A Justification of the Modulation Approximation to the 3D Full Water Wave Problem

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We consider modulational solutions to the 3D inviscid incompressible irrotational infinite depth water wave problem, neglecting surface tension. For such solutions, it is well known that one formally expects the modulation to be a profile traveling at group velocity and governed by a 2D hyperbolic cubic nonlinear Schrödinger equation. In this paper we justify this fact by providing rigorous error estimates in Sobolev spaces. We reproduce the multiscale calculation to derive an approximate wave packet-like solution to the evolution equations with mild quadratic nonlinearities constructed by Sijue Wu. Then we use the energy method along with the method of normal forms to provide suitable a priori bounds on the difference between the true and approximate solutions.

Original languageEnglish (US)
Pages (from-to)369-443
Number of pages75
JournalCommunications in Mathematical Physics
Issue number1
StatePublished - Jan 1 2015
Externally publishedYes


ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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