Finding and Analyzing the Solutions of Singular Nonlinear Traveling Wave Equations: A Dynamical Systems Approach

This overview paper reviews a dynamical systems approach to studying the singular nonlinear traveling wave equations. First, the notion of exact peakon, periodic peakon, pseudo-peakon, as well as compacton solutions for the generalized Camassa–Holm (CH) equation and the Degasperis–Procesi (DP) equation, is introduced. Based on the method of dynamical systems and the theory of singular traveling wave equations, the exact explicit parametric representations of the solutions of the above-mentioned equations are derived. These solutions show that peakon is a limit solution of a family of periodic peakons or a limit solution of a family of pseudo-peakons in different limit senses, whereas the pseudo-peakon and pseudo-periodic peakon families are smooth classical solutions in two different time scales. Second, nonlinear wave equation models are introduced to show that there exist various exact explicit peakon solutions, which are different from the peakon solution given by the generalized CH equation and the DP equation. Third, the so-called “peakon equations” discussed in some studies actually have no peakons. Corresponding to these “peakon equations”, their traveling wave systems are singular traveling wave systems of the second kind, which cannot have peakon solutions. Finally, an application example is presented to illustrate our theory and methodology using a model of nonlinear elastodynamics of materials with strong ellipticity.