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

*Symmetry reductions and exact solutions of shallow water wave equations*

### P.A. Clarkson ¹, E.L. Mansfield

¹Institute of Mathematics and Statistics,

University of Kent at Canterbury,

Canterbury, CT2 7NF, U.K.

Abstract

In this paper we study symmetry reductions and exact solutions of the shallow
water wave (SWW) equation

u_{xxxt} + \alpha u_x u_{xt} + \beta u_t u_{xx} - u_{xt} - u_{xx} =0 .....(1)

where \alpha and \beta are arbitrary, nonzero, constants, which is derivable
using the so-called Boussinesq approximation. Two special cases of this equation
have been discussed in the literature; the case \alpha=2\beta by
Ablowitz, Kaup, Newell and Segur [*Stud. Appl. Math.*, 53 (1974) 249]
and the case
\alpha=\beta by Hirota and Satsuma
[* J. Phys. Soc. Japan,* 40 (1976) 611].
Further (1) is known to be solvable by inverse scattering in these
two special cases and the Painlev\'e test suggest that it is not for other
choices of the parameters.
A catalogue of symmetry reductions is obtained using the
classical Lie method and the nonclassical method due to Bluman and Cole
[*J. Math. Mech. *, 18 (1969) 1025]. The classical Lie method yields
symmetry reductions of (1) expressible in terms of the first and third
\p\ transcendents and Weierstrass elliptic functions. The nonclassical
method yields a plethora of exact solutions of (1) with \alpha=\beta which
possess a rich variety of qualitative behaviours. These solutions all like a
two-soliton solution for t<0 but differ radically for t>0 and may be viewed
as a nonlinear superposition of two solitons, one travelling to the left with
arbitrary speed and the other to the right with equal and opposite speed.
These families of solutions have important implications with regard to the
numerical analysis of SWW and suggests that solving (1) numerically could pose
some fundamental difficulties. In particular, one would not be able to
distinguish the solutions in an initial value problem since an exponentially
small change in the initial conditions can result in completely different
qualitative behaviours.

Further, we show that there is an analogous nonlinear superposition of
solutions for two 2+1-dimensional generalisations of the SWW equation (1)
with \alpha=\beta. This yields solutions expressible as the sum of two
solutions of the Korteweg-de Vries equation.

This paper is a revised version of our paper
[* Acta Appl. Math.*, 39 (1995) 245-276].

Conference paper.

*Last modified Mon Apr 8 16:24:54 GB-Eire 1996
(DBD)*