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

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.