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

目的是促进对非线性偏微分方程（PDEs）的理解；特别是描述与流体动力学、非线性光学和等离子体物理学中产生的物理过程相关的波动现象的演化方程。我主要在两个方向上工作：1。海浪理论。数学分析和应用数学的许多方面最初都是由流体动力学的研究引起的，特别是水体中的水流和波浪。反过来，数学对于理解地球海洋的动力学非常重要，对于预测海浪和洋流及其对天气和气候的影响至关重要。我对海浪的研究计划有两个部分：(i)对自由表面水波的偏微分方程进行数学分析。（ii）具有应用数学观点的项目；重要的课题包括大振幅非线性波的相互作用和波浪在粗糙的海底地形上的传播。光学和等离子体中的非线性波：非线性薛定谔方程（NLS）是出现在许多物理领域的典型方程。它作为波的包络动力学模型无处不在，并经常用于光学、等离子体和流体。在量子物理学中，它作为约束势中的多体玻色子系统和稀气体中的玻色-爱因斯坦凝聚的平均场方程出现。我的研究计划涉及物理相关的NLS型方程，如色散Alfven波的导数NLS方程和Langmuir湍流的Zakharov系统。我的工作集中在非线性动力学的两个中心现象：(i)孤立波的存在和稳定性；（ii）“自聚焦”或“波塌缩”与解决方案及其对应的爆破、适位性和长时间动力学有关。

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.