  In this paper we study a shallow water equation derivable using the
Boussinesq approximation, which includes as two special cases, one equation
discussed by Ablowitz et. al. [Stud. Appl. Math., 53 (1974) 249--315] and one
by Hirota and Satsuma [J. Phys. Soc. Japan}, 40 (1976) 611--612]. A catalogue
of classical and nonclassical symmetry reductions, and a Painleve analysis, are
given. Of particular interest are families of solutions found containing a rich
variety of qualitative behaviours. Indeed we exhibit and plot a wide variety of
solutions all of which look like a two-soliton for t>0 but differ radically for
t<0. These families arise as nonclassical symmetry reduction solutions and
solutions found using the singular manifold method. This example shows that
nonclassical symmetries and the singular manifold method do not, in general,
yield the same solution set. We also obtain symmetry reductions of the shallow
water equation solvable in terms of solutions of the first, third and fifth
Painleve equations. We give evidence that the variety of solutions found which
exhibit ``nonlinear superposition'' is not an artefact of the equation being
linearisable since the equation is solvable by inverse scattering. These
solutions have important implications with regard to the numerical analysis for
the shallow water equation we study, which 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.
