JETP Letters

, Volume 104, Issue 12, pp 868–872 | Cite as

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

  • V. P. Ruban
Methods of Theoretical Physics


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=C 1 and M · D̂N=C 0=0. The equation of the evolution of the tangential vector M(t) appears to be closed and has integrals of motion M ·D̂M=C 2 and (|M| − M· ĜΩ) = C, with the matrix Ĝ = 2(ÎTrD̂ − D̂)−1. Crossing of the respective isosurfaces specifies trajectories in the phase space.


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© Pleiades Publishing, Inc. 2016

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsRussian Academy of SciencesChernogolovka, Moscow regionRussia

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