Title

Smooth And Non-Smooth Traveling Wave Solutions Of Some Generalized Camassa-Holm Equations

Keywords

Generalized Camassa-Holm equations; Homoclinic and heteroclinic orbits; Traveling waves

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 and cuspon solutions. One of the considered GCH equations supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. The second equation does not support singular traveling waves and the last one supports four-segmented, non-smooth M-wave solutions.Moreover, smooth traveling waves of the three GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic orbits of their traveling-wave equations, corresponding to pulse (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 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 these pulse and front solutions to the GCH NLPDEs. © 2013 Elsevier B.V.

Publication Date

6-1-2014

Publication Title

Communications in Nonlinear Science and Numerical Simulation

Volume

19

Issue

6

Number of Pages

1746-1769

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1016/j.cnsns.2013.10.029

Socpus ID

84890794700 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84890794700

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