Deterministic and probabilistic dynamics of nonlinear dispersive PDEs

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Nonlinear dispersive partial differential equations (PDEs) such as the nonlinear wave equations and the nonlinear Schrodinger equations, are time evolution equations modelling wave phenomena. They appear ubiquitously in various branches of physics and engineering such as nonlinear optics, plasma physics, water waves and telecommunication systems. Their validity is widely recognised and supported by numerical and experimental evidences.On the one hand, the mathematical theoretical research of nonlinear dispersive PDEs is important for applied sciences since it has provided solid foundations for the verification and applicability of these models. On the other hand, this theoretical research has proved to be very valuable for mathematics itself. Indeed, over the last thirty years, nonlinear dispersive PDEs have presented very difficult and interesting challenges, motivating the development of many new ideas and techniques in mathematical analysis. One of the sources of richness of nonlinear dispersive PDEs is that each subclass of equations poses its own difficulties, thus requiring the elaboration of specific tools.The aim of this proposal is to explore the dynamics of nonlinear dispersive PDEs using mathematical analysis from both deterministic and probabilistic points of view. In the deterministic setting, this proposal focuses on (i) constructing special solutions to a class of nonlinear Schrodinger equations and (ii) proving the long-time existence of solutions to an equation from plasma physics with non-constant vorticity. The principal investigator (PI) plans to combine PDE techniques with tools from harmonic analysis and spectral theory.In the traditional (deterministic) study of nonlinear evolution equations, one aims to construct solutions to a given PDE for all initial data. In applications, however, one is often content with understanding the behaviour of typical solutions, neglecting rare pathological behaviours. This point of view can be made rigorous by employing probability theory and has led to exciting developments over the last decade. In particular, it has allowed us to go beyond the limits of deterministic analysis. One aspect of this proposal is to investigate dynamics of nonlinear dispersive PDEs from a probabilistic point of view. More specifically, the PI will focus on constructing well-defined dynamics with rough and random initial data by incorporating ideas and tools from probability theory and the very active field of singular stochastic PDEs.