Vortices in Bose—Einstein Condensates pp 123-156 | Cite as

# Three-Dimensional Rotating Condensate

Chapter

## 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:
Where r=(
Which yields ρ0

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

(1)

*x*,*y*,*z*), Ω_{ε}is parallel to the*z*axis, ρ0 ⊂x^{2}+α^{2}y^{2}+β^{2}z^{2}.*D*is the ellipsoid {ρTF > 0}={x^{2}+α^{2}ty^{2}+β^{2}z^{2}< ρ0}, and ρ0 is determined by$$
(r)
$$

(1)

^{5/2}= 15αβ/8π. If β is small, as in the experiments, this gives rise to an elongated domain*D*along the*z*direction.## Keywords

Convergence Result Critical Velocity Lagrange Equation Vortex Core Vortex Line
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Birkhäuser Boston 2006