Variform Exact One-Peakon Solutions for Some Singular Nonlinear Traveling Wave Equations of the First Kind

In this paper, we consider variform exact peakon solutions for four nonlinear wave equations. We show that under different parameter conditions, one nonlinear wave equation can have different exact one-peakon solutions and different nonlinear wave equations can have different explicit exact one-peakon solutions. Namely, there are various explicit exact one-peakon solutions, which are different from the one-peakon solution pe−α|x−ct|. In fact, when a traveling system has a singular straight line and a curve triangle surrounding a periodic annulus of a center under some parameter conditions, there exists peaked solitary wave solution (peakon).