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

Nathan Totz, Sijue Wu

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We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data of the form ε B (ε α)e ikα for k > 0, we show that the modulation of the solution is a profile traveling at group velocity and governed by a focusing cubic nonlinear Schrödinger 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 times of order O(ε -2) provided the initial data differs from the wave packet by at most O(ε 3/2) in Sobolev spaces. These results are obtained by directly applying modulational analysis to the evolution equation with no quadratic nonlinearity constructed in Wu (Invent Math 117(1):45-135, 2009) and by the energy method.

Original languageEnglish (US)
Pages (from-to)817-883
Number of pages67
JournalCommunications in Mathematical Physics
Issue number3
StatePublished - Mar 1 2012
Externally publishedYes


ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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