Séminaire: Problèmes Spectraux en Physique Mathématique

Année 2022-2023

Pour tout renseignement complémentaire, veuillez contacter les organisateurs, Hakim Boumaza, Mathieu Lewin ou Stéphane Nonnenmacher.

Prochain séminaire le lundi 3 avril 2023 après-midi à l'Institut Henri Poincaré, en salle Grisvard (ex-314, 3e étage)

14h - 15h Annalaura Stingo (Ecole Polytechnique)
Global stability of the Kaluza-Klein theories

The Kaluza-Klein theories represent the classical mathematical approach to the unification of general relativity with electromagnetism and more generally with gauge fields. In these theories, general relativity is considered in 1+3+d dimensions and in the simplest case d=1 dimensional gravity is compactified on a circle to obtain at low energies a (3+1)-dimensional Einstein-Maxwell-Scalar systems.  In this talk I will discuss the problem of the classical global stability of Kaluza-Klein theories when d=1. This is a joint work with C. Huneau and Z. Wyatt.

 15h15 - 16h15 Julien Ricaud (Ecole Polytechnique) Spectral stability in the nonlinear Dirac equation with Soler-type nonlinearity

This talk concerns the (generalized) Soler model: a nonlinear (massive) Dirac equation with a nonlinearity taking the form of a space dependent mass. The equation admits standing wave solutions and they are generally expected to be stable (i.e., small perturbations in the initial conditions stay small) based on numerical simulations. However, contrarily to the nonlinear Schrödinger equation for example, there are very few results in this direction. The results that I will discuss concern the simpler question of spectral stability (and instability), i.e., the absence (or presence) of exponentially growing solutions to the linearized equation around a solitary wave. As in the case of the nonlinear Schrödinger equation, this is equivalent to the presence or absence of "unstable eigenvalues" of a non-selfadjoint operator with a particular block structure. I will highlight the differences and similarities with the Schrödinger case, present some results for the one-dimensional case, and discuss open problems.

This is joint work with Danko Aldunate, Edgardo Stockmeyer, and Hanne Van Den Bosch.

Prochaines séances :

15 mai 2023: Angeliki Menegaki
19 juin 2023: Cyril Labbé

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Dernière mise à jour: 21 mars 2023
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