The focusing singularity of the Davey-Stewartson equations for gravity-capillary surface waves

We analyze the self-focusing of gravity-capillary surface waves modeled by the Davey-Stewartson equations. This system governs the coupling of the wave amplitude to the induced mean flow and is an anisotropic two-dimensional Schrödinger equation with a non local cubic nonlinearity. With accurate numerical simulations based on dynamic rescaling and asymptotic analysis we show that the dynamics of singularity formation is critical, as in the usual two-dimensional cubic Schrödinger equation, which is the deep water limit of the Davey-Stewartson equations. We also derive a sharp upper bound for the initial amplitude of the wave that prevents singularity formation.