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Bifurcations of traveling wave solutions for a generalized Camassa-Holm equation
Keywords:Generalized Camassa-Holm equation, Bifurcation theory, Peakon, Solitary wave solution, Kink and Anti-kink wave solutions
Abstract:
      In this paper, the traveling solutions for a generalized Camassa-Holm equation u_t ?uxxt = 1/2 (p+1)(p+2)u^pu_x? 1/2p(p?1)u^p?2u^3_x?2pu^{p?1}u_xu_xx?u^pu_xxx 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 Camassa- Holm equation is 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.