  We consider three novel PDEs associated with the integrable generalizations
of the short pulse equation classified recently by Hone {\it et al} (2018 {\it
Lett. Math. Phys.} {\bf 108} 927-947). In particular, we obtain a variety of
exact solutions by means of a direct method analogous to that used for solving
the short pulse equation. The main results reported here are the parametric
representations of the multisoliton solutions. These solutions include cusp
solitons, unbounded solutions with finite slope and breathers. In addition, the
cusped periodic wave solutions are constructed from the cusp solitons by means
of a simple procedure. As for non-periodic solutions, smooth breather solutions
are of particular interest from the perspective of applications to real
physical phenomena. The cycloid reduced from the periodic traveling wave with
cusps is also worth remarking in connection with Gerstner's trochoidal solution
in deep gravity waves. A number of works are left for future study, some of
which will be addressed in concluding remarks.
