The modulational regime of three-dimensional water waves and the Davey-Stewartson system

Nonlinear modulation of gravity-capillary waves travelling principally in one direction at the surface of a three-dimensional fluid leads to the Davey-Stewartson system for the wave amplitude and the induced mean flow. In this paper, we present a rigorous derivation of the system and show that the resulting wavepacket satisfies the water wave equations at leading order-with precise bounds for the remainder. Key steps in the analysis are the analyticity of the Dirichlet-Neumann operator with respect to the surface elevation that defines the fluid domain, precise bounds for the Taylor remainders and the description of individual terms in the Taylor series as pseudo-differential operators and their estimates under multiple scale expansions.