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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=712380
• 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.

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。 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)从解的存在性和正则性的经典问题出发，继续对解的演变进行更详细的相空间分析，对自由表面水波的偏微分方程组进行数学分析；(Ii)从应用数学的角度进行项目：主题包括大振幅非线性波相互作用和粗糙海底地形上的波传播。 至少150年来，海浪模型一直是一个活跃的研究领域。在更广泛的层面上，由于在更大的科学界内建立对海浪的更好了解的重要性，特别是由于相对较少了解的自然灾害，如地震引发的海啸和无赖海浪的发生，这一专题获得了更大的兴趣。 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Alfvén波的导数NLS方程和等离子体中朗缪尔湍流的Zakharov系统。我的工作集中在非线性动力学的两个中心现象上：(I)与解的爆破有关的自聚焦或波崩溃，以及与之对应的健康和长时间动力学；(Ii)孤立波的动力学，它们的长期稳定性和所谓的孤子分辨，它指的是解随着时间的推移分解成分离的孤子和辐射部分的有限和的性质。 我的建议结合了来自物理问题的动机和来自现代分析的技术。它涉及几种方法，从数学分析，包括动力系统，调和分析和谱理论，到形式渐近展开和数值模拟。