Inverse Scattering Theory

逆散射理论

• 批准号: 9706644
• 负责人: Xin Zhou
• 金额: $ 8.04万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706644&HistoricalAwards=false
• 关键词: Inverse; Scattering; Theory

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.

主要研究方向包括逆散射变换的Sobolev空间双射性、黎曼曲面上黎曼-希尔伯特问题相关的可积系统的长时间渐近性、聚焦NLS的零色散问题、可积系统的无限维摄动理论、某些可积统计模型—随机矩阵—正交多项式等与逆散射理论相关的问题。逆散射理论为分析所谓可积系统所描述的许多具有数学和科学重要性的非线性现象提供了一种方法。许多这样的系统起源于统计物理学和非线性光学。例如，聚焦NLS方程是光纤中孤子传输的一个模型，可用于工业上新一代的信息传输系统。该项目旨在进一步发展由PI及其合作者发起的某些数学工具。这些工具允许人们获得关于可积系统解的非常详细的信息和对相关非线性现象的更深入的理解。