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In the present work, we motivate and explore the dynamics of a dissipativevariant of the nonlinear Schr{\"o}dinger equation under the impact of externalrotation. As in the well established Hamiltonian case, the rotation gives riseto the formation of vortices. We show, however, that the most unstable modeleading to this instability scales with an appropriate power of the chemicalpotential $\mu$ of the system, increasing proportionally to $\mu^{2/3}$. Theprecise form of the relevant formula, obtained through our asymptotic analysis,provides the most unstable mode as a function of the atomic density and thetrap strength. We show how these unstable modes typically nucleate a largenumber of vortices in the periphery of the atomic cloud. However, through apattern selection mechanism, prompted by symmetry-breaking, only few isolatedvortices are pulled in sequentially from the periphery towards the bulk of thecloud resulting in highly symmetric stable vortex configurations with far fewervortices than the original unstable mode. These results may be of relevance tothe experimentally tractable realm of finite temperature atomic condensates.

Vortex Nucleation in a Dissipative Variant of the Nonlinear Schr\"odinger Equation under Rotation