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

Wolf-Patrick Düll (Universität Stuttgart) #

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. The rigorous justification of the Nonlinear Schr"odinger approximation for the water wave problem was an open problem for a long time. In recent years, the validity of this approximation has been proven by several authors only for the case without surface tension.

In this talk, we present the first rigorous justification of the Nonlinear Schrödinger approximation for the two-dimensional water wave problem which is valid for the cases with and without surface tension by proving error estimates over a physically relevant timespan in the arc length formulation of the water wave problem. Our 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.

References #

[1] W.-P. Düll. Validity of the nonlinear Schrödinger approximation for the two-dimensional water wave problem with and without surface tension in the arc length formulation. Arch. Ration. Mech. Anal. 239 (2021), no. 2, 831-914.