Exact traveling-wave solutions to bi-directional wave equations

Abstract

In this paper, we present several systematic ways to find exact traveling-wave solutions of the systems

\eta_t+u_x+(u\eta)_x+au_{xxx}-b\eta_{xxt}=0,

u_t+\eta_x+uu_x+c\eta_{xxx}-du_{xxt}=0,

where $a, b, c$ and $d$ are real constants. These systems, derived by Bona, Chen and Saut for describing small-amplitude long waves in a water channel, are formally equivalent to the classical Boussinesq system and correct through first order with regard to a small parameter characterizing the typical amplitude to depth ratio. Exact solutions for a large class of systems are presented. The existence of the exact traveling-wave solutions is in general extremely helpful in the theoretical and numerical study of the systems.