Mathematical theory of water waves and plasmas

该项目的学术目的是研究出现在流体、等离子体、半导体和物理科学其他分支的最基本理论中的波的数学模型。将研究具有涡度(涡旋)的精确水波的物理特性，例如精确的海浪模型。特别是，表面下的压力，单个粒子的路径，以及停滞的存在和位置将被调查。将探讨盐度和温度变化引起的层化对水波的影响。将强调在许多科学理论中出现的边际稳定的能量守恒波。特别是，将在动力学理论的范围内研究聚变反应堆或太阳风等带电粒子系统稳定性的电磁效应。此外，还将研究半导体流体动力学模型中的态的稳定性。数学分析方法将是调查中使用的主要工具。还将采用高性能的数值计算。培养受过应用科学问题精确数学分析培训的人员，包括研究生、本科生和博士后研究员，将是该项目的一个重要成果。此外，这些研究工作还导致了向理工科学生教授现代数学思想和技术进步的新的教学方法的发展。严谨的数学使我们有可能在等离子体波、机械振动和许多其他物理现象中进行稳定的数值计算，并了解其定性特征。某些精确波的存在及其稳定性或不稳定性影响着我们对自然现象的理解。对水波的研究可以加深我们对海浪和海流、船舶安全、盐度对漩涡形成的影响以及风对无赖波形成的影响的了解。它可以阐明颗粒物是如何在海洋中移动的。对等离子体的研究可以解释太阳风等天体物理等离子体中的哪些构型是稳定的，因此很可能被看到。半导体分析可以提高我们对微型半导体器件的理解，从而影响它们的设计。

The intellectual purpose of the project is to study mathematical models of waves that occur in the most fundamental theories of fluids, of plasmas, of semiconductors and of other branches of physical science. The physical properties of exact water waves with vorticity (eddies), such as precise models of ocean waves, will be studied. In particular, the pressure beneath the surface, the paths of individual particles, and the existence and location of stagnation will be investigated. The effects of stratification, due to salinity and temperature variations, on the water waves will be explored. Energy conserving waves of marginal stability, which occur in many scientific theories, will be emphasized. In particular, electric and magnetic effects on the stability of systems of charged particles, such as in fusion reactors or in the solar wind, will be studied in the context of kinetic theory. Moreover, the stability of states in the hydrodynamic model of semiconductors will be studied. Methods of mathematical analysis will be the primary tools employed in the investigations. High-performance numerical computations will also be employed. The development of personnel, including graduate and undergraduate students and postdoctoral fellows, who are trained in the precise mathematical analysis of applied scientific problems, will be an important outgrowth of the project. Furthermore, such research work leads to the development of new pedagogical approaches to the teaching of modern mathematical ideas and technical advances to science and engineering students. The rigorous mathematics makes it possible both to perform stable numerical computations in, and to understand the qualitative features of, plasma waves, mechanical vibrations and many other physical phenomena. The existence of certain kinds of exact waves and their stability or instability has an impact on our understanding of natural phenomena. The research on water waves may improve our understanding of ocean waves and currents, ship safety, the effect of salinity on the formation of whirlpools, and the effect of wind on the formation of rogue waves. It could illuminate how particulate matter moves through the ocean. The research on plasmas could explain which configurations in astrophysical plasma such as the solar wind are stable and therefore likely to be seen. The semiconductor analysis could improve our understanding of miniature semiconductor devices and thus influence their design.