Application of Bifurcation Method to the Generalized Zakharov Equations

Ming Song

doi:10.1155/2012/308326

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 blow‐up wave solutions and solitary wave solutions.

Highlights

The Zakharov equations iut uxx − uv 0, 1.1 vtt − vxx |u|20, xx are of the fundamental models governing dynamics of nonlinear waves in one-dimensional systems
We assume that the traveling wave solutions of 1.3 is of the form u x, t eiηφ ξ, v x, t ψ ξ, η px qt, ξ k x − 2pt, 2.1 where φ ξ and ψ ξ are real functions, p, q, and k are real constants
From the qualitative theory of dynamical systems, we know that a smooth solitary wave solution of a partial differential system corresponds to a smooth homoclinic orbit of a traveling wave equation

Summary

Introduction

0, xx are 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. Layeni obtained the exact traveling wave solutions of 1.2 by using the new rational auxiliary equation method. Song and Liu obtained a number of traveling wave solutions of 1.2 by using bifurcation method of dynamical systems. We aim to apply the bifurcation method of dynamical systems 18–22 to study the phase portraits for the corresponding traveling wave system of 1.3.

Phase Portraits and Qualitative Analysis

Conclusion