Understanding Peakons, Periodic Peakons and Compactons via a Shallow Water Wave Equation

In this paper, a shallow water wave model is used to introduce the concepts of peakon, periodic peakon and compacton. Traveling wave solutions of the shallow water equation are presented. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. By using the method of dynamical systems, bifurcation diagrams and explicit exact parametric representations of the solutions are given, including solitary wave solution, periodic wave solution, peakon solution, periodic peakon solution and compacton solution under different parameter conditions.