Three-Dimensional Rotating Condensate

Abstract

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:

$$ E_\varepsilon (u) = \int_\mathcal{D} {\left\{ {\frac{1} {2}|\nabla u|^2 - \Omega r^ \bot \cdot (iu,\nabla u) + \frac{1} {{4_\varepsilon ^2 }}\left( {|u|^2 - \rho {\rm T}F(r)} \right)^2 } \right\} dxdydz,} $$

(1)

Where r=(x, y, z), Ω_ε is parallel to the z axis, ρ0 ⊂x²+α²y²+β²z². D is the ellipsoid {ρTF > 0}={x²+α²ty²+β²z² < ρ0}, and ρ0 is determined by

$$ (r) $$

(1)

Which yields ρ0^5/2= 15αβ/8π. If β is small, as in the experiments, this gives rise to an elongated domain D along the z direction.