Nonlinear Partial Differential Equations: 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.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 starting from the classical questions of existence and regularity of solutions and continuing with a more detailed phase space analysis of the evolution of solutions; (ii) projects with an applied mathematics perspective: topics include large amplitude nonlinear wave interactions and wave propagation over rough bottom topography. Modeling of ocean waves has been an active area of research for at least 150 years. At a broader level, the topic has gained an increased interest due to the importance in establishing a better understanding of ocean waves within the larger scientific community, in particular because of the relatively poorly understood natural hazards such as seismically generated tsunamis and the occurrence of rogue waves.2. Nonlinear waves in optics and plasmas. The nonlinear Schrödinger (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 Alfvén waves and the Zakharov system for Langmuir turbulence in plasmas. My work concentrates on two central phenomena of nonlinear dynamics: (i) self-focusing or wave collapse associated to the blow-up of solutions, and its counterpart, wellposedness and long-time dynamics; (ii) the dynamics of solitary waves, their long-time stability and the so-called soliton resolution that refers to the property that the solution decomposes into a finite sum of separated solitons and a radiative part as time goes to infinity.My proposal combines motivation from physical problems and techniques from modern analysis. It involves several approaches, ranging from mathematical analysis including dynamical systems, harmonic analysis and spectral theory, to formal asymptotic expansions and numerical simulations.