Evolution of wind-generated shallow water waves in a Benney-Luke equation

Abstract: In papers, denoted by MG24, MG25 in this text, we used the Korteweg-de Vries (KdV) equation and its two-dimensional extension, the Kadomtsev-Petviashvili (KP) equation to describe the evolution of wind-driven water wave packets in shallow water. Both equations were modified to include the effect of wind forcing, modelled using the Miles critical level instability theory. In this paper that is extended to a Benney-Luke (BL) equation, similarly modified for wind forcing. The motivation is that the BL equation is isotropic in the horizontal space variables, unlike the KP model, and noting that the KdV model is one-dimensional. The modified BL equation is studied using wave modulation theory as in MG24, MG25, and with comprehensive numerical simulations. Despite the very different spatial structure the results show that under the right initial conditions and parameter settings, solitary wave trains emerge as in MG24, MG25.