Quantum vortex dynamics under the tangent representation of the local induction approximation

Abstract

AbstractWe derive the local induction approximation (LIA) for a quantum vortex filament in the arclength coordinate frame where the tangent vector is the unknown function. The equation for the tangent vector to the filament is then converted to a potential form, which ends up being a type of nonlinear Schrödinger equation that governs the tangential LIA model (T-LIA). Such a formulation was previously derived by Umeki for the standard fluid model (in the absence of superfluid friction terms). While it is challenging to generalize many of the exact solutions found for the standard LIA to the quantum LIA model, we demonstrate that the T-LIA model facilitates this generalization nicely. Indeed, under the T-LIA model, we are able to construct a variety of solutions. The Hasimoto solution related to elastica is one of the fundamental solutions present for the standard fluid model; however, using the T-LIA model, we are able to demonstrate the existence of such a solution, thereby extending the Hasimoto solution to the superfluid case. In the zero-temperature limit, purely self-similar solutions are shown to exist for the T-LIA model. As the superfluid warms (so that the influence of the normal flow is no longer negligible), the analogue to the self-similar solution is a new class of solutions, which depend on the similarity variable as well as a time-dependent additive scaling. In other words, the self-similar structures gradually deform as the magnitude of the normal-fluid velocity increases, which makes complete physical sense. When dealing with small deviations from the central axis of alignment, we can describe such solutions analytically. There exists a family of helical vortex filaments in the presence of a normal fluid impinging on the vortex, in complete agreement with the previously studied results for the LIA model. Finally, a number of soliton solutions are shown to exist in different regimes of the T-LIA model.