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C. Garcia-Azpeitia and D.E. Pelinovsky

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Bifurcations of multi-vortex congurations in rotating Bose-Einstein condensates

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Milan Journal of Mathematics 85 (2017), 331-367

**Abstract:**

We analyze global bifurcations along the family of radially symmetric vortices in the
Gross-Pitaevskii equation with a symmetric harmonic potential and a chemical potential
under the steady rotation with some frequency
. The families are constructed in the small-amplitude
limit when the chemical potential is close to an eigenvalue of the Schrodinger
operator for a quantum harmonic oscillator. We show that for small frequencies
, the Hessian operator
at the radially symmetric vortex of charge m_{0} has m_{0}(m_{0} + 1)/2 pairs of negative
eigenvalues. When the frequency
is increased, 1 + m_{0}(m_{0} - 1)/2 global bifurcations
happen. Each bifurcation results in the disappearance of a pair of negative eigenvalues in
the Hessian operator at the radially symmetric vortex. The distributions of vortices in the
bifurcating families are analyzed by using symmetries of the Gross-Pitaevskii equation and
the zeros of Hermite-Gauss eigenfunctions. The vortex congurations that can be found in
the bifurcating families are the asymmetric vortex (m_{0} = 1), the asymmetric vortex pair
(m_{0} = 2), and the vortex polygons (m_{0} > 2).

**Keywords:**

Gross-Pitaevskii equation, rotating vortices, harmonic potentials,
Lyapunov-Schmidt reductions, bifurcations and symmetries.