Solutions of Gross-Pitaevskii equations beyond the hydrodynamic approximation: Application to the vortex problem

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Abstract
We develop a multiscale technique to describe excitations of a Bose-Einstein condensate, whose characteristic scales are comparable to the healing length, thus going beyond the conventional hydrodynamical approximation. As an application of the theory, we derive an approximate explicit vortex and other solutions. The dynamical stability of the vortex is discussed on the basis of the mathematical framework developed here, the result being that its stability is granted at least up to times on the order of seconds, which is the condensate lifetime. Our analytical results are confirmed by numerical simulations.
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