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

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Its 9300638 The research supported by this award falls into three related areas in the mathematical theory of nonlinear integrable equations, that is equations which admit a Lax-pair representation. The work derives from the first use of inverse scattering methods in solving nonlinear equations and has evolved into an important branch of mathematical physics. The analytic aspects of inverse scattering include a wide range of problems arising in the study of the solutions of integrable equations. For example, a large class of explicit quasiperiodical finite-gap solutions, as well as the comprehensive description of the long-time behavior of the solution of the Cauchy problem for an integrable system, can be obtained by inverse scattering. This project focuses on the application of the Riemann-Hilbert problem technique to the asymptotic analysis of the correlation functions of the quantum exactly solvable models. Continuing work will also be done on the development of the isomonodromy technique for the investigation of the Painleve-type ordinary differential equations and their application to the matrix model of 2D quantum gravity. A third line of investigation will be concerned with boundary value problems of integrable equations within the framework of the finite-gap integration and Riemann-Hilbert problem approach. Differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. The work represented in this project has close relationships with theoretical physics as well as the traditional synergisms with studies of wave phenomena ***