Bifurcations, Exact Peakon, Periodic Peakons and Solitary Wave Solutions of the Cubic Camassa–Holm Type Equation

For the cubic Camassa–Holm type equation, by using the techniques from dynamical systems and singular traveling wave theory developed by Li and Chen [2007] to analyze its corresponding traveling wave system, it was found that under different parameter conditions, its bifurcation portraits exhibit all possible exact explicit bounded solutions (solitary wave solutions, periodic wave solutions, peakon as well as periodic peakons). A total of 19 explicit exact parametric representations of the traveling wave system of the Camassa–Holm type equation are presented.