Nonlinear Quantum Graphs – NQG

Nonlinear Quantum Graphs are metric graphs, i.e. graphs for which edges are considered with a metric structure, equipped with a nonlinear Schrödinger equation. They are of interest both from the physical point of view, as they are simplified model of the dynamics in elongated structures, and the mathematical point of view, as their analysis poses new and interesting challenges. If nonlinear quantum graphs now attract much attention abroad it is not yet the case in France. The goal of this project is to gather French researchers sharing an interest in this topic and possessing complementary skills. We aim at establishing ourselves as a leading team in the analysis of nonlinear quantum graphs, both from the numerical and the theoretical point of view. An important part of the project will be the development of the topic via the training of young researchers.

As when the non-linear Schrödinger equation is placed on the entire space the study of stationary solutions will be the subject of a special attention. The questions that are asked regarding the existence and behavior of these solutions (as their orbital or asymptotic stability), depend on the type of graph considered (compact, non-compact, periodic, of tree type, ...), the choice of vertex conditions (Kirchhoff, Dirac, ...), the type of nonlinearity (mass subcritical, mass supercritical, local or non-local, focusing or defocusing), etc.

We intend to attack the analysis of nonlinear quantum graphs with a cross-fertilization of the numerical and analytical approach. Among the directions we are interested in, let us mention :
- The relations between various variational characterizations of solutions.
- The existence of prescribed mass solutions for metric graphs in the mass supercritical case ;
- The existence of sign-changing solutions on metric graphs ;
- The dynamics around standing waves ;
- The development of novel and efficient numerical methods for the problems listed above, their analysis and their implementation as an open source and easy to use library.