Inverse Scattering Theory

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9706644 Zhou The project includes the study of Sobolev space bijectivity of inverse scattering transforms, long time asymptotics for integrable systems related to Riemann-Hilbert problems on Riemann surfaces, the zero dispersion problem for focusing NLS, infinite dimensional perturbation theory for integrable systems, certain integrable statistical models--random matrices--orthogonal polynomials, and other problems related to inverse scattering theory. The inverse scattering theory provides a method for analyzing many nonlinear phenomena of mathematical and scientific importance described by so-called integrable systems. Many of these systems originated from statistical physics and nonlinear optics. For instance, the focusing NLS equation is a model for soliton transmission in optical fibres, which may be adopted by industry for the new generation of information transmission systems. This project aims at further developments of certain mathematical tools originated by the PI and his collaborators. These tools allow one to obtain very detailed information on the solutions of integrable systems and a deeper understanding of the related nonlinear phenomena.