Exact traveling-wave solutions to
bi-directional wave equations

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.

Min Chen (chen@math.purdue.edu)