Exact Solutions of Travelling Wave Model via Dynamical System Method

Heng Wang,Longwei Chen,Hongjiang Liu

doi:10.1155/2016/9290734

Abstract

By using the method of dynamical system, the exact travelling wave solutions of the coupled nonlinear Schrödinger-Boussinesq equations are studied. Based on this method, the bounded exact travelling wave solutions are obtained which contain solitary wave solutions and periodic travelling wave solutions. The solitary wave solutions and periodic travelling wave solutions are expressed by the hyperbolic functions and the Jacobian elliptic functions, respectively. The results show that the presented findings improve the related previous conclusions. Furthermore, the numerical simulations of the solitary wave solutions and the periodic travelling wave solutions are given to show the correctness of our results.

Highlights

In laser and plasma physics, the significant problems under interactions between a nonlinear real Boussinesq field and a nonlinear complex Schrodinger field have been raised [1]
We show that the hyperbolic function solutions and the Jacobian elliptic function solutions we found in this paper are different from the solutions presented by other authors before
We obtain additional travelling wave solutions of (1) as follows:

Summary

Introduction

In laser and plasma physics, the significant problems under interactions between a nonlinear real Boussinesq field and a nonlinear complex Schrodinger field have been raised [1]. In [4], the approximate solutions and conservation law for the nonlinear Schrodinger-Boussinesq equations have been studied. We consider the following coupled nonlinear Schrodinger-Boussinesq equations [5]: iEt + Exx + γE = NE, 3Ntt. where γ, δ are real parameters. The study of travelling wave solutions of the coupled Schrodinger-Boussinesq equations has attracted much attention of physicists and mathematicians. In [7], Farah and Pastor used the (G󸀠/G)expansion method to construct travelling wave solutions for the equations. We notice that the previous authors did not study the nonlinear dynamics of (1) and did not find all possible travelling wave solutions. It is essential to study the nonlinear dynamics of (1) and find all possible travelling wave solutions of (1).

The Dynamical System Method

Conclusion