Solitary and Periodic Wave Solutions for Several Short Wave Model Equations

Abstract

We study the periodic and solitary wave solutions to several short wave model equations arising from a so-called $\beta$-family equation for $\beta=1,2,4$. These are integrable cases which possess Lax pair and multi-soliton solutions. By phase plane analysis, either the loop or cuspon type solutions are predicted. Then, by introducing a hodograph, or reciprocal, transformation, a coupled system is derived for each $\beta$. Applying a travelling wave setting, we are able to find the periodic solutions exactly expressed in terms of Jacobi Elliptic functions. In the limiting cases of modulus k=1, they all converge to the known solitary waves