Nonlinear Partial Differential Equations; Applications to Wave Propagation in Fluids, Optics and Plasmas

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The objective is to advance the understanding of nonlinear partial differential equations (PDEs); in particular, evolution equations that describe wave phenomena relevant to physical processes arising in fluid dynamics, nonlinear optics and plasma physics. I work in two main directions:1. The theory of ocean waves. Many aspects of mathematical analysis and applied mathematics were originally motivated by the study of fluid dynamics, and in particular currents and waves in bodies of water. In turn, mathematics is important to understand the dynamics of the earth's oceans, and is central to prediction of ocean waves and currents and their effect on weather and climate. My research proposal on ocean waves has two components: (i) mathematical analysis of the PDEs for free surface water waves. (ii) projects with an applied mathematics perspective; important topics include large amplitude nonlinear wave interactions and wave propagation over rough bottom topography.2. Nonlinear waves in optics and plasmas: The nonlinear Schroedinger (NLS) equation is a canonical equation that appears in many fields of physics. It arises ubiquitously as a model for the envelope dynamics of waves and is used frequently in optics, plasmas and fluids. In quantum physics, it arises as a mean field equation for a many-body boson system in a confining potential and for Bose-Einstein condensation in dilute gases. My research proposal concerns NLS type equations of physical relevance such as the Derivative NLS equation for dispersive Alfven waves and the Zakharov system for Langmuir turbulence. My work concentrates on two central phenomena of nonlinear dynamics: (i) existence and stability of solitary waves; (ii) `self-focusing' or `wave collapse' associated to the blow-up of solutions and its counterpart, wellposedness and long-time dynamics.