Smooth and non-smooth traveling wave solutions of some generalized Camassa-Holm equations

Abstract

In this paper we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of a recently-derived integrable family of generalized Camassa-Holm (GCH) equations. A recent, novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of three of the GCH NLPDEs, i.e. the possible non-smooth peakon, cuspon and compacton solutions. Two of the GCH equations do not support singular traveling waves. The third equation supports four-segmented, non-smooth $M$-wave solutions, while the fourth supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. Moreover, smooth traveling waves of the four GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic and heteroclinic orbits of their traveling-wave equations, corresponding to pulse and front (kink or shock) solutions respectively of the original PDEs. We perform many numerical tests in different parameter regime to pinpoint real saddle equilibrium points of the corresponding GCH equations, as well as ensure simultaneous convergence and continuity of the multi-infinite series solutions for the homoclinic and heteroclinic orbits anchored by these saddle points. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. We also show the traveling wave nature of thesepulse and front solutions to the GCH NLPDEs.