Other Trapping Potentials

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


In this chapter, we are interested in the minimizers of the energy
$$ E_\varepsilon (u) = \int_\mathcal{D} {\left\{ {\frac{1} {2}|\nabla u|^2 - \Omega r^ \bot \cdot (iu,\nabla u) + \frac{1} {{4_\varepsilon ^2 }}\left( {|u|^2 - \rho {\rm T}F(r)} \right)^2 } \right\} dxdy,} $$
for varoius function ρTF(r). As before, 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 D = {ρTF > 0} and ρTF(r) describes respectively a nonradial harmonic confinement and a quartic trapping potential, that is, the model case are
$$ \rho _{TF} (r) = \rho _0 - x^2 - \alpha ^2 y^2 with \alpha \ne 1 and \rho _0 s.t. \int_D {\rho _{TF} } = 1 $$
$$ (b - 1)r^2 $$
In case (4.3), for certain values of b and k, the domain D becomes an annulus, and this changes the pattern of vortices.


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© Birkhäuser Boston 2006

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