Exact explicit traveling wave solutions for the CDF equation

The traveling wave solutions of the Calogero–Degasperis–Fokas (CDF) equation are studied by using the approach of dynamical systems and the theory of bifurcations. With the aid of Maple, all bifurcations and phase portraits in the parametric space are obtained. All possible explicit parametric representations of the bounded traveling wave solutions (solitary wave solutions, kink and anti-kink wave solutions, periodic wave solutions and breaking wave solutions) are given. Moreover, the reason for appearance of breaking waves and for persistence of smoothness of smooth traveling waves is explained.