Schrödinger equations and applications – SchEq

The Schrödinger equations play an important role in modeling physical phenomena on the one hand, and constitute on the other hand an important mathematical object. They represent an extremely active research field. The first aim of this project is to advance the understanding of the links between Schrödinger-type equations and some physical phenomena such as vortex dynamics in fluids and in super-fluids, as well as propagation of signals in quantum wires. The second aim is the study of theoretical questions like the large time behavior of nonlinear solutions of the Schrödinger equation on manifolds, in critical situations and in the case of high frequency regimes. We decompose the pro ject into two main parts, each one having several sub-parts: 1) Applications of Schrödinger Equation • Vortex dynamics in fluid mechanics: We aim to investigate the link between singular vorticity dynamics in fluids on the one hand geometric flow and underlying dispersive equations on the other hand. The formation of singularities in finite time will be of particular interest. • Vortex dynamics in superfluids: We plan to describe interactions between special solutions to the Gross-Pitaevskii equation, in particular, between sums of traveling waves, or between two-dimensional traveling waves and vortex solutions. • Quantum wires: The propagation of signals in quantum wires is related to the Schrödinger equation on graphs. Our purpose is to advance the nonlinear time-dependent study, which has been initiated only a few years ago and is poorly represented in France. 2) Profiles for large time behavior of NLS • NLS on manifolds: In this part of the project we would like to continue the investigations of the properties of the nonlinear Schrödinger equation on manifolds, with a particular interest for the blow-up in the focusing case, and for the large time behavior in the defocusing case on compact or product manifolds. Treating an equation on a manifold implies a deeper understanding of the Euclidean tools and the introduction of new tools to deal with non-flatness issues. • High frequency regimes for NLS: We want to understand better the large time semiclassical limit of nonlinear Schrödinger (or Hartree) equations. The high frequency regime for systems of nonlinear Schrödinger equations is motivated by models from Physics, and requires a new approach.