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

非线性偏微分方程；

• 批准号: 46179-2013
• 负责人: Sulem, Catherine
• 金额: $ 2.48万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=568932
• 关键词: Nonlinear; Partial; Differential; Equations; Applications

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.

其目标是促进对非线性偏微分方程（PDE）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出现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，特别是水流和水体中的波浪。反过来，数学对于理解地球海洋的动力学非常重要，并且对于预测海浪和洋流及其对天气和气候的影响至关重要。我关于海浪的研究计划有两个组成部分：（i）自由水面波浪偏微分方程的数学分析。(ii)项目与应用数学的角度;重要的课题包括大振幅非线性波的相互作用和波传播粗糙的底部地形。 2.光学和等离子体中的非线性波：非线性薛定谔（NLS）方程是出现在许多物理领域的正则方程。它作为波的包络动力学的模型无处不在，并且经常用于光学，等离子体和流体。在量子物理学中，它作为一个平均场方程出现，用于限制势中的多体玻色子系统和稀气体中的玻色-爱因斯坦凝聚。我的研究计划涉及NLS型方程的物理相关性，如色散阿尔芬波的导数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：（一）孤立波的存在性和稳定性;（二）“自聚焦”或“波崩溃”与爆破的解决方案和它的对应物，适定性和长时间动态。