Stability Theory of Nonlinear Waves

NSF奖摘要- DMS-0071838数学科学：非线性波的稳定性理论摘要0071838施特劳斯这个项目研究各种波的数学模型，这些波出现在等离子体理论、半导体理论、流体理论、机械振动理论和物理科学的其他分支中。 严格的数学使得在这些物理现象中进行稳定的数值计算和理解其定性特征成为可能。 稳定性和不稳定性现象进行了研究，特别是在带电粒子的动力学理论。 半导体材料的数值模拟为混合量子动力学模型的研究提供了理论基础。 各种其他类型的波，如孤立波在流体和固体中的平衡的稳定性问题，也进行了探讨。 能量守恒波的边际稳定性，这发生在许多这些科学理论，强调。 数学分析方法是研究的主要工具。本项目的目的之一是分析用于制造计算机芯片的半导体材料的结构。 随着芯片变得越来越小，量子效应变得越来越重要。 由于整个设备的量子力学的模拟在计算上是不可行的，该项目研究了混合模型相结合的宏观动力学理论与局部区域的量子描述的设备的大部分。 该项目的另一个目标是研究物理等离子体的稳定性，这对地球电离层的屏蔽效应以及聚变反应堆都很重要。 人员的发展，包括研究生和本科生，谁是在应用科学问题的精确数学分析的培训，是该项目的一个重要成果。

NSF Award Abstract - DMS-0071838Mathematical Sciences: Stability Theory of Nonlinear WavesAbstract0071838 StraussThis project studies mathematical models of various kinds of waves that occur in the theories of plasmas, of semiconductors, of fluids, of mechanical vibrations and of other branches of physical science. The rigorous mathematics makes it possible both to make stable numerical computations in, and to understand the qualitative features of, such physical phenomena. Stability and instability phenomena are investigated, particularly in the kinetic theory of charged particles. Numerical simulation of semiconducting materials provides the rationale for a study of hybrid quantum-kinetic modeling. Stability problems for a variety of other kinds of waves, such as solitary waves in fluids and equilibria in solids, are also explored. Energy conserving waves of marginal stability, which occur in many of these scientific theories, are emphasized. Methods of mathematical analysis are the primary tool employed in the investigations.One purpose of this project is to analyze the structure of semiconducting materials that are used to manufacture computer chips. As chips become smaller, quantum effects become increasingly important. Since simulation of the quantum mechanics of an entire device is computationally infeasible, this project investigates hybrid models combining macroscopic kinetic theory for the bulk of a device with quantum descriptions in localized regions. Another goal of the project is to study the stability of physical plasmas, important in the shielding effect of the earth's ionosphere as well as in fusion reactors. The development of personnel, including graduate and undergraduate students, who are trained in the precise mathematical analysis of applied scientific problems, is an important outgrowth of the project.