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

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

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
Volume335
Issue number1
DOIs
StatePublished - Jan 1 2015
Externally publishedYes

Fingerprint

Cubic equation
A Priori Bounds
water waves
Group Velocity
Wave Packet
Water Waves
Energy Method
Surface Tension
Justification
Justify
Sobolev Spaces
Normal Form
Evolution Equation
Error Estimates
Nonlinear Equations
Approximate Solution
Modulation
Nonlinearity
modulation
Approximation

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

A Justification of the Modulation Approximation to the 3D Full Water Wave Problem. / Totz, Nathan.

In: Communications in Mathematical Physics, Vol. 335, No. 1, 01.01.2015, p. 369-443.

Research output: Contribution to journalArticle

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