Generating Functions, Polynomials and Vortices with Alternating Signs in Bose-Einstein Condensates

TL;DR

This paper develops generating functions for vortices with alternating signs in Bose-Einstein condensates, incorporating vortex precession and extending classical vortex equilibrium solutions to more complex configurations.

Contribution

It introduces modified generating functions satisfying a Tkachenko differential equation that models vortex precession and extends vortex equilibrium solutions to larger and more complex arrangements.

Findings

Constructed generating functions for alternating sign vortices.

Extended vortex equilibrium solutions to larger vortex numbers.

Derived conditions for vortex configurations and generalized equations in 2D.

Abstract

In this work, we construct suitable generating functions for vortices of alternating signs in the realm of Bose-Einstein condensates. In addition to the vortex-vortex interaction included in earlier fluid dynamics constructions of such functions, the vortices here precess around the center of the trap. This results in the generating functions of the vortices of positive charge and of negative charge satisfying a modified, so-called, Tkachenko differential equation. From that equation, we reconstruct collinear few-vortex equilibria obtained in earlier work, as well as extend them to larger numbers of vortices. Moreover, particular moment conditions can be derived e.g. about the sum of the squared locations of the vortices for arbitrary vortex numbers. Furthermore, the relevant differential equation can be generalized appropriately in the two-dimensional complex plane and allows the construction e.g. of polygonal vortex ring and multi-ring configurations, as well as ones with rings surrounding a vortex at the center that are again connected to earlier bibliography.