New Directions in the Asymptotics of Nonlinear Waves

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Nonlinear wave equations give a mathematical description for many phenomena in optics, fluid dynamics, and a variety of other physical systems. Such a description is instrumental for predicting behavior and designing engineering devices, such as optical switches for information transmission. Even when it is not possible to completely determine the solutions to such equations, the system can often be understood satisfactorily by considering an appropriate approximation, such as long time or small dispersion. A rich and maturing asymptotic theory has been developed for the somewhat idealized integrable wave equations. However, in order to provide accurate predictions for more realistic physical systems, it is necessary to extend the current models to account for issues including coupling or interference, higher-order corrections, and boundaries. By taking advantage of recent mathematical advances, this project will improve available methods to better model these three effects. Specific applications of the models studied include the development of all-optical switches, Raman scattering, flux propagation in superconducting Josephson junctions, and hydrodynamic turbulence. The research aims to enhance the physical applicability of current models of nonlinear wave propagation through the following three projects: (1) Extension of the small-dispersion theory to multicomponent systems, including the three-wave resonant interaction equations, to develop better mathematical models of coupled systems. (2) Establishing results on the long-time behavior and onset of instabilities in near-integrable equations, such as Hamiltonian perturbations of the sine-Gordon equation, in order to better incorporate higher-order physical effects that may not be negligible. (3) Extending the unified transform method to understand small-dispersion behavior on finite or semi-finite domains. Analysis of the defocusing nonlinear Schrodinger, massive Thirring, and related equations will improve models where boundary effects are important.