The Interaction of Wind-Generated Gravity Water Wave Groups with Capillary-Gravity Wave Groups: Coupled Nonlinear Schrödinger Equations

The nonlinear Schrödinger equation is a well-known and much-studied canonical equation for weakly nonlinear wave groups. In the context of wind-generated water waves, an additional wind forcing term can be added. It is asymptotically derived for the slowly varying amplitude of a sinusoidal wave, with a dominant wavenumber and frequency. When two such wave groups are present, the asymptotic outcome is two coupled nonlinear Schrödinger equations, with coupling through a nonlinear term in each equation. In this article we examine this scenario, in a one horizontal space dimension setting, when one wave is a gravity wind-driven water wave, and the other is a capillary-gravity wave. For this coupled system, we present an analysis of modulation instability and a suite of numerical simulations analogous to those presented previously for the forced nonlinear Schrödinger equation.