New Challenges in Nonlinear PDEs.

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This project deals with research at the interface of harmonic analysis, nonlinear partial differential equations, geometry, and probability. On the one hand, it is concerned with the study of dispersive nonlinear wave phenomena from a nondeterministic viewpoint. In the last two decades enormous progress has been made in settling questions on existence of solutions to dispersive equations, their long-time behavior, and singularity formation. The thrust of this body of work has focused primarily on deterministic aspects of wave phenomena, where sophisticated tools from nonlinear Fourier analysis, geometry, and analytic number theory have played crucial roles. Yet there remain some fundamental obstacles. A natural approach to overcome them is to consider evolution equations from a nondeterministic point of view and to incorporate into the analysis tools from probability. Some of the issues to be investigated in this project are the role of randomization in the well-posedness theory, the almost sure (in the sense of probability) global-in-time existence of solutions, the existence and dynamical properties of associated Gibbs measures, and the behavior of statistical ensembles under gauge transformations. On the other hand, the principal investigator will study the existence and long-time dynamics of special types of solutions to certain hyperbolic (or nonelliptic) nonlinear Schrodinger equations and systems. The aim is to develop a rigorous mathematical analysis of models arising in connection with the theory of vortex filaments, ferromagnetism, and current work in nonlinear optics (e.g., examining the evolution of optical pulses in normally dispersive optical media), models that have attracted the attention of the physics community. Wave phenomena in physics such as light, sound, and gravity are mathematically modeled using partial differential equations. Nonlinear wave models arise in quantum mechanics, ferromagnetism, vibrating systems, semiconductors, and optical fibers. Dispersive equations model important wave propagation phenomena in nature. Their solutions are waves that spread out in space as time evolves while conserving energy or mass. The best known dispersive equations are the nonlinear Schrodinger equations that govern the motion of quantum particles (e.g., electrons), the macroscopic dynamics of the Bose-Einstein condensate, and signals in fiber optics. This project focuses on the rigorous mathematical analysis of dispersive equations that arise naturally in physics and engineering. The synergy of Fourier analysis, probability, geometry, and analytic number theory provides a well-adapted and powerful toolbox to study the nonlinear effects that allow waves to interact and produce new, modified propagation patterns. The problems that the principal investigator will study are of particular interest in the study of long internal gravity waves in deep stratified fluids, the theory of vortex filaments and aerodynamics, and in current work on nonlinear fiber optics that is of fundamental importance in today's telecommunication systems and internet traffic. The ubiquitous role of mathematics is to lay the foundations through rigorous research for the best predictions, based on which the technological advances and engineering applications we enjoy every day, can be efficiently enabled. The training of students and junior researchers is an integral part of the project.