The principal objective of the project is the study of dynamics of vortex filaments (slender filaments where vorticity is concentrated). The research is focused on the vortex filament equation (VFE) that describes the motion of the vortex filament, assuming that the motion of a vortex filament in an incompressible, inviscid fluid is due to its own induction. The VFE is related to the nonlinear Schroedinger equation via a Hasimoto map. The project will take advantage of this relation to investigate stability of special solutions (Hasimoto solitons) for VFE, and to extend the analysis to the models that incorporate physical effects initially neglected by the idealized VFE model.
Hasimoto solitons take the form of localized loops traveling along the vortex filament. Solitary waves of vortices in turbulent fluids have been observed both in laboratory and numerical experiments. While the most commonly encountered phenomena involving vortex filaments are waterspouts and tornadoes, the study of vortex filaments has fascinating applications to other aspects of science such as in superfluidity and superconductivity. In specific applications, Hasimoto solitons occur, for example, in the context of superconductivity, of tornadoes, of particle transport in a fluid by solitons, and of laser-matter interaction. Although they have been studied abundantly in the literature not much is known about their stability. The main purpose of this project is studying solutions' stability properties, which is of fundamental importance, since only stable solutions can be realized and observed experimentally. The project involves a broad participation of undergraduate and graduate students. In the process of doing research the students will learn sophisticated numerical and mathematical methods needed to carry this research through.
