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MiS Preprint
34/2008
Diffractive geometric optics for Bloch wave packets
Gregoire Allaire, Mariapia Palombaro and Jeffrey Rauch
Abstract
We study, for times of order $1/\e$, solutions of wave equations which are $\Op(\e^2)$ modulations of an $\e$ periodic wave equation. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order $\e$. We construct accurate approximate solutions of three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrödinger equation given by the quadratic approximation of the Bloch dispersion relation at the plane wave. A ray average hypothesis of small divisor type guarantees stability. We introduce techniques related to those developed in nonlinear geometric optics which lead to new results even on times scales $t=\Op(1)$. A pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems. The leading term yields $H^1$ asymptotics when the envelopes are only $H^1$.
