Mathematical Sciences: Theory and Applications of Nonlinear Wave Equations

沃尔特克雷格 NSF DMS 9501514 首席研究员将在以下四个领域开展研究计划： 偏微分方程理论，这些都很有趣 数学分析中的问题，所有这些问题的动机都是问题， 数学物理学和应用数学。第一个主题是 继续发展Kolmogorov、Arnold和Moser方法（KAM） 适用于无限多自由度的系统。这 包括非线性演化方程，如非线性波 方程，非线性薛定谔方程，Korteweg方程 方程，以及自由表面的水波方程组， 流体。这项工作的目的是了解稳定的运动， 这些非线性演化方程，并描述一些主要的 无限维相空间的不变结构， 他们被摆姿势。第二个项目进一步涉及水波系统， 在最重要的物理尺度下的渐近尺度极限 政权。这种分析对波的描述设定了严格的界限 在自由表面上用调制理论和长波理论描述 Boussinesq和Korteweg deVries方程。几个新要素 的分析已经介绍了这是有用的数值 流体动力学问题的建模。 第三个研究项目涉及的奇点的演变 薛定谔方程，在线性和非线性的情况下。目标 是了解宇宙中奇点的位置和结构 基本解，它是微局部信息， 高能粒子的经典轨迹最后 第四个主题涉及量子力学逆谱问题， 其目的是理解光谱变换的三个原则， 散射理论，Floquet理论和随机理论的设置 潜力，并量化这些设置之间的相似性。 由于数学是科学的语言， 是通过物理方程的解的性质来理解的，这些物理方程大部分是偏微分方程。 研究者的主要兴趣在于方程， 描述保守现象，这是最常见的哈密顿 系统.本项目建议书中提出的问题在物理和工程科学中都具有核心重要性， 从海洋表面的流体运动到半导体的非线性量子力学， 装置.所有四个项目的目标都是了解重要的 这些方程的解的各个方面，所有这些都是相关的 对系统的理解，其中许多系统也存在 数学分析中非常具有挑战性的问题。一个观察是， 值得注意的是，人们在系统中发现了相关的结构， 描述了非常不同的现象。例如，在第一个项目中， 非线性量子的描述 水表面的力学和波动现象;这变得很清楚 只在两个系统的数学分析中。此外 数学分析在某些情况下导致了 物理系统的建模，以及新的 预测波浪现象的计算程序，这是一个 联邦战略中与海洋和气候模拟有关的主题 关注全球变化和环境。

Walter Craig NSF DMS 9501514 The principal investigator will pursue a research program in four areas of the theory of partial differential equations, all of which are interesting problems in mathematical analysis, and all of them motivated by problems in mathematical physics and applied mathematics. The first topic is the continued development of methods of Kolmogorov, Arnold and Moser (KAM) that are suited to systems with infinitely many degrees of freedom. This includes nonlinear evolution equations such as the nonlinear wave equation, nonlinear Schrodinger equation, versions of the Korteweg deVries equation, and the water waves system of equations for the free surface of a fluid. The intent of the work is to understand the stable motions of these nonlinear evolution equations, and to describe some of the principal invariant structures of the infinite dimensional phase spaces in which they are posed. The second project concerns further the water wave system, and its asymptotic scaling limits in the most physically important scaling regimes. This analysis sets rigorous bounds on the descriptions of waves in free surfaces by modulation theory and by long wave theories described by the Boussinesq and the Korteweg deVries equations. Several new elements of analysis have already been introduced which are useful in numerical modeling of the fluid dynamical problem. The third research project concerns the evolution of singularities of Schrodinger's equation, in both the linear and nonlinear cases. The goal is to understand the location and structure of the singularities of the fundamental solution, which is microlocal information and is related to the classical trajectories of the high energy particles. Finally, the fourth subject concerns the quantum mechanical inverse spectral problem, the object being to understand the spectral transform in the three principal settings of scattering theory, Floquet theory and the theory of random potentia ls, and to quantify the similarities between these settings. As mathematics is the language of the sciences, physical phenomena are understood through the properties of solutions of the equations of physics, which are for the most part partial differential equations. The investigator's principal interests are in the equations which describe conservative phenomena, which are most often Hamiltonian systems. The problems that are addressed in this project proposal are all of central importance in the physical and engineering sciences, and govern a remarkable variety of systems, from fluid motions of the ocean surface to the nonlinear quantum mechanics of semiconductor devices. The goals of all four projects are to understand important aspects of the solutions of these equations, all of which are relevant to the understanding of the systems, and many of them which also present very challenging problems in mathematical analysis. One observation is that it is remarkable that one finds related structure in systems which describe very different phenomena. For example in the first project there is a relation between the description of nonlinear quantum mechanics and wave phenomena in water surfaces; this becomes clear only in the mathematical analysis of the two systems. Furthermore the mathematical analysis has in some cases led to improvements in the modeling of the physical systems, and to the implementation of new computational procedures for predicting wave phenomena, which is a topic relevant to ocean and climate modeling in the Federal strategic interest area of global change and the environment.