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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 wave solutions for a generalized Camassa-Holm equation $u_t-u_{xxt}=\frac{1}{2}(p+1)(p+2)u^pu_x-\frac{1}{2}p(p-1)u^{p-2}u_x^3-2pu^{p-1}u_xu_{xx}-u^pu_{xxx}$ 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 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.