Mathematical Sciences: Theory and Applications of Nonlinear Wave Equations

首席研究员将在偏微分方程理论的四个领域进行研究项目，这些领域都是数学分析中有趣的问题，所有这些问题都是由数学物理和应用数学中的问题所激发的。第一个主题是Kolmogorov， Arnold和Moser （KAM）方法的持续发展，这些方法适用于具有无限多个自由度的系统。这包括非线性演化方程，如非线性波动方程，非线性薛定谔方程，Korteweg deVries方程的版本，以及流体自由表面的水波方程组。这项工作的目的是理解这些非线性演化方程的稳定运动，并描述它们所处的无限维相空间的一些主要不变结构。第二个项目进一步关注水波系统，及其在最重要的物理标度体系中的渐近标度极限。这一分析对由调制理论和由Boussinesq和Korteweg deVries方程描述的长波理论在自由表面上的波的描述设置了严格的界限。已经引入了一些新的分析元素，它们对流体动力学问题的数值模拟很有用。第三个研究项目涉及薛定谔方程在线性和非线性情况下奇点的演化。目标是了解基本解的奇异点的位置和结构，这是微局部信息，与高能粒子的经典轨迹有关。最后，第四个主题涉及量子力学逆光谱问题，目的是了解散射理论，Floquet理论和随机势理论三种主要设置下的光谱变换，并量化这些设置之间的相似性。由于数学是科学的语言，物理现象是通过物理方程的解的性质来理解的，这些方程大部分是偏微分方程。研究者的主要兴趣在于描述保守现象的方程，这些方程通常是哈密顿系统。在本项目提案中所解决的问题在物理和工程科学中都是至关重要的，并且管理着各种各样的系统，从海洋表面的流体运动到半导体器件的非线性量子力学。这四个项目的目标是理解这些方程的解的重要方面，所有这些都与系统的理解有关，其中许多在数学分析中也提出了非常具有挑战性的问题。一个观察是，在描述非常不同现象的系统中发现相关的结构是很了不起的。例如，在第一个项目中，非线性量子力学的描述与水面波动现象之间存在关系；这一点只有在对这两个系统进行数学分析时才变得清楚。此外，在某些情况下，数学分析导致了物理系统建模的改进，并导致了预测波浪现象的新计算程序的实施，这是一个与全球变化和环境联邦战略利益领域的海洋和气候建模相关的主题。

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.