Peakons and periodic cusp wave solutions in a generalized Camassa–Holm equation

By using the bifurcation theory of planar dynamical systems to a generalized Camassa–Holm equationmt+c0ux+umx+2mux=-γuxxxmt+c0ux+umx+2mux=-γuxxxwith m = u − α2uxx, α ≠ 0, c0, γ are constant, which is called CH-r equation, the existence of peakons and periodic cusp wave solutions is obtained. The analytic expressions of the peakons and periodic cusp wave solutions are given and numerical simulation results show the consistence with the theoretical analysis at the same time.