The unsteady evolution of localized unidirectional deep water wave groups

Abstract: We study the evolution of localized wave groups in unidirectional water wave envelope equations (nonlinear Schrodinger (NLS) and modified NLS (MNLS)). These localizations of energy can lead to disastrous extreme responses (rogue waves). Previous studies have focused on the role of energy distribution in the frequency domain in the formation of extreme waves. We analytically quantify the role of spatial localization, introducing a novel technique to reduce the underlying PDE dynamics to a simple ODE for the wave packet amplitude. We use this reduced model to show how the scale- invariant symmetries of NLS break down when the additional terms in MNLS are included, inducing a critical scale for the occurrence of extreme waves.