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Craig, Walter; Schanz, Ulrich; Sulem, Catherine

The modulational regime of three-dimensional water waves and the Davey-Stewartson system. (English) Zbl 0892.76008

Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, No. 5, 615-667 (1997).
Summary: 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.

Cited in 72 Documents

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics
35C10 Series solutions to PDEs

Keywords:

analyticity of Dirichlet-Neumann operator; gravity-capillary waves; wave amplitude; wavepacket; surface elevation; Taylor series; pseudo-differential operators; multiple scale expansions
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