Bifurcations of Multi-Vortex Configurations in Rotating Bose–Einstein Condensates

Abstract

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 µ under the steady rotation with frequency Ω. The families are constructed in the smallamplitude limit when the chemical potential µ is close to an eigenvalue of the Schrödinger operator for a quantum harmonic oscillator. We show that for Ω near 0, the Hessian operator at the radially symmetric vortex of charge m0 ∈ N has m0(m0 +1)/2 pairs of negative eigenvalues. When the parameter Ω 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 configurations 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).

On the characterization of vortex configurations in the steady rotating Bose–Einstein condensates

“…The critical point (a, 0) with a > 0 and ω > ω 0 (ε) related by equation (2.4) is a saddle point of E 1 with one negative and one zero eigenvalues. This conclusion agrees with the full bifurcation analysis of the GP equation (1.1) given in [11,25].…”

“…The critical point (a, 0) with a > 0 and ω > ω 0 (ε) related by equation (7) is a saddle point of E 1 with one negative and one zero eigenvalues. This conclusion agrees with the full bifurcation analysis of the GP equation (1) given in [12,28]. The zero eigenvalue for the asymmetric vortex with a > 0 is related to the rotational invariance of the vortex configuration, which can be placed at any (ξ 0 , η 0 ) = a(cos α, sin α) with arbitrary α ∈ [0, 2π].…”

“…Bifurcations of radially symmetric vortices with charge m ∈ N and dipole solutions were studied in [24] with the help of the equivariant degree theory. Bifurcations of multi-vortex configurations in the parameter continuation with respect to the rotation frequency ω were considered in [25]. Existence and stability of stationary states were analysed in [26] with the resonant normal forms.…”

“…In [25], it was shown that the asymmetric pair of two vortices of charge one bifurcates from the symmetric vortex of charge two and that this vortex pair shares the instability of the symmetric vortex of charge two in the small-amplitude limit. This instability is due to the fact that the vortex pair is a saddle point of total energy above the bifurcation threshold in the smallamplitude limit.…”

Stability and instability properties of rotating Bose–Einstein condensates

“…In summary, these results show that G Ω , in general, will be a more complicated set than G 0 . Moreover, G Ω should also be distinguished from the set of rotationally symmetric vortex solutions studied in, e.g., [17].…”