Other Trapping Potentials

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

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.

Keywords

Critical Velocity Angular Speed Model Case Schwarz Inequality Trapping Potential 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2006

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