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Bifurcations and exact traveling wave solutions for a generalized Camassa-Holm equation. (English) Zbl 1270.34086
Summary: In this paper, we study all possible traveling wave solutions of an integrable system with both quadratic and cubic nonlinearities: m t =bu x +1 2k 1 [m(u 2 -u x 2 )] x +1 2k 2 (2mu x +m x u),m=u-u xx , where b,k 1 and k 2 are arbitrary constants. We call this model a generalized Camassa-Holm equation since it is kind of a cubic generalization of the Camassa-Holm (CH) equation: m t +m x u+2mu x =0. In the paper, we show that the traveling wave system of this generalized Camassa-Holm equation is actually a singular dynamical system of the second class. We apply the method of dynamical systems to analyze the dynamical behavior of the traveling wave solutions and their bifurcations depending on the parameters of the system. Some exact solutions such as smooth soliton solutions, kink and anti-kink wave solutions, M-shape and W-shape wave profiles of the breaking wave solutions are obtained. To guarantee the existence of those solutions, some constraint parameter conditions are given.
MSC:
34C23Bifurcation (ODE)
34C07Theory of limit cycles of polynomial and analytic vector fields
34A05Methods of solution of ODE
34C37Homoclinic and heteroclinic solutions of ODE
34C05Location of integral curves, singular points, limit cycles (ODE)