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

TL;DR

This paper rigorously justifies the approximation of small amplitude 3D water waves by a nonlinear Schrödinger equation, providing error estimates and confirming the validity of the modulation approximation over relevant time scales.

Contribution

It offers a rigorous mathematical validation of the modulation approximation for 3D water waves using energy methods and direct modulational analysis.

Findings

Solution exists and maintains wave packet form over NLS time scales.

Provides explicit error bounds between true and approximate solutions.

Validates the use of NLS as an effective model for 3D water wave modulation.

Abstract

We consider small amplitude wave packet-like solutions to the 3D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Formal multiscale calculations suggest that the modulation of such a solution is described by a profile traveling at group velocity and governed by a hyperbolic cubic nonlinear Schr\"odinger equation. In this paper we show that, given wave packet initial data, the corresponding solution exists and retains the form of a wave packet on natural NLS time scales. Moreover, we give rigorous error estimates between the true and formal solutions on the appropriate time scale in Sobolev spaces using the energy method. The proof proceeds by directly applying modulational analysis to the formulation of the 3D water wave problem developed by Sijue Wu.