This project is devoted to the analysis of nonlinear partial differential equations arising in the mathematical description of collective quantum dynamics, with applications to, e.g., Bose-Einstein condensates. The mathematical equations under consideration will mostly be dispersive, and hence conservative, but several non-conservative (or dissipative) extensions will also be studied. By doing so, connections between equations of nonlinear Schrödinger type and complex Ginzburg-Landau type models will become apparent. In the present project the PI will: (1) study the influence of fast rotating potentials, experimentally used to create quantum vortices, on the possibility of finite time blow-up, and establish connections to recently found singular ring type solutions; (2) give a time-dependent description of vortex creation by means of adiabatic perturbation theory; (3) establish rigorous connections between nonlinear Schrödinger equations and other dispersive models; (4) explore the possibility of dispersive blow-up for the considered equations; (5) analyze the long-time behavior of solutions to Schrödinger type equations with additional nonlinear damping terms and other dissipative regularizations. The techniques to be deployed will range from numerical simulations over formal analytical methods to fully rigorous mathematical investigations based on, e.g., virial estimates, multi-scale techniques, Lyapunov-type functionals, and concentration compactness methods.
This project aims to find a rigorous mathematical description of certain collective phenomena in many-body quantum mechanics. Quantum mechanics is one of the two fundamental building blocks in modern physics and it concerns the study of mechanical systems whose dimensions are close to, or even smaller than, the atomic scale. The mathematical formulation pioneered by E. Schrödinger, P. Dirac, and J. von Neumann is to a large extent based on partial differential equations for complex-valued wave functions, which describe the state of the particles in a probabilistic interpretation. It is the solution of these equations which can be seen as one of the main driving forces in the early stages of the development of the theory. The present project lies at the intersection between the mathematical analysis of such equations and their application in theoretical physics. It will help gain a deeper mathematical understanding of the qualitative behavior of the different classes of nonlinear equations involved and their relations with each other. Answers to the questions raised will have a strong impact as proven statements in a wide range of the mathematical sciences, and can be seen as complimentary to the vast amount of non-rigorous results available in the physics literature. Most of the proposed research will be conducted through national and international collaborations with colleagues working in mathematical analysis, theoretical physics, and numerical simulations.
