A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem.

Abstract

We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet-like initial data, we show that the modulation of the solution is a profile traveling at group velocity and governed by a focusing cubic nonlinear Schrodinger equation, with rigorous error estimates in Sobolev spaces. As a consequence, we establish existence of solutions of the water wave problem in Sobolev spaces for time intervals of sufficient length to discern the NLS evolution of the modulation, provided the initial data is suitably close to a wave packet in Sobolev spaces. These results are obtained by directly applying modulational analysis to the evolution equation with no quadratic nonlinearity constructed in Professor Sijue Wu's paper, "Almost Global Well-Posedness of the 2D Full Water Wave Problem," and by the energy method.