Vortices in Bose—Einstein Condensates

Volume 67 of the series Progress in Nonlinear Differential Equations and Their Applications pp 123-156

Three-Dimensional Rotating Condensate

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In this chapter, we are interested in a three-dimensional rotating condensate, in a setting similar to that of the experiments. In particular, we want to justify the observations of the bent vortices. Thus we want to study the shape of vortices in minimizers of the following energy:
$$ E_\varepsilon (u) = \int_\mathcal{D} {\left\{ {\frac{1} {2}|\nabla u|^2 - \Omega r^ \bot \cdot (iu,\nabla u) + \frac{1} {{4_\varepsilon ^2 }}\left( {|u|^2 - \rho {\rm T}F(r)} \right)^2 } \right\} dxdydz,} $$
Where r=(x, y, z), Ωε is parallel to the z axis, ρ0 ⊂x22y22z2. D is the ellipsoid {ρTF > 0}={x22ty22z2 < ρ0}, and ρ0 is determined by
$$ (r) $$
Which yields ρ05/2= 15αβ/8π. If β is small, as in the experiments, this gives rise to an elongated domain D along the z direction.