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We consider vortex dynamics in the context of Bose-Einstein Condensates (BEC)with a rotating trap, with or without anisotropy. Starting with theGross-Pitaevskii (GP) partial differential equation (PDE), we derive a novelreduced system of ordinary differential equations (ODEs) that describes stableconfigurations of multiple co-rotating vortices (vortex crystals). Thisdescription is found to be quite accurate quantitatively especially in the caseof multiple vortices. In the limit of many vortices, BECs are known to formvortex crystal structures, whereby vortices tend to arrange themselves in ahexagonal-like spatial configuration. Using our asymptotic reduction, we derivethe effective vortex crystal density and its radius. We also obtain anasymptotic estimate for the maximum number of vortices as a function ofrotation rate. We extend considerations to the anisotropic trap case,confirming that a pair of vortices lying on the long (short) axis is linearlystable (unstable), corroborating the ODE reduction results with full PDEsimulations. We then further investigate the many-vortex limit in the case ofstrong anisotropic potential. In this limit, the vortices tend to alignthemselves along the long axis, and we compute the effective one-dimensionalvortex density, as well as the maximum admissible number of vortices. Detailednumerical simulations of the GP equation are used to confirm our analyticalpredictions.

Multi-vortex crystal lattices in Bose-Einstein Condensates with a rotating trap