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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

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