Mathematical Sciences: Inverse Scattering Theory

数学科学：逆散射理论

• 批准号: 9204804
• 负责人: Xin Zhou
• 金额: $ 4.4万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-05-15 至 1994-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9204804&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Inverse; Scattering; Theory

The mathematical research supported by this award continues work on the development of a nonlinear harmonic analysis on integrable equations in connection with inverse scattering theory. Central to the project is a steepest decent method for analyzing the asymptotics of oscillatory Riemann-Hilbert problems of the kind that arise in the theory of integrable nonlinear equations. To date the method has been applied to the modified Korteweg-deVries equation and the defocusing nonlinear Schrodinger equation. The steepest descent method mentioned above gives long time asymptotics of solutions to Cauchy problems integrable by the inverse scattering method. A great deal of work has been done along the same lines, but much is formal and very little is rigorous. This project will first seek to analyze the zero dispersion limit problems for Korteweg-deVries equations, that is, to study directly the corresponding oscillatory Riemann- Hilbert problem. It is expected that the new method will not require any spectral assumptions, and that, once worked out, will apply to a broad class of problems. A second line of research will consider Hamiltonians arising as transverse Ising models at the critical transverse magnetic field. The corresponding autocorrelation function can be computed from the solution of the associated Riemann-Hilbert problem. Preliminary calculations give results different from previous formal results of others. Work will be done to justify these discoveries and to continue work on a class of related integrable statistical models to compute the large time behavior of the autocorrelation function.

该奖项支持的数学研究仍在继续 工作的非线性谐波分析的发展， 逆散射可积方程 理论该项目的核心是一个最陡峭的体面的方法， 分析振动Riemann-Hilbert问题的渐近性 在可积非线性理论中出现的那种 方程 到目前为止，该方法已应用于修改后的 Korteweg-deVries方程与散焦非线性 薛定谔方程 上面提到的最速下降法给出了长时间 可积柯西问题解的渐近性 逆散射法 做了大量的工作 沿着相同的路线，但大部分是正式的，很少是正式的 严谨 这个项目将首先寻求分析零 Korteweg-deVries方程的色散极限问题， 直接研究相应的振荡黎曼 希尔伯特问题 预计新方法将不会 需要任何光谱假设，而且，一旦制定出来，将 适用于广泛的问题。 第二条研究线将考虑哈密顿量的产生 作为横向伊辛模型在临界横向磁场 领域 相应的自相关函数可以是 从相关的黎曼-希尔伯特的解计算 问题. 初步计算得出的结果与 其他人的正式成果。 我们将努力证明 这些发现并继续研究一类相关的 可积统计模型来计算大时间行为 的自相关函数。