TY - JOUR

T1 - A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem

AU - Totz, Nathan

AU - Wu, Sijue

N1 - Funding Information:
The authors are supported in part by NSF grant DMS-0800194.

PY - 2012/3

Y1 - 2012/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84863279498&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84863279498&partnerID=8YFLogxK

U2 - 10.1007/s00220-012-1422-2

DO - 10.1007/s00220-012-1422-2

M3 - Article

AN - SCOPUS:84863279498

VL - 310

SP - 817

EP - 883

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -