C. Garcia-Azpeitia and D.E. Pelinovsky

Bifurcations of multi-vortex con gurations in rotating Bose-Einstein condensates


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 m0 has m0(m0 + 1)/2 pairs of negative eigenvalues. When the frequency is increased, 1 + m0(m0 - 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 con gurations that can be found in the bifurcating families are the asymmetric vortex (m0 = 1), the asymmetric vortex pair (m0 = 2), and the vortex polygons (m0 > 2).

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