# Nathan Totz : A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem

In this joint work with Sijue Wu (U. Mich.), we consider solutions to the 2D
inviscid infinite depth water wave problem neglecting surface tension which
are to leading order wave packets of the form $\alpha + \epsilon B(\epsilon
\alpha, \epsilon t, \epsilon^2 t)e^{i(k\alpha + \omega t)}$ for $k > 0$.
Multiscale calculations formally suggest that such solutions have
modulations $B$ that evolve on slow time scales according to a focusing
cubic NLS equation. Justifying this rigorously is a real problem, since
standard existence results do not yield solutions which exist for long
enough to see the NLS dynamics. Nonetheless, given initial data within
$O(\epsilon^{3/2})$ of such wave packets in $L^2$ Sobolev space, we show
that there exists a unique solution to the water wave problem which remains
within $O(\epsilon^{3/2})$ to the approximate solution for times of order
$O(\epsilon^{-2})$. This is done by using a version of the evolution
equations for the water wave problem developed by Sijue Wu with no quadratic nonlinearity.

See arXiv:1101.0545

**Category**: Applied Math and Analysis**Duration**: 01:34:52**Date**: September 19, 2011 at 4:25 PM**Views**: 143-
**Tags:**seminar, Applied Math And Analysis Seminar

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