In this lecture I shall discuss 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 Boussinesq approximation. Two special cases of this equation have been discussed in the literature; the case $\alpha=2\beta$ by Ablowitz {\sl et al.\/}[\ref{Stud. Appl. Math.}{53}{1974}{249}] and the case $\alpha=\beta$ by Hirota and Satsuma [\ref{J. Phys. Soc. Japan}{40}{1976}{611}]. Further equation (1) is known to be solvable by inverse scattering in these two special cases and the Painlev\'e test suggests that (1) is not integrable for other choices of the parameters.

A catalogue of classical and nonclassical symmetry reductions and exact solutions of the SWW equation (1) will given. Of particular interest are families of solutions possessing a rich variety of qualitative behaviours in the special case $\alpha=\beta$. For this special case I shall exhibit and illustrate a wide variety of solutions which arise as nonclassical reduction solutions, all of which look like a two-soliton solution as $t\to-\infty$, yet are radically different as $t\to\infty$. These results have important implications with regard to numerical analysis and suggest that solving (1) with $\alpha=\beta$ numerically could pose some fundamental difficulties: an exponentially small change in the initial data can yield a fundamentally different solution as $t\to\infty$. How can any numerical scheme in current use cope with such behaviour?

I shall also compare and contrast the application of the nonclassical method due to Bluman \& Cole [\ref{J. Math. Mech.}{18}{1969}{1025}] and the direct method due to Clarkson \& Kruskal [\ref{J. Math. Phys.}{30}{1989}{2201}] for both the SWW equation (1) and the system

    u_{xxt} + \alpha u u_t + \beta v u_x - u_t - u_x = 0,
v_x = u_t.


Finally, I shall exhibit analogous nonlinear superposition of solutions for some $2+1$-dimensional equations including a $2+1$-dimensional generalisation of (1) with $\alpha=\beta$ and a $2+1$-dimensional Sine-Gordon equation derived by Konopelchenko \& Rogers [\ref{Phys. Lett.}{158A}{1991}{391}].