Conservation Laws and Web-Solutions for the Benney--Luke Equation

Abstract: A long wave multi-dimensional approximation of shallow water waves is the bi-directional Benney-Luke equation. It yields the well-known Kadomtsev-Petviashvili equation in a quasi one-directional limit. A direct perturbation method is developed; it uses the underlying conservation laws to determine the slow evolution of parameters of two space dimensional, non-decaying web-type solutions to the Benney-Luke equation. New numerical simulations, based on windowing methods which are effective for non-decaying data, are presented. These simulations support the analytical results and elucidate the relationship between the Kadomtsev-Petviashvilli and the Benney-Luke equations and are also used to obtain amplitude information regarding particular web solutions. Additional dissipative perturbations to the Benney-Luke equation are also studied.