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.

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