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.

目的是提高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的物理过程相关的波浪现象的进化方程。****** 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，特别是水体中的电流和波动的动机。反过来，数学对于了解地球海洋的动态非常重要，并且对海浪和电流的预测及其对天气和气候的影响至关重要。我对海浪的研究建议有两个组成部分：（i）从经典的存在和解决方案的经典问题开始的自由地表水波的数学分析，并继续对解决方案的演变进行更详细的相空间分析； （ii）具有应用数学视角的项目：主题包括大型放大器非线性波相互作用和在粗糙的底部地形上的波传播。 ***海浪建模至少150年来一直是研究的积极研究领域。在更广泛的层面上，由于对更大的科学界内的海浪有更好的了解，尤其是因为相对了解的自然危害（例如地震产生的海啸和流氓浪潮的发生），该主题已经增加了兴趣。光学和等离子体中的非线性波。非线性schrödinger（NLS）方程是出现在许多物理领域的规范方程。它无处不在，作为波浪动力学的模型，经常用于光学，等离子体和流体。在量子物理学中，它是多体玻色子系统的平均田间等效性，该系统具有限制的潜力和稀释气体中的玻色 - 内斯坦凝结。我的研究建议涉及物理相关性的NLS类型方程，例如用于分散性Alfvén波的衍生NLS方程和Plains 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上：（i）与解决方案的爆炸相关的自我关注或波浪崩溃，及其对应物，良好的和长期的动力学； （ii）固体波的动力学，它们的长期稳定性和所谓的孤子分辨率，它指的是溶液将分解为有限的分离固体总和的特性，并且随着时间的推移是无限的。它涉及几种方法，包括数学分析，包括动态系统，谐波分析和光谱理论，到形式的不对称扩展和数值模拟。