Two-Dimensional Model for otating Condensate

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 67)


In this chapter, we want to study the shape of the minimizers u=uεH01D, 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 −r2D 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.


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© Birkhäuser Boston 2006

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