A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem
Abstract
We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data of the form {ɛ B(ɛ α)e^{ik α}} 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(ɛ^{2})} provided the initial data differs from the wave packet by at most {O(ɛ^{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):45135, 2009) and by the energy method.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 March 2012
 DOI:
 10.1007/s0022001214222
 arXiv:
 arXiv:1101.0545
 Bibcode:
 2012CMaPh.310..817T
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 doi:10.1007/s0022001214222