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

TL;DR

This paper rigorously justifies that the evolution of 2D water waves with wave packet initial data can be approximated by a focusing cubic nonlinear Schrödinger equation, with precise error bounds in Sobolev spaces.

Contribution

It provides a rigorous mathematical validation of the modulation approximation linking water wave solutions to the NLS equation in Sobolev spaces.

Findings

Approximate solutions follow the NLS equation with controlled error.

Existence of water wave solutions in Sobolev spaces for NLS times.

Initial data close to small amplitude wave packets ensures valid approximation.

Abstract

We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data, we show that the modulation of the solution is a profile traveling at group velocity and governed by a focusing cubic nonlinear Schrodinger 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 in the NLS regime provided the initial data is suitably close to a wave packet of sufficiently small amplitude in Sobolev spaces.