| **Proceedings of the Conference on Nonlinear Coherent Structures in Physics and Biology,** Heriot-Watt University, Edinburgh, July 95 | |

*On a Shallow Water Wave System*

### T.J. Priestley ¹, P.A. Clarkson

¹Institute of Mathematics and Statistics,

University of Kent, Canterbury, CT2 7NF, U.K.

Abstract

In this paper we discuss symmetry reductions
for the shallow water wave system
$$\eqalign{& u_{xxt}+\alpha u u_t+\beta v u_x -u_t -u_x =0\cr
&v_x=u_t\cr}\eqno(*) $$
where $\alpha$ and $\beta$ are arbitrary, non-zero, constants.
This arises in shallow water
theory as a scalar equation in non-local form by setting
$v=\partial^{-1}_x u_t$,
where $(\partial^{-1}_x f)(x)=\int_x^{\infty} f(y) \,{\rm d}y$,
using the so-called
Boussinesq approximation; it has also been discussed in local form
obtained by setting
$u=U_x, v=U_t$. Two special cases of these scalar equations
have been discussed in the literature, namely
$\alpha=\beta$ and $\alpha=2\beta$.
The inverse scattering problem has been solved
in both these cases, further they have both been studied
using Hirota's bi-linear technique.
A full
catalogue of classical and nonclassical reductions have
been given for the general local scalar equation.
We derive symmetry reductions of $(*)$ using
the classical Lie method of infinitesimal transformations,
the nonclassical method due to Bluman & Cole
and the direct method due to Clarkson & Kruskal.
We observe a large increase in the complexity of
the calculation of the nonclassical method for the system $(*)$
as compared with the associated scalar equation and also the need for
unusual techniques in both the direct and nonclassical method.
Conference paper.

*Last modified Mon Apr 8 16:25:00 GB-Eire 1996
(DBD)*