Symmetries and exact solutions for a 2+1-dimensional shallow water wave equation

Classical and nonclassical reductions of a 2 + 1-dimensional shallow water wave equation are classified. Using these reductions, we derive some exact solutions, including solutions expressed as the nonlinear superposition of solutions of a generalised variable-coefficient Korteweg-de Vries equation. Many of the reductions obtained involve arbitrary functions and so the associated families of solutions have a rich variety 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 overdetermined systems of partial differential equations whose solutions yield the reductions were analysed and solved using the MAPLE package diffgrob2, which we describe briefly.