Localization of high frequency waves propagating in a locally periodic medium - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Proceedings of the Royal Society of Edinburgh: Section A, Mathematics Year : 2010

Localization of high frequency waves propagating in a locally periodic medium

Abstract

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.
Fichier principal
Vignette du fichier
gregluis-rev2.pdf (354.01 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-00784055 , version 1 (05-02-2020)

Identifiers

  • HAL Id : hal-00784055 , version 1

Cite

Grégoire Allaire, Luis Friz. Localization of high frequency waves propagating in a locally periodic medium. Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, 2010, 140A, pp.897-926. ⟨hal-00784055⟩
99 View
77 Download

Share

Gmail Facebook X LinkedIn More