Other Trapping Potentials

Abstract

In this chapter, we are interested in the minimizers of the energy

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

(1)

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

$$ \rho _{TF} (r) = \rho _0 - x^2 - \alpha ^2 y^2 with \alpha \ne 1 and \rho _0 s.t. \int_D {\rho _{TF} } = 1 $$

(1)

$$ (b - 1)r^2 $$

(1)

In case (4.3), for certain values of b and k, the domain D becomes an annulus, and this changes the pattern of vortices.