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D.E. Pelinovsky and P.G. Kevrekidis

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Bifurcations of asymmetric vortices in symmetric harmonic traps

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Applied Mathematics Research eXpress, Vol. 2013, No. 1, pp. 127–164 (2013)

**Abstract:**

We show that the rotating symmetric vortices located at the center of a two-dimensional
harmonic potential undertake a pitchfork bifurcation with radial symmetry. This bifurcation
leads to the family of vortices, which precess constantly along an orbit enclosing the center of
symmetry. The radius of the orbit depends on the precessional frequency and the chemical
potential. We show that both symmetric and asymmetric vortices are spectrally and orbitally
stable with respect to small time-dependent perturbations for sufficiently large precessional
frequencies. At the same time, the symmetric vortex is a local minimizer of energy in the
parameter region where the asymmetric vortex exists, whereas the latter corresponds to a saddle
point of energy.

**Keywords:**

Gross-Pitaevskii equation, harmonic potential, local bifurcation,
energy functionals, vortex configurations, existence and stability, Lyapunov-Schmidt reduction method.