Integrable Dynamics of Knotted Vortex Filaments

Abstract

The dynamics of vortex filaments has provided for almost a century one of the most interesting connections between differ ential geometry and soliton equations, and an example in which knotted curves arise as solutions of differential equations possessing an infinite family of symmetries and a remarkably rich geometrical structure. These lectures discuss several aspects of the integrable dynamics of closed vortex filaments in an Eulerian fluid, including its Hamiltonian formulation, the construction of a large class of special solutions, and the role of the Floquet spectrum in characterizing the geometric and topological properties of the evolving curves.