INTEGRABLE SYSTEMS OF PDE WITH QUASI-PERIODIC INITIAL DATA.

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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.