Exact traveling-wave solutions to bidirectional wave equations

In this paper, we present several systematic ways to find exact traveling-wave solutions of the systems eta(i) + u(x) + (u eta)(x) + au(xxx) - b eta(xxi) = 0 u(i) + eta(x) + uu(x) + c eta(xxx) - du(xxi) = 0 where a, b, c, and d are real constants. These systems, derived by Bona, Saut and Toland 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.