Hamiltonian partial differential equations and dynamical systems, and their applications

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My research program involves the analysis of solutions of partial differential equations (PDEs) that describe wave evolution in a broad spectrum of physical phenomena. I am working on problems of ocean wave dynamics, techniques of Hamiltonian mechanics and their applications to vortex dynamics, nonlinear Schrodinger equations including ones relevant to photonics, and many other problems in nonlinear waves. My research mostly involves theoretical analysis,***contributing to the understanding of PDEs. However in the past it has had an impact on other disciplines, where advances lend themselves to more efficient numerical simulations or to a deeper conceptual understanding of experiments. And I often collaborate with physical scientists on projects in areas having to do with wave phenomena. *** *This proposal is for research on the properties of solutions of specific PDEs that arise in the physical sciences, which describe wave phenomena such as free surface water waves, nonlinear optics, and continuum mechanics. This includes dispersive evolution equations such as the nonlinear Schrodinger and Klein - Gordon equations, vortex filament evolution for the Euler equations, and problems of water waves. I also plan to continue my prior analysis of the incompressible Navier - Stokes equations.*** ***This program has three principal objectives: ***(1) To address Hamiltonian PDEs from the point of view of a phase space analysis, extending the refined analytic techniques of Hamiltonian systems, such as KAM theory, normal forms, and Nekhoroshev stability, to the dynamics of PDEs in the appropriate infinite dimensional setting in which they are naturally posed.***(2) To study free surface water waves in several settings, and to use their property of a Hamiltonian PDE to describe nonlinear interactions, solitary wave collisions, and asymptotic scaling regimes such as using homogenization techniques to describe waves over a variable bathymetry.***(3) To consider aspects of regularity theory for the Navier - Stokes equations, including the ramifications of upper bounds on spectral behavior, and the recent microlocal lower bounds on the singular set.***I always involve postdoctoral fellows and graduate students in my research projects. In addition, I am intending to address problems that have a potential value to other areas of science, and I plan to continue to seek out interesting collaborations with non-mathematician physical scientists, as I have successfully pursued in the past.**