Vortices and Bound States

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This research project is concerned with the mathematical modeling and analysis of certain types of behaviors in superconductivity and nonlinear optics. Many problems arising in physics can be modeled by nonlinear partial differential equations, and some of these differential equations generate solutions with interesting structures, such as pulse-like solutions that model light traveling through fiber optics or vortex solutions that arise in fluids and condensed matter physics. This research aims to understand some of these structures. In the first case, the behavior of large numbers of vortices in superconducting materials (a typical feature of certain classes of superconductors) will be studied. In the second case, how vortices are generated in superconductors will be examined. The third project will consider numerical algorithms that are used to compute pulse-like solutions that arise in nonlinear optics.This research will focus on three problems arising in applied mathematics. The first project will study the mean-field behavior of asymptotically large numbers of vortices arising in superconductivity. The analysis requires handling asymptotically large numbers of vortices, each of which has asymptotically large energy. To do so, refined estimates will be developed for tracking concentrations of energy. The project on the understanding of vortex nucleation in the time-dependent setting should help to understand the transition to the mixed-vortex state in Type-II superconductors. Finally, the project on iterative methods for computing bound states will study why certain methods are unconditionally convergent, which should lead to better algorithm design.