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

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.