Vortices in Bose—Einstein Condensates

Volume 67 of the series Progress in Nonlinear Differential Equations and Their Applications pp 29-77

Two-Dimensional Model for otating Condensate

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In this chapter, we want to study the shape of the minimizers u=uε H01D, C of
$$ E_\varepsilon (u) = \int_\mathcal{D} {\left\{ {\frac{1} {2}|\nabla u|^2 - \Omega r^ \bot \cdot (iu,\nabla u) + \frac{1} {{4_\varepsilon ^2 }}\left( {|u|^2 - \rho {\rm T}F(r)} \right)^2 } \right\} dxdy,} $$
Where r = (x, y), r = (−y, x), (iu,∇u)= i(ū∇u - (u∈ū)/2, ε is a small parameter, and Ω is the given rotational velocity. We assume that ρTF(r)= ρ0 −r2 D is the disc of radius R0= √ρ0 in R2 (so that ρTF = 0 on ∂D, and ∫D ρTF = 1, which prescribes the value of ρ0. The issue is to determine the number and location of vortices according to the value of Ω.