Exact solutions of various Boussinesq systems

It was shown in [1,2] that surface water waves in a water tunnel can be described by systems of the form 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, (1) 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. (C) 1998 Elsevier Science Ltd. All rights reserved.