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

Abstract

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.

On some configurations of oppositely charged trapped vortices in the plane

“…We consider the different layers of approximation, starting with Equation (2), which is the one also tackled via our algebraic techniques. At that level, as is established in [8] (and also found herein) the positions of the vortices are cube roots of unity for both the positive and negative charges, displaced by π/3 with respect to each other. The radius of the solutions, as shown also in Table 3 (for the complex (3,3) roots) is √ 2/2 ≈ 0.707.…”

“…There is a history of connections between the theory of different types of polynomials and the study of vortices in fluids [35,6,5]. Recent years have seen an attempt to extend such considerations to the more complex (in that it bears an external trapping potential) setting of BECs, including extensions of relevant multi-vortex configurations [7] and associated polynomial generating function techniques [8]. It is along these lines of associating the equations for the vortex positions (and their conjugates) with a system of polynomial equations and using symbolic algebraic techniques to tackle the latter that we proceed in the present study.…”

Existence, Stability, and Symmetry of Relative Equilibria with a Dominant Vortex

“…Moreover, we are currently exploring the application of these methods to Bose-Einstein Condensate point vortex models, see e.g. [5] in which equilibria were described as roots of generating polynomials.…”

Computing stationary solutions of the two-dimensional Gross–Pitaevskii equation with deflated continuation

“…3(d). Explicit algebraic conditions for such states have been obtained via a generating function approach in the large-density Thomas-Fermi limit [42], as have the rectilinear vortex states of Fig. 1.…”