Diffractive Geometric Optics for Bloch Wave Packets

Autor:	Grégoire Allaire, Jeffrey Rauch, Mariapia Palombaro
Rok vydání:	2011
Předmět:	Mathematics (miscellaneous) Divisor Mechanical Engineering Dispersion relation Mathematical analysis Plane wave Order (ring theory) Type (model theory) Wave equation Analysis WKB approximation Bloch wave Mathematics
Zdroj:	Archive for Rational Mechanics and Analysis. 202:373-426
ISSN:	1432-0673 0003-9527
DOI:	10.1007/s00205-011-0452-9
Popis:	We study, for times of order \({1/\varepsilon}\), solutions of wave equations which are \({\fancyscript{O}(\varepsilon^2)}\) modulations of an \({\varepsilon}\) periodic wave equation. The solutions are of slowly varying amplitude type built on Bloch plane waves with wavelength of order \({\varepsilon}\). We construct accurate approximate solutions of the three scale WKB type. The leading profile is both transported at the group velocity and dispersed by a Schrodinger equation given by the quadratic approximation of the Bloch dispersion relation at the plane wave. A ray average hypothesis of the small divisor type guarantees stability. We introduce techniques related to those developed in nonlinear geometric optics which lead to new results even on time scales \({t=\fancyscript{O}(1)}\). A pair of asymptotic solutions yield accurate approximate solutions of oscillatory initial value problems. The leading term yields H1 asymptotics when the envelopes are only H1.
Databáze:	OpenAIRE
Externí odkaz:	https://explore.openaire.eu/search/publication?articleId=doi_dedup___::a6ff6ccc305eaface9935ebf0cff85e1 https://doi.org/10.1007/s00205-011-0452-9 Zobrazit plný text záznamu Plný text ve formátu PDF