Séminaire

Titre : Séminaire commun IDEFIX-MEDISIM-POEMS
Contact : Sarah Al Humaikani
Date : 17/10/2024
Lieu : Amphi R112, ENSTA Paris
Résumé de F. Cakoni: Many imaging methods in inverse scattering, such as the generalized linear sampling method, rely on the ability to superimpose scattering data so that the resulting scattered field corresponds to that of a point source. In the frequency domain, for compactly supported inhomogeneities, this leads to solving two elliptic PDEs in a bounded region with a prescribed difference in Cauchy data. In particular, the case of zero scattered field leads to the study of the transmission eigenvalue problem and the regularity of free boundaries. In this talk, we introduce this concept for linear (possibly anisotropic) media and review key results on the transmission eigenvalue problem, its resolvent, and non-scattering phenomena. We then present recent results on the scattering problem for a nonlinear medium with compact support that exhibits second-harmonic generation. When such a medium is probed with monochromatic light beams at a frequency ω, it generates additional waves at the frequency 2ω. The response of the medium is governed by a system of two coupled semilinear partial differential equations for the electric fields at frequencies ω and 2ω. We explore whether there are scenarios in which the generated 2ω wave remains localized within the support of the medium, effectively rendering the nonlinear interaction with the probing wave invisible to an outside observer. This part is a joint work with Narek Hovsepyan, Matti Lassas and Michael.

Résumé de J.L. Jaramillo: The characterisation of Scattering Resonances or Quasi-Normal Modes of a resonating system can be cast as a (proper) eigenvalue problem of a non-selfadjoint operator. In this setting, the potential spectral instability associated with non-selfadjoint operators, and more generally with non-normal operators, naturally raises the question about the structural stability of such complex resonant frequencies. An important technical aspect here involved concerns the key role of singular differential operators, where their principal part is multiplied by a function vanishing at the boundaries. In this talk we will present the problem, illustrating it in the context of Black Hole spacetimes in gravitational physics. The phenomenon is however generic in linear scattering problems, with applications ranging from optics to hydrodynamics, acoustics or condensed matter.