Smooth travelling wave solutions in a generalized Degasperis–Procesi equation

Abstract

In this paper, by using a dynamical system-based approach, namely, the singular travelling wave method, we classify all the travelling wave solutions including periodic wave solutions, periodic peakon solutions, solitary wave solutions and compacton solutions etc. in the generalized Degasperis–Procesi (gDP) equation. All these travelling waves are smooth and classical solutions. The parameter conditions for the existence of these travelling waves are also obtained. Then we combine geometric singular perturbation theory with an “explicit”Melnikov method to analyze the persistence of solitary wave solutions under singular perturbation, i.e., the existence of solitary wave solutions in a perturbed gDP equation. By calculating the representations of the (unperturbed) homoclinic orbits and the associated Melnikov integral explicitly, the persistence of solitary wave solutions is shown and the speed of the wave is determined (to leading order).