Generating functions, polynomials and vortices with alternating signs in Bose–Einstein condensates

Abstract

In this work, we construct suitable generating functions for vortices of alternating signs in the realm of quasi-two-dimensional Bose–Einstein condensates in the large density (so-called Thomas–Fermi) limit, where the vortices can be treated as effective particles. 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 for the construction e.g. of polygonal vortex ring and multi-ring configurations, as well as ones with rings surrounding a central vortex that are again connected to earlier bibliography and associated numerical computations.