Mathematical Sciences: Inverse Scattering Theory

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The mathematical research supported by this award continues work on the development of a nonlinear harmonic analysis on integrable equations in connection with inverse scattering theory. Central to the project is a steepest decent method for analyzing the asymptotics of oscillatory Riemann-Hilbert problems of the kind that arise in the theory of integrable nonlinear equations. To date the method has been applied to the modified Korteweg-deVries equation and the defocusing nonlinear Schrodinger equation. The steepest descent method mentioned above gives long time asymptotics of solutions to Cauchy problems integrable by the inverse scattering method. A great deal of work has been done along the same lines, but much is formal and very little is rigorous. This project will first seek to analyze the zero dispersion limit problems for Korteweg-deVries equations, that is, to study directly the corresponding oscillatory Riemann- Hilbert problem. It is expected that the new method will not require any spectral assumptions, and that, once worked out, will apply to a broad class of problems. A second line of research will consider Hamiltonians arising as transverse Ising models at the critical transverse magnetic field. The corresponding autocorrelation function can be computed from the solution of the associated Riemann-Hilbert problem. Preliminary calculations give results different from previous formal results of others. Work will be done to justify these discoveries and to continue work on a class of related integrable statistical models to compute the large time behavior of the autocorrelation function.