Exact solutions of various Boussinesq systems

Abstract

It was shown in [1,2] that surface water waves in a water tunnel can be described by systems of the form η t+u x+(uη) x+au xxx−bη xxt=0, u t+η x+uu x+cη xxx−du xxt=0 , where a, b, c, and d are real constants. In this paper, we show that to find an exact traveling-wave solution of the system, it is suffice to find a solution of an ordinary differential equation, and the solution of the ordinary differential equation in a prescribed form can be found by solving a system of nonlinear algebraic equation. The exact solutions for some of the systems are presented at the end of the paper.