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

Nathan Totz, Sijue Wu

Research output: Contribution to journalArticle

35 Citations (Scopus)

Abstract

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
Volume310
Issue number3
DOIs
StatePublished - Mar 1 2012
Externally publishedYes

Fingerprint

Sobolev space
water waves
Water Waves
Justification
Sobolev Spaces
Modulation
Wave Packet
modulation
wave packets
Approximation
approximation
Cubic equation
energy methods
Group Velocity
Energy Method
Surface Tension
group velocity
nonlinear equations
Evolution Equation
Error Estimates

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem. / Totz, Nathan; Wu, Sijue.

In: Communications in Mathematical Physics, Vol. 310, No. 3, 01.03.2012, p. 817-883.

Research output: Contribution to journalArticle

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