In this continuing project we explore the implications of integrability for the Vortex Filament Equation, which is the simplest model of the self-induced motion of a slender vortex filament in an ideal fluid, and which is known to be equivalent to the focusing nonlinear Schrodinger equation (NLS). The goals of our research are to relate geometric and topological properties -- in particular, knot types -- of large classes of closed filaments to the Floquet spectra of the corresponding periodic NLS solutions, and to understand how the knot type of a filament can change as it evolves. The methods we employ include algebro-geometric constructions of multi-phase solutions, Backlund transformations, isoperiodic deformations, and perturbation theory.
Our work is part of a larger trend of making connections between differential equations, which usually model physical phenomena, and knot theory, which is traditionally a part of pure mathematics. Here, we study a simplified model of vortex motion, which possesses a rich class of knotted loops among its solutions, and whose mathematical structure is well understood, in order to develop the mathematical tools necessary to effectively address questions such as knot formation, stability of knotted structures, and knot classification.
