# Validity of the Nonlinear Schr\"odinger Approximation for the Two-Dimensional Water Wave Problem With and Without Surface Tension in the Arc Length Formulation

## TL;DR

This paper rigorously justifies the nonlinear Schr"odinger approximation for 2D water waves with and without surface tension, providing uniform error estimates over relevant timescales in the arc length formulation.

## Contribution

It offers a rigorous proof of the NLS approximation's validity for 2D water waves with and without surface tension, including uniform error bounds.

## Key findings

- Error estimates are uniform with respect to surface tension strength.
- The NLS approximation is valid over physically relevant timescales.
- The proof applies to both cases with and without surface tension.

## Abstract

We consider the two-dimensional water wave problem in an infinitely long canal of finite depth both with and without surface tension. In order to describe the evolution of the envelopes of small oscillating wave packet-like solutions to this problem the Nonlinear Schr\"odinger equation can be derived as a formal approximation equation. In recent years, the validity of this approximation has been proven by several authors for the case without surface tension. In this paper, we rigorously justify the Nonlinear Schr\"odinger approximation for the cases with and without surface tension by proving error estimates over a physically relevant timespan in the arc length formulation of the two-dimensional water wave problem. The error estimates are uniform with respect to the strength of the surface tension, as the height of the wave packet and the surface tension go to zero.