Traveling Wave Solutions for the Generalized Zakharov Equations

Ming Song,Zhengrong Liu

doi:10.1155/2012/747295

Abstract

We use the bifurcation method of dynamical systems to study the traveling wave solutions for the generalized Zakharov equations. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow‐up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended.

Highlights

The Zakharov equations iut uxx − uv 0, 1.1 vtt − vxx |u|20, xx which is one of the fundamental models governing dynamics of nonlinear waves in one-dimensional systems
The equations can be derived from a hydrodynamic description of the plasma 2, 3
Some important effects such as transit-time damping and ion nonlinearities, which are implied by the fact that the values used for the ion damping have been anomalously large from the point of view of linear ionacoustic wave dynamics, have been ignored in 1.1

Summary

Introduction

0, xx which is one of the fundamental models governing dynamics of nonlinear waves in one-dimensional systems. Some important effects such as transit-time damping and ion nonlinearities, which are implied by the fact that the values used for the ion damping have been anomalously large from the point of view of linear ionacoustic wave dynamics, have been ignored in 1.1 This is equivalent to saying that 1.1 is a simplified model of strong Langmuir turbulence. Javidi and Golbabai used the He’s variational iteration method to obtain solitary wave solutions of 1.2. Borhanifar et al obtained the generalized solitary solutions and periodic solutions of 1.2 by using the Exp-function method. The aim of this paper is to study the traveling wave solutions and their limits for 1.2 by using the bifurcation method and qualitative theory of dynamical systems 17– 24.

Phase Portraits and Qualitative Analysis

Traveling Wave Solutions and Their Relations

Conclusion