INTEGRABLE SYSTEMS OF PDE WITH QUASI-PERIODIC INITIAL DATA

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The study of the conductance of electrons lies at the heart of condensed-matter physics. The classical theory of electronic conductivity was built on the idea of free electrons scattered by positive ions. A key concept was the mean free path, the average length an electron travels before it collides with an ion.******According to classical theory, electronic conductivity should be directly proportional to the mean free path. 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 was described via the fundamental Schrodinger Equation. It turned out, however, that to develop a mathematically correct theory of "localization" of electrons, caused by a high level of impurities, posed a deep problem -- one that renowned physicist Phil Anderson successfully confronted in the late 1950s. Anderson's discovery, which carries his name, helped earn him the 1977 Nobel Prize. Mathematicians realized, starting with ground-breaking works by Yakov G. Sinai, that the theory of Anderson Localization relates 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 the presence of random impurities, but also for different other types of structures such as quasi-periodic ones. The latter leads to an analysis of spectrum of the quasi-periodic Schrodinger Equation. The analysis relies on a number of classical fundamental domains of mathematics such as complex variables, Fourier transform, and Dynamical systems. The theory of quasi-periodic Schrodinger Equations has been extensively developed in the last forty years by mathematicians working at Princeton University, the Institute for Advanced Study, the University of Chicago, Caltech, Irvine University, several Paris universities, ETH Zurich, and universities in Brazil, Japan, and Israel.******One of applications of the theory addresses the so-called completely integrable non-linear differential equations. In the late 1960s, mathematicians discovered the connection between these non-linear equations and the spectral theory of linear differential equations. The discovery allowed mathematicians to integrate the former with various initial data. However, no theory was developed for quasi-periodic initial data, i.e. 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. *** **