In this 2-part talk, we will discuss the construction and properties of closed solutions of the Vortex Filament Equation (VFE), (also known as Localized Induction Approximation), the simplest model of the self-induced motion of vortex filaments evolution in an ideal flow.

The VFE is related to the cubic focusing Nonlinear Schrödinger (NLS) equation by the Hasimoto transformation, and we exploit this connection to construct finite-gap solutions of the VFE. These originate from the classical construction of finite-gap quasiperiodic NLS solutions based on the Baker-Akhiezer eigenfunction for a given set of algebro-geometric data on a hyperelliptic Riemann surface, but this data must be carefully chosen to achieve closure on the VFE side. 
The key steps and main ideas will be illustrated as we build a family of finite-gap solutions of increasing complexity, consisting of torus knots and their iterated cables which are close to multiply-covered vortex rings, with each iteration step associated with incrementing the genus of the Riemann surface. The knot type of any of these solutions is recoverable from the Floquet spectrum of the underlying NLS potential and is therefore preserved during the VFE evolution, making this family a natural candidate for building filaments of more complex topology.

If time allows, we will discuss how the Baker-Akhiezer eigenfunction also encodes the linear stability properties of the associated vortex filaments, and how these results generalize to solutions of higher-order flows in the VFE hierarchy.

Authors: Annalisa Calini and Thomas Ivey