Rogue Waves on a Vortex Filament

The presence of a standing soliton on a vortex filament has been demonstrated theoretically by Hasimoto [1] for classical fluids, by Van Gorder [2] for quantum fluids as well as experimentally by Hopfinger and Browand [3] for turbulent flow in a rotating container. This was done by mapping solutions of the local induction approximation (LIA) governing the motion of a thin vortex filament to those of nonlinear scalar PDEs such as the cubic nonlinear Schrödinger equation (NLS) and the complex Ginsburg-Landau equation (GLE) using the Hasimoto transformation and its recently developed quantum analogue. Unlike solitons that maintain a constant shape over time, rogue waves are another class of solutions to the NLS, that possess greater variability in their shape and can thus be used to study the propagation of temporary disturbances along vortex filaments. The purpose of this dissertation is therefore to study the formation and propagation of such rogue waves along vortex filaments in both classical and quantum fluids. This has not been done previously in the presence of quantum effects such as mutual friction and a coupled normal fluid flow. While rogue waves have recently been observed experimentally in water tanks [4], models such as the Peregrine soliton [5] and the more generalised Akhmediev breather wave [6] have also been investigated as solutions of the NLS in previous literature. By employing the Hasimoto transformation as done for solitons, LIA solutions for the motion of both classical and quantum vortex filament solutions are obtained for complex rogue wave models such as breathers. It is found that the solutions obtained do indeed demonstrate the transient behaviour that is typically associated with rogue waves, particularly as the vortex filament transitions from the classical to the quantum regime. The results obtained not only shed light on the emergence of highly localised asymptotically-decaying disturbances along vortex filaments, but also bear significant implications for the onset of quantum turbulence and chaotic motion in superfluids