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.
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