Validity of the Nonlinear Schrödinger Approximation for the Two-Dimensional Water Wave Problem With and Without Surface Tension in the Arc Length Formulation

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 Schrödinger 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 Schrödinger 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.

This paper is a minor revision of the previous submission arXiv:1907.03014v1. In its present form, the paper has been accepted for publication in Archive for Rational Mechanics and Analysis