Analytical, Numerical and Integrable systems approaches for nonlinear dispersive PDEs

This mathematical project is concerned with the careful study of certain partial differential equations which are omnipresent in applications for wave phenomena, in hydrodynamics, nonlinear optics (signal propagation in optical fibers), plasma physics and Bose-Einstein condensates. An important common feature of these situations is that dissipation is dominated by dispersion, loosely speaking the spreading of the propagated waves packets. The main features of these equations are that their solutions can develop zones of rapid modulated oscillations called dispersive shock waves, that they may have stable traveling wave solutions called solitons (which may be interesting for the transport of information), playing an important role in the long time dynamics, and that the time of existence of a smooth solution can be finite before a blow-up, i.e., a breakdown of the applicability of the studied model. It is the main goal of the project to provide a deep analytical understanding of the main features of these equations in order to develop novel numerical tools for applications. To this end we use hybrid approaches, i.e., a mixture of analytic and numerical techniques. This allows to treat otherwise inaccessible situations such as rapid oscillations or blow-up reliably. The idea is to implement a-priori analytical information on the solutions and to reduce the complex problems to more standard ones which are then treated with state of the art numerical methods. This will in turn allow to obtain more analytic information on these important situations which will be not only mathematically interesting. Some of the numerical techniques (for d-bar problems) to be developed during the project will be for medical imaging in tomography. The project is unique in the combination of techniques from various domains of mathematics, in particular analysis and integrable systems combined with state of the art numerical tools. This novel approach will allow both breakthrough results on the analytical side as well as new numerical tools which can then be used for instance in nonlinear optics and in the computation of Bose-Einstein condensates.

Nonlinear dispersive partial differential equations: Nonlinear dispersive partial differential equations such as the celebrated Korteweg-de Vries and the nonlinear Schrodinger equation are omnipresent in applications whenever dispersion dominates dissipation, for instance in hydrodynamics, nonlinear optics, plasma physics, Bose-Einstein condensates, Though remarkable progress has been made during the last thirty years, the theory of dispersive PDEs is much less developed than, for example, the theory of diffusion, in particular its description of the relevant dynamics, despite the importance of the former in applications. The reasons for the mathematical difficulties in treating these equations are that the solutions can have zones of rapid modulated oscillations, so-called dispersive shocks, that the solutions may have various regimes of completely different "character" (stable "solitonic" structures, dispersion, ...) and that the solutions can develop singularities in finite time for smooth initial data. Roughly speaking, one could say that nonlinear dispersive equations share some properties of hyperbolic and parabolic equations. Analytical and Numerical approaches for nonlinear dispersive partial differential equations: Many important issues in hydrodynamics, nonlinear and nano-optics, Bose-Einstein condensates and medical imaging are related mathematically to nonlinear dispersive partial differential equations (PDEs). In this project these PDEs, mainly in higher dimensions, were studied with a unique innovative combination of analytic, geometric, and numerical approaches and techniques from the theory of integrable systems, also applied to non-integrable PDEs. The goal was to use the predictive power of numerics for breakthroughs on the analytical side, and analytical insight into the equations to generate innovative numerical schemes able to address challenges in applications. The numerical approaches to so-called d-bar problems appearing in the context of integrable PDEs in two spatial dimensions will help to develop more efficient approaches in electrical impedance tomography. Of special interest is the asymptotic description of dispersive shock waves (zones of rapid modulated oscillations in the solutions as in the semi-classical limit of the Schrodinger equation), of blow-ups (a loss of regularity of the solutions in finite time), and the construction of exact solutions, mainly solitons and breathers, and their stability. An important point in this context was the development of efficient numerical techniques of high accuracy. Main results: Efficient numerical and analytical techniques have been developed for nonlinear dispersive PDEs that will be directly applicable to electrical impedance tomography, a form of medical imaging not exposing patients to potentially harmful radiation. The goal is that further progress along the lines explored in this project will allow for a general use of this technology to discover aneurisms and cancer. High performance computing on low cots GPUs has been studied in this context.