TY - JOUR

T1 - A Justification of the Modulation Approximation to the 3D Full Water Wave Problem

AU - Totz, Nathan

N1 - Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2015/4

Y1 - 2015/4

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84925543177&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84925543177&partnerID=8YFLogxK

U2 - 10.1007/s00220-014-2259-7

DO - 10.1007/s00220-014-2259-7

M3 - Article

AN - SCOPUS:84925543177

VL - 335

SP - 369

EP - 443

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -