Some exact solutions of the local induction equation for the motion of a vortex in a Bose–Einstein condensate with a Gaussian density profile


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Abstract

The dynamics of a vortex filament in a Bose–Einstein condensate whose equilibrium density in the reference frame rotating at the angular velocity Ω is Gaussian with the quadratic form r·D̂r has been considered. It has been shown that the equation of motion of the filament in the local-induction approximation permits a class of exact solutions in the form R(β, t) = βM(t) + N(t) of a straight vortex, where β is the longitudinal parameter and is the time. The vortex slips over the surface of an ellipsoid, which follows from the conservation laws N · D̂N=C1 and M · D̂N=C0=0. The equation of the evolution of the tangential vector M(t) appears to be closed and has integrals of motion M ·D̂M=C2 and (|M| − M· ĜΩ) = C, with the matrix Ĝ = 2(ÎTrD̂ − D̂)−1. Crossing of the respective isosurfaces specifies trajectories in the phase space.

About the authors

V. P. Ruban

Landau Institute for Theoretical Physics

Author for correspondence.
Email: ruban@itp.ac.ru
Russian Federation, Chernogolovka, Moscow region, 142432

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