Stability and Existence of Nonlinear Waves

非线性波的稳定性和存在性

• 批准号: 0405066
• 负责人: Walter Strauss
• 金额: --
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405066&HistoricalAwards=false
• 关键词: Stability; Existence; Nonlinear; Waves

ABSTRACT: 0405066Walter Strauss, Brown UniversityStability and existence of nonlinear wavesThe purpose of this project is to study mathematical models of nonlinear waves that occur in the most fundamentaltheories of fluids, of plasmas, of semiconductors and of other branches of physical science. In particular, the mathematical construction of exact periodic and solitary traveling water waves with vorticity will be proven, both with and without surfacetension. Stability and instability phenomena will be investigatedfor motions of ideal fluids and physical plasmas. Electric and magnetic effects on the stability of charged particles will be studiedin the context of the kinetic theory of plasmas. Stability problems for a variety of other kinds of waves will also be explored.Energy conserving waves of marginal stability, which occur in many of these scientific theories, will be emphasized. Methods of mathematical analysis are the primary tool employed in the investigations. The rigorous mathematicsmakes it feasible to make stable numerical computations and to understand thequalitative features of the nonlinear waves.Nonlinear waves are encountered everywhere in the natural world. In thisproject we study several kinds of such nonlinear waves that are well described by mathematics, but where the mathematical equations are extremely difficult to solve.One of our objectives is to study water waves that may occur in the ocean, and to understand how they can form eddies or whirlpools and how they can become turbulent.The knowledge of specific kinds of solutions to the mathematical equations has an impact on our fundamental understanding of ocean waves and currents. Another objective is to study semiconducting materials from which computing devices are manufactured. We will analyze the basic equations that describe how the electrons can move inside and at the edges of the material. A third objective is to understand the behavior of charged particles in the vicinity of the earth's magnetic field, which affect satellitecommunications and the health of astronauts. A fourth objective of the project is the training of graduate and undergraduate students and postdoctoral fellows in the precise mathematical analysis of applied scientific problems such as nonlinear waves.

摘要：0405066沃尔特·施特劳斯，布朗大学非线性波的稳定性和存在性这个项目的目的是研究非线性波的数学模型，这些非线性波发生在流体、等离子体、半导体和物理科学的其他分支的最基本的理论中。特别是，精确的周期性和孤立的行波与涡量的数学结构将被证明，无论有和没有表面张力。稳定性和不稳定性现象将研究理想流体和物理等离子体的运动。将在等离子体动力学理论的范围内研究电和磁对带电粒子稳定性的影响。本课程也将探讨其他各种波的稳定性问题，并将强调在许多科学理论中出现的边际稳定性的能量守恒波。数学分析方法是调查中采用的主要工具。非线性波在自然界中随处可见，严格的几何学使得进行稳定的数值计算和了解非线性波的定性特征成为可能。 在这个项目中，我们研究了几种可以用数学很好地描述的非线性波，但其中的数学方程非常难以求解。我们的目标之一是研究可能发生在海洋中的水波，了解它们是如何形成漩涡或漩涡池的，以及它们是如何变成湍流的。了解海浪和洋流。另一个目标是研究制造计算设备的半导体材料。 我们将分析描述电子如何在材料内部和边缘移动的基本方程。第三个目标是了解带电粒子在地球磁场附近的行为，这会影响卫星通信和宇航员的健康。该项目的第四个目标是培训研究生、本科生和博士后研究员，对非线性波等应用科学问题进行精确的数学分析。