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