Stability of Nonlinear Waves and Spectral-Scattering Problems Using Krein Signature and Pontryagin Spaces

使用 Kerin 签名和 Pontryagin 空间的非线性波稳定性和光谱散射问题

• 批准号: 0705563
• 负责人: Richard Kollar
• 金额: $ 8.9万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0705563&HistoricalAwards=false
• 关键词: Stability; Nonlinear; Waves; Spectral; Scattering

The goal of this project is to study existence and stability of nonlinear waves and spectrum of scattering problems arising in integrable systems. Particular problems studied are: extension of the Evans function technique for detection of unstable eigenvalues to three-dimensional and non-local problems in Bose-Einstein condensates; stability of nonlinear waves in strongly coupled Korteweg-de Vries (KdV) equations in a regime when local perturbation techniques fail; and application of Krein signature and Pontryagin spaces to the theory of integrable systems, particularly the study of spectral problems originated in the inverse scattering theory associated with the nonlinear Schrodinger and the Sine-Gordon equations.Mathematical analysis of existence and stability of nonlinear waves in mathematical models in nonlinear optics, condensed matter physics, or chemical processes in human brain, has far-reaching consequences for applications as analytical results often guide future experiments in physics, chemistry, or medicine. A typical feature which distinguishes the field of nonlinear waves from many other fields of pure mathematics is a case study, many problems demonstrate very similar features but they do not rely on any common theory. In the recent years a particular theme appeared recurrently in two close fields - nonlinear waves and integrable systems. It is useful to consider a problem in a space which allows existence of states with 'negative' energy. These states then may lead to instabilities of coherent structures. For stable structures these negative energy states must be either not active or completely eliminated. The aim of the project is to gain more insight into this topic by solving interesting particular applied problems and apply it to build and simplify the general theory.

本计画的目标是研究可积系统中非线性波的存在性与稳定性以及散射问题的频谱。研究的具体问题有：将Evans函数法推广到三维玻色-爱因斯坦凝聚体的非定域问题：强耦合Korteweg-de弗里斯（KdV）方程中非线性波在定域扰动法失效时的稳定性以及Krein签名和Pontryagin空间在可积系统理论中的应用，特别是对起源于与非线性薛定谔方程和Sine-Gordon方程相关的逆散射理论的谱问题的研究。非线性光学、凝聚态物理、或人类大脑中的化学过程，对应用具有深远的影响，因为分析结果通常指导未来的物理，化学或医学实验。一个典型的特点，区分领域的非线性波从许多其他领域的纯数学是一个案例研究，许多问题表现出非常相似的特点，但他们不依赖于任何共同的理论。近年来，在两个密切相关的领域--非线性波和可积系统中，一个特殊的主题反复出现。在允许存在具有“负”能量的状态的空间中考虑问题是有用的。这些状态可能导致相干结构的不稳定性。对于稳定的结构，这些负能态必须要么不活跃，要么完全消除。该项目的目的是通过解决有趣的特定应用问题来深入了解这个主题，并将其应用于构建和简化一般理论。