INTEGRABLE SYSTEMS OF PDE WITH QUASI-PERIODIC INITIAL DATA.

对电子电导的研究属于凝聚态物理的核心。经典的电子导电性理论是建立在自由电子被正离子分散在金属晶格位置的思想之上的。该描述中的一个关键概念是平均自由程，即电子在与离子碰撞之前移动的平均长度。根据经典理论，电子电导率应与平均自由程成正比。量子力学解释了为什么电子显然不会从占据规则晶格位置的离子中散射：电子的波动特性导致电子从理想晶体中衍射。只有当电子从晶体的缺陷处散射时，电阻才会出现。有了量子力学的修正，经典模型仍然可以使用，但在新的图像中，电子在杂质之间曲折移动。杂质越多，平均自由程越小，电导率越低。数学上，这种现象是通过基本的薛定谔方程来描述的。事实证明，要建立一个数学上正确的、与实验数据一致的“高水平杂质导致的局域电子”理论是一个非常深刻的问题。这个问题是著名物理学家p .安德森在50年代末成功解决的问题。安德森以他的名字命名的这一发现为他赢得了1977年的诺贝尔奖。从西奈开创性的工作开始，数学家们认识到安德森局部化理论与许多数学结构有关，并深深植根于现代数学的问题中。据了解，这种现象不仅出现在随机杂质中，而且出现在其他不同类型的结构中，例如准周期结构。后者导致了准周期薛定谔方程谱的分析。分析依赖于一些经典的数学基本领域，如复变量、傅立叶变换、动力系统。准周期薛定谔方程理论在过去四十年中得到了普林斯顿大学和高等研究院、芝加哥大学、加州理工学院、欧文大学、巴黎大学、苏黎世联邦理工学院、巴西、日本、以色列等国数学家的广泛发展。然而，最难的问题，特别是从波动解到安德森局域解的相变，仍然没有解决。另一方面，在过去十年中，人们认识到该理论已经发展到足以建立各种应用。其中一个应用是处理所谓的具有准周期初始数据的完全可积非线性微分方程。这些方程，比如Korteweg-de-Vries方程，Todda晶格同样来自一些基本现象的物理学。60年代后期，数学家发现了这些非线性方程与谱理论的线性微分方程（如薛定谔方程）之间的联系。这样就可以通过所谓的Lax对方法将前者与各种初始数据进行积分。然而，对于准周期初始数据，即由不同周期的几个周期函数组成的数据，没有建立理论。该建议的主要目标是为这些初始数据制订综合方法。部分拟议的研究计划也针对原始的安德森模型。本部分的主要目的是研究Anderson模型在条带域的局部化长度。这个问题现在已经成为世界上许多顶尖专家关注的焦点。

The study of the conductance of electrons belongs to the very heart of condensed-matter physics. The classical theory of electronic conductivity was built on the idea of free electrons scattered by positive ions in metal lattice sites. A key concept in that description was the mean free path, the average length an electron travels before it collides with an ion. According to classical theory, the electronic conductivity should be directly proportional to the mean free path. The quantum mechanics explained why electrons apparently do not scatter from ions that occupy regular lattice sites: the wave character of an electron causes the electron to diffract from an ideal crystal. Resistance appears only when electrons scatter from imperfections in the crystal. With that quantum mechanical revision, the classical model can still be used, but in the new picture an electron is zigzagging between impurities. The more the impurities, the smaller the mean free path and the lower the conductivity. Mathematically, the phenomenon is described via the fundamental Schrodinger Equation. It turns out that to develop a mathematically correct theory of this "localized electrons due to high level of impurities" which would agree with experimental data is a very deep problem. This problem was the problem which famous physicist P.Anderson successfully confronted in the late 50th. Anderson's discovery which carries his name was one of the citations which earned him 1977 Nobel Prize. It was realized by mathematicians, starting with ground breaking works by Sinai that the theory of Anderson Localization is related to many mathematical structures and has deep roots in the problems of modern mathematics. It was understood that the phenomenon exhibits itself not only in presence of random impurities but also for different other types of structures such as quasi-periodic ones. The latter leads to analysis of spectrum of quasi- periodic Schrodinger Equation. The analysis relies on a number of classical fundamental domains of mathematics such as complex variables, Fourier transform, Dynamical systems. The theory of quasi-periodic Schrodinger Equations was extensively developed in last forty years by mathematicians working in Princeton University and the Institute for Advanced Study, University of Chicago, Caltech, Irvine University, Paris universities, Zurich ETH, Brazil, Japan, Israel . Still, the hardest problems, particularly, the phase transitions from fluctuating solutions to Anderson localized ones remain not solved. On the other hand, it was understood in last ten years that the theory is already developed enough to establish various applications. One of these applications addresses the so called completely integrable non-linear differential equations with quasi-periodic initial data. These equations, like for instance Korteweg-de-Vries equation, Todda Lattice once again come from physics of some fundamental phenomena. In late sixties mathematicians discovered the connection between these non-linear equations and spectral theory of linear differential equations such as Scrodinger equation. That allowed to integrate the former with various initial data via so called method of Lax pair. However, no theory was developed for quasi-periodic initial data, i.e. for data composed from several periodic functions with different periods. The main objective of the proposal is to develop methods of integration for such initial data. Part of the proposed research program targets also the original Anderson Model. The main objective for this part is to study the localization length of the Anderson Model in a strip domain. This problem is now in the center of attention for a number of leading experts around the world.