# American Institute of Mathematical Sciences

August  2016, 21(6): 2057-2071. doi: 10.3934/dcdsb.2016036

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

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

Received  April 2015 Revised  April 2016 Published  June 2016

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.
Citation: Yuqian Zhou, Qian Liu. Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2057-2071. doi: 10.3934/dcdsb.2016036
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