Preprint 34/2008

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 formula7, solutions of wave equations which are formula9 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 formula15. A pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems. The leading term yields formula17 asymptotics when the envelopes are only formula17.

07.06.2018, 02:11