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

Jing Chen,Li Liu,Ling Liu

doi:10.1155/2014/974050

Abstract

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.

Highlights

The Klein-Gordon equation [1] is a relativistic version of the Schrodinger equation
It is attributed to the classical u4 field theory in the physics of elementary particles and fields, and it can describe the propagation of dislocations within crystals and the propagation of magnetic flux on a Josephson line, and so on
The following proposition reveals the relationship between the exact solutions of the (n + 1)-dimensional KGZ equation (3) and two nonlinear ordinary differential equations (ODEs) along with a set of partial differential equations (PDEs)

Summary

Introduction

The Klein-Gordon equation (sometimes called KleinGordon-Fock equation) [1] is a relativistic version of the Schrodinger equation. One extension of the nonlinear Klein-Gordon equation is the (1 + 1)-dimensional KleinGordon-Zakharov (KGZ) equation [3, 4]: uV. The high-dimensional extension of KGZ equation is important in real applications, so in this paper we would like to investigate the (n + 1)-dimensional KGZ equation: Δu (3). Wang [9] extended the separation transformation method proposed in [10,11,12] to the (N + 1)dimensional coupled nonlinear Klein-Gordon equations. Liu et al [13] and we [14] further extended the separation transformation method to various high-dimensional nonlinear soliton equations and obtained explicitly many exact solutions with arbitrary functions.

Separation Transformation and Its Application

Conclusion