  In this paper we employ three recent analytical approaches to investigate the
possible classes of traveling wave solutions of some members of a family of
so-called short-pulse equations (SPE). A recent, novel application of
phase-plane analysis is first employed to show the existence of breaking kink
wave solutions in certain parameter regimes. Secondly, smooth traveling waves
are derived using a recent technique to derive convergent multi-infinite series
solutions for the homoclinic (heteroclinic) orbits of the traveling-wave
equations for the SPE equation, as well as for its generalized version with
arbitrary coefficients. These correspond to pulse (kink or shock) solutions
respectively of the original PDEs.
  Unlike the majority of unaccelerated convergent series, high accuracy is
attained with relatively few terms. And finally, variational methods are
employed to generate families of both regular and embedded solitary wave
solutions for the SPE PDE. The technique for obtaining the embedded solitons
incorporates several recent generalizations of the usual variational technique
and it is thus topical in itself. One unusual feature of the solitary waves
derived here is that we are able to obtain them in analytical form (within the
assumed ansatz for the trial functions). Thus, a direct error analysis is
performed, showing the accuracy of the resulting solitary waves. Given the
importance of solitary wave solutions in wave dynamics and information
propagation in nonlinear PDEs, as well as the fact that not much is known about
solutions of the family of generalized SPE equations considered here, the
results obtained are both new and timely.
