Extension on Bifurcations of Traveling Wave Solutions for a Two-Component Fornberg-Whitham Equation

Zhenshu Wen

doi:10.1155/2012/704931

Abstract

Fan et al. studied the bifurcations of traveling wave solutions for a two-component Fornberg-Whitham equation. They gave a part of possible phase portraits and obtained some uncertain parametric conditions for solitons and kink (antikink) solutions. However, the exact explicit parametric conditions have not been given for the existence of solitons and kink (antikink) solutions. In this paper, we study the bifurcations for the two-component Fornberg-Whitham equation in detalis, present all possible phase portraits, and give the exact explicit parametric conditions for various solutions. In addition, not only solitons and kink (antikink) solutions, but also peakons and periodic cusp waves are obtained. Our results extend the previous study.

Highlights

In 2011, Fan et al 1 introduced the following two-component Fornberg-Whitham equation ut uxxt − ux − uux 3uxuxx uuxxx ρx, 1.1 ρt − ρu x, where u u x, t denotes the height of the water surface above a horizontal bottom, and ρ ρ x, t indicts the horizontal velocity field
We state our results about solitons, kink antikink solutions, peakons, and periodic cusp waves for the first component of system 1.1
Based on a previous paper 1, we further study the bifurcations of traveling wave solutions for the two-component Fornberg-Whitham equation, present all possible phase portraits determinately, and show all the exact explicit parametric conditions under which there exist solitons and kink or antikink solutions for 1.1

Summary

Introduction

In 2011, Fan et al 1 introduced the following two-component Fornberg-Whitham equation ut uxxt − ux − uux 3uxuxx uuxxx ρx, 1.1 ρt − ρu x, where u u x, t denotes the height of the water surface above a horizontal bottom, and ρ ρ x, t indicts the horizontal velocity field. They studied the bifurcations of traveling wave solutions for 1.1 through obtaining some uncertain parametric conditions for solitons, kink antikink solutions, and further gave some expressions of those solutions. We obtain explicit peakons and periodic cusp waves for 1.1 , which were not included in 1

Bifurcations of Phase Portraits

Main Results and the Theoretic Derivations of Main Results

Conclusions