Kelly Epperson, College of Charleston


On a Model of Vortex Filament Motion: Closed Solutions and Knot Energies

Abstract: We conduct a numerical study of a completely integrable PDE (the Vortex Filament Equation) modeling the evolution of a vortex filament in a perfect fluid. Several of the conserved quantities of the Vortex Filament Equation can be interpreted as global geometric invariants (including total length, total squared curvature, and total torsion), and are used as diagnostic tools in the numerical experiments. For the numerical integration, a pseudo-spectral method with Runge-Kutta time discretization is used, and tested on several examples of closed knotted vortex filaments. In the second part of this work, we introduce two knot energy functionals (the well-known Mšbius energy, and the Beta Function of a knot), and dicuss how they can be used to detect changes in knot type, and to characterize the degree of complexity of a knot. We conclude with mentioning some open-ended questions relating the Beta Function of a knot to the conserved quantities of the Vortex Filament Equation.

Advisor: Annalisa Calini (CofC)