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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635594
• 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.

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。我在两个主要方向上工作：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）对自由地表水波的数学分析。 （ii）具有应用数学观点的项目；重要的主题包括大幅度非线性波相互作用和在粗糙的底部地形上的波传播2。光学和等离子体中的非线性波：非线性schroedinger（NLS）方程是一个典型的方程，出现在许多物理领域。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，它是在稀释气体中具有限制电势和玻色仁凝结中的多体玻色子系统的平均场方程。我的研究建议涉及NLS类型的物理相关性方程，例如分散性Alfven波的衍生NLS方程和Langmuir湍流的Zakharov系统。我的工作集中在两个非线性动力学的中心现象上：（i）孤立波的存在和稳定性； （ii）与解决方案的爆炸及其对应物，良好性和长期动态相关的“自我关注”或“波浪崩溃”。