The NLS approximation for two dimensional deep gravity waves

TL;DR

This paper demonstrates that small wave packet solutions of two-dimensional deep gravity water waves can be effectively approximated by the cubic nonlinear Schrödinger equation over natural cubic time scales.

Contribution

It establishes the validity of the NLS approximation for 2D deep gravity water waves in holomorphic coordinates with small initial data.

Findings

NLS provides a good approximation for small wave packets

The approximation holds over cubic time scales

The analysis uses holomorphic coordinate formulation

Abstract

This article is concerned with infinite depth gravity water waves in two space dimensions. We consider this system expressed in position-velocity potential holomorphic coordinates. Our goal is to study this problem with small wave packet data, and to show that this is well approximated by the cubic nonlinear Schr\"odinger equation (NLS) on the natural cubic time scale.