Integrable Systems and Random Matrices

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Abstract The proposal describes three sets of problems that the PI plans to investigate during the funding period. First, the PI will continue her study of the Ablowitz-Ladik (AL) equation, and address questions related to generalized orbits and multi-Hamiltonian structures for the finite system, as well as their connection to the analogous Toda lattice problems. She is also interested in finding the solution of AL with periodic boundary conditions, and in the study of critical phenomena for the NLS equation, viewed as a continuum limit of the AL equation. Second, building on her work on matrix models for general beta-ensembles, the PI plans to investigate the asymptotic properties of these models via the approach of Adler and van Moerbeke. This describes various eigenvalue statistics of the model as solutions of completely integrable systems; so far, only the cases with beta equal to 1, 2 or 4 have been studied, and general beta-ensembles have never been used in this context. Finally, the PI plans to investigate, jointly with P. Deift, the question of long-time asymptotics for solutions of the water-wave problem with rough data, in the small amplitude/long wavelength regime. In their investigation, they will treat the problem as a perturbation of a completely integrable PDE, the KdV equation, and use the associated scattering transform and Riemann-Hilbert techniques to control the perturbation. In particular, a first step in this project is the rigorous treatment of the long-time asymptotics for the KdV equation with Sobolev initial data. The research described in this proposal concerns classical problems in two of the most active fields in mathematics, random matrix theory and integrable systems. One of the most fascinating scientific developments over the last fifty years has been the discovery that a wide variety of mathematical and physical phenomena are modeled by the eigenvalues of a random matrix. In particular, random matrix theory describes the scattering of neutrons off large nuclei, the statistics of the zeros of the Riemann zeta function on the critical line in the complex plane, as well as problems in the "real world", such as the bus scheduling in the city of Cavalierness in Mexico, or distances between cars on the freeway. The goal of the PI's proposed research is to describe the asymptotic properties of certain matrix ensembles which model some of the phenomena described above. Another part of the proposal is concerned with studying the properties of two remarkable evolution equations: The first is the Ablowitz-Ladik (AL) equation, which is a discrete version of the well-known nonlinear Schroedinger equation (NLS). Beyond their theoretical interest, both of the aforementioned equations have numerous scientific applications, one of the most important of which is in optics. The PI approaches the study of the AL equation using the methods from the theories of orthogonal polynomials and completely integrable systems. Finally, the PI proposes to study the water wave equation in a regime which can be used to model tsunamis, by further developing the method of nonlinear stationary phase used in the treatment of Riemann-Hilbert problems.