RIGOROUS JUSTIFICATION OF MODULATION APPROXIMATIONS TO
THE FULL WATER WAVE PROBLEM
NATHAN TOTZ
Abstract
We consider solutions to the infinite depth water wave problem in 2D and 3D neglecting surface
tension which are to leading order wave packets with small O() amplitude and slow spatial decay
that are balanced. Multiscale calculations formally suggest that such solutions have modulations
that evolve on O(−2 ) time scales according to a version of a cubic NLS equation depending on
dimension. Justifying this rigorously is a real problem, since standard existence results do not
yield solutions to the water wave problem that exist for long enough to see the NLS dynamics.
Nonetheless, given initial data suitably close to such a wave packet in L2 Sobolev space, we show
that there exists a unique solution to the water wave problem which remains within o() to the formal
approximation on the natural NLS time scales. The key ingredient in the proof is a formulation of
the evolution equations for the water wave problem developed by Sijue Wu (U Mich.) with either
no quadratic nonlinearities (in 2D) or mild quadratic nonlinearities that can be eliminated using
the method of normal forms (in 3D).
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