In this talk we discuss the existence of prescribed L^2 norm solutions to nonlinear Schrödinger equations set on metric graphs. A common strategy employed to find such a solution is to search for a constrained critical point of the associated energy functional. Some geometric properties of the functional vary depending on the exponent in the nonlinear term of the equation. In the so-called...
This talk builds on the one of L. Jeanjean. In the mass-supercritical case we consider nonlinear Schrödinger equations on a noncompact metric graph with a localized nonlinearity. We show, for any prescribed L^2 norm, the existence of infinitely many solutions having this norm.
The usual procedure to obtain one solution is to prove that the associated energy functional possesses, on the...
In this talk, we will study qualitative properties of solutions to the Nonlinear Schrödinger Equation -u'' + au = |u|ᵖ⁻²u on compact metric graphs.
Among the questions of interest are:
- if the graph possesses a symmetry, what about its positive solutions? What about action ground states? Can one identify and describe symmetry-breaking situations depending on parameters (lengths of edges,...
Dans cet exposé, je présenterai mes travaux en collaboration avec Christophe BESSE et Stefan LE COZ autour de la simulation numérique pour les graphes quantiques (nonlinéaires). En particulier, je parlerai de la librairie Python GRAFIDI qui permet de résoudre numériquement des problèmes tels que le calcul d'états stationnaires ou bien encore l'évolution d'une solution de l'équation de...
La connaissance de certaines propriétés spectrales de l’opérateur de Laplace sur des graphes a de nombreuses conséquences, en particulier dans l’étude de la stabilité asymptotique des petites ondes stationnaires pour NLS sur les graphes. L'objectif principal de mon exposé sera de rappeler quelques anciens résultats sur des graphes finis et de présenter quelques nouveaux résultats pour des...
We shall talk about computational methods for multiscale partial differential equations, and in particular the multiscale finite element method (MsFEM). This is a finite element type method that performs a Galerkin approximation of the PDE on a problem-dependent basis. We shall discuss two aspects of these methods based on our recent work with Rutger Biezemans, Claude Le Bris, and Frédéric...
