Mathematical Sciences: Classical And Quantum Integrable Systems

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DMS-9501233 Sattinger The isospectral flows of a second order energy dependent Schrodinger operator will be investigated. The associated inverse scattering problem cannot be resolved by the Gel'fand-Levitan-Marchenko method; but instead must be treated by solving a Riemann-Hilbert problem on an appropriate Riemann surface. The flows themselves divide into two cases. The two distinct flows have global solutions only when the initial data is restricted to suitable invariant submanifolds. The asymptotic behavior of the solutions will be investigated using the recent methods developed by Deift and Zhou for analyzing long time asymptotics of solutions to Riemann-Hilbert problems. The nature of the break-down of the solutions for inapprmpriate data will be investigated. The Hamiltonian for the three wave interaction contains a cubic interaction term and a kinetic energy term that allows unbounded positive and negative energies, as in quantum electrodynamics. The model thus contains a number of interesting features, including creation and annihilation processes. The model serves as a paradigm for such processes in elementary particle physics. Modern techniques for analyzing nonlinear waves will be applied to models of physical interest. The first model consists of equations approximating waves in fluids. These models allow for singularities to develop, similar to waves breaking in the ocean. An attempt will be made to analyze the structure of the singularities. The second model consists of quantum versions of classical equations that arise in nonlinear optics. The system exhibits processes similar to those in elementary particle interactions, in which particles are created and destroyed.