Mathematical Sciences: Some Analytical Aspects of the Theory of the Integrable Systems

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DMS-9501559 PI: Its Its will investigate several topics in mathematical physics. These include Riemann-Hilbert approach to the asymptotic analysis of the correlation functions of quantum exactly solvable models, the isomonodromy method in the theory of Painleve equations and its application to the inverse monodromy problems, and Riemann-Hilbert scheme of the asymptotic analysis of the matrix models and orthogonal polynomials. The research involves the theory of nonlinear integrable equations, which was first used to integrate the Korteweg-de Vries equation via the so called inverse scattering method. During the last twenty years this method has been developed into an important branch of mathematical physics and is one of the principal sources of new analytical and algebraic ideas of many branches of modern mathematics, including quantum group theory and the theory of special functions.