THE FOCUSING SINGULARITY OF THE DAVEY-STEWARTSON EQUATIONS FOR GRAVITY-CAPILLARY SURFACE-WAVES

We analyze the self-focusing of gravity-capillary surface waves modele d by the Davey-Stewartson equations. This system governs the coupling of the wave amplitude to the induced mean flow and is an anisotropic t wo-dimensional Schrodinger equation with a non local cubic nonlinearit y. 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 Schrodinger equati on, which is the deep water limit of the Davey-Stewartson equations. W e also derive a sharp upper bound for the initial amplitude of the wav e that prevents singularity formation.