Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation

Yuqian  Zhou; Qian  Liu

doi:10.3934/dcdsb.2016036

Article Contents

2016, Volume 21, Issue 6: 2057-2071. Doi: 10.3934/dcdsb.2016036

This issue Previous Article Global boundedness and decay for a multi-dimensional chemotaxis-haptotaxis system with nonlinear diffusion Next Article

Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation

Yuqian Zhou^1, and
Qian Liu^2,

1.
School of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225
2.
School of Computer Science and Technology, Southwest University for Nationalities, Chengdu, Sichuan 610041

Received: April 2015

Revised: April 2016

Published: August 2016

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Abstract

In this paper, the Lie symmetry analysis is performed on the KBK equation. By constructing its one-dimensional optimal system, we obtain four classes of reduced equations and corresponding group-invariant solutions. Particularly, the traveling wave equation, as an important reduced equation, is investigated in detail. Treating it as a singular perturbation system in $\mathbb{R}^3$, we study the phase space geometry of its reduced system on a two-dimensional invariant manifold by using the dynamical system methods such as tracking the unstable manifold of the saddle, studying the equilibria at infinity and discussing the homoclinic bifurcation and Poincaré bifurcation. Correspongding wavespeed conditions are determined to guarantee the existence of various bounded traveling waves of the KBK equation.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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