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金属晶格中的正离子散射自由电子的思想之上的。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的平均长度。根据经典理论，电子电导率应该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绕射。只有当电子从晶体中的缺陷散射时，才会出现电阻。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，电子在杂质之间呈之字形移动。杂质越多，平均自由程越小，电导率越低。在数学上，这一现象可用基本薛定谔方程来描述。事实证明，建立一个与实验数据相一致的、数学上正确的理论来解释这种“由于高水平杂质引起的局域电子”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。安德森的发现以他的名字命名，是为他赢得1977年诺贝尔奖的引文之一。数学家们从西奈的开创性工作开始认识到，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问题。据了解，这种现象不仅表现在随机杂质的存在上，而且还表现在其他不同类型的结构上，如准周期结构。后者导致了准周期薛定谔方程的谱分析。分析依赖于一些经典的数学基础领域，如复变量、傅里叶变换、动力系统。准周期薛定谔方程理论是由普林斯顿大学、芝加哥大学高级研究所、加州理工大学、欧文大学、巴黎大学、苏黎世理工大学、巴西、日本和以色列的数学家在过去的40年里广泛发展起来的。尽管如此，最困难的问题，特别是从波动解到Anderson局部化解的相变问题仍然没有得到解决。另一方面，人们在过去十年中了解到，这一理论已经发展到足以建立各种应用的地步。其中一个应用是研究具有准周期初值的所谓完全可积的非线性微分方程组。这些方程，就像Korteweg-de-Vries方程，Todda晶格一样，同样来自一些基本现象的物理学。60年代后期，数学家们发现了这些非线性方程与线性微分方程谱理论之间的联系，如斯克罗丁格方程。这使得前者可以通过所谓的松散对方法与各种初始数据相结合。然而，对于准周期初始数据，即由几个不同周期的周期函数组成的数据，还没有发展出理论。该提案的主要目标是为这类初始数据制定综合方法。拟议的研究计划的一部分也针对最初的安德森模型。这一部分的主要目的是研究Anderson模型在条形区域中的局部化长度。这个问题现在是世界各地一些顶尖专家关注的中心。

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.