Copyright © 2009 Jiangbo Zhou and Lixin Tian. 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 employ the bifurcation theory of planar dynamical systems to investigate the exact travelling wave solutions of a generalized Degasperis-Procesi equation ut−uxxt+4uux+γ(u−uxx)x=3uxuxx+uuxxx. The implicit expression of smooth soliton solutions is given. The explicit expressions of peaked soliton solutions and periodic cuspon solutions are also obtained. Further, we show the relationship among the smooth soliton solutions, the peaked soliton solutions, and the periodic cuspon solutions. The physical relevance of the found solutions and the reason why these solutions can exist in this equation are also given.