Inverse Scattering Theory

9706644周该项目包括研究逆散射变换的索波列夫空间双射性，与黎曼曲面上的黎曼-希尔伯特问题相关的可积系统的长时间渐近性，聚焦NLS的零散问题，可积系统的无限维微扰理论，某些可积统计模型--随机矩阵--正交多项式，以及其他与逆散射理论相关的问题。逆散射理论提供了一种分析由所谓可积系统描述的具有重要数学和科学意义的非线性现象的方法。这些系统中的许多起源于统计物理和非线性光学。例如，聚焦NLS方程是孤子在光纤中传输的模型，可用于工业上的新一代信息传输系统。这个项目旨在进一步开发由PI和他的合作者发起的某些数学工具。这些工具使人们能够获得关于可积系统解的非常详细的信息，并对相关的非线性现象有更深的理解。

9706644 Zhou The project includes the study of Sobolev space bijectivity of inverse scattering transforms, long time asymptotics for integrable systems related to Riemann-Hilbert problems on Riemann surfaces, the zero dispersion problem for focusing NLS, infinite dimensional perturbation theory for integrable systems, certain integrable statistical models--random matrices--orthogonal polynomials, and other problems related to inverse scattering theory. The inverse scattering theory provides a method for analyzing many nonlinear phenomena of mathematical and scientific importance described by so-called integrable systems. Many of these systems originated from statistical physics and nonlinear optics. For instance, the focusing NLS equation is a model for soliton transmission in optical fibres, which may be adopted by industry for the new generation of information transmission systems. This project aims at further developments of certain mathematical tools originated by the PI and his collaborators. These tools allow one to obtain very detailed information on the solutions of integrable systems and a deeper understanding of the related nonlinear phenomena.