Bifurcations of traveling wave solutions for a generalized Camassa-Holm equation

In this paper, the traveling wave solutions for a generalized CamassaHolm equation ut−uxxt = 12 (p+1)(p+2)u ux− 1 2p(p−1)u ux−2puuxuxx− uuxxx are investigated. By using the bifurcation method of dynamical systems, three major results for this equation are highlighted. First, there are one or two singular straight lines in the two-dimensional system under some different conditions. Second, all the bifurcations of the generalized CamassaHolm equation are given for p either positive or negative integer. Third, we prove that the corresponding traveling wave system of this equation possesses peakon, smooth solitary wave solution, kink and anti-kink wave solution, and periodic wave solutions.

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