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Mathematical Problems in Engineering
Volume 2015, Article ID 610979, 19 pages
http://dx.doi.org/10.1155/2015/610979
Research Article

Traveling Wave Solutions of a Generalized Camassa-Holm Equation: A Dynamical System Approach

Department of Mathematics, Huzhou University, Huzhou, Zhejiang 313000, China

Received 1 August 2015; Accepted 14 September 2015

Academic Editor: Maria Gandarias

Copyright © 2015 Lina Zhang and Tao Song. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate a generalized Camassa-Holm equation : . We show that the equation can be reduced to a planar polynomial differential system by transformation of variables. We treat the planar polynomial differential system by the dynamical systems theory and present a phase space analysis of their singular points. Two singular straight lines are found in the associated topological vector field. Moreover, the peakon, peakon-like, cuspon, smooth soliton solutions of the generalized Camassa-Holm equation under inhomogeneous boundary condition are obtained. The parametric conditions of existence of the single peak soliton solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for single peak soliton, kink wave, and kink compacton solutions of the equation.