Traveling wave solutions of a generalized Camassa-Holm equation: a dynamical system approach (Q1666138)

Summary: We investigate a generalized Camassa-Holm equation \(C(3,2, 2)\): \(u_t + k u_x + \gamma_1 u_{x x t} + \gamma_2(u^3)_x + \gamma_3 u_x(u^2)_{x x} + \gamma_3 u(u^2)_{x x x} = 0\). We show that the \(C(3,2, 2)\) equation can be reduced to a planar polynomial differential system by transformation of variables. We treat the planar polynomial differential system by the dynamical systems theory and present a phase space analysis of their singular points. Two singular straight lines are found in the associated topological vector field. Moreover, the peakon, peakon-like, cuspon, smooth soliton solutions of the generalized Camassa-Holm equation under inhomogeneous boundary condition are obtained. The parametric conditions of existence of the single peak soliton solutions are given by using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for single peak soliton, kink wave, and kink compacton solutions of the \(C(3,2, 2)\) equation.