

2018-01-01

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂tϕ +∑k=1mαk∂xkΠk(ϕ ) =0 (1 ≤k ≤m ≤6 ), αk are constant coefficients, Πk(ϕ ) are some known differential functions of ϕ . As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K (n ,n ) -Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009), 10.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009), 10.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

Evolutionary algorithm based heuristic scheme for nonlinear heat transfer equations.

Science.gov (United States)

Ullah, Azmat; Malik, Suheel Abdullah; Alimgeer, Khurram Saleem

2018-01-01

In this paper, a hybrid heuristic scheme based on two different basis functions i.e. Log Sigmoid and Bernstein Polynomial with unknown parameters is used for solving the nonlinear heat transfer equations efficiently. The proposed technique transforms the given nonlinear ordinary differential equation into an equivalent global error minimization problem. Trial solution for the given nonlinear differential equation is formulated using a fitness function with unknown parameters. The proposed hybrid scheme of Genetic Algorithm (GA) with Interior Point Algorithm (IPA) is opted to solve the minimization problem and to achieve the optimal values of unknown parameters. The effectiveness of the proposed scheme is validated by solving nonlinear heat transfer equations. The results obtained by the proposed scheme are compared and found in sharp agreement with both the exact solution and solution obtained by Haar Wavelet-Quasilinearization technique which witnesses the effectiveness and viability of the suggested scheme. Moreover, the statistical analysis is also conducted for investigating the stability and reliability of the presented scheme.

Evolutionary algorithm based heuristic scheme for nonlinear heat transfer equations.

Directory of Open Access Journals (Sweden)

Azmat Ullah

Full Text Available In this paper, a hybrid heuristic scheme based on two different basis functions i.e. Log Sigmoid and Bernstein Polynomial with unknown parameters is used for solving the nonlinear heat transfer equations efficiently. The proposed technique transforms the given nonlinear ordinary differential equation into an equivalent global error minimization problem. Trial solution for the given nonlinear differential equation is formulated using a fitness function with unknown parameters. The proposed hybrid scheme of Genetic Algorithm (GA with Interior Point Algorithm (IPA is opted to solve the minimization problem and to achieve the optimal values of unknown parameters. The effectiveness of the proposed scheme is validated by solving nonlinear heat transfer equations. The results obtained by the proposed scheme are compared and found in sharp agreement with both the exact solution and solution obtained by Haar Wavelet-Quasilinearization technique which witnesses the effectiveness and viability of the suggested scheme. Moreover, the statistical analysis is also conducted for investigating the stability and reliability of the presented scheme.

Modified harmonic balance method for the solution of nonlinear jerk equations

Science.gov (United States)

Rahman, M. Saifur; Hasan, A. S. M. Z.

2018-03-01

In this paper, a second approximate solution of nonlinear jerk equations (third order differential equation) can be obtained by using modified harmonic balance method. The method is simpler and easier to carry out the solution of nonlinear differential equations due to less number of nonlinear equations are required to solve than the classical harmonic balance method. The results obtained from this method are compared with those obtained from the other existing analytical methods that are available in the literature and the numerical method. The solution shows a good agreement with the numerical solution as well as the analytical methods of the available literature.

Simplified Model of Nonlinear Landau Damping

International Nuclear Information System (INIS)

Yampolsky, N.A.; Fisch, N.J.

2009-01-01

The nonlinear interaction of a plasma wave with resonant electrons results in a plateau in the electron distribution function close to the phase velocity of the plasma wave. As a result, Landau damping of the plasma wave vanishes and the resonant frequency of the plasma wave downshifts. However, this simple picture is invalid when the external driving force changes the plasma wave fast enough so that the plateau cannot be fully developed. A new model to describe amplification of the plasma wave including the saturation of Landau damping and the nonlinear frequency shift is proposed. The proposed model takes into account the change of the plasma wave amplitude and describes saturation of the Landau damping rate in terms of a single fluid equation, which simplifies the description of the inherently kinetic nature of Landau damping. A proposed fluid model, incorporating these simplifications, is verified numerically using a kinetic Vlasov code.

Damped nonlinear Schrodinger equation

International Nuclear Information System (INIS)

Nicholson, D.R.; Goldman, M.V.

1976-01-01

High frequency electrostatic plasma oscillations described by the nonlinear Schrodinger equation in the presence of damping, collisional or Landau, are considered. At early times, Landau damping of an initial soliton profile results in a broader, but smaller amplitude soliton, while collisional damping reduces the soliton size everywhere; soliton speeds at early times are unchanged by either kind of damping. For collisional damping, soliton speeds are unchanged for all time

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.

Numerical study of fractional nonlinear Schrodinger equations

KAUST Repository

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.

Nonlocal and nonlinear dispersion in a nonlinear Schrodinger-type equation: exotic solitons and short-wavelength instabilities

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...

Spectral transform and solvability of nonlinear evolution equations

International Nuclear Information System (INIS)

Degasperis, A.

1979-01-01

These lectures deal with an exciting development of the last decade, namely the resolving method based on the spectral transform which can be considered as an extension of the Fourier analysis to nonlinear evolution equations. Since many important physical phenomena are modeled by nonlinear partial wave equations this method is certainly a major breakthrough in mathematical physics. We follow the approach, introduced by Calogero, which generalizes the usual Wronskian relations for solutions of a Sturm-Liouville problem. Its application to the multichannel Schroedinger problem will be the subject of these lectures. We will focus upon dynamical systems described at time t by a multicomponent field depending on one space coordinate only. After recalling the Fourier technique for linear evolution equations we introduce the spectral transform method taking the integral equations of potential scattering as an example. The second part contains all the basic functional relationships between the fields and their spectral transforms as derived from the Wronskian approach. In the third part we discuss a particular class of solutions of nonlinear evolution equations, solitons, which are considered by many physicists as a first step towards an elementary particle theory, because of their particle-like behaviour. The effect of the polarization time-dependence on the motion of the soliton is studied by means of the corresponding spectral transform, leading to new concepts such as the 'boomeron' and the 'trappon'. The rich dynamic structure is illustrated by a brief report on the main results of boomeron-boomeron and boomeron-trappon collisions. In the final section we discuss further results concerning important properties of the solutions of basic nonlinear equations. We introduce the Baecklund transform for the special case of scalar fields and demonstrate how it can be used to generate multisoliton solutions and how the conservation laws are obtained. (HJ)

Non-linear partial differential equations an algebraic view of generalized solutions

CERN Document Server

Rosinger, Elemer E

1990-01-01

A massive transition of interest from solving linear partial differential equations to solving nonlinear ones has taken place during the last two or three decades. The availability of better computers has often made numerical experimentations progress faster than the theoretical understanding of nonlinear partial differential equations. The three most important nonlinear phenomena observed so far both experimentally and numerically, and studied theoretically in connection with such equations have been the solitons, shock waves and turbulence or chaotical processes. In many ways, these phenomen

Reduction of the equation for lower hybrid waves in a plasma to a nonlinear Schroedinger equation

Science.gov (United States)

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.

KINETIC-J: A computational kernel for solving the linearized Vlasov equation applied to calculations of the kinetic, configuration space plasma current for time harmonic wave electric fields

Science.gov (United States)

Green, David L.; Berry, Lee A.; Simpson, Adam B.; Younkin, Timothy R.

2018-04-01

We present the KINETIC-J code, a computational kernel for evaluating the linearized Vlasov equation with application to calculating the kinetic plasma response (current) to an applied time harmonic wave electric field. This code addresses the need for a configuration space evaluation of the plasma current to enable kinetic full-wave solvers for waves in hot plasmas to move beyond the limitations of the traditional Fourier spectral methods. We benchmark the kernel via comparison with the standard k →-space forms of the hot plasma conductivity tensor.

The application of He's exp-function method to a nonlinear differential-difference equation

International Nuclear Information System (INIS)

Dai Chaoqing; Cen Xu; Wu Shengsheng

2009-01-01

This paper applies He's exp-function method, which was originally proposed to find new exact travelling wave solutions of nonlinear partial differential equations (NPDEs) or coupled nonlinear partial differential equations (CNPDEs), to a nonlinear differential-difference equation, and some new travelling wave solutions are obtained.

Nonlinear Fredholm Integral Equation of the Second Kind with Quadrature Methods

Directory of Open Access Journals (Sweden)

M. Jafari Emamzadeh

2010-06-01

Full Text Available In this paper, a numerical method for solving the nonlinear Fredholm integral equation is presented. We intend to approximate the solution of this equation by quadrature methods and by doing so, we solve the nonlinear Fredholm integral equation more accurately. Several examples are given at the end of this paper

On the integrability of the generalized Fisher-type nonlinear diffusion equations

International Nuclear Information System (INIS)

Wang Dengshan; Zhang Zhifei

2009-01-01

In this paper, the geometric integrability and Lax integrability of the generalized Fisher-type nonlinear diffusion equations with modified diffusion in (1+1) and (2+1) dimensions are studied by the pseudo-spherical surface geometry method and prolongation technique. It is shown that the (1+1)-dimensional Fisher-type nonlinear diffusion equation is geometrically integrable in the sense of describing a pseudo-spherical surface of constant curvature -1 only for m = 2, and the generalized Fisher-type nonlinear diffusion equations in (1+1) and (2+1) dimensions are Lax integrable only for m = 2. This paper extends the results in Bindu et al 2001 (J. Phys. A: Math. Gen. 34 L689) and further provides the integrability information of (1+1)- and (2+1)-dimensional Fisher-type nonlinear diffusion equations for m = 2

The Volterra's integral equation theory for accelerator single-freedom nonlinear components

International Nuclear Information System (INIS)

Wang Sheng; Xie Xi

1996-01-01

The Volterra's integral equation equivalent to the dynamic equation of accelerator single-freedom nonlinear components is given, starting from which the transport operator of accelerator single-freedom nonlinear components and its inverse transport operator are obtained. Therefore, another algorithm for the expert system of the beam transport operator of accelerator single-freedom nonlinear components is developed

Study of the heavy ions (Au+Au at 150 AMeV) collisions with the FOPI detector. Comparison with the Landau-Vlasov model; Etude des collisions d`ions lourds AU+AU a 150 A.MeV avec le detecteur FOPI. Comparaison avec le modele de Landau-Vlasov

Energy Technology Data Exchange (ETDEWEB)

Boussange, S

1995-09-15

In this thesis, heavy ions (Au+Au) collisions experiments are made at 150 AMeV.In the first part, a general study of the nuclear matter equation is presented. Then the used Landau-Vlasov theoretical model is describe. The third part presents the FOPI experience and the details of how to obtain this theoretical predictions (filter, cuts, corrections, possible centrality selections).At the end, experimental results and comparisons with the Landau-Vlasov model are presented. (TEC). 105 refs., 96 figs., 14 tabs.

Current interactions from the one-form sector of nonlinear higher-spin equations

Science.gov (United States)

Gelfond, O. A.; Vasiliev, M. A.

2018-06-01

The form of higher-spin current interactions in the sector of one-forms is derived from the nonlinear higher-spin equations in AdS4. Quadratic corrections to higher-spin equations are shown to be independent of the phase of the parameter η = exp ⁡ iφ in the full nonlinear higher-spin equations. The current deformation resulting from the nonlinear higher-spin equations is represented in the canonical form with the minimal number of space-time derivatives. The non-zero spin-dependent coupling constants of the resulting currents are determined in terms of the higher-spin coupling constant η η bar . Our results confirm the conjecture that (anti-)self-dual nonlinear higher-spin equations result from the full system at (η = 0) η bar = 0.

Equating TIMSS Mathematics Subtests with Nonlinear Equating Methods Using NEAT Design: Circle-Arc Equating Approaches

Science.gov (United States)

Ozdemir, Burhanettin

2017-01-01

The purpose of this study is to equate Trends in International Mathematics and Science Study (TIMSS) mathematics subtest scores obtained from TIMSS 2011 to scores obtained from TIMSS 2007 form with different nonlinear observed score equating methods under Non-Equivalent Anchor Test (NEAT) design where common items are used to link two or more test…

Finding all solutions of nonlinear equations using the dual simplex method

Science.gov (United States)

Yamamura, Kiyotaka; Fujioka, Tsuyoshi

2003-03-01

Recently, an efficient algorithm has been proposed for finding all solutions of systems of nonlinear equations using linear programming. This algorithm is based on a simple test (termed the LP test) for nonexistence of a solution to a system of nonlinear equations using the dual simplex method. In this letter, an improved version of the LP test algorithm is proposed. By numerical examples, it is shown that the proposed algorithm could find all solutions of a system of 300 nonlinear equations in practical computation time.

Differential constraints and exact solutions of nonlinear diffusion equations

International Nuclear Information System (INIS)

Kaptsov, Oleg V; Verevkin, Igor V

2003-01-01

The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining equations used in the search for classical Lie symmetries

Allowable graphs of the nonlinear Schrödinger equation and their ...

Indian Academy of Sciences (India)

Bich Nguyen

2017-11-20

Nov 20, 2017 ... Non-linear Schrödinger equation; graphs; characteristic polynomial; .... Allowable graphs of the NLS and their applications. 795 ...... nonlinear Schroödinger equation, J. Algebra Appl. 16 (2017) 37 pp., https://doi.org/10.1142/.

Partially integrable nonlinear equations with one higher symmetry

International Nuclear Information System (INIS)

Mikhailov, A V; Novikov, V S; Wang, J P

2005-01-01

In this letter, we present a family of second order in time nonlinear partial differential equations, which have only one higher symmetry. These equations are not integrable, but have a solution depending on one arbitrary function. (letter to the editor)

Bifurcation of positive solutions to scalar reaction-diffusion equations with nonlinear boundary condition

Science.gov (United States)

Liu, Ping; Shi, Junping

2018-01-01

The bifurcation of non-trivial steady state solutions of a scalar reaction-diffusion equation with nonlinear boundary conditions is considered using several new abstract bifurcation theorems. The existence and stability of positive steady state solutions are proved using a unified approach. The general results are applied to a Laplace equation with nonlinear boundary condition and bistable nonlinearity, and an elliptic equation with superlinear nonlinearity and sublinear boundary conditions.

Role of statistical linearization in the solution of nonlinear stochastic equations

International Nuclear Information System (INIS)

Budgor, A.B.

1977-01-01

The solution of a generalized Langevin equation is referred to as a stochastic process. If the external forcing function is Gaussian white noise, the forward Kolmogarov equation yields the transition probability density function. Nonlinear problems must be handled by approximation procedures e.g., perturbation theories, eigenfunction expansions, and nonlinear optimization procedures. After some comments on the first two of these, attention is directed to the third, and the method of statistical linearization is used to demonstrate a relation to the former two. Nonlinear stochastic systems exhibiting sustained or forced oscillations and the centered nonlinear Schroedinger equation in the presence of Gaussian white noise excitation are considered as examples. 5 figures, 2 tables

On nonlinear differential equation with exact solutions having various pole orders

International Nuclear Information System (INIS)

Kudryashov, N.A.

2015-01-01

We consider a nonlinear ordinary differential equation having solutions with various movable pole order on the complex plane. We show that the pole order of exact solution is determined by values of parameters of the equation. Exact solutions in the form of the solitary waves for the second order nonlinear differential equation are found taking into account the method of the logistic function. Exact solutions of differential equations are discussed and analyzed

Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate elliptic equations

Directory of Open Access Journals (Sweden)

Espen R. Jakobsen

2002-05-01

Full Text Available Using the maximum principle for semicontinuous functions [3,4], we prove a general ``continuous dependence on the nonlinearities'' estimate for bounded Holder continuous viscosity solutions of fully nonlinear degenerate elliptic equations. Furthermore, we provide existence, uniqueness, and Holder continuity results for bounded viscosity solutions of such equations. Our results are general enough to encompass Hamilton-Jacobi-Bellman-Isaacs's equations of zero-sum, two-player stochastic differential games. An immediate consequence of the results obtained herein is a rate of convergence for the vanishing viscosity method for fully nonlinear degenerate elliptic equations.

Integrable peakon equations with cubic nonlinearity

International Nuclear Information System (INIS)

Hone, Andrew N W; Wang, J P

2008-01-01

We present a new integrable partial differential equation found by Vladimir Novikov. Like the Camassa-Holm and Degasperis-Procesi equations, this new equation admits peaked soliton (peakon) solutions, but it has nonlinear terms that are cubic, rather than quadratic. We give a matrix Lax pair for V Novikov's equation, and show how it is related by a reciprocal transformation to a negative flow in the Sawada-Kotera hierarchy. Infinitely many conserved quantities are found, as well as a bi-Hamiltonian structure. The latter is used to obtain the Hamiltonian form of the finite-dimensional system for the interaction of N peakons, and the two-body dynamics (N = 2) is explicitly integrated. Finally, all of this is compared with some analogous results for another cubic peakon equation derived by Zhijun Qiao. (fast track communication)

Nonclassical Symmetries for Nonlinear Partial Differential Equations via Compatibility

International Nuclear Information System (INIS)

El-Sabbagh, Mostafa F.; Ahmad, Ali T.

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 illustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries. (general)

Bright and dark soliton solutions for some nonlinear fractional differential equations

International Nuclear Information System (INIS)

Guner, Ozkan; Bekir, Ahmet

2016-01-01

In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space–time fractional modified Benjamin–Bona–Mahoney (mBBM) equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions: amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional derivatives are described in the modified Riemann–Liouville sense. (paper)

Any order approximate analytical solution of the nonlinear Volterra's integral equation for accelerator dynamic systems

International Nuclear Information System (INIS)

Liu Chunliang; Xie Xi; Chen Yinbao

1991-01-01

The universal nonlinear dynamic system equation is equivalent to its nonlinear Volterra's integral equation, and any order approximate analytical solution of the nonlinear Volterra's integral equation is obtained by exact analytical method, thus giving another derivation procedure as well as another computation algorithm for the solution of the universal nonlinear dynamic system equation

Approximate Solution of Nonlinear Klein-Gordon Equation Using Sobolev Gradients

Directory of Open Access Journals (Sweden)

Nauman Raza

2016-01-01

Full Text Available The nonlinear Klein-Gordon equation (KGE models many nonlinear phenomena. In this paper, we propose a scheme for numerical approximation of solutions of the one-dimensional nonlinear KGE. A common approach to find a solution of a nonlinear system is to first linearize the equations by successive substitution or the Newton iteration method and then solve a linear least squares problem. Here, we show that it can be advantageous to form a sum of squared residuals of the nonlinear problem and then find a zero of the gradient. Our scheme is based on the Sobolev gradient method for solving a nonlinear least square problem directly. The numerical results are compared with Lattice Boltzmann Method (LBM. The L2, L∞, and Root-Mean-Square (RMS values indicate better accuracy of the proposed method with less computational effort.

New exact solutions for two nonlinear equations

International Nuclear Information System (INIS)

Wang Quandi; Tang Minying

2008-01-01

In this Letter, we investigate two nonlinear equations given by u t -u xxt +3u 2 u x =2u x u xx +uu xxx and u t -u xxt +4u 2 u x =3u x u xx +uu xxx . Through some special phase orbits we obtain four new exact solutions for each equation above. Some previous results are extended

Entire solutions of nonlinear differential-difference equations.

Science.gov (United States)

Li, Cuiping; Lü, Feng; Xu, Junfeng

2016-01-01

In this paper, we describe the properties of entire solutions of a nonlinear differential-difference equation and a Fermat type equation, and improve several previous theorems greatly. In addition, we also deduce a uniqueness result for an entire function f(z) that shares a set with its shift [Formula: see text], which is a generalization of a result of Liu.

Nonlinear stability of source defects in the complex Ginzburg–Landau equation

International Nuclear Information System (INIS)

Beck, Margaret; Nguyen, Toan T; Sandstede, Björn; Zumbrun, Kevin

2014-01-01

In an appropriate moving coordinate frame, source defects are time-periodic solutions to reaction–diffusion equations that are spatially asymptotic to spatially periodic wave trains whose group velocities point away from the core of the defect. In this paper, we rigorously establish nonlinear stability of spectrally stable source defects in the complex Ginzburg–Landau equation. Due to the outward transport at the far field, localized perturbations may lead to a highly non-localized response even on the linear level. To overcome this, we first investigate in detail the dynamics of the solution to the linearized equation. This allows us to determine an approximate solution that satisfies the full equation up to and including quadratic terms in the nonlinearity. This approximation utilizes the fact that the non-localized phase response, resulting from the embedded zero eigenvalues, can be captured, to leading order, by the nonlinear Burgers equation. The analysis is completed by obtaining detailed estimates for the resolvent kernel and pointwise estimates for Green's function, which allow one to close a nonlinear iteration scheme. (paper)

Universal and integrable nonlinear evolution systems of equations in 2+1 dimensions

International Nuclear Information System (INIS)

Maccari, A.

1997-01-01

Integrable systems of nonlinear partial differential equations (PDEs) are obtained from integrable equations in 2+1 dimensions, by means of a reduction method of broad applicability based on Fourier expansion and spatio endash temporal rescalings, which is asymptotically exact in the limit of weak nonlinearity. The integrability by the spectral transform is explicitly demonstrated, because the corresponding Lax pairs have been derived, applying the same reduction method to the Lax pair of the initial equation. These systems of nonlinear PDEs are likely to be of applicative relevance and have a open-quotes universalclose quotes character, inasmuch as they may be derived from a very large class of nonlinear evolution equations with a linear dispersive part. copyright 1997 American Institute of Physics

The presentation of explicit analytical solutions of a class of nonlinear evolution equations

International Nuclear Information System (INIS)

Feng Jinshun; Guo Mingpu; Yuan Deyou

2009-01-01

In this paper, we introduce a function set Ω m . There is a conjecture that an arbitrary explicit travelling-wave analytical solution of a real constant coefficient nonlinear evolution equation is necessarily a linear (or nonlinear) combination of the product of some elements in Ω m . A widespread applicable approach for solving a class of nonlinear evolution equations is established. The new analytical solutions to two kinds of nonlinear evolution equations are described with the aid of the guess.

Exact non-linear equations for cosmological perturbations

Energy Technology Data Exchange (ETDEWEB)

Gong, Jinn-Ouk [Asia Pacific Center for Theoretical Physics, Pohang 37673 (Korea, Republic of); Hwang, Jai-chan [Department of Astronomy and Atmospheric Sciences, Kyungpook National University, Daegu 41566 (Korea, Republic of); Noh, Hyerim [Korea Astronomy and Space Science Institute, Daejeon 34055 (Korea, Republic of); Wu, David Chan Lon; Yoo, Jaiyul, E-mail: jinn-ouk.gong@apctp.org, E-mail: jchan@knu.ac.kr, E-mail: hr@kasi.re.kr, E-mail: clwu@physik.uzh.ch, E-mail: jyoo@physik.uzh.ch [Center for Theoretical Astrophysics and Cosmology, Institute for Computational Science, Universität Zürich, CH-8057 Zürich (Switzerland)

2017-10-01

We present a complete set of exact and fully non-linear equations describing all three types of cosmological perturbations—scalar, vector and tensor perturbations. We derive the equations in a thoroughly gauge-ready manner, so that any spatial and temporal gauge conditions can be employed. The equations are completely general without any physical restriction except that we assume a flat homogeneous and isotropic universe as a background. We also comment briefly on the application of our formulation to the non-expanding Minkowski background.

A Generalized Halanay Inequality for Stability of Nonlinear Neutral Functional Differential Equations

Directory of Open Access Journals (Sweden)

Wansheng Wang

2010-01-01

Full Text Available This paper is devoted to generalize Halanay's inequality which plays an important rule in study of stability of differential equations. By applying the generalized Halanay inequality, the stability results of nonlinear neutral functional differential equations (NFDEs and nonlinear neutral delay integrodifferential equations (NDIDEs are obtained.

Cross-constrained problems for nonlinear Schrodinger equation with harmonic potential

Directory of Open Access Journals (Sweden)

Runzhang Xu

2012-11-01

Full Text Available This article studies a nonlinear Schodinger equation with harmonic potential by constructing different cross-constrained problems. By comparing the different cross-constrained problems, we derive different sharp criterion and different invariant manifolds that separate the global solutions and blowup solutions. Moreover, we conclude that some manifolds are empty due to the essence of the cross-constrained problems. Besides, we compare the three cross-constrained problems and the three depths of the potential wells. In this way, we explain the gaps in [J. Shu and J. Zhang, Nonlinear Shrodinger equation with harmonic potential, Journal of Mathematical Physics, 47, 063503 (2006], which was pointed out in [R. Xu and Y. Liu, Remarks on nonlinear Schrodinger equation with harmonic potential, Journal of Mathematical Physics, 49, 043512 (2008].

Numerical Simulations of Light Bullets, Using The Full Vector, Time Dependent, Nonlinear Maxwell Equations

Science.gov (United States)

Goorjian, Peter M.; Silberberg, Yaron; Kwak, Dochan (Technical Monitor)

1995-01-01

This paper will present results in computational nonlinear optics. An algorithm will be described that solves the full vector nonlinear Maxwell's equations exactly without the approximations that we currently made. Present methods solve a reduced scalar wave equation, namely the nonlinear Schrodinger equation, and neglect the optical carrier. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of 'light bullet' like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and can take into account such quantum effects as Karr and Raman interactions. The present approach is robust and should permit modeling 2-D and 3-D optical soliton propagation, scattering, and switching directly from the full-vector Maxwell's equations.

Exact solutions of some nonlinear partial differential equations using ...

Indian Academy of Sciences (India)

Nonlinear partial differential equations (NPDEs) are encountered in various ... such as physics, mechanics, chemistry, biology, mathematics and engineering. ... In §3, this method is applied to the generalized forms of Klein–Gordon equation,.

An analytical method for solving exact solutions of the nonlinear Bogoyavlenskii equation and the nonlinear diffusive predator–prey system

Directory of Open Access Journals (Sweden)

Md. Nur Alam

2016-06-01

Full Text Available In this article, we apply the exp(-Φ(ξ-expansion method to construct many families of exact solutions of nonlinear evolution equations (NLEEs via the nonlinear diffusive predator–prey system and the Bogoyavlenskii equations. These equations can be transformed to nonlinear ordinary differential equations. As a result, some new exact solutions are obtained through the hyperbolic function, the trigonometric function, the exponential functions and the rational forms. If the parameters take specific values, then the solitary waves are derived from the traveling waves. Also, we draw 2D and 3D graphics of exact solutions for the special diffusive predator–prey system and the Bogoyavlenskii equations by the help of programming language Maple.

Nonlinear delta f Simulations of Collective Effects in Intense Charged Particle Beams

CERN Document Server

Hong Qi

2003-01-01

A nonlinear delta(f) particle simulation method based on the Vlasov-Maxwell equations has been recently developed to study collective processes in high-intensity beams, where space-charge and magnetic self-field effects play a critical role in determining the nonlinear beam dynamics. Implemented in the Beam Equilibrium, Stability and Transport (BEST) code [H. Qin, R.C. Davidson, and W.W. Lee, Physical Review -- Special Topics on Accelerator and Beams 3 (2000) 084401; 3 (2000) 109901.], the nonlinear delta(f) method provides a low-noise and self-consistent tool for simulating collective interactions and nonlinear dynamics of high-intensity beams in modern and next-generation accelerators and storage rings, such as the Spallation Neutron Source and heavy ion fusion drivers. A wide range of linear eigenmodes of high-intensity charged-particle beams can be systematically studied using the BEST code. Simulation results for the electron-proton two-stream instability in the Proton Storage Ring experiment [R. Macek, ...

Mathematic study and numerical implantation of the Vlasov-Darwin model

International Nuclear Information System (INIS)

Sonnendrucker, E.

1994-12-01

Numerical simulation of some phenomena in plasma physics, or more generally in electromagnetism, can be more easily done using approximate models of Maxwell equations such as the Darwin model in which the transverse part of the displacement current in the Ampere equation is neglected, or such as the static model in which the time derivatives are neglected. In this note, the Darwin model is presented first, and then an asymptotic analysis of Maxwell equations is given with limit conditions of perfect conductor on one part of the side, and Silver-Muller absorbing conditions on the other part. This allows to obtain a variational formulation for the Darwin model which is a good approximation of Maxwell equations. A variational formulation for the quasi-static model is also obtained. In a second part this implantation is described using a 2-D finite element method coupled with a particulate method for the Vlasov equations which leads to numerical results allowing a determination of the different models application. (J.S.). 2 refs

Large-time asymptotic behaviour of solutions of non-linear Sobolev-type equations

International Nuclear Information System (INIS)

Kaikina, Elena I; Naumkin, Pavel I; Shishmarev, Il'ya A

2009-01-01

The large-time asymptotic behaviour of solutions of the Cauchy problem is investigated for a non-linear Sobolev-type equation with dissipation. For small initial data the approach taken is based on a detailed analysis of the Green's function of the linear problem and the use of the contraction mapping method. The case of large initial data is also closely considered. In the supercritical case the asymptotic formulae are quasi-linear. The asymptotic behaviour of solutions of a non-linear Sobolev-type equation with a critical non-linearity of the non-convective kind differs by a logarithmic correction term from the behaviour of solutions of the corresponding linear equation. For a critical convective non-linearity, as well as for a subcritical non-convective non-linearity it is proved that the leading term of the asymptotic expression for large times is a self-similar solution. For Sobolev equations with convective non-linearity the asymptotic behaviour of solutions in the subcritical case is the product of a rarefaction wave and a shock wave. Bibliography: 84 titles.

A bivariate Chebyshev spectral collocation quasilinearization method for nonlinear evolution parabolic equations.

Science.gov (United States)

Motsa, S S; Magagula, V M; Sibanda, P

2014-01-01

This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.

A Bivariate Chebyshev Spectral Collocation Quasilinearization Method for Nonlinear Evolution Parabolic Equations

Directory of Open Access Journals (Sweden)

S. S. Motsa

2014-01-01

Full Text Available This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs. The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.

Piecewise-linear and bilinear approaches to nonlinear differential equations approximation problem of computational structural mechanics

OpenAIRE

Leibov Roman

2017-01-01

This paper presents a bilinear approach to nonlinear differential equations system approximation problem. Sometimes the nonlinear differential equations right-hand sides linearization is extremely difficult or even impossible. Then piecewise-linear approximation of nonlinear differential equations can be used. The bilinear differential equations allow to improve piecewise-linear differential equations behavior and reduce errors on the border of different linear differential equations systems ...

Nonlinear evolution equations having a physical meaning

International Nuclear Information System (INIS)

Nakach, R.

1976-06-01

The non stationary self-similar solutions of the nonlinear evolution equations which can be solved by the inverse scattering method are studied. It turns out, as shown by means of several examples, that when the L linear operator associated with these equations, is of second order and only then, the self-similar solutions can be expressed in terms of the various Painleve's transcendents [fr

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

Institute of Scientific and Technical Information of China (English)

无

2011-01-01

The initial value problem of a nonlinear fractional differential equation is discussed in this paper. Using the nonlinear alternative of Leray-Schauder type and the contraction mapping principle,we obtain the existence and uniqueness of solutions to the fractional differential equation,which extend some results of the previous papers.

Response to Comment on 'On Higher-Order Corrections to Gyrokinetic Vlasov-Poisson Equations in the Long Wavelength Limit [Phys. Plasmas 16,044506 (2009)]'

International Nuclear Information System (INIS)

Lee, W.W.; Kolesnikov, R.A.

2009-01-01

We show in this Response that the nonlinear Poisson's equation in our original paper derived from the drift kinetic approach can be verified by using the nonlinear gyrokinetic Poisson's equation of Dubin et al. (Phys. Fluids 26, 3524 (1983)). This nonlinear contribution in φ 2 is indeed of the order of k # perpendicular# 4 in the long wavelength limit and remains finite for zero ion temperature, in contrast to the nonlinear term by Parra and Catto (Plasma Phys. Control. Fusion 50, 065014 (2008)), which is of the order of k # perpendicular# 2 and diverges for T i → 0. For comparison, the leading term for the gyrokinetic Poisson's equation in this limit is of the order of k # perpendicular# 2 φ.

Nonlinear wave equations, formation of singularities

CERN Document Server

John, Fritz

1990-01-01

This is the second volume in the University Lecture Series, designed to make more widely available some of the outstanding lectures presented in various institutions around the country. Each year at Lehigh University, a distinguished mathematical scientist presents the Pitcher Lectures in the Mathematical Sciences. This volume contains the Pitcher lectures presented by Fritz John in April 1989. The lectures deal with existence in the large of solutions of initial value problems for nonlinear hyperbolic partial differential equations. As is typical with nonlinear problems, there are many results and few general conclusions in this extensive subject, so the author restricts himself to a small portion of the field, in which it is possible to discern some general patterns. Presenting an exposition of recent research in this area, the author examines the way in which solutions can, even with small and very smooth initial data, "blow up" after a finite time. For various types of quasi-linear equations, this time de...

An introduction to geometric theory of fully nonlinear parabolic equations

International Nuclear Information System (INIS)

Lunardi, A.

1991-01-01

We study a class of nonlinear evolution equations in general Banach space being an abstract version of fully nonlinear parabolic equations. In addition to results of existence, uniqueness and continuous dependence on the data, we give some qualitative results about stability of the stationary solutions, existence and stability of the periodic orbits. We apply such results to some parabolic problems arising from combustion theory. (author). 24 refs

Complex nonlinear Lagrangian for the Hasegawa-Mima equation

International Nuclear Information System (INIS)

Dewar, R.L.; Abdullatif, R.F.; Sangeetha, G.G.

2005-01-01

The Hasegawa-Mima equation is the simplest nonlinear single-field model equation that captures the essence of drift wave dynamics. Like the Schroedinger equation it is first order in time. However its coefficients are real, so if the potential φ is initially real it remains real. However, by embedding φ in the space of complex functions a simple Lagrangian is found from which the Hasegawa-Mima equation may be derived from Hamilton's Principle. This Lagrangian is used to derive an action conservation equation which agrees with that of Biskamp and Horton. (author)

Soliton solutions of some nonlinear evolution equations with time ...

Indian Academy of Sciences (India)

Abstract. In this paper, we obtain exact soliton solutions of the modified KdV equation, inho- mogeneous nonlinear Schrödinger equation and G(m, n) equation with variable coefficients using solitary wave ansatz. The constraint conditions among the time-dependent coefficients turn out as necessary conditions for the ...

Backward stochastic differential equations from linear to fully nonlinear theory

CERN Document Server

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.

Picone-type inequalities for nonlinear elliptic equations and their applications

Directory of Open Access Journals (Sweden)

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.

Quantum theory from a nonlinear perspective Riccati equations in fundamental physics

CERN Document Server

Schuch, Dieter

2018-01-01

This book provides a unique survey displaying the power of Riccati equations to describe reversible and irreversible processes in physics and, in particular, quantum physics. Quantum mechanics is supposedly linear, invariant under time-reversal, conserving energy and, in contrast to classical theories, essentially based on the use of complex quantities. However, on a macroscopic level, processes apparently obey nonlinear irreversible evolution equations and dissipate energy. The Riccati equation, a nonlinear equation that can be linearized, has the potential to link these two worlds when applied to complex quantities. The nonlinearity can provide information about the phase-amplitude correlations of the complex quantities that cannot be obtained from the linearized form. As revealed in this wide ranging treatment, Riccati equations can also be found in many diverse fields of physics from Bose-Einstein-condensates to cosmology. The book will appeal to graduate students and theoretical physicists interested in ...

Modified wave operators for nonlinear Schrodinger equations in one and two dimensions

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Nakao Hayashi

2004-04-01

Full Text Available We study the asymptotic behavior of solutions, in particular the scattering theory, for the nonlinear Schr"{o}dinger equations with cubic and quadratic nonlinearities in one or two space dimensions. The nonlinearities are summation of gauge invariant term and non-gauge invariant terms. The scattering problem of these equations belongs to the long range case. We prove the existence of the modified wave operators to those equations for small final data. Our result is an improvement of the previous work [13

An effective method for finding special solutions of nonlinear differential equations with variable coefficients

International Nuclear Information System (INIS)

Qin Maochang; Fan Guihong

2008-01-01

There are many interesting methods can be utilized to construct special solutions of nonlinear differential equations with constant coefficients. However, most of these methods are not applicable to nonlinear differential equations with variable coefficients. A new method is presented in this Letter, which can be used to find special solutions of nonlinear differential equations with variable coefficients. This method is based on seeking appropriate Bernoulli equation corresponding to the equation studied. Many well-known equations are chosen to illustrate the application of this method

Multiple solutions to some singular nonlinear Schrodinger equations

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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.

Stochastic effects on the nonlinear Schroedinger equation

International Nuclear Information System (INIS)

Flessas, G P; Leach, P G L; Yannacopoulos, A N

2004-01-01

The aim of this article is to provide a brief review of recent advances in the field of stochastic effects on the nonlinear Schroedinger equation. The article reviews rigorous and perturbative results. (review article)

Application of Exp-function method for (2 + 1)-dimensional nonlinear evolution equations

International Nuclear Information System (INIS)

Bekir, Ahmet; Boz, Ahmet

2009-01-01

In this paper, the Exp-function method is used to construct solitary and soliton solutions of (2 + 1)-dimensional nonlinear evolution equations. (2 + 1)-dimensional breaking soliton (Calogero) equation, modified Zakharov-Kuznetsov and Konopelchenko-Dubrovsky equations are chosen to illustrate the effectiveness of the method. The method is straightforward and concise, and its applications are promising. The Exp-function method presents a wider applicability for handling nonlinear wave equations.

Semiclassical soliton ensembles for the focusing nonlinear Schrödinger equation

CERN Document Server

Kamvissis, Spyridon; Miller, Peter D

2003-01-01

This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrödinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing

Solving Nonlinear Partial Differential Equations with Maple and Mathematica

CERN Document Server

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

Nonlinear dynamics in the Einstein-Friedmann equation

International Nuclear Information System (INIS)

Tanaka, Yosuke; Mizuno, Yuji; Ohta, Shigetoshi; Mori, Keisuke; Horiuchi, Tanji

2009-01-01

We have studied the gravitational field equations on the basis of general relativity and nonlinear dynamics. The space component of the Einstein-Friedmann equation shows the chaotic behaviours in case the following conditions are satisfied: (i)the expanding ratio: h=x . /x max = +0.14) for the occurrence of the chaotic behaviours in the Einstein-Friedmann equation (0 ≤ λ ≤ +0.14). The numerical calculations are performed with the use of the Microsoft EXCEL(2003), and the results are shown in the following cases; λ = 2b = +0.06 and +0.14.

Oscillation criteria for third order delay nonlinear differential equations

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E. M. Elabbasy

2012-01-01

via comparison with some first differential equations whose oscillatory characters are known. Our results generalize and improve some known results for oscillation of third order nonlinear differential equations. Some examples are given to illustrate the main results.

Solution of linear and nonlinear matrix systems. Application to a nonlinear diffusion equation

International Nuclear Information System (INIS)

Bonnet, M.; Meurant, G.

1978-01-01

Different methods of solution of linear and nonlinear algebraic systems are applied to the nonlinear system obtained by discretizing a nonlinear diffusion equation. For linear systems, methods in general use of alternating directions type or Gauss Seidel's methods are compared to more recent ones of the type of generalized conjugate gradient; the superiority of the latter is shown by numerical examples. For nonlinear systems, a method on nonlinear conjugate gradient is studied as also Newton's method and some of its variants. It should be noted, however that Newton's method is found to be more efficient when coupled with a good method for solution of the linear system. To conclude, such methods are used to solve a nonlinear diffusion problem and the numerical results obtained are to be compared [fr

Solution of linear and nonlinear matrix systems. Application to a nonlinear diffusion equation

International Nuclear Information System (INIS)

Bonnet, M.; Meurant, G.

1978-01-01

The object of this study is to compare different methods of solving linear and nonlinear algebraic systems and to apply them to the nonlinear system obtained by discretizing a nonlinear diffusion equation. For linear systems the conventional methods of alternating direction type or Gauss Seidel's methods are compared to more recent ones of the type of generalized conjugate gradient; the superiority of the latter is shown by numerical examples. For nonlinear systems, a method of nonlinear conjugate gradient is studied together with Newton's method and some of its variants. It should be noted, however, that Newton's method is found to be more efficient when coupled with a good method for solving the linear system. As a conclusion, these methods are used to solve a nonlinear diffusion problem and the numerical results obtained are compared [fr

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.

Nonlinear anisotropic parabolic equations in Lm

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Fares Mokhtari

2014-01-01

Full Text Available In this paper, we give a result of regularity of weak solutions for a class of nonlinear anisotropic parabolic equations with lower-order term when the right-hand side is an Lm function, with m being ”small”. This work generalizes some results given in [2] and [3].

From nonlinear Schrödinger hierarchy to some (2+1)-dimensional nonlinear pseudodifferential equations

Science.gov (United States)

Yang, Xiao; Du, Dianlou

2010-08-01

The Poisson structure on CN×RN is introduced to give the Hamiltonian system associated with a spectral problem which yields the nonlinear Schrödinger (NLS) hierarchy. The Hamiltonian system is proven to be Liouville integrable. Some (2+1)-dimensional equations including NLS equation, Kadomtesev-Petviashvili I (KPI) equation, coupled KPI equation, and modified Kadomtesev-Petviashvili (mKP) equation, are decomposed into Hamilton flows via the NLS hierarchy. The algebraic curve, Abel-Jacobi coordinates, and Riemann-Jacobi inversion are used to obtain the algebrogeometric solutions of these equations.

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.

Oscillation criteria for third order nonlinear delay differential equations with damping

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Said R. Grace

2015-01-01

Full Text Available This note is concerned with the oscillation of third order nonlinear delay differential equations of the form \\[\\label{*} \\left( r_{2}(t\\left( r_{1}(ty^{\\prime}(t\\right^{\\prime}\\right^{\\prime}+p(ty^{\\prime}(t+q(tf(y(g(t=0.\\tag{\\(\\ast\\}\\] In the papers [A. Tiryaki, M. F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007, 54-68] and [M. F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear functional differential equations, Applied Math. Letters 23 (2010, 756-762], the authors established some sufficient conditions which insure that any solution of equation (\\(\\ast\\ oscillates or converges to zero, provided that the second order equation \\[\\left( r_{2}(tz^{\\prime }(t\\right^{\\prime}+\\left(p(t/r_{1}(t\\right z(t=0\\tag{\\(\\ast\\ast\\}\\] is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation (\\(\\ast\\ oscillates if equation (\\(\\ast\\ast\\ is nonoscillatory. We also establish results for the oscillation of equation (\\(\\ast\\ when equation (\\(\\ast\\ast\\ is oscillatory.

Superdiffusions and positive solutions of nonlinear partial differential equations

CERN Document Server

Dynkin, E B

2004-01-01

This book is devoted to the applications of probability theory to the theory of nonlinear partial differential equations. More precisely, it is shown that all positive solutions for a class of nonlinear elliptic equations in a domain are described in terms of their traces on the boundary of the domain. The main probabilistic tool is the theory of superdiffusions, which describes a random evolution of a cloud of particles. A substantial enhancement of this theory is presented that can be of interest for everybody who works on applications of probabilistic methods to mathematical analysis.

Topological soliton solutions for some nonlinear evolution equations

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Ahmet Bekir

2014-03-01

Full Text Available In this paper, the topological soliton solutions of nonlinear evolution equations are obtained by the solitary wave ansatz method. Under some parameter conditions, exact solitary wave solutions are obtained. Note that it is always useful and desirable to construct exact solutions especially soliton-type (dark, bright, kink, anti-kink, etc. envelope for the understanding of most nonlinear physical phenomena.

An ansatz for solving nonlinear partial differential equations in mathematical physics.

Science.gov (United States)

Akbar, M Ali; Ali, Norhashidah Hj Mohd

2016-01-01

In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin-Bona-Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems.

Multi-soliton management by the integrable nonautonomous nonlinear integro-differential Schrödinger equation

International Nuclear Information System (INIS)

Zhang, Yu-Juan; Zhao, Dun; Luo, Hong-Gang

2014-01-01

We consider a wide class of integrable nonautonomous nonlinear integro-differential Schrödinger equation which contains the models for the soliton management in Bose–Einstein condensates, nonlinear optics, and inhomogeneous Heisenberg spin chain. With the help of the nonisospectral AKNS hierarchy, we obtain the N-fold Darboux transformation and the N-fold soliton-like solutions for the equation. The soliton management, especially the synchronized dispersive and nonlinear management in optical fibers is discussed. It is found that in the situation without external potential, the synchronized dispersive and nonlinear management can keep the integrability of the nonlinear Schrödinger equation; this suggests that in optical fibers, the synchronized dispersive and nonlinear management can control and maintain the propagation of a multi-soliton. - Highlights: • We consider a unified model for soliton management by an integrable integro-differential Schrödinger equation. • Using Lax pair, the N-fold Darboux transformation for the equation is presented. • The multi-soliton management is considered. • The synchronized dispersive and nonlinear management is suggested

Approximate Solutions of Nonlinear Partial Differential Equations by Modified q-Homotopy Analysis Method

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Shaheed N. Huseen

2013-01-01

Full Text Available A modified q-homotopy analysis method (mq-HAM was proposed for solving nth-order nonlinear differential equations. This method improves the convergence of the series solution in the nHAM which was proposed in (see Hassan and El-Tawil 2011, 2012. The proposed method provides an approximate solution by rewriting the nth-order nonlinear differential equation in the form of n first-order differential equations. The solution of these n differential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.

Nonlinear heat conduction equations with memory: Physical meaning and analytical results

Science.gov (United States)

Artale Harris, Pietro; Garra, Roberto

2017-06-01

We study nonlinear heat conduction equations with memory effects within the framework of the fractional calculus approach to the generalized Maxwell-Cattaneo law. Our main aim is to derive the governing equations of heat propagation, considering both the empirical temperature-dependence of the thermal conductivity coefficient (which introduces nonlinearity) and memory effects, according to the general theory of Gurtin and Pipkin of finite velocity thermal propagation with memory. In this framework, we consider in detail two different approaches to the generalized Maxwell-Cattaneo law, based on the application of long-tail Mittag-Leffler memory function and power law relaxation functions, leading to nonlinear time-fractional telegraph and wave-type equations. We also discuss some explicit analytical results to the model equations based on the generalized separating variable method and discuss their meaning in relation to some well-known results of the ordinary case.

Exact solutions of nonlinear fractional differential equations by (G′/G)-expansion method

International Nuclear Information System (INIS)

Bekir Ahmet; Güner Özkan

2013-01-01

In this paper, we use the fractional complex transform and the (G′/G)-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie's modified Riemann—Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations

Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations

Science.gov (United States)

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.

Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations

International Nuclear Information System (INIS)

Indekeu, Joseph O; Smets, Ruben

2017-01-01

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. (paper)

Inverse scattering solution of non-linear evolution equations in one space dimension: an introduction

International Nuclear Information System (INIS)

Alvarez-Estrada, R.F.

1979-01-01

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

The Use of Nonlinear Constitutive Equations to Evaluate Draw Resistance and Filter Ventilation

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Eitzinger B

2014-12-01

Full Text Available This study investigates by nonlinear constitutive equations the influence of tipping paper, cigarette paper, filter, and tobacco rod on the degree of filter ventilation and draw resistance. Starting from the laws of conservation, the path to the theory of fluid dynamics in porous media and Darcy's law is reviewed and, as an extension to Darcy's law, two different nonlinear pressure drop-flow relations are proposed. It is proven that these relations are valid constitutive equations and the partial differential equations for the stationary flow in an unlit cigarette covering anisotropic, inhomogeneous and nonlinear behaviour are derived. From these equations a system of ordinary differential equations for the one-dimensional flow in the cigarette is derived by averaging pressure and velocity over the cross section of the cigarette. By further integration, the concept of an electrical analog is reached and discussed in the light of nonlinear pressure drop-flow relations. By numerical calculations based on the system of ordinary differential equations, it is shown that the influence of nonlinearities cannot be neglected because variations in the degree of filter ventilation can reach up to 20% of its nominal value.

An Integrable Discrete Generalized Nonlinear Schrödinger Equation and Its Reductions

International Nuclear Information System (INIS)

Li Hong-Min; Li Yu-Qi; Chen Yong

2014-01-01

An integrable discrete system obtained by the algebraization of the difference operator is studied. The system is named discrete generalized nonlinear Schrödinger (GNLS) equation, which can be reduced to classical discrete nonlinear Schrödinger (NLS) equation. Furthermore, all of the linear reductions for the discrete GNLS equation are given through the theory of circulant matrices and the discrete NLS equation is obtained by one of the reductions. At the same time, the recursion operator and symmetries of continuous GNLS equation are successfully recovered by its corresponding discrete ones. (general)

A family of analytical solutions of a nonlinear diffusion-convection equation

Science.gov (United States)

Hayek, Mohamed

2018-01-01

Despite its popularity in many engineering fields, the nonlinear diffusion-convection equation has no general analytical solutions. This work presents a family of closed-form analytical traveling wave solutions for the nonlinear diffusion-convection equation with power law nonlinearities. This kind of equations typically appears in nonlinear problems of flow and transport in porous media. The solutions that are addressed are simple and fully analytical. Three classes of analytical solutions are presented depending on the type of the nonlinear diffusion coefficient (increasing, decreasing or constant). It has shown that the structure of the traveling wave solution is strongly related to the diffusion term. The main advantage of the proposed solutions is that they are presented in a unified form contrary to existing solutions in the literature where the derivation of each solution depends on the specific values of the diffusion and convection parameters. The proposed closed-form solutions are simple to use, do not require any numerical implementation, and may be implemented in a simple spreadsheet. The analytical expressions are also useful to mathematically analyze the structure and properties of the solutions.

Nonlinear evolution equations for waves in random media

International Nuclear Information System (INIS)

Pelinovsky, E.; Talipova, T.

1994-01-01

The scope of this paper is to highlight the main ideas of asymptotical methods applying in modern approaches of description of nonlinear wave propagation in random media. We start with the discussion of the classical conception of ''mean field''. Then an exactly solvable model describing nonlinear wave propagation in the medium with fluctuating parameters is considered in order to demonstrate that the ''mean field'' method is not correct. We develop new asymptotic procedures of obtaining the nonlinear evolution equations for the wave fields in random media. (author). 16 refs

Explicit high-order non-canonical symplectic particle-in-cell algorithms for Vlasov-Maxwell systems

International Nuclear Information System (INIS)

Xiao, Jianyuan; Liu, Jian; He, Yang; Zhang, Ruili; Qin, Hong; Sun, Yajuan

2015-01-01

Explicit high-order non-canonical symplectic particle-in-cell algorithms for classical particle-field systems governed by the Vlasov-Maxwell equations are developed. The algorithms conserve a discrete non-canonical symplectic structure derived from the Lagrangian of the particle-field system, which is naturally discrete in particles. The electromagnetic field is spatially discretized using the method of discrete exterior calculus with high-order interpolating differential forms for a cubic grid. The resulting time-domain Lagrangian assumes a non-canonical symplectic structure. It is also gauge invariant and conserves charge. The system is then solved using a structure-preserving splitting method discovered by He et al. [preprint http://arxiv.org/abs/arXiv:1505.06076 (2015)], which produces five exactly soluble sub-systems, and high-order structure-preserving algorithms follow by combinations. The explicit, high-order, and conservative nature of the algorithms is especially suitable for long-term simulations of particle-field systems with extremely large number of degrees of freedom on massively parallel supercomputers. The algorithms have been tested and verified by the two physics problems, i.e., the nonlinear Landau damping and the electron Bernstein wave

Explicit high-order non-canonical symplectic particle-in-cell algorithms for Vlasov-Maxwell systems

Energy Technology Data Exchange (ETDEWEB)

Xiao, Jianyuan [School of Nuclear Science and Technology and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China; Key Laboratory of Geospace Environment, CAS, Hefei, Anhui 230026, China; Qin, Hong [School of Nuclear Science and Technology and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China; Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543, USA; Liu, Jian [School of Nuclear Science and Technology and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China; Key Laboratory of Geospace Environment, CAS, Hefei, Anhui 230026, China; He, Yang [School of Nuclear Science and Technology and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China; Key Laboratory of Geospace Environment, CAS, Hefei, Anhui 230026, China; Zhang, Ruili [School of Nuclear Science and Technology and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China; Key Laboratory of Geospace Environment, CAS, Hefei, Anhui 230026, China; Sun, Yajuan [LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100190, China

2015-11-01

Explicit high-order non-canonical symplectic particle-in-cell algorithms for classical particle-field systems governed by the Vlasov-Maxwell equations are developed. The algorithms conserve a discrete non-canonical symplectic structure derived from the Lagrangian of the particle-field system, which is naturally discrete in particles. The electromagnetic field is spatially discretized using the method of discrete exterior calculus with high-order interpolating differential forms for a cubic grid. The resulting time-domain Lagrangian assumes a non-canonical symplectic structure. It is also gauge invariant and conserves charge. The system is then solved using a structure-preserving splitting method discovered by He et al. [preprint arXiv: 1505.06076 (2015)], which produces five exactly soluble sub-systems, and high-order structure-preserving algorithms follow by combinations. The explicit, high-order, and conservative nature of the algorithms is especially suitable for long-term simulations of particle-field systems with extremely large number of degrees of freedom on massively parallel supercomputers. The algorithms have been tested and verified by the two physics problems, i.e., the nonlinear Landau damping and the electron Bernstein wave. (C) 2015 AIP Publishing LLC.

Taylor's series method for solving the nonlinear point kinetics equations

International Nuclear Information System (INIS)

Nahla, Abdallah A.

2011-01-01

Highlights: → Taylor's series method for nonlinear point kinetics equations is applied. → The general order of derivatives are derived for this system. → Stability of Taylor's series method is studied. → Taylor's series method is A-stable for negative reactivity. → Taylor's series method is an accurate computational technique. - Abstract: Taylor's series method for solving the point reactor kinetics equations with multi-group of delayed neutrons in the presence of Newtonian temperature feedback reactivity is applied and programmed by FORTRAN. This system is the couples of the stiff nonlinear ordinary differential equations. This numerical method is based on the different order derivatives of the neutron density, the precursor concentrations of i-group of delayed neutrons and the reactivity. The r th order of derivatives are derived. The stability of Taylor's series method is discussed. Three sets of applications: step, ramp and temperature feedback reactivities are computed. Taylor's series method is an accurate computational technique and stable for negative step, negative ramp and temperature feedback reactivities. This method is useful than the traditional methods for solving the nonlinear point kinetics equations.

Numerical method for the nonlinear Fokker-Planck equation

International Nuclear Information System (INIS)

Zhang, D.S.; Wei, G.W.; Kouri, D.J.; Hoffman, D.K.

1997-01-01

A practical method based on distributed approximating functionals (DAFs) is proposed for numerically solving a general class of nonlinear time-dependent Fokker-Planck equations. The method relies on a numerical scheme that couples the usual path-integral concept to the DAF idea. The high accuracy and reliability of the method are illustrated by applying it to an exactly solvable nonlinear Fokker-Planck equation, and the method is compared with the accurate K-point Stirling interpolation formula finite-difference method. The approach is also used successfully to solve a nonlinear self-consistent dynamic mean-field problem for which both the cumulant expansion and scaling theory have been found by Drozdov and Morillo [Phys. Rev. E 54, 931 (1996)] to be inadequate to describe the occurrence of a long-lived transient bimodality. The standard interpretation of the transient bimodality in terms of the flat region in the kinetic potential fails for the present case. An alternative analysis based on the effective potential of the Schroedinger-like Fokker-Planck equation is suggested. Our analysis of the transient bimodality is strongly supported by two examples that are numerically much more challenging than other examples that have been previously reported for this problem. copyright 1997 The American Physical Society

Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach

International Nuclear Information System (INIS)

Darmani, G.; Setayeshi, S.; Ramezanpour, H.

2012-01-01

In this paper an efficient computational method based on extending the sensitivity approach (SA) is proposed to find an analytic exact solution of nonlinear differential difference equations. In this manner we avoid solving the nonlinear problem directly. By extension of sensitivity approach for differential difference equations (DDEs), the nonlinear original problem is transformed into infinite linear differential difference equations, which should be solved in a recursive manner. Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained. Numerical examples are employed to show the effectiveness of the proposed approach. (general)

Inverse operator method for solutions of nonlinear dynamical equations and some typical applications

International Nuclear Information System (INIS)

Fang Jinqing; Yao Weiguang

1993-01-01

The inverse operator method (IOM) is described briefly. We have realized the IOM for the solutions of nonlinear dynamical equations by the mathematics-mechanization (MM) with computers. They can then offer a new and powerful method applicable to many areas of physics. We have applied them successfully to study the chaotic behaviors of some nonlinear dynamical equations. As typical examples, the well-known Lorentz equation, generalized Duffing equation and two coupled generalized Duffing equations are investigated by using the IOM and the MM. The results are in good agreement with those given by Runge-Kutta method. So the IOM realized by the MM is of potential application valuable in nonlinear physics and many other fields

Stabilization of solutions to higher-order nonlinear Schrodinger equation with localized damping

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Eleni Bisognin

2007-01-01

Full Text Available We study the stabilization of solutions to higher-order nonlinear Schrodinger equations in a bounded interval under the effect of a localized damping mechanism. We use multiplier techniques to obtain exponential decay in time of the solutions of the linear and nonlinear equations.

On kinetic description of electromagnetic processes in a quantum plasma

International Nuclear Information System (INIS)

Tyshetskiy, Yu.; Vladimirov, S. V.; Kompaneets, R.

2011-01-01

A nonlinear kinetic equation for nonrelativistic quantum plasma with electromagnetic interaction of particles is obtained in the Hartree's mean-field approximation. It is cast in a convenient form of Vlasov-Boltzmann-type equation with ''quantum interference integral'', which allows for relatively straightforward modification of existing classical Vlasov codes to incorporate quantum effects (quantum statistics and quantum interference of overlapping particles wave functions), without changing the bulk of the codes. Such modification (upgrade) of existing Vlasov codes may provide a direct and effective path to numerical simulations of nonlinear electrostatic and electromagnetic phenomena in quantum plasmas, especially of processes where kinetic effects are important (e.g., modulational interactions and stimulated scattering phenomena involving plasma modes at short wavelengths or high-order kinetic modes, dynamical screening and interaction of charges in quantum plasma, etc.) Moreover, numerical approaches involving such modified Vlasov codes would provide a useful basis for theoretical analyses of quantum plasmas, as quantum and classical effects can be easily separated there.

Equivalence transformations and differential invariants of a generalized nonlinear Schroedinger equation

International Nuclear Information System (INIS)

Senthilvelan, M; Torrisi, M; Valenti, A

2006-01-01

By using Lie's invariance infinitesimal criterion, we obtain the continuous equivalence transformations of a class of nonlinear Schroedinger equations with variable coefficients. We construct the differential invariants of order 1 starting from a special equivalence subalgebra E χ o . We apply these latter ones to find the most general subclass of variable coefficient nonlinear Schr?dinger equations which can be mapped, by means of an equivalence transformation of E χ o , to the well-known cubic Schroedinger equation. We also provide the explicit form of the transformation

Traveling solitary wave solutions to evolution equations with nonlinear terms of any order

International Nuclear Information System (INIS)

Feng Zhaosheng

2003-01-01

Many physical phenomena in one- or higher-dimensional space can be described by nonlinear evolution equations, which can be reduced to ordinary differential equations such as the Lienard equation. Thus, to study those ordinary differential equations is of significance not only in mathematics itself, but also in physics. In this paper, a kind of explicit exact solutions to the Lienard equation is obtained. The applications of the solutions to the nonlinear RR-equation and the compound KdV-type equation are presented, which extend the results obtained in the previous literature

Optimal analytic method for the nonlinear Hasegawa-Mima equation

Science.gov (United States)

Baxter, Mathew; Van Gorder, Robert A.; Vajravelu, Kuppalapalle

2014-05-01

The Hasegawa-Mima equation is a nonlinear partial differential equation that describes the electric potential due to a drift wave in a plasma. In the present paper, we apply the method of homotopy analysis to a slightly more general Hasegawa-Mima equation, which accounts for hyper-viscous damping or viscous dissipation. First, we outline the method for the general initial/boundary value problem over a compact rectangular spatial domain. We use a two-stage method, where both the convergence control parameter and the auxiliary linear operator are optimally selected to minimize the residual error due to the approximation. To do the latter, we consider a family of operators parameterized by a constant which gives the decay rate of the solutions. After outlining the general method, we consider a number of concrete examples in order to demonstrate the utility of this approach. The results enable us to study properties of the initial/boundary value problem for the generalized Hasegawa-Mima equation. In several cases considered, we are able to obtain solutions with extremely small residual errors after relatively few iterations are computed (residual errors on the order of 10-15 are found in multiple cases after only three iterations). The results demonstrate that selecting a parameterized auxiliary linear operator can be extremely useful for minimizing residual errors when used concurrently with the optimal homotopy analysis method, suggesting that this approach can prove useful for a number of nonlinear partial differential equations arising in physics and nonlinear mechanics.

Forward-backward equations for nonlinear propagation in axially invariant optical systems

International Nuclear Information System (INIS)

Ferrando, Albert; Zacares, Mario; Fernandez de Cordoba, Pedro; Binosi, Daniele; Montero, Alvaro

2005-01-01

We present a general framework to deal with forward and backward components of the electromagnetic field in axially invariant nonlinear optical systems, which include those having any type of linear or nonlinear transverse inhomogeneities. With a minimum amount of approximations, we obtain a system of two first-order equations for forward and backward components, explicitly showing the nonlinear couplings among them. The modal approach used allows for an effective reduction of the dimensionality of the original problem from 3+1 (three spatial dimensions plus one time dimension) to 1+1 (one spatial dimension plus one frequency dimension). The new equations can be written in a spinor Dirac-like form, out of which conserved quantities can be calculated in an elegant manner. Finally, these equations inherently incorporate spatiotemporal couplings, so that they can be easily particularized to deal with purely temporal or purely spatial effects. Nonlinear forward pulse propagation and nonparaxial evolution of spatial structures are analyzed as examples

Soliton solutions of coupled nonlinear Klein-Gordon equations

International Nuclear Information System (INIS)

Alagesan, T.; Chung, Y.; Nakkeeran, K.

2004-01-01

The coupled nonlinear Klein-Gordon equations are analyzed for their integrability properties in a systematic manner through Painleve test. From the Painleve test, by truncating the Laurent series at the constant level term, the Hirota bilinear form is identified, from which one-soliton solutions are derived. Then, the results are generalized to the two, three and N-coupled Klein-Gordon equations

Tensor-GMRES method for large sparse systems of nonlinear equations

Science.gov (United States)

Feng, Dan; Pulliam, Thomas H.

1994-01-01

This paper introduces a tensor-Krylov method, the tensor-GMRES method, for large sparse systems of nonlinear equations. This method is a coupling of tensor model formation and solution techniques for nonlinear equations with Krylov subspace projection techniques for unsymmetric systems of linear equations. Traditional tensor methods for nonlinear equations are based on a quadratic model of the nonlinear function, a standard linear model augmented by a simple second order term. These methods are shown to be significantly more efficient than standard methods both on nonsingular problems and on problems where the Jacobian matrix at the solution is singular. A major disadvantage of the traditional tensor methods is that the solution of the tensor model requires the factorization of the Jacobian matrix, which may not be suitable for problems where the Jacobian matrix is large and has a 'bad' sparsity structure for an efficient factorization. We overcome this difficulty by forming and solving the tensor model using an extension of a Newton-GMRES scheme. Like traditional tensor methods, we show that the new tensor method has significant computational advantages over the analogous Newton counterpart. Consistent with Krylov subspace based methods, the new tensor method does not depend on the factorization of the Jacobian matrix. As a matter of fact, the Jacobian matrix is never needed explicitly.

Soliton Resolution for the Derivative Nonlinear Schrödinger Equation

Science.gov (United States)

Jenkins, Robert; Liu, Jiaqi; Perry, Peter; Sulem, Catherine

2018-05-01

We study the derivative nonlinear Schrödinger equation for generic initial data in a weighted Sobolev space that can support bright solitons (but exclude spectral singularities). Drawing on previous well-posedness results, we give a full description of the long-time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. At leading order and in space-time cones, the solution has the form of a multi-soliton whose parameters are slightly modified from their initial values by soliton-soliton and soliton-radiation interactions. Our analysis provides an explicit expression for the correction dispersive term. We use the nonlinear steepest descent method of Deift and Zhou (Commun Pure Appl Math 56:1029-1077, 2003) revisited by the {\\overline{partial}} -analysis of McLaughlin and Miller (IMRP Int Math Res Pap 48673:1-77, 2006) and Dieng and McLaughlin (Long-time asymptotics for the NLS equation via dbar methods. Preprint, arXiv:0805.2807, 2008), and complemented by the recent work of Borghese et al. (Ann Inst Henri Poincaré Anal Non Linéaire, https://doi.org/10.1016/j.anihpc.2017.08.006, 2017) on soliton resolution for the focusing nonlinear Schrödinger equation. Our results imply that N-soliton solutions of the derivative nonlinear Schrödinger equation are asymptotically stable.

Dissipative behavior of some fully non-linear KdV-type equations

Science.gov (United States)

Brenier, Yann; Levy, Doron

2000-03-01

The KdV equation can be considered as a special case of the general equation u t+f(u) x-δg(u xx) x=0, δ>0, where f is non-linear and g is linear, namely f( u)= u2/2 and g( v)= v. As the parameter δ tends to 0, the dispersive behavior of the KdV equation has been throughly investigated (see, e.g., [P.G. Drazin, Solitons, London Math. Soc. Lect. Note Ser. 85, Cambridge University Press, Cambridge, 1983; P.D. Lax, C.D. Levermore, The small dispersion limit of the Korteweg-de Vries equation, III, Commun. Pure Appl. Math. 36 (1983) 809-829; G.B. Whitham, Linear and Nonlinear Waves, Wiley/Interscience, New York, 1974] and the references therein). We show through numerical evidence that a completely different, dissipative behavior occurs when g is non-linear, namely when g is an even concave function such as g( v)=-∣ v∣ or g( v)=- v2. In particular, our numerical results hint that as δ→0 the solutions strongly converge to the unique entropy solution of the formal limit equation, in total contrast with the solutions of the KdV equation.

A mixed finite element method for nonlinear diffusion equations

KAUST Repository

Burger, Martin; Carrillo, José ; Wolfram, Marie-Therese

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.

Two-dimensional nonlinear equations of supersymmetric gauge theories

International Nuclear Information System (INIS)

Savel'ev, M.V.

1985-01-01

Supersymmetric generalization of two-dimensional nonlinear dynamical equations of gauge theories is presented. The nontrivial dynamics of a physical system in the supersymmetry and supergravity theories for (2+2)-dimensions is described by the integrable embeddings of Vsub(2/2) superspace into the flat enveloping superspace Rsub(N/M), supplied with the structure of a Lie superalgebra. An equation is derived which describes a supersymmetric generalization of the two-dimensional Toda lattice. It contains both super-Liouville and Sinh-Gordon equations

Taming the nonlinearity of the Einstein equation.

Science.gov (United States)

Harte, Abraham I

2014-12-31

Many of the technical complications associated with the general theory of relativity ultimately stem from the nonlinearity of Einstein's equation. It is shown here that an appropriate choice of dynamical variables may be used to eliminate all such nonlinearities beyond a particular order: Both Landau-Lifshitz and tetrad formulations of Einstein's equation are obtained that involve only finite products of the unknowns and their derivatives. Considerable additional simplifications arise in physically interesting cases where metrics become approximately Kerr or, e.g., plane waves, suggesting that the variables described here can be used to efficiently reformulate perturbation theory in a variety of contexts. In all cases, these variables are shown to have simple geometrical interpretations that directly relate the local causal structure associated with the metric of interest to the causal structure associated with a prescribed background. A new method to search for exact solutions is outlined as well.

Exact solutions for the cubic-quintic nonlinear Schroedinger equation

International Nuclear Information System (INIS)

Zhu Jiamin; Ma Zhengyi

2007-01-01

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

Exact solutions of some nonlinear partial differential equations using ...

Indian Academy of Sciences (India)

The functional variable method is a powerful solution method for obtaining exact solutions of some nonlinear partial differential equations. In this paper, the functional variable method is used to establish exact solutions of the generalized forms of Klein–Gordon equation, the (2 + 1)-dimensional Camassa–Holm ...

Inverse periodic problem for the discrete approximation of the Schroedinger nonlinear equation

International Nuclear Information System (INIS)

Bogolyubov, N.N.; Prikarpatskij, A.K.; AN Ukrainskoj SSR, Lvov. Inst. Prikladnykh Problem Mekhaniki i Matematiki)

1982-01-01

The problem of numerical solution of the Schroedinger nonlinear equation (1) iPSIsub(t) = PSIsub(xx)+-2(PSI)sup(2)PSI. The numerical solution of nonlinear differential equation supposes its discrete approximation is required for the realization of the computer calculation process. Tor the equation (1) there exists the following discrete approximation by variable x(2) iPSIsub(n, t) = (PSIsub(n+1)-2PSIsub(n)+PSIsub(n-1))/(Δx)sup(2)+-(PSIsub(n))sup(2)(PSIsub(n+1)+PSIsub(n-1)), n=0, +-1, +-2... where PSIsub(n)(+) is the corresponding value of PSI(x, t) function in the node and divisions with the equilibrium step Δx. The main problem is obtaining analytically exact solutions of the equations (2). The analysis of the equation system (2) is performed on the base of the discrete analogue of the periodic variant of the inverse scattering problem method developed with the aid of nonlinear equations of the Korteweg-de Vries type. Obtained in explicit form are analytical solutions of the equations system (2). The solutions are expressed through the Riemann THETA-function [ru

Nonlinear MHD-equations: symmetries, solutions and conservation laws

International Nuclear Information System (INIS)

Samokhin, A.V.

1985-01-01

To investigate stability and nonlinear effects in a high-temperature plasma the system of two scalar nonlinear equations is considered. The algebra of classical symmetries of this system and a certain natural part of its conservation laws are described. It is shown that first, with symmetries one can derive invariant (self-similar) solutions, second, acting with symmetry on the known solution the latter can be included into parametric family

Hidden regularity for a strongly nonlinear wave equation

International Nuclear Information System (INIS)

Rivera, J.E.M.

1988-08-01

The nonlinear wave equation u''-Δu+f(u)=v in Q=Ωx]0,T[;u(0)=u 0 ,u'(0)=u 1 in Ω; u(x,t)=0 on Σ= Γx]0,T[ where f is a continuous function satisfying, lim |s| sup →+∞ f(s)/s>-∞, and Ω is a bounded domain of R n with smooth boundary Γ, is analysed. It is shown that there exist a solution for the presented nonlinear wave equation that satisfies the regularity condition: |∂u/∂ η|ε L 2 (Σ). Moreover, it is shown that there exist a constant C>0 such that, |∂u/∂ η|≤c{ E(0)+|v| 2 Q }. (author) [pt

Solitary wave solutions to nonlinear evolution equations in ...

Indian Academy of Sciences (India)

1Computer Engineering Technique Department, Al-Rafidain University College, Baghdad, ... applied to extract solutions are tan–cot method and functional variable approaches. ... Consider the nonlinear partial differential equation in the form.

Nonlinear analysis of a reaction-diffusion system: Amplitude equations

Energy Technology Data Exchange (ETDEWEB)

Zemskov, E. P., E-mail: zemskov@ccas.ru [Russian Academy of Sciences, Dorodnicyn Computing Center (Russian Federation)

2012-10-15

A reaction-diffusion system with a nonlinear diffusion term is considered. Based on nonlinear analysis, the amplitude equations are obtained in the cases of the Hopf and Turing instabilities in the system. Turing pattern-forming regions in the parameter space are determined for supercritical and subcritical instabilities in a two-component reaction-diffusion system.

A hierarchy of systems of nonlinear equations

International Nuclear Information System (INIS)

Falkensteiner, P.; Grosse, H.

1985-01-01

Imposing isospectral invariance for the one-dimensional Dirac operator yields an infinite hierarchy of systems of chiral invariant nonlinear partial differential equations. The same system is obtained through a Lax pair construction and finally a formulation in terms of Kac-Moody generators is given. (Author)

Explicit solutions of two nonlinear dispersive equations with variable coefficients

International Nuclear Information System (INIS)

Lai Shaoyong; Lv Xiumei; Wu Yonghong

2008-01-01

A mathematical technique based on an auxiliary equation and the symbolic computation system Matlab is developed to construct the exact solutions for a generalized Camassa-Holm equation and a nonlinear dispersive equation with variable coefficients. It is shown that the variable coefficients of the derivative terms in the equations cause the qualitative change in the physical structures of the solutions

The breakdown of the weakly-nonlinear regime for kinetic instabilities

Science.gov (United States)

Sanz-Orozco, David; Berk, Herbert; Wang, Ge

2017-10-01

The evolution of marginally-unstable waves that interact resonantly with populations of energetic particles is governed by a well-known cubic integro-differential equation for the mode amplitude. One of the outcomes predicted by the equation is the so-called ``explosive'' regime, where the amplitude grows indefinitely, eventually taking the equation outside of its domain of validity. Beyond this point, only full Vlasov simulations will accurately describe the evolution of the mode amplitude. In this work, we study the breakdown of the cubic equation in detail. We find that, while the cubic equation is still valid, the distribution function of the energetic particles locally flattens or ``folds'' in phase space. This feature is unexpected in view of the assumptions of the theory that are given in. We also derive fifth-order terms in the wave equation, which not only give us a more accurate description of the marginally-unstable modes, but they also allow us to predict the breakdown of the cubic equation. Our findings allow us to better understand the transition between weakly-nonlinear modes and the long-term chirping modes that ultimately emerge.

Exact Solutions of Five Complex Nonlinear Schrödinger Equations by Semi-Inverse Variational Principle

International Nuclear Information System (INIS)

Najafi Mohammad; Arbabi Somayeh

2014-01-01

In this paper, we establish exact solutions for five complex nonlinear Schrödinger equations. The semi-inverse variational principle (SVP) is used to construct exact soliton solutions of five complex nonlinear Schrödinger equations. Many new families of exact soliton solutions of five complex nonlinear Schrödinger equations are successfully obtained. (general)

Chaos synchronization of nonlinear Bloch equations

International Nuclear Information System (INIS)

Park, Ju H.

2006-01-01

In this paper, the problem of chaos synchronization of Bloch equations is considered. A novel nonlinear controller is designed based on the Lyapunov stability theory. The proposed controller ensures that the states of the controlled chaotic slave system asymptotically synchronizes the states of the master system. A numerical example is given to illuminate the design procedure and advantage of the result derived

Global well posedness of the relativistic Vlasov-Yukawa system with small data

International Nuclear Information System (INIS)

Ha, Seung-Yeal; Lee, Ho

2007-01-01

In this paper, we present an existence theory and uniform L 1 -stability estimate for classical solutions with small data to the Vlasov-Yukawa system. The Vlasov-Yukawa system corresponds to a short-range correction of the Vlasov-Poisson system appearing in plasma physics and astrophysics. For the existence and stability of classical solutions, we crucially use dispersion estimates due to the smallness of data

Nonlinear model of a rotating hub-beams structure: Equations of motion

Science.gov (United States)

Warminski, Jerzy

2018-01-01

Dynamics of a rotating structure composed of a rigid hub and flexible beams is presented in the paper. A nonlinear model of a beam takes into account bending, extension and nonlinear curvature. The influence of geometric nonlinearity and nonconstant angular velocity on dynamics of the rotating structure is presented. The exact equations of motion and associated boundary conditions are derived on the basis of the Hamilton's principle. The simplification of the exact nonlinear mathematical model is proposed taking into account the second order approximation. The reduced partial differential equations of motion together with associated boundary conditions can be used to study natural or forced vibrations of a rotating structure considering constant or nonconstant angular speed of a rigid hub and an arbitrary number of flexible blades.

Combined solitary-wave solution for coupled higher-order nonlinear Schroedinger equations

International Nuclear Information System (INIS)

Tian Jinping; Tian Huiping; Li Zhonghao; Zhou Guosheng

2004-01-01

Coupled nonlinear Schroedinger equations model several interesting physical phenomena. We used a trigonometric function transform method based on a homogeneous balance to solve the coupled higher-order nonlinear Schroedinger equations. We obtained four pairs of exact solitary-wave solutions including a dark and a bright-soliton pair, a bright- and a dark-soliton pair, a bright- and a bright-soliton pair, and the last pair, a combined bright-dark-soliton pair

Analytic continuation of solutions of some nonlinear convolution partial differential equations

Directory of Open Access Journals (Sweden)

Hidetoshi Tahara

2015-01-01

Full Text Available The paper considers a problem of analytic continuation of solutions of some nonlinear convolution partial differential equations which naturally appear in the summability theory of formal solutions of nonlinear partial differential equations. Under a suitable assumption it is proved that any local holomorphic solution has an analytic extension to a certain sector and its extension has exponential growth when the variable goes to infinity in the sector.

Numerical simulation of nonlinear continuity equations by evolving diffeomorphisms

KAUST Repository

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.

New variable separation approach: application to nonlinear diffusion equations

International Nuclear Information System (INIS)

Zhang Shunli; Lou, S Y; Qu Changzheng

2003-01-01

The concept of the derivative-dependent functional separable solution (DDFSS), as a generalization to the functional separable solution, is proposed. As an application, it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach. As a consequence, a complete list of canonical forms for such equations which admit the DDFSS is obtained and some exact solutions to the resulting equations are described

Nonlinear tunneling of optical soliton in 3 coupled NLS equation with symbolic computation

Energy Technology Data Exchange (ETDEWEB)

Mani Rajan, M.S., E-mail: senthilmanirajanofc@gmail.com [Department of Physics, Anna University, Madurai Region, Ramanathapuram (India); Mahalingam, A. [Department of Physics, Anna University, Chennai - 600 025 (India); Uthayakumar, A. [Department of Physics, Presidency College, Chennai - 600 005 (India)

2014-07-15

We investigated the soliton solution for N coupled nonlinear Schrödinger (CNLS) equations. These equations are coupled due to the cross-phase-modulation (CPM). Lax pair of this system is obtained via the Ablowitz–Kaup–Newell–Segur (AKNS) scheme and the corresponding Darboux transformation is constructed to derive the soliton solution. One and two soliton solutions are generated. Using two soliton solutions of 3 CNLS equation, nonlinear tunneling of soliton for both with and without exponential background has been discussed. Finally cascade compression of optical soliton through multi-nonlinear barrier has been discussed. The obtained results may have promising applications in all-optical devices based on optical solitons, study of soliton propagation in birefringence fiber systems and optical soliton with distributed dispersion and nonlinearity management. -- Highlights: •We consider the nonlinear tunneling of soliton in birefringence fiber. •3-coupled NLS (CNLS) equation with variable coefficients is considered. •Two soliton solutions are obtained via Darboux transformation using constructed Lax pair. •Soliton tunneling through dispersion barrier and well are investigated. •Finally, cascade compression of soliton has been achieved.

Dispersive optical soliton solutions for the hyperbolic and cubic-quintic nonlinear Schrödinger equations via the extended sinh-Gordon equation expansion method

Science.gov (United States)

Seadawy, Aly R.; Kumar, Dipankar; Chakrabarty, Anuz Kumar

2018-05-01

The (2+1)-dimensional hyperbolic and cubic-quintic nonlinear Schrödinger equations describe the propagation of ultra-short pulses in optical fibers of nonlinear media. By using an extended sinh-Gordon equation expansion method, some new complex hyperbolic and trigonometric functions prototype solutions for two nonlinear Schrödinger equations were derived. The acquired new complex hyperbolic and trigonometric solutions are expressed by dark, bright, combined dark-bright, singular and combined singular solitons. The obtained results are more compatible than those of other applied methods. The extended sinh-Gordon equation expansion method is a more powerful and robust mathematical tool for generating new optical solitary wave solutions for many other nonlinear evolution equations arising in the propagation of optical pulses.

Nonlinear ordinary differential equations analytical approximation and numerical methods

CERN Document Server

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...

A nonlinear wave equation in nonadiabatic flame propagation

International Nuclear Information System (INIS)

Booty, M.R.; Matalon, M.; Matkowsky, B.J.

1988-01-01

The authors derive a nonlinear wave equation from the diffusional thermal model of gaseous combustion to describe the evolution of a flame front. The equation arises as a long wave theory, for values of the volumeric heat loss in a neighborhood of the extinction point (beyond which planar uniformly propagating flames cease to exist), and for Lewis numbers near the critical value beyond which uniformly propagating planar flames lose stability via a degenerate Hopf bifurcation. Analysis of the equation suggests the possibility of a singularity developing in finite time

Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach spaces

Institute of Scientific and Technical Information of China (English)

LI Shoufu

2005-01-01

A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.

New Solutions of Three Nonlinear Space- and Time-Fractional Partial Differential Equations in Mathematical Physics

International Nuclear Information System (INIS)

Yao Ruo-Xia; Wang Wei; Chen Ting-Hua

2014-01-01

Motivated by the widely used ansätz method and starting from the modified Riemann—Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper. (general)

The Gaussian radial basis function method for plasma kinetic theory

Energy Technology Data Exchange (ETDEWEB)

Hirvijoki, E., E-mail: eero.hirvijoki@chalmers.se [Department of Applied Physics, Chalmers University of Technology, SE-41296 Gothenburg (Sweden); Candy, J.; Belli, E. [General Atomics, PO Box 85608, San Diego, CA 92186-5608 (United States); Embréus, O. [Department of Applied Physics, Chalmers University of Technology, SE-41296 Gothenburg (Sweden)

2015-10-30

Description of a magnetized plasma involves the Vlasov equation supplemented with the non-linear Fokker–Planck collision operator. For non-Maxwellian distributions, the collision operator, however, is difficult to compute. In this Letter, we introduce Gaussian Radial Basis Functions (RBFs) to discretize the velocity space of the entire kinetic system, and give the corresponding analytical expressions for the Vlasov and collision operator. Outlining the general theory, we also highlight the connection to plasma fluid theories, and give 2D and 3D numerical solutions of the non-linear Fokker–Planck equation. Applications are anticipated in both astrophysical and laboratory plasmas. - Highlights: • A radically new method to address the velocity space discretization of the non-linear kinetic equation of plasmas. • Elegant and physically intuitive, flexible and mesh-free. • Demonstration of numerical solution of both 2-D and 3-D non-linear Fokker–Planck relaxation problem.

Forced oscillation of hyperbolic equations with mixed nonlinearities

Directory of Open Access Journals (Sweden)

Yutaka Shoukaku

2012-04-01

Full Text Available In this paper, we consider the mixed nonlinear hyperbolic equations with forcing term via Riccati inequality. Some sufficient conditions for the oscillation are derived by using Young inequality and integral averaging method.

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.

Properties of some nonlinear Schroedinger equations motivated through information theory

International Nuclear Information System (INIS)

Yuan, Liew Ding; Parwani, Rajesh R

2009-01-01

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 η = 1/4 for the interpolation parameter, where leading order energy shifts vanish, implies the preservation of existing supersymmetry in nonlinearised supersymmetric quantum mechanics. Physically, η might be encoding relativistic effects.

Convergence criteria for systems of nonlinear elliptic partial differential equations

International Nuclear Information System (INIS)

Sharma, R.K.

1986-01-01

This thesis deals with convergence criteria for a special system of nonlinear elliptic partial differential equations. A fixed-point algorithm is used, which iteratively solves one linearized elliptic partial differential equation at a time. Conditions are established that help foresee the convergence of the algorithm. Under reasonable hypotheses it is proved that the algorithm converges for such nonlinear elliptic systems. Extensive experimental results are reported and they show the algorithm converges in a wide variety of cases and the convergence is well correlated with the theoretical conditions introduced in this thesis

Analysis and classification of nonlinear dispersive evolution equations in the potential representation

International Nuclear Information System (INIS)

Eichmann, U.A.; Draayer, J.P.; Ludu, A.

2002-01-01

A potential representation for the subset of travelling solutions of nonlinear dispersive evolution equations is introduced. The procedure involves reduction of a third-order partial differential equation to a first-order ordinary differential equation. The potential representation allows us to deduce certain properties of the solutions without the actual need to solve the underlying evolution equation. In particular, the paper deals with the so-called K(n, m) equations. Starting from their respective potential representations it is shown that these equations can be classified according to a simple point transformation. As a result, e.g., all equations with linear dispersion join the same equivalence class with the Korteweg-deVries equation being its representative, and all soliton solutions of higher order nonlinear equations are thus equivalent to the KdV soliton. Certain equations with both linear and quadratic dispersions can also be treated within this equivalence class. (author)

Traveling waves and conservation laws for highly nonlinear wave equations modeling Hertz chains

Science.gov (United States)

Przedborski, Michelle; Anco, Stephen C.

2017-09-01

A highly nonlinear, fourth-order wave equation that models the continuum theory of long wavelength pulses in weakly compressed, homogeneous, discrete chains with a general power-law contact interaction is studied. For this wave equation, all solitary wave solutions and all nonlinear periodic wave solutions, along with all conservation laws, are derived. The solutions are explicitly parameterized in terms of the asymptotic value of the wave amplitude in the case of solitary waves and the peak of the wave amplitude in the case of nonlinear periodic waves. All cases in which the solution expressions can be stated in an explicit analytic form using elementary functions are worked out. In these cases, explicit expressions for the total energy and total momentum for all solutions are obtained as well. The derivation of the solutions uses the conservation laws combined with an energy analysis argument to reduce the wave equation directly to a separable first-order differential equation that determines the wave amplitude in terms of the traveling wave variable. This method can be applied more generally to other highly nonlinear wave equations.

Creation and annihilation of solitons in the string nonlinear equation

International Nuclear Information System (INIS)

Aguero G, M.A.; Espinosa G, A.A.; Martinez O, J.

1997-01-01

Starting from the cubic-quintic Schroedinger equation it is obtained the nonlinear string equation. This system supports regular and singular solitons. It is shown that two singular solitons could be generated after the interaction of two regular solitons and viceversa. (Author)

Stability of Nonlinear Neutral Stochastic Functional Differential Equations

Directory of Open Access Journals (Sweden)

Minggao Xue

2010-01-01

Full Text Available Neutral stochastic functional differential equations (NSFDEs have recently been studied intensively. The well-known conditions imposed for the existence and uniqueness and exponential stability of the global solution are the local Lipschitz condition and the linear growth condition. Therefore, the existing results cannot be applied to many important nonlinear NSFDEs. The main aim of this paper is to remove the linear growth condition and establish a Khasminskii-type test for nonlinear NSFDEs. New criteria not only cover a wide class of highly nonlinear NSFDEs but they can also be verified much more easily than the classical criteria. Finally, several examples are given to illustrate main results.

Strongly enhanced flow effect from Landau-Vlasov versus Vlasov-Uehling-Uhlenbeck approach

International Nuclear Information System (INIS)

Gregoire, C.; Remaud, B.; Sebille, F.; Schuck, P.

1988-01-01

The simulation of the collision integral in the Landau-Vlasov approach for heavy ion collisions is examined. It turns out that quantities like the nucleon mean free path can be compared with parallel ensemble models. Convergency of results with time step and sampling is clearly established. Quadratic quantities, like the internal pressure, are found to be strongly underestimated in parallel ensemble models

A general nonlinear evolution equation for irreversible conservative approach to stable equilibrium

International Nuclear Information System (INIS)

Beretta, G.P.

1986-01-01

This paper addresses a mathematical problem relevant to the question of nonequilibrium and irreversibility, namely, that of ''designing'' a general evolution equation capable of describing irreversible but conservative relaxtion towards equilibrium. The objective is to present an interesting mathematical solution to this design problem, namely, a new nonlinear evolution equation that satisfies a set of very stringent relevant requirements. Three different frameworks are defined from which the new equation could be adopted, with entirely different interpretations. Some useful well-known mathematics involving Gram determinants are presented and a nonlinear evolution equation is given which meets the stringent design specifications

On realization of nonlinear systems described by higher-order differential equations

NARCIS (Netherlands)

van der Schaft, Arjan

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

Existence Results for Some Nonlinear Functional-Integral Equations in Banach Algebra with Applications

Directory of Open Access Journals (Sweden)

Lakshmi Narayan Mishra

2016-04-01

Full Text Available In the present manuscript, we prove some results concerning the existence of solutions for some nonlinear functional-integral equations which contains various integral and functional equations that considered in nonlinear analysis and its applications. By utilizing the techniques of noncompactness measures, we operate the fixed point theorems such as Darbo's theorem in Banach algebra concerning the estimate on the solutions. The results obtained in this paper extend and improve essentially some known results in the recent literature. We also provide an example of nonlinear functional-integral equation to show the ability of our main result.

Nonlinear partial differential equations of second order

CERN Document Server

Dong, Guangchang

1991-01-01

This book addresses a class of equations central to many areas of mathematics and its applications. Although there is no routine way of solving nonlinear partial differential equations, effective approaches that apply to a wide variety of problems are available. This book addresses a general approach that consists of the following: Choose an appropriate function space, define a family of mappings, prove this family has a fixed point, and study various properties of the solution. The author emphasizes the derivation of various estimates, including a priori estimates. By focusing on a particular approach that has proven useful in solving a broad range of equations, this book makes a useful contribution to the literature.

Infinite sets of conservation laws for linear and nonlinear field equations

International Nuclear Information System (INIS)

Mickelsson, J.

1984-01-01

The relation between an infinite set of conservation laws of a linear field equation and the enveloping algebra of the space-time symmetry group is established. It is shown that each symmetric element of the enveloping algebra of the space-time symmetry group of a linear field equation generates a one-parameter group of symmetries of the field equation. The cases of the Maxwell and Dirac equations are studied in detail. Then it is shown that (at least in the sense of a power series in the 'coupling constant') the conservation laws of the linear case can be deformed to conservation laws of a nonlinear field equation which is obtained from the linear one by adding a nonlinear term invariant under the group of space-time symmetries. As an example, our method is applied to the Korteweg-de Vries equation and to the massless Thirring model. (orig.)

Geometric integration of the Vlasov-Maxwell system with a variational particle-in-cell scheme

Energy Technology Data Exchange (ETDEWEB)

Squire, J.; Tang, W. M. [Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States); Qin, H. [Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States); Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026 (China)

2012-08-15

A fully variational, unstructured, electromagnetic particle-in-cell integrator is developed for integration of the Vlasov-Maxwell equations. Using the formalism of discrete exterior calculus [Desbrun et al., e-print arXiv:math/0508341 (2005)], the field solver, interpolation scheme, and particle advance algorithm are derived through minimization of a single discrete field theory action. As a consequence of ensuring that the action is invariant under discrete electromagnetic gauge transformations, the integrator exactly conserves Gauss's law.

Initial-value problem for the Gardner equation applied to nonlinear internal waves

Science.gov (United States)

Rouvinskaya, Ekaterina; Kurkina, Oxana; Kurkin, Andrey; Talipova, Tatiana; Pelinovsky, Efim

2017-04-01

The Gardner equation is a fundamental mathematical model for the description of weakly nonlinear weakly dispersive internal waves, when cubic nonlinearity cannot be neglected. Within this model coefficients of quadratic and cubic nonlinearity can both be positive as well as negative, depending on background conditions of the medium, where waves propagate (sea water density stratification, shear flow profile) [Rouvinskaya et al., 2014, Kurkina et al., 2011, 2015]. For the investigation of weakly dispersive behavior in the framework of nondimensional Gardner equation with fixed (positive) sign of quadratic nonlinearity and positive or negative cubic nonlinearity {eq1} partial η/partial t+6η( {1± η} )partial η/partial x+partial ^3η/partial x^3=0, } the series of numerical experiments of initial-value problem was carried out for evolution of a bell-shaped impulse of negative polarity (opposite to the sign of quadratic nonlinear coefficient): {eq2} η(x,t=0)=-asech2 ( {x/x0 } ), for which amplitude a and width x0 was varied. Similar initial-value problem was considered in the paper [Trillo et al., 2016] for the Korteweg - de Vries equation. For the Gardner equation with different signs of cubic nonlinearity the initial-value problem for piece-wise constant initial condition was considered in detail in [Grimshaw et al., 2002, 2010]. It is widely known, for example, [Pelinovsky et al., 2007], that the Gardner equation (1) with negative cubic nonlinearity has a family of classic solitary wave solutions with only positive polarity,and with limiting amplitude equal to 1. Therefore evolution of impulses (2) of negative polarity (whose amplitudes a were varied from 0.1 to 3, and widths at the level of a/2 were equal to triple width of solitons with the same amplitude for a 1) was going on a universal scenario with the generation of nonlinear Airy wave. For the Gardner equation (1) with the positive cubic nonlinearity coefficient there exist two one-parametric families of

Self-similar solutions of the modified nonlinear schrodinger equation

International Nuclear Information System (INIS)

Kitaev, A.V.

1986-01-01

This paper considers a 2 x 2 matrix linear ordinary differential equation with large parameter t and irregular singular point of fourth order at infinity. The leading order of the monodromy data of this equation is calculated in terms of its coefficients. Isomonodromic deformations of the equation are self-similar solutions of the modified nonlinear Schrodinger equation, and therefore inversion of the expressions obtained for the monodromy data gives the leading term in the time-asymptotic behavior of the self-similar solution. The application of these results to the type IV Painleve equation is considered in detail

Variational method for the derivative nonlinear Schroedinger equation with computational applications

Energy Technology Data Exchange (ETDEWEB)

Helal, M A [Mathematics Department, Faculty of Science, Cairo University (Egypt); Seadawy, A R [Mathematics Department, Faculty of Science, Beni-Suef University (Egypt)], E-mail: mahelal@yahoo.com, E-mail: aly742001@yahoo.com

2009-09-15

The derivative nonlinear Schroedinger equation (DNLSE) arises as a physical model for ultra-short pulse propagation. In this paper, the existence of a Lagrangian and the invariant variational principle (i.e. in the sense of the inverse problem of calculus of variations through deriving the functional integral corresponding to a given coupled nonlinear partial differential equations) for two-coupled equations describing the nonlinear evolution of the Alfven wave with magnetosonic waves at a much larger scale are given and the functional integral corresponding to those equations is derived. We found the solutions of DNLSE by choice of a trial function in a region of a rectangular box in two cases, and using this trial function, we find the functional integral and the Lagrangian of the system without loss. Solution of the general case for the two-box potential can be obtained on the basis of a different ansatz where we approximate the Jost function using polynomials of order n instead of the piecewise linear function. An example for the third order is given for illustrating the general case.

Application of He’s Variational Iteration Method to Nonlinear Helmholtz Equation and Fifth-Order KDV Equation

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 c...

Multi-symplectic variational integrators for nonlinear Schrödinger equations with variable coefficients

International Nuclear Information System (INIS)

Liao Cui-Cui; Cui Jin-Chao; Liang Jiu-Zhen; Ding Xiao-Hua

2016-01-01

In this paper, we propose a variational integrator for nonlinear Schrödinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrödinger equations with variable coefficients, cubic nonlinear Schrödinger equations and Gross–Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space. (paper)

The soliton solution of BBGKY quantum kinetic equations chain for different type particles system

International Nuclear Information System (INIS)

Rasulova, M.Yu.; Avazov, U.; Hassan, T.

2006-12-01

In the present paper on the basis of BBGKY chain of quantum kinetic equations the chain of equations for correlation matrices is derived, describing the evolution of a system of different types particles, which interact by pair potential. The series, which is the solution of this chain of equations for correlation matrices, is suggested. Using this series the solution of the last chain of equations is reduced to a solution of a set of homogeneous and nonhomogeneous von-Neumann's kinetic equations (analogue of Vlasov equations for quantum case). The first and second equations of this set of equations coincide with the first and second kinetic equations of the set, which is used in plasma physics. For an potential in the form of Dirac delta function, the solution of von-Neumann equation is defined through soliton solution of nonlinear Schrodinger equations. Based on von-Neumann equation one can define all terms of series, which is a solution of a chain of equations for correlation matrices. On the basis of these correlation matrices for a system of different types of particles we can define exact solution of BBGKY chain of quantum kinetic equations

Numerical bifurcation analysis of a class of nonlinear renewal equations

NARCIS (Netherlands)

Breda, Dimitri; Diekmann, Odo; Liessi, Davide; Scarabel, Francesca

2016-01-01

We show, by way of an example, that numerical bifurcation tools for ODE yield reliable bifurcation diagrams when applied to the pseudospectral approximation of a one-parameter family of nonlinear renewal equations. The example resembles logistic-and Ricker-type population equations and exhibits

Traveling wave solutions for two nonlinear evolution equations with nonlinear terms of any order

International Nuclear Information System (INIS)

Feng Qing-Hua; Zhang Yao-Ming; Meng Fan-Wei

2011-01-01

In this paper, based on the known first integral method and the Riccati sub-ordinary differential equation (ODE) method, we try to seek the exact solutions of the general Gardner equation and the general Benjamin—Bona—Mahoney equation. As a result, some traveling wave solutions for the two nonlinear equations are established successfully. Also we make a comparison between the two methods. It turns out that the Riccati sub-ODE method is more effective than the first integral method in handling the proposed problems, and more general solutions are constructed by the Riccati sub-ODE method. (general)

Nonlinear evolution equations and solving algebraic systems: the importance of computer algebra

International Nuclear Information System (INIS)

Gerdt, V.P.; Kostov, N.A.

1989-01-01

In the present paper we study the application of computer algebra to solve the nonlinear polynomial systems which arise in investigation of nonlinear evolution equations. We consider several systems which are obtained in classification of integrable nonlinear evolution equations with uniform rank. Other polynomial systems are related with the finding of algebraic curves for finite-gap elliptic potentials of Lame type and generalizations. All systems under consideration are solved using the method based on construction of the Groebner basis for corresponding polynomial ideals. The computations have been carried out using computer algebra systems. 20 refs

Embedded solitons in the third-order nonlinear Schroedinger equation

International Nuclear Information System (INIS)

Pal, Debabrata; Ali, Sk Golam; Talukdar, B

2008-01-01

We work with a sech trial function with space-dependent soliton parameters and envisage a variational study for the nonlinear Schoedinger (NLS) equation in the presence of third-order dispersion. We demonstrate that the variational equations for pulse evolution in this NLS equation provide a natural basis to derive a potential model which can account for the existence of a continuous family of embedded solitons supported by the third-order NLS equation. Each member of the family is parameterized by the propagation velocity and co-efficient of the third-order dispersion

Existence of Solutions of Nonlinear Integrodifferential Equations of ...

Indian Academy of Sciences (India)

Abstract. In this paper we prove the existence of mild and strong solutions of a nonlinear integrodifferential equation of Sobolev type with nonlocal condition. The results are obtained by using semigroup theory and the Schauder fixed point theorem.

Localized solutions of non-linear Klein--Gordon equations

International Nuclear Information System (INIS)

Werle, J.

1977-05-01

Nondissipative, stationary solutions for a class of nonlinear Klein-Gordon equations for a scalar field were found explicitly. Since the field is different from zero only inside a sphere of definite radius, the solutions are called quantum droplets

Semi-classical analysis for nonlinear Schrödinger equations

CERN Document Server

Carles, Remi

2008-01-01

These lecture notes review recent results on the high-frequency analysis of nonlinear Schrödinger equations in the presence of an external potential. The book consists of two relatively independent parts: WKB analysis, and caustic crossing. In the first part, the basic linear WKB theory is constructed and then extended to the nonlinear framework. The most difficult supercritical case is discussed in detail, together with some of its consequences concerning instability phenomena. Applications of WKB analysis to functional analysis, in particular to the Cauchy problem for nonlinear Schrödinger e

New extended (G'/G)-expansion method to solve nonlinear evolution equation: the (3 + 1)-dimensional potential-YTSF equation.

Science.gov (United States)

Roshid, Harun-Or-; Akbar, M Ali; Alam, Md Nur; Hoque, Md Fazlul; Rahman, Nizhum

2014-01-01

In this article, a new extended (G'/G) -expansion method has been proposed for constructing more general exact traveling wave solutions of nonlinear evolution equations with the aid of symbolic computation. In order to illustrate the validity and effectiveness of the method, we pick the (3 + 1)-dimensional potential-YTSF equation. As a result, abundant new and more general exact solutions have been achieved of this equation. It has been shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in applied mathematics, engineering and mathematical physics.

Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria

International Nuclear Information System (INIS)

Frieman, E.A.; Chen, L.

1981-10-01

A nonlinear gyrokinetic formalism for low-frequency (less than the cyclotron frequency) microscopic electromagnetic perturbations in general magnetic field configurations is developed. The nonlinear equations thus derived are valid in the strong-turbulence regime and contain effects due to finite Larmor radius, plasma inhomogeneities, and magentic field geometries. The specific case of axisymmetric tokamaks is then considered, and a model nonlinear equation is derived for electrostatic drift waves. Also, applying the formalism to the shear Alfven wave heating sceme, it is found that nonlinear ion Landau damping of kinetic shear-Alfven waves is modified, both qualitatively and quantitatively, by the diamagnetic drift effects. In particular, wave energy is found to cascade in wavenumber instead of frequency

Vlasov simulations of kinetic Alfvén waves at proton kinetic scales

Energy Technology Data Exchange (ETDEWEB)

Vásconez, C. L. [Dipartimento di Fisica, Università della Calabria, I-87036 Cosenza (Italy); Observatorio Astronómico de Quito, Escuela Politécnica Nacional, Quito (Ecuador); Valentini, F.; Veltri, P. [Dipartimento di Fisica, Università della Calabria, I-87036 Cosenza (Italy); Camporeale, E. [Centrum Wiskunde and Informatica, Amsterdam (Netherlands)

2014-11-15

Kinetic Alfvén waves represent an important subject in space plasma physics, since they are thought to play a crucial role in the development of the turbulent energy cascade in the solar wind plasma at short wavelengths (of the order of the proton gyro radius ρ{sub p} and/or inertial length d{sub p} and beyond). A full understanding of the physical mechanisms which govern the kinetic plasma dynamics at these scales can provide important clues on the problem of the turbulent dissipation and heating in collisionless systems. In this paper, hybrid Vlasov-Maxwell simulations are employed to analyze in detail the features of the kinetic Alfvén waves at proton kinetic scales, in typical conditions of the solar wind environment (proton plasma beta β{sub p} = 1). In particular, linear and nonlinear regimes of propagation of these fluctuations have been investigated in a single-wave situation, focusing on the physical processes of collisionless Landau damping and wave-particle resonant interaction. Interestingly, since for wavelengths close to d{sub p} and β{sub p} ≃ 1 (for which ρ{sub p} ≃ d{sub p}) the kinetic Alfvén waves have small phase speed compared to the proton thermal velocity, wave-particle interaction processes produce significant deformations in the core of the particle velocity distribution, appearing as phase space vortices and resulting in flat-top velocity profiles. Moreover, as the Eulerian hybrid Vlasov-Maxwell algorithm allows for a clean almost noise-free description of the velocity space, three-dimensional plots of the proton velocity distribution help to emphasize how the plasma departs from the Maxwellian configuration of thermodynamic equilibrium due to nonlinear kinetic effects.

Nonlinear reaction-diffusion equations with delay: some theorems, test problems, exact and numerical solutions

Science.gov (United States)

Polyanin, A. D.; Sorokin, V. G.

2017-12-01

The paper deals with nonlinear reaction-diffusion equations with one or several delays. We formulate theorems that allow constructing exact solutions for some classes of these equations, which depend on several arbitrary functions. Examples of application of these theorems for obtaining new exact solutions in elementary functions are provided. We state basic principles of construction, selection, and use of test problems for nonlinear partial differential equations with delay. Some test problems which can be suitable for estimating accuracy of approximate analytical and numerical methods of solving reaction-diffusion equations with delay are presented. Some examples of numerical solutions of nonlinear test problems with delay are considered.

From the hypergeometric differential equation to a non-linear Schrödinger one

International Nuclear Information System (INIS)

Plastino, A.; Rocca, M.C.

2015-01-01

We show that the q-exponential function is a hypergeometric function. Accordingly, it obeys the hypergeometric differential equation. We demonstrate that this differential equation can be transformed into a non-linear Schrödinger equation (NLSE). This NLSE exhibits both similarities and differences vis-a-vis the Nobre–Rego-Monteiro–Tsallis one. - Highlights: • We show that the q-exponential is a hypergeometric function. • It thus obeys the hypergeometric differential equation (HDE). • We show that the HDE can be cast as a non-linear Schrödinger equation. • This is different from the Nobre, Rego-Monteiro, Tsallis one.

Small-χ singlet structure functions from the nonlinear GLR equation

International Nuclear Information System (INIS)

Kim, V.T.; Ryskin, M.G.

1991-06-01

The effect of absorptive corrections in the nonlinear GLR evolution equation is considered. A simple method how to estimate the corrections numerically is described. In the case of the parametrization based on semihard hadron phenomenology developed earlier a visible difference between linear and nonlinear evolution is expected at HERA energies. (orig.)

Analysis of the magnetohydrodynamic equations and study of the nonlinear solution bifurcations

International Nuclear Information System (INIS)

Morros Tosas, J.

1989-05-01

The nonlinear saturation of a plasma magnetohydrodynamic instabilities is studied, by means of a bifurcation theory. The work includes: an accurate mathematical method to study the MHD equations, in which the physical content is clear; and the study of the nonlinear solutions of the branch bifurcations, applied to different unstable plasma models. A scalar function representation is proposed for the MHD equations. This representation is characterized by a reference steady magnetic field and by a velocity field, which allow to write the equations for the scalar functions. An approximation method, leading to the obtention of the reduced equations applied in the instability study, is given. The cylindrical or toroidal plasmas are studied by using the nonlinear solutions bifurcation. Concerning the cylindrical plasma, the representation leads to a reduced system which enables the analytical calculations: two different steady bifurcation solutions are obtained. In the case of the toroidal plasma, an appropriate reduced equations system, is obtained. A qualitative approach of the Kink-type steady solution bifurcation, in a toroidal geometry, is performed [fr

Inverse operator theory method mathematics-mechanization for the solutions of nonlinear equations and some typical applications in nonlinear physics

International Nuclear Information System (INIS)

Fang Jinqing; Yao Weiguang

1992-12-01

Inverse operator theory method (IOTM) has developed rapidly in the last few years. It is an effective and useful procedure for quantitative solution of nonlinear or stochastic continuous dynamical systems. Solutions are obtained in series form for deterministic equations, and in the case of stochastic equation it gives statistic measures of the solution process. A very important advantage of the IOTM is to eliminate a number of restrictive and assumption on the nature of stochastic processes. Therefore, it provides more realistic solutions. The IOTM and its mathematics-mechanization (MM) are briefly introduced. They are used successfully to study the chaotic behaviors of the nonlinear dynamical systems for the first time in the world. As typical examples, the Lorentz equation, generalized Duffing equation, two coupled generalized Duffing equations are investigated by the use of the IOTM and the MM. The results are in good agreement with ones by the Runge-Kutta method (RKM). It has higher accuracy and faster convergence. So the IOTM realized by the MM is of potential application valuable in nonlinear science

New Exact Solutions for New Model Nonlinear Partial Differential Equation

OpenAIRE

Maher, A.; El-Hawary, H. M.; Al-Amry, M. S.

2013-01-01

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.

Nonlinear q-Generalizations of Quantum Equations: Homogeneous and Nonhomogeneous Cases—An Overview

Directory of Open Access Journals (Sweden)

Fernando D. Nobre

2017-01-01

Full Text Available Recent developments on the generalizations of two important equations of quantum physics, namely the Schroedinger and Klein–Gordon equations, are reviewed. These generalizations present nonlinear terms, characterized by exponents depending on an index q, in such a way that the standard linear equations are recovered in the limit q → 1 . Interestingly, these equations present a common, soliton-like, traveling solution, which is written in terms of the q-exponential function that naturally emerges within nonextensive statistical mechanics. In both cases, the corresponding well-known Einstein energy-momentum relations, as well as the Planck and the de Broglie ones, are preserved for arbitrary values of q. In order to deal appropriately with the continuity equation, a classical field theory has been developed, where besides the usual Ψ ( x → , t , a new field Φ ( x → , t must be introduced; this latter field becomes Ψ * ( x → , t only when q → 1 . A class of linear nonhomogeneous Schroedinger equations, characterized by position-dependent masses, for which the extra field Φ ( x → , t becomes necessary, is also investigated. In this case, an appropriate transformation connecting Ψ ( x → , t and Φ ( x → , t is proposed, opening the possibility for finding a connection between these fields in the nonlinear cases. The solutions presented herein are potential candidates for applications to nonlinear excitations in plasma physics, nonlinear optics, in structures, such as those of graphene, as well as in shallow and deep water waves.

New non-linear modified massless Klein-Gordon equation

Energy Technology Data Exchange (ETDEWEB)

Asenjo, Felipe A. [Universidad Adolfo Ibanez, UAI Physics Center, Santiago (Chile); Universidad Adolfo Ibanez, Facultad de Ingenieria y Ciencias, Santiago (Chile); Hojman, Sergio A. [Universidad Adolfo Ibanez, UAI Physics Center, Santiago (Chile); Universidad Adolfo Ibanez, Departamento de Ciencias, Facultad de Artes Liberales, Santiago (Chile); Universidad de Chile, Departamento de Fisica, Facultad de Ciencias, Santiago (Chile); Centro de Recursos Educativos Avanzados, CREA, Santiago (Chile)

2017-11-15

The massless Klein-Gordon equation on arbitrary curved backgrounds allows for solutions which develop ''tails'' inside the light cone and, therefore, do not strictly follow null geodesics as discovered by DeWitt and Brehme almost 60 years ago. A modification of the massless Klein-Gordon equation is presented, which always exhibits null geodesic propagation of waves on arbitrary curved spacetimes. This new equation is derived from a Lagrangian which exhibits current-current interaction. Its non-linearity is due to a self-coupling term which is related to the quantum mechanical Bohm potential. (orig.)

Hypocoercivity for linear kinetic equations conserving mass

KAUST Repository

Dolbeault, Jean; Mouhot, Clé ment; Schmeiser, Christian

2015-01-01

We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $ L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed. - See more at: http://www.ams.org/journals/tran/2015-367-06/S0002-9947-2015-06012-7/#sthash.ChjyK6rc.dpuf

Hypocoercivity for linear kinetic equations conserving mass

KAUST Repository

Dolbeault, Jean

2015-02-03

We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $ L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed. - See more at: http://www.ams.org/journals/tran/2015-367-06/S0002-9947-2015-06012-7/#sthash.ChjyK6rc.dpuf

Projection-iteration methods for solving nonlinear operator equations