Nonlinear Schrödinger-type equations

cms We are mainly interested in the analytical and numerical study of multiscale phenomena in linear and nonlinear Schroedinger-type equations. The corresponding asymptotic analysis is based on semiclassical expansions, homogenization limits, adiabatic approximations, etc. Applications arise in the modelling of

  • Bose-Einstein condensates,
  • electron dynamics in crystals and nanostructures,
  • nonlinear wave propagation.

Classical and quantum kinetic equations

cms Main mathematical topics under investigation are mean free path (scaling) limits such as generalized energy transport systems, concentration effects in bosonic Boltzmann equations, Wigner transforms, quantum Fokker-Planck equations and open quantum systems, kinetic equations with non-standard scattering laws, homogenisation limits and long time as

ymptotics.

The main application areas are

  • Bose-Einstein condensates (interaction of thermalised and non-thermalised particles),
  • quantum scattering in semiconductors,
  • collective behaviour of swimming mirco-organismns (squirmers) and chemotaxis,
  • kinetics and mean field models for human crowding behaviour, Pareto economics and opinion formation in human societies.

Reaction-diffusion/convection systems

cms Entropy methods establishing explicit functional inequalities relating entropy and entropy dissipation for nonlinear systems are modern tools to derive a-priori estimates and quantitative rates

for the large time asymptotics in model systems describing

  • reversible chemical kinetics,
  • drift and diffusion of charge carries in semiconductors.

Free boundary problems

cms Research focuses on regularity and geometry of free boundaries, fully nonlinear PDEs and level set methods. Applications arise in

  • superconductivity,
  • mathematical biology.

Socio-Economic Models

cms In recent years, the understanding of Social and Economic phenomena using mathematical models has gained a lot of attention. In our group, we study in particular the motion of human crowd in different situations. Furthermore, we study a new free boundary problem proposed by J.M. Lasy and P.-L. Lions. This model is based on the so-called mean field games approach. For more details see the publication section on this homepage.

PDE Imaging

cms In PDE Imaging we are working on the development, analysis and efficient numerical implementation of PDE-based imaging approaches and their applications. Our current research revolves around the following objectives:

  • Gain a better understanding of nonlinear PDEs and non-smooth variational approaches in imaging by a careful mathematical analysis.
  • Propose new models within this class taking into account the insights gained from our analysis and the needs dictated from applications.
  • Develop efficient and reliable implementations for the numerical solution of these PDEs to make them attractive for applications and for real-time user interaction.
  • Supply applications, like medical imaging tools or security software, with efficient imaging algorithms. We hold interdisciplinary projects with astronomers, computer scientists, engineers, neuroscientists, physicists and art conservators.