Stability and Existence of Nonlinear Waves

摘要：沃尔特·施特劳斯，布朗大学非线性波的稳定性和存在这个项目的目的是研究在流体、等离子体、半导体和其他物理科学的最基本理论中出现的非线性波的数学模型。特别地，我们将证明在有表面尺寸和无表面尺寸的情况下，具有涡度的精确周期和孤立行波的数学构造。我们将研究理想流体和物理等离子体运动的稳定性和不稳定性现象。带电粒子稳定性的电磁效应将在等离子体运动理论的背景下进行研究。还将探讨各种其他类型的波的稳定性问题。重点讨论在许多这些科学理论中出现的边缘稳定的能量守恒波。数学分析方法是调查中使用的主要工具。严谨的数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。非线性波在自然界中随处可见。在这个项目中，我们研究了几种这样的非线性波，它们用数学很好地描述了，但其中的数学方程非常难解。我们的目标之一是研究海洋中可能出现的水波，并了解它们如何形成涡旋或漩涡，以及它们如何变成湍流。对数学方程特定类型的解的知识对我们对海浪和海流的基本理解有影响。另一个目标是研究用来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运动的基本方程。第三个目标是了解地球磁场附近带电粒子的行为，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动等应用科学问题进行精确的数学分析。

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.