  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.
