The APDE seminar on Monday, 03/02, will be given by Wolf-Patrick Düll in Evans 939 from 4:10 to 5pm.

Title: Validity of the nonlinear Schrödinger approximation for the two-dimensional water wave problem 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 Schrödinger equation can be derived as a formal approximation equation.

The rigorous justification of the Nonlinear Schrödinger 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.