January 20 Math Colloquium
by Stephane Lafortune, College of Charleston
Stability of waves propagating on vortex filaments
By the term vortex filament we mean a mass of whirling fluid or air (e.g. a whirlpool or whirlwind) concentrated along a slender tube. The most spectacular and well-known example of a vortex filament is a tornado. A waterspout and dust devil are other examples. In its simplest form, the self-induced dynamics of a vortex filament in a perfect fluid is governed by the Vortex Filament Equation (VFE). What makes the VFE particularly interesting to study is its connection to integrability, more precisely to the nonlinear Schrodinger equation (NLS). Indeed, a relation between the solutions of the VFE and the NLS is known as the Hasimoto map. An immediate consequence of this connection with integrability is the existence of large classes of special solutions of the VFE such as solitons, which take the form of localized loops traveling along the filament, and their periodic counterparts, which take the form of circular vortex rings and knotted vortex filaments.
Even though VFE solutions appear in several applications, it seems that their stability has deserved little attention. A stability analysis is crucial since it enables one to distinguish the solutions that can be seen experimentally from the ones that cannot.
Naively, one would think that due to the connection between the VFE and NLS, performing a stability analysis of the solutions of the VFE should be doable in a straightforward manner. However, the nature of this connection is such that it is not a priori clear of how to extract information about the stability properties of the solutions of the VFE from all what is known about the stability of NLS solutions.
During this talk, I will present a formalism to study the stability of solutions of the VFE. A particular attention will be given to the periodic case. This work was performed with Annalisa Calini and the undergraduate student Scotty Keith. This work also benefited tremendously from numerous discussions with Thomas Ivey.
