Bifurcation of Traveling Wave Solutions for a Two-Component Generalizedθ-Equation

Zhenshu Wen

doi:10.1155/2012/597431

Abstract

We study the bifurcation of traveling wave solutions for a two-component generalizedθ-equation. We show all the explicit bifurcation parametric conditions and all possible phase portraits of the system. Especially, the explicit conditions, under which there exist kink (or antikink) solutions, are given. Additionally, not only solitons and kink (antikink) solutions, but also peakons and periodic cusp waves with explicit expressions, are obtained.

Highlights

In 2008, Liu 1 introduced a class of nonlocal dispersive models, that is, θ-equations, as follows: ut − uxxt uux 1 − θ uxuxx θuuxxx, x ∈ R, t > 0, 1.1 where u x, t denotes the velocity field at time t in the spatial x direction
We study the bifurcation of traveling wave solutions for the following system: ut − uxxt uux
0, x ∈ R, t > 0, which is a special form of system 1.2 through taking θ 2/5 and σ 1, by employing the bifurcation method and qualitative theory of dynamical systems 3–7

Summary

Introduction

In 2008, Liu 1 introduced a class of nonlocal dispersive models, that is, θ-equations, as follows: ut − uxxt uux 1 − θ uxuxx θuuxxx, x ∈ R, t > 0, 1.1 where u x, t denotes the velocity field at time t in the spatial x direction. Ni 2 further investigated the cauchy problem for the following twocomponent generalized θ-equations: ut − uxxt uux − 1 − θ uxuxx − θuuxxx σρρx 0, x ∈ R, t > 0, 1.2 ρt θρxu 1 − 2θ ρux 0, x ∈ R, t > 0, where σ takes 1 or −1. This system includes two components u x, t and ρ x, t. We give all the explicit bifurcation parametric conditions for various solutions and all possible phase portraints of the system, from which solitons and kink antikink solutions, and peakons and periodic cusp waves are obtained

Bifurcation of Phase Portraits

Main Results and the Theoretic Derivations of Main Results

Conclusions