The investigation of the travelling wave solutions to nonlinear evolution equations play an important role in the mathematical physics.
The aim of this talk is to give a more systematic account for the classification of the traveling wave solutions and the understanding for
the concepts of peakons, periodic peakons, pseudo-peakons and compactons created by the singular nonlinear traveling equations. By using
bifurcation theory method of dynamical systems and ideas from the singular perturbation theory, we show that periodic peakons are two time
scale smooth classical solutions. Peakon is a limit solution in the two senses.