By Annalisa Calini, College of Charleston

The Vortex Filament Equation or VFE, a model for the self-induced motion of a vortex filament in an ideal fluid, is a simple but important example of integrable geometric evolution equation for space curves. Its connection with the cubing focusing Nonlinear Schrödinger equation through the well-known Hasimoto transformation allows the use of many of the tools of soliton theory to construct and investigate finite-gap and soliton solutions.  I will discuss linear and nonlinear stability properties of some of these solutions, including filaments in the shape of torus knots and solitons on vortex filaments. This is joint work with with Tom Ivey and Stephane Lafortune.