Diffractive geometric optics for Bloch wave packets

Gregoire Allaire, Mariapia Palombaro, and Jeffrey Rauch

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Submission date: 16. Apr. 2008
Pages: 52
Bibtex
MSC-Numbers: 35B40, 35B27, 35L30, 35B34, 35J10
Keywords and phrases: geometric optics, diffractive geometric optics, Bloch waves, diffraction, homogenization
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Abstract:
We study, for times of order , solutions of wave equations which are modulations of an periodic wave equation. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order . 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 . A pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems. The leading term yields asymptotics when the envelopes are only .