Symmetry Reductions and Exact Solutions of Shallow Water Wave Equations
  In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation $$u_{xxxt} + \alpha u_x u_{xt} + \beta u_t u_{xx} - u_{xt} - u_{xx} = 0,\eqno(1)$$ where $\alpha$ and $\beta$ are arbitrary, nonzero, constants, which is derivable using the so-called Boussinesq approximation. Two special cases of this equation, or the equivalent nonlocal equation obtained by setting $u_x=U$, have been discussed in the literature. The case $\alpha=2\beta$ was discussed by Ablowitz, Kaup, Newell and Segur [{\it Stud.\ Appl.\ Math.}, {\bf53} (1974) 249], who showed that this case was solvable by inverse scattering through a second order linear problem. This case and the case $\alpha=\beta$ were studied by Hirota and Satsuma [{\it J.\ Phys.\ Soc.\ Japan}, {\bf40} (1976) 611] using Hirota's bi-linear technique. Further the case $\alpha=\beta$ is solvable by inverse scattering through a third order linear problem. In this paper a catalogue of symmetry reductions is obtained using the classical Lie method and the nonclassical method due to Bluman and Cole [{\it J.\ Math.\ Mech.\/}, {\bf 18} (1969) 1025]. The classical Lie method yields symmetry reductions of (1) expressible in terms of the first, third and fifth \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. 
Peter A. Clarkson and Elizabeth L. Mansfield (Department of   Mathematics, University of Exeter, Exeter, U.K.)