CAREER: Long-time asymptotics of completely integrable systems with connections to random matrices and partial differential equations

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

The research side of this project focuses on understanding the long-time asymptotics for a diverse set of models from analysis and mathematical physics. In particular, the principal investigator will concentrate on three projects: (1) the continued study of a discrete version of the defocusing nonlinear Schroedinger equation, in particular, its connections with orthogonal polynomials on the unit circle, Lie-Poisson algebras, and its continuum limits; (2) the study of properties (e.g., expected diagonalization times) of certain random matrix ensembles when considered as initial data for integrable flows such as the Toda lattice and the QR-algorithm; and (3) the study, using Riemann-Hilbert methods, of long-time asymptotics for integrable systems, namely, the Toda lattice and the Korteweg-deVries equation, and for certain nonintegrable perturbations of these systems. The research topics of this project fall in the wide areas of mathematical physics and partial differential equations, with a particular emphasis on questions related to such well-established mathematical areas as completely integrable systems, random matrices, and their applications to certain numerical algorithms. Partial differential equations emerge in a variety of physical contexts as mathematical models for the time-evolution of certain physical quantities. A particular class of such equations are the so-called completely integrable systems, which are characterized by the fact they satisfy a sufficiently large (in a sense which can be made precise) number of conservation laws. The theory of completely integrable systems has a long and distinguished history. Over the past thirty years, in particular, interest in the field has been fueled on the one hand by the fact that many of the equations known to be completely integrable are obtained as models of fluid dynamics, and on the other hand by the close connections that have been discovered with many fields of pure mathematics, such as Lie algebras or symplectic and Poisson geometry. One of the fundamental goals of the current project is to expand and deepen the understanding of these connections and to widen the class of models to which the techniques of completely integrable theory apply. Hence one aims to describe as fully as possible the solutions to these models and hopefully to gain thereby insights into real-life phenomena. An essential part of the proposal is its educational component, which is centered around the organization of an annual summer school for advanced graduate students and recent Ph.D.'s on various topics of current research interest at the juncture of analysis, partial differential equations, and numerical analysis. All talks in the summer school will be delivered on preassigned articles by the participants. Beyond its broad mathematical impact, the project will allow the principal investigator to expand other educational activities as well. She will support the work of graduate students through advanced courses, seminars, and student research workshops, and she will work with local chapters of the Association for Women In Mathematics (AWM) in the Chicago area to promote the advancement of women in mathematical careers.