A Tale of Two Distributions: From Few To Many Vortices In Quasi-Two-Dimensional Bose-Einstein Condensates

Motivated by the recent successes of particle models in capturing the precession and interactions of vortex structures in quasi-two-dimensional Bose-Einstein condensates, we revisit the relevant systems of ordinary differential equations. We consider the number of vortices $N$ as a parameter and explore the prototypical configurations ("ground states") that arise in the case of few or many vortices. In the case of few vortices, we modify the classical result of Havelock [Phil. Mag. ${\bf 11}$, 617 (1931)] illustrating that vortex polygons in the form of a ring are unstable for $N \geq7$. Additionally, we reconcile this modification with the recent identification of symmetry breaking bifurcations for the cases of $N=2,\dots,5$. We also briefly discuss the case of a ring of vortices surrounding a central vortex (so-called $N+1$ configuration). We finally examine the opposite limit of large $N$ and illustrate how a coarse-graining, continuum approach enables the accurate identification of the radial distribution of vortices in that limit.Comment: 15 pages, 2 figure

An Adiabatic Invariant Approach to Transverse Instability: Landau Dynamics of Soliton Filaments

Assume a lower-dimensional solitonic structure embedded in a higher dimensional space, e.g., a 1D dark soliton embedded in 2D space, a ring dark soliton in 2D space, a spherical shell soliton in 3D space etc. By extending the Landau dynamics approach [Phys. Rev. Lett. {\bf 93}, 240403 (2004)], we show that it is possible to capture the transverse dynamical modes (the "Kelvin modes") of the undulation of this "soliton filament" within the higher dimensional space. These are the transverse stability/instability modes and are the ones potentially responsible for the breakup of the soliton into structures such as vortices, vortex rings etc. We present the theory and case examples in 2D and 3D, corroborating the results by numerical stability and dynamical computations.Comment: 5 pages, 3 figure