SYMMETRIES AND EXACT-SOLUTIONS FOR A 2-DIMENSIONAL SHALLOW-WATER WAVE-EQUATION(1)

Classical and nonclassical reductions of a 2 + 1-dimensional shallow w ater wave equation are classified. Using these reductions, we derive s ome exact solutions, including solutions expressed as the nonlinear su perposition of solutions of a generalised variable-coefficient Kortewe g-de Vries equation. Many of the reductions obtained involve arbitrary functions and so the associated families of solutions have a rich var iety of qualitative behaviours. This suggests that solving the initial value problem for the 2 + 1-dimensional shallow water equation under discussion could pose some fundamental difficulties. The nonlinear ove rdetermined systems of partial differential equations whose solutions yield the reductions were analysed and solved using the MAPLE package diffgrob2, which we describe briefly.