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

其目的是促进对非线性偏微分方程组的理解，特别是描述与流体力学、非线性光学和等离子体物理中的物理过程相关的波动现象的演化方程。我的工作主要有两个方向： 1.海浪理论。数学分析和应用数学的许多方面最初是由流体动力学的研究推动的，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态是重要的，对于预测海浪和洋流及其对天气和气候的影响是至关重要的。我对海浪的研究建议有两个部分：(I)自由表面水波偏微分方程组的数学分析。(2)具有应用数学观点的项目；重要专题包括大幅度非线性波相互作用和粗糙海底地形上的波传播。 2.光学和等离子体中的非线性波：非线性薛定谔(NLS)方程是一个出现在许多物理领域的正则方程。它是波包络动力学的一种普遍存在的模型，广泛应用于光学、等离子体和流体等领域。在量子物理学中，它是作为禁闭势中的多体玻色子系统和稀薄气体中的玻色-爱因斯坦凝聚的平均场方程出现的。我的研究建议涉及NLS型物理关联方程，如色散Alfven波的导数NLS方程和朗缪尔湍流的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)与解及其对应的爆破有关的“自聚焦”或“波崩溃”，适定性和长时间动力学。

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.