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H 1 A new complex spectrum associated to invisibility in waveguides

Auteurs : Bonnet-Ben Dhia, Anne-Sophie (Auteur de la Conférence)
CIRM (Editeur )

Résumé : We consider an acoustic waveguide modeled as follows:

$\left \{\begin {matrix} \Delta u+k^2(1+V)u=0& in & \Omega= \mathbb{R} \times]0,1[\\ \frac{\partial u}{\partial y}=0& on & \partial \Omega \end{matrix}\right.$

where $u$ denotes the complex valued pressure, k is the frequency and $V \in L^\infty(\Omega)$ is a compactly supported potential.
It is well-known that they may exist non trivial solutions $u$ in $L^2(\Omega)$, called trapped modes. Associated eigenvalues $\lambda = k^2$ are embedded in the essential spectrum $\mathbb{R}^+$. They can be computed as the real part of the complex spectrum of a non-self-adjoint eigenvalue problem, defined by using the so-called Perfectly Matched Layers (which consist in a complex dilation in the infinite direction) [1].
We show here that it is possible, by modifying in particular the parameters of the Perfectly Matched Layers, to define new complex spectra which include, in addition to trapped modes, frequencies where the potential $V$ is, in some sense, invisible to one incident wave.
Our approach allows to extend to higher dimension the results obtained in [2] on a 1D model problem.

Codes MSC :
35B40 - Asymptotic behavior of solutions of PDE
35J05 - Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
35Q35 - PDEs in connection with fluid mechanics
41A60 - Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
47H10 - Fixed-point theorems
65N30 - Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
76Q05 - Hydro- and aero-acoustics

 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 15/06/17 Date de captation : 08/06/17 Collection : Research talks ; Partial Differential Equations ; Mathematical Physics Format : MP4 Durée : 00:45:04 Domaine : PDE ; Mathematical Physics Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2017-06-08_Bonnet-Ben_Dhia.mp4 Informations sur la rencontre Nom de la rencontre : Mathematical aspects of physics with non-self-adjoint operators / Les aspects mathématiques de la physique avec les opérateurs non-auto-adjointsOrganisateurs de la rencontre : Krejcirik, David ; Siegl, PetrDates : 05/06/17 - 09/06/17 Année de la rencontre : 2017 URL Congrès : http://conferences.cirm-math.fr/1596.htmlCitation Data DOI : 10.24350/CIRM.V.19181803 Cite this video as: Bonnet-Ben Dhia, Anne-Sophie (2017). A new complex spectrum associated to invisibility in waveguides. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19181803 URI : http://dx.doi.org/10.24350/CIRM.V.19181803

### Voir aussi

Bibliographie

1. [1] Hein, S., & Koch, W. (2008). Acoustic resonances and trapped modes in pipes and tunnels. Journal of Fluid Mechanics, 605, 401-428 - https://doi.org/10.1017/S002211200800164X

2. [2] Hernandez-Coronado, H., Krejcirik, D., & Siegl, P. (2011). Perfect transmission scattering as a $\mathcal{PT}$-symmetric spectral problem. Physics Letters A, 375(22), 2149-2152 - https://doi.org/10.1016/j.physleta.2011.04.021

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