# Parametric solutions of the generalized short pulse equations

## TL;DR

This paper introduces parametric solutions for three new integrable PDEs related to the short pulse equation, including multisoliton, cusp, breather, and cycloid solutions, expanding the understanding of their mathematical and physical properties.

## Contribution

It provides explicit parametric multisoliton solutions for three novel integrable PDEs associated with the generalized short pulse equations.

## Key findings

- Derived exact multisoliton solutions including cusped and breather types
- Constructed cusp solitons and periodic wave solutions from them
- Connected cycloid solutions to classical gravity wave solutions

## Abstract

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.