Localization of high frequency waves propagating in a locally periodic medium

Autor: Allaire, Grégoire, Friz, Luis
Přispěvatelé: Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Shape reconstruction and identification (DeFI ), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Department of Basic Sciences [Concepción], Universidad del Bio Bio [Concepción] (UBB), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
Jazyk: angličtina
Rok vydání: 2010
Předmět:
Zdroj: Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Cambridge University Press (CUP), 2010, 140A, pp.897-926
Proceedings of the Royal Society of Edinburgh: Section A, Mathematics
Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, 2010, 140A, pp.897-926
ISSN: 0308-2105
1473-7124
Popis: International audience; We study the homogenization and localization of high frequency waves in a locally periodic media with period ε. We consider initial data that are localized Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave at a given frequency ξ and of a smooth envelope function whose support is concentrated at a point x with length scale √ ε. We assume that (ξ, x) is a stationary point in the phase space of the Hamiltonian λ(ξ, x), i.e., of the corresponding Bloch eigenvalue. Upon rescaling at size √ ε we prove that the solution of the wave equation is approximately the sum of two terms with opposite phases which are the product of the oscillating Bloch wave and of two limit envelope functions which are the solution of two Schrödinger type equations with quadratic potential. Furthermore, if the full Hessian of the Hamiltonian λ(ξ, x) is positive definite, then localization takes place in the sense that the spectrum of each homogenized Schrödinger equation is made of a countable sequence of finite multiplicity eigenvalues with exponentially decaying eigenfunctions.
Databáze: OpenAIRE
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