Abstract:

In this paper, we first study the exact peakon, periodic peakon, pseudo-peakon as well as the compacton solutions for the generalized Camassa-Holm equation and the Degasperis-Procesi equation. Based on the method of dynamical systems and the theory of singular traveling wave equations, the exact explicit parametric representations of the above mentioned solutions are derived. These solutions tell us that peakon is a limit solution of a family of periodic peakons or a limit solution of a family of pseudo-peakons under two classes of limit senses. The pseudo-peakon and pseudo-periodic peakon family are smooth classical solutions with two time scales.

Second, we use some nonlinear wave equation models to show that there exist various exact explicit peakon solutions, which are different from the peakon solutions given by the generalized Camassa-Holm equation and the Degasperis-Procesi equation.

Third, we point out that the so called ``peakon equations'' in some references have no peakons. Corresponding to these ``peakon equations'', their traveling systems are the singular traveling systems of the second kind, which can not have the peakon solution.

Key words: Nonlinear wave equation, Traveling wave solution, Bifurcation, Pseudo-peakon, Periodic peakon, Compacton, Exact solution, Peakon