Spectral Problems and Inverse Spectral Problems

谱问题和逆谱问题

• 批准号: 0003268
• 负责人: Percy Deift
• 金额: --
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2001-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0003268&HistoricalAwards=false
• 关键词: Spectral; Problems; Inverse

The Principal Investigator will consider a variety of problems concerning the asymptotic behavior of finite and infinite dimensional systems containing external parameters such as space and time, as the parameters become large. In particular, the Principal Investigator will consider the long-time behavior of solutions of the perturbed focusing Nonlinear Schrodinger equation, with initial data that is close to a solution. In addition, the Principal Investigator will consider statistical problems arising in the theory of permutations of N numbers as N becomes large, as well as questions concerning the rational approximation of special numbers such as z (5) where z is the Riemann zeta function. The Principal Investigator will also consider the computation of various physical constants arising in integrable models. In all the above problems, a key analytical role will be played by the steepest descent method for Riemann-Hilbert problems introduced by Xin Zhou and the Principal Investigator in 1993. Much of the work proposed by the Principal Investigator has a strong interdisciplinary flavor. For example, the statistical problems mentioned above for permutations of N numbers, are intimately related to a model for the condensation of a supersaturated liquid on a substrate and also to a version of solitaire ("patience sorting"), and also to the problem of re-ordering a library in which books have been improperly shelved. In addition, a key objective of the Principle Investigator will be to prove "universality" for a variety of physical systems. For example, motivated by earlier work with Xin Zhou, the Principal Investigator plans to show that solutions of the perturbed focusing Non-Linear Schrodinger equation behave just like solutions of the unperturbed focusing Non-linear Schrodinger equation, once the "scales" of the problem are properly adjusted. This is a key step in the development of models for a wide variety of physical phenomena, in particular for phenomena arising in the transmission of signals along optical fibers.

当参数变大时，首席调查员将考虑与包含外部参数(如空间和时间)的有限维和无限维系统的渐近行为有关的各种问题。特别是，首席调查者将考虑具有接近解的初始数据的扰动聚焦非线性薛定谔方程的解的长期行为。此外，首席调查员将考虑当N变大时N数排列理论中出现的统计问题，以及关于特殊数的有理逼近的问题，例如z(5)，其中z是Riemann Zeta函数。首席调查员还将考虑在可积模型中产生的各种物理常数的计算。在所有这些问题中，最陡下降法将在所有这些问题中发挥关键的分析作用。首席调查员提出的许多工作具有很强的跨学科色彩。例如，上面提到的N个排列的统计问题，与过饱和液体在底物上凝结的模型密切相关，也与纸牌的一个版本(“耐心排序”)密切相关，也与图书被不当搁置的图书馆重新排序的问题密切相关。此外，首席调查员的一个关键目标将是证明各种物理系统的“普遍性”。例如，受早先与周欣合作的启发，首席调查者计划证明，一旦问题的“尺度”得到适当调整，扰动聚焦非线性薛定谔方程的解就像未扰动聚焦非线性薛定谔方程的解一样。这是为各种物理现象建立模型的关键一步，特别是对沿光纤传输信号时出现的现象。