Scotty Keith, College of Charleston


Stability of Vortex Filaments in the Localized Induction Approximation


Abstract: In its simplest form, the self-induced dynamics of a vortex filament in a perfect fluid is governed by the Vortex Filament Equation (VFE), a nonlinear partial differential equation that is related to the cubic, focussing Nonlinear Schroedinger (NLS) equation via a change of variables known as the Hasimoto map.
The NLS equation is a fundamental example of soliton equation, with solitary wave solutions constructed using inverse scattering techniques. At the heart of the inverse scattering method for the NLS equation is the AKNS system: an eigenvalue problem and an evolution equation for a vector-valued eigenfunction, whose solvability condition is the NLS equation itself.
The squared eigenfunctions of the AKNS system play a central role in linear stability studies of solutions of the NLS equation, as they provide a large set of solutions of the linearization of the NLS equation about a given solution. Using the squared eigenfunctions of the AKNS system and the relation between the VFE and NLS equations, we construct solutions of the linearized VFE equation and relate the stability properties of vortex filaments to those of the associated NLS potentials. We illustrate this approach in the case of multiply-covered circles, corresponding to plane wave potentials: the simplest spatially independent solutions of the NLS equation.

Advisors: Annalisa Calini, Stephane Lafortune (College of Charleston)