A note on soliton solutions of Klein-Gordon-Zakharov equation by variational approach

2016 ◽  
Vol 5 (3) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Fatima Riaz ◽  
Asmat Ara
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Variational Approach ◽  
Soliton Solutions ◽  
Unknown Parameters ◽  
Zakharov Equation ◽  
Solitary Solution ◽  
Linear Partial Differential Equations ◽  
Course Of Action ◽  
Klein Gordon

AbstractIn this paper, an investigation has been made to validate the variational approach to obtain soliton solutions of the Klein-Gordon-Zakharov (KGZ) equations. It is evident that to resolve the non-linear partial differential equations are quite complex and difficult. The presented approach is capable of achieving the condition for continuation of the solitary solution of KGZ equation as well as the initial solutions selected in soliton form including various unknown parameters can be resolute in the solution course of action. The procedure of attaining the solution reveals that the scheme is simple and straightforward.

scholarly journals Soliton Solutions of the Klein-Gordon-Zakharov Equation with Power Law Nonlinearity

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Mehmet Ekici ◽  
Durgun Duran ◽  
Abdullah Sonmezoglu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Power Law ◽  
Elliptic Function ◽  
Soliton Solutions ◽  
Mathematical Physics ◽  
Zakharov Equation ◽  
Trial Equation Method ◽  
Klein Gordon

We introduce a new version of the trial equation method for solving nonintegrable partial differential equations in mathematical physics. Some exact solutions including soliton solutions and rational and elliptic function solutions to the Klein-Gordon-Zakharov equation with power law nonlinearity in (1 + 2) dimensions are obtained by this method.

scholarly journals Separation Transformation and a Class of Exact Solutions to the Higher-Dimensional Klein-Gordon-Zakharov Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Jing Chen ◽  
Ling Liu ◽  
Li Liu
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Langmuir Wave ◽  
Nonlinear Ordinary Differential Equations ◽  
Partial Differential ◽  
Zakharov Equation ◽  
Ion Acoustic Wave ◽  
Klein Gordon

The separation transformation method is extended to then+1-dimensional Klein-Gordon-Zakharov equation describing the interaction of the Langmuir wave and the ion acoustic wave in plasma. We first reduce then+1-dimensional Klein-Gordon-Zakharov equation to a set of partial differential equations and two nonlinear ordinary differential equations of the separation variables. Then the general solutions of the set of partial differential equations are given and the two nonlinear ordinary differential equations are solved by extendedF-expansion method. Finally, some new exact solutions of then+1-dimensional Klein-Gordon-Zakharov equation are proposed explicitly by combining the separation transformation with the exact solutions of the separation variables. It is shown that, for the case ofn≥2, there is an arbitrary function in every exact solution, which may reveal more nontrivial nonlinear structures in the high-dimensional Klein-Gordon-Zakharov equation.

Lump and interaction solutions to linear PDEs in 2+1 dimensions via symbolic computation

2019 ◽  
Vol 33 (36) ◽  
pp. 1950457 ◽  
Author(s):  
Wen-Xiu Ma
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Symbolic Computation ◽  
Rogue Wave ◽  
Soliton Solutions ◽  
Partial Differential ◽  
Interaction Solutions ◽  
Linear Partial Differential Equations ◽  
Linear Pdes

The aim of this paper is to show that there exist lump solutions and interaction solutions to linear partial differential equations in 2[Formula: see text]+[Formula: see text]1 dimensions. Through symbolic computations with Maple, we exhibit a great variety of exact solutions to a class of (2[Formula: see text]+[Formula: see text]1)-dimensional linear partial differential equations, and present a specific example which possesses lump, lump-kink and lump-soliton solutions. This supplements the study on lump, rogue wave and breather solutions and their interaction solutions to nonlinear integrable equations.

Exact, multiple soliton solutions of the double sine Gordon equation

1978 ◽  
Vol 359 (1699) ◽  
pp. 479-495 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Gordon Equation ◽  
Dimensional Space ◽  
Nonlinear Partial Differential Equations ◽  
Soliton Solutions ◽  
Sine Gordon Equation ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Method Of Solution

Exact, particular solutions of the double sine Gordon equation in n dimensional space are constructed. Under certain restrictions these solutions are N solitons, where N ≤ 2 q — 1 and q is the dimensionality of spacetime. The method of solution, known as the base equation technique, relates solutions of nonlinear partial differential equations to solutions of linear partial differential equations. This method is reviewed and its applicability to the double sine Gordon equation shown explicitly. The N soliton solutions have the remarkable property that they collapse to a single soliton when the wave vectors are parallel.

Multi-parameter Singular Integrals. (AM-189)

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Partial Differential Equations ◽  
Graduate Students ◽  
General Theory ◽  
Differential Equations ◽  
Singular Integrals ◽  
Main Tool ◽  
Classical Theorem ◽  
Quantitative Version ◽  
Linear Partial Differential Equations

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and applications to linear partial differential equations. It then outlines the theory of multi-parameter Carnot–Carathéodory geometry, where the main tool is a quantitative version of the classical theorem of Frobenius. The book then gives several examples of multi-parameter singular integrals arising naturally in various problems. The final chapter of the book develops a general theory of singular integrals that generalizes and unifies these examples. This is one of the first general theories of multi-parameter singular integrals that goes beyond the product theory of singular integrals and their analogs. This book will interest graduate students and researchers working in singular integrals and related fields.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

A method for constructing solutions of homogeneous partial differential equations: localized waves

1992 ◽  
Vol 437 (1901) ◽  
pp. 673-692 ◽  
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Gordon Equation ◽  
Partial Differential ◽  
Wave Solutions ◽  
Propagation Axis ◽  
Potential Applications ◽  
Klein Gordon ◽  
Localized Waves

We introduce a method for constructing solutions of homogeneous partial differential equations. This method can be used to construct the usual, well-known, separable solutions of the wave equation, but it also easily gives the non-separable localized wave solutions. These solutions exhibit a degree of focusing about the propagation axis that is dependent on a free parameter, and have many important potential applications. The method is based on constructing the space-time Fourier transform of a function so that it satisfies the transformed partial differential equation. We also apply the method to construct localized wave solutions of the wave equation in a lossy infinite medium, and of the Klein-Gordon equation. The localized wave solutions of these three equations differ somewhat, and we discuss these differences. A discussion of the properties of the localized waves, and of experiments to launch them, is included in the Appendix.

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