Nonlinear Partial Differential Equations: Wave Propagation in Fluids, Optics and Plasmas

非线性偏微分方程：流体、光学和等离子体中的波传播

• 批准号: RGPIN-2018-04536
• 负责人: Sulem, Catherine
• 金额: $ 2.99万
• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=688323
• 关键词: Nonlinear; Partial; Differential; Equations; Wave

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.

目标是推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产生的物理过程相关的波动现象的演化方程。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。我对海浪的研究计划有两个组成部分：（i）从解的存在性和规律性的经典问题开始，对自由表面水波的偏微分方程进行数学分析，并继续对解的演化进行更详细的相空间分析;（ii）应用数学视角的项目：主题包括大振幅非线性波浪相互作用和波浪在粗糙海底地形上的传播。* 海浪建模是一个活跃的研究领域至少有150年。在更广泛的层面上，由于在更大的科学界更好地了解海浪的重要性，特别是因为对地震引起的海啸和流氓波的发生等自然灾害的了解相对较少，该主题已获得越来越多的兴趣。2.光学与等离子体中的非线性波。非线性薛定谔（NLS）方程是出现在物理学许多领域的正则方程。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。我的研究建议涉及NLS型方程的物理相关性，如色散阿尔文波的导数NLS方程和Zakharov系统的朗缪尔湍流等离子体。我的工作集中在非线性动力学的两个中心现象：（i）与解的爆破相关的自聚焦或波塌缩，以及与之对应的适定性和长时间动力学;（ii）孤立波的动力学，它的长期稳定性，称为孤子分辨率，它指的是解分解为分离孤子的有限和和以及随时间的辐射部分的性质会变成无穷大我的建议结合了物理问题的动机和现代分析的技术。它涉及几种方法，从数学分析，包括动力系统，谐波分析和谱理论，正式渐近展开和数值模拟。