# Analysis Seminar

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

**Speaker:**
Wolf-Patrick DUELL, Universitaet Stuttgart

**Location:**
Warren Weaver Hall 1302

**Date:**
Thursday, February 27, 2020, 11 a.m.

**Synopsis:**

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 ̀ˆodinger equation can be derived as a formal approximation equation.

The rigorous justification of the Nonlinear Schr ̀ˆodinger 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 Schr ̀ˆodinger

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.