Three-Dimensional Rotating Condensate
Progress in nonlinear differential equations and their applications · 2006
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: 1 $$ E_\varepsilon (u) = \int_\mathcal{D} {\left\{ {\frac{1} {2}| abla u|^2 - \Omega r^ \bot \cdot (iu, abla u) + \frac{1} {{4_\varepsilon ^2 }}\left( {|u|^2 - \rho {\rm T}F(r)} \right)^2 } \right\} dxdydz,} $$ Where r=(x, y, z), Ωε is parallel to the z axis, ρ0 ⊂x2+α2y2+β2z2. D is the ellipsoid {ρTF > 0}={x2+α2ty2+β2z2 < ρ0}, and ρ0 is determined by 1 $$ (r) $$ 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.