Traveling waves in a generalized Camassa–Holm equation involving dual-power law nonlinearities

Abstract

This article focuses on traveling waves including peakons, periodic peakons, compactons and solitary waves as well as the persistence of some of these waves in a generalized Camassa–Holm (gCH) equation involving dual-power law nonlinearities. Firstly, we analyze the global phase portraits of the associated traveling wave system by a dynamical system-based approach. All the traveling waves are classified and their corresponding parameter conditions are obtained. It is found that the gCH equation with dual-power law nonlinearities can admit different peakons under different parameter conditions. Secondly, we pay attention to the persistence of solitary waves, i.e., the existence of solitary waves in a perturbed gCH equation by using geometric singular perturbation theory (GSPT) and an “explicit” Melnikov method. An elementary method to calculate the unperturbed homoclinic orbits and the associated Melnikov integral in an “explicit” way is developed in this article.