Spectral Problems and Inverse Spectral Problems

主要研究者将考虑各种问题的渐近行为的有限和无限维系统包含外部参数，如空间和时间，参数变得很大。 特别是，主要研究者将考虑扰动聚焦非线性薛定谔方程的解的长期行为，初始数据接近解。 此外，主要研究者将考虑随着N变大，N数排列理论中出现的统计问题，以及关于特殊数（如z（5），其中z是黎曼zeta函数）的有理逼近的问题。 主要研究者还将考虑在可积模型中产生的各种物理常数的计算。 在上述所有问题中，由Xin Zhou和首席研究员于1993年提出的Riemann-Hilbert问题的最速下降法将起到关键的分析作用。首席研究员提出的许多工作都具有强烈的跨学科色彩。 例如，上面提到的N数排列的统计问题，与过饱和液体在基底上的冷凝模型密切相关，也与单人纸牌游戏（“耐心排序”）的版本密切相关，也与重新排序图书的问题密切相关。 此外，主要研究者的一个关键目标将是证明各种物理系统的“普遍性”。 例如，受周欣早期工作的启发，首席研究员计划表明，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，特别是信号沿沿着光纤传输中出现的现象建立模型的关键步骤。

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.