RUI: Topology and Stability of Integrable Vortex Filament Motion

College Of Charleston, Charleston SC

Investigators

Abstract

We study periodic solutions of the Vortex Filament Equation (VFE), a nonlinear partial differential equation that models the self-induced dynamics of a vortex line in an ideal fluid. The VFE is closely related to the cubic focusing Nonlinear Schroedinger (NLS) equation, a canonical soliton equation arising as model of nonlinear wave propagation in a variety of situations (from water waves in deep water to nonlinear optical media). Our research has several goals including: a deeper understanding of the topological properties of closed vortex filaments associated to the class of finite-gap solutions of the NLS equation (the periodic analogues of solitons); relating the linear stability of periodic NLS solutions to the linear stability of the corresponding vortex filaments; and studying perturbations of the vortex filament flow that preserve knot energy functionals. Our work concerns a canonical model of the dynamics of filamentary structures in fluids (such as smoke rings traveling through the air, or loops of intense vorticity generated by a strong underwater current). Despite its simplicity, this model has a rich class of solutions that realize many interesting topological features: we find knotted vortex loops of increasing complexity, and observe topological changes such as strand crossing and self-intersections. Many areas of mathematics are brought together to address questions such as the topological properties of knotted filamentary structures, their stability, and their energetics. This work also continues to integrate research and education, by providing undergraduate students with research experiences aimed at better preparing them for graduate programs, and by supporting graduate students involved in advanced research at a primarily undergraduate institution.