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

本课题的目的是研究非线性波的存在性和稳定性，以及可积系统中的散射谱问题。研究的具体问题有：将检测不稳定特征值的Evans函数技术推广到玻色-爱因斯坦凝聚体的三维非局部问题；局部摄动失效时强耦合Korteweg-de Vries （KdV）方程非线性波的稳定性以及Krein签名和Pontryagin空间在可积系统理论中的应用，特别是在与非线性薛定谔方程和sin - gordon方程相关的逆散射理论中产生的光谱问题的研究。在非线性光学、凝聚态物理或人脑化学过程的数学模型中对非线性波的存在和稳定性进行数学分析，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。将非线性波领域与许多其他纯数学领域区分开来的一个典型特征是一个案例研究，许多问题表现出非常相似的特征，但它们不依赖于任何共同的理论。近年来，在非线性波和可积系统这两个密切相关的领域中反复出现了一个特殊的主题。考虑一个允许存在负能量状态的空间中的问题是有用的。这些状态可能导致相干结构的不稳定。对于稳定的结构，这些负能态要么不活跃，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。

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.