Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1610.09119 · Journal of Experimental and Theoretical Physics Letters

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

Published Dec 1, 2016• 4 citations
arXivHTMLPDF

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=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.

DOI
10.1134/s0021364016240115

Related papers

Dynamics of straight vortex filaments in a Bose–Einstein condensate with the Gaussian density profile

V. P. Ruban

The dynamics of interacting quantized vortex filaments in a rotating Bose–Einstein condensate existing in the Thomas–Fermi regime at zero temperature and obeying the Gross–Pitaevskii equation has been considered in the hydrodynamic “nonelastic” approximation. A noncanonical Hamilton equation of motion for the macroscopically averaged vorticity has been derived for a smoothly inhomogeneous array of filaments (vortex lattice) taking into account spatial nonuniformity of the equilibrium density of the condensate, which is determined by the trap potential. The minimum of the corresponding Hamiltonian describes the static configuration of the deformed vortex lattice against the preset density background. The condition of minimum can be reduced to a nonlinear second-order partial differential vector equation for which some exact and approximate solutions are obtained. It has been shown that if the condensate density has an anisotropic Gaussian profile, the equation of motion for the averaged vorticity has solutions in the form of a vector exhibiting a nontrivial time dependence, but homogeneous in space. An integral representation has also been obtained for the matrix Green function that determines the nonlocal Hamiltonian of a system of several quantized vortices of an arbitrary shape in a Bose–Einstein condensate with the Gaussian density. In particular, if all filaments are straight and oriented along one of the principal axes of the ellipsoid, we have a finitedimensional reduction that can describe the dynamics of the system of pointlike vortices against an inhomogeneous background. A simple approximate expression is proposed for the 2D Green function with an arbitrary density profile and is compared numerically with the exact result in the Gaussian case. The corresponding approximate equations of motion, describing the long-wavelength dynamics of interacting vortex filaments in condensates with a density depending only on transverse coordinates, have been derived.

arXiv:1612.00165•Journal of Experimental and Theoretical Physics•Jun 1, 2017•11 citations

Hamilton’s equations of motion of a vortex filament in the rotating Bose–Einstein condensate and their “soliton” solutions

V. P. Ruban

The equation of motion of a quantized vortex filament in a trapped Bose–Einstein condensate [A. A. Svidzinsky and A. L. Fetter, Phys. Rev. A 62, 063617 (2000)] has been generalized to the case of an arbitrary anharmonic anisotropic rotating trap and presented in the variational form. For condensate density profiles of the form ρ = f(x 2 + y 2 + ReΨ(x + iy)) in the presence of the plane of symmetry y = 0, the solutions x(z) describing stationary vortices of U and S types coming to the surface and solitary waves have been found in quadratures. Analogous three-dimensional configurations of the vortex filament uniformly moving along the z axis have also been found in strictly cylindrical geometry. The dependence of solutions on the form of the function f(q) has been analyzed.

arXiv:1609.03747•Journal of Experimental and Theoretical Physics Letters•Jun 1, 2016•6 citations

Non-stationary vortex ring in a Bose-Einstein condensate with Gaussian density

V. P. Ruban

The local induction equation, approximately describing dynamics of a quantized vortex filament in a trapped Bose-Einstein condensate in the Thomas-Fermi regime on a spatially nonuniform density background $ρ({\bf r})$ and taking dimensionless form ${\mathbf R}_t=\varkappa {\mathbf b}+[\nabla\lnρ({\mathbf R})\times {\boldsymbol τ}]$ (where $\varkappa$ is a local curvature of the filament, ${\mathbf b}$ is the unit binormal vector, and ${\boldsymbol τ}$ is the unit tangent vector), is shown to admit a finite-dimensional reduction if the density profile is an isotropic Gaussian, $ρ\propto\exp(-|{\bf r}|^2/2)$. The reduction corresponds to a geometrically perfect vortex ring centered at position ${\bf A}(t)$, with orientation and size both determined by a vector ${\bf B}(t)$. Parameters ${\bf A}$ and ${\bf B}$ exhibit the same dynamics as velocity and position of a Newtonian particle do in 3D: $\dot {\bf A}={\bf B}/|{\bf B}|^2-{\bf B}$, and $\dot {\bf B}={\bf A}$.

arXiv:1709.05097•Sep 15, 2017•0 citations

Velocity of vortices in inhomogeneous Bose–Einstein condensates

Halvor Møll Nilsen, Gordon Baym, C. J. Pethick

We derive, from the Gross-Pitaevskii equation, an exact expression for the velocity of any vortex in a Bose-Einstein condensate, in equilibrium or not, in terms of the condensate wave function at the center of the vortex. In general, the vortex velocity is a sum of the local superfluid velocity, plus a correction related to the density gradient near the vortex. A consequence is that in rapidly rotating, harmonically trapped Bose-Einstein condensates, unlike in the usual situation in slowly rotating condensates and in hydrodynamics, vortices do not move with the local fluid velocity. We indicate how Kelvin's conservation of circulation theorem is compatible with the velocity of the vortex center being different from the local fluid velocity. Finally, we derive an exact wave function for a single vortex near the rotation axis in a weakly interacting system, from which we derive the vortex precession rate.

arXiv:cond-mat/0508195•Proceedings of the National Academy of Sciences•May 15, 2006•32 citations

Dynamics of quantum vortices in a quasi-two-dimensional Bose–Einstein condensate with two “holes”

V. P. Ruban

The dynamics of interacting quantum vortices in a quasi-two-dimensional spatially inhomogeneous Bose–Einstein condensate, whose equilibrium density vanishes at two points of the plane with a possible presence of an immobile vortex with a few circulation quanta at each point, has been considered in a hydrodynamic approximation. A special class of density profiles has been chosen, so that it proves possible to calculate analytically the velocity field produced by point vortices. The equations of motion have been given in a noncanonical Hamiltonian form. The theory has been generalized to the case where the condensate forms a curved quasi-two-dimensional shell in the three-dimensional space.

arXiv:1702.07775•Journal of Experimental and Theoretical Physics Letters•Apr 1, 2017•7 citations

Quantum vortices in the Bose–Einstein condensate. Reconnection and motion equations

Sergey K. Nemirovskii

The paper describes several features of vortex filaments in a weakly non-ideal Bose-Einstein condensate (BEC) with a counterflow. The study focuses on a process of reconnection and equations of motion. A primary motivation of this study is a fact that quantum vortex dynamics for superfluid helium was obtained previously at a phenomenological level. But some critical problems can be solved through a study of a corresponding problem in BEC. By considering a vortex filament as a crossing line for surfaces where the real and imaginary parts of a macroscopic wave function ψ ( r , t ) turn into zero, we derived the equations of motion for a vortex filament. It is remarkable that the equation structure coincides completely with a structure of equation for vortex filament dynamics in superfluid helium that has been obtained from other (phenomenological) foundations.

Thermophysics and Aeromechanics•Jul 25, 2026•0 citations
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.
The dynamics of interacting quantized vortex filaments in a rotating Bose–Einstein condensate existing in the Thomas–Fermi regime at zero temperature and obeying the Gross–Pitaevskii equation has been considered in the hydrodynamic “nonelastic” approximation. A noncanonical Hamilton equation of motion for the macroscopically averaged vorticity has been derived for a smoothly inhomogeneous array of filaments (vortex lattice) taking into account spatial nonuniformity of the equilibrium density of the condensate, which is determined by the trap potential. The minimum of the corresponding Hamiltonian describes the static configuration of the deformed vortex lattice against the preset density background. The condition of minimum can be reduced to a nonlinear second-order partial differential vector equation for which some exact and approximate solutions are obtained. It has been shown that if the condensate density has an anisotropic Gaussian profile, the equation of motion for the averaged vorticity has solutions in the form of a vector exhibiting a nontrivial time dependence, but homogeneous in space. An integral representation has also been obtained for the matrix Green function that determines the nonlocal Hamiltonian of a system of several quantized vortices of an arbitrary shape in a Bose–Einstein condensate with the Gaussian density. In particular, if all filaments are straight and oriented along one of the principal axes of the ellipsoid, we have a finitedimensional reduction that can describe the dynamics of the system of pointlike vortices against an inhomogeneous background. A simple approximate expression is proposed for the 2D Green function with an arbitrary density profile and is compared numerically with the exact result in the Gaussian case. The corresponding approximate equations of motion, describing the long-wavelength dynamics of interacting vortex filaments in condensates with a density depending only on transverse coordinates, have been derived.
The equation of motion of a quantized vortex filament in a trapped Bose–Einstein condensate [A. A. Svidzinsky and A. L. Fetter, Phys. Rev. A 62, 063617 (2000)] has been generalized to the case of an arbitrary anharmonic anisotropic rotating trap and presented in the variational form. For condensate density profiles of the form ρ = f(x 2 + y 2 + ReΨ(x + iy)) in the presence of the plane of symmetry y = 0, the solutions x(z) describing stationary vortices of U and S types coming to the surface and solitary waves have been found in quadratures. Analogous three-dimensional configurations of the vortex filament uniformly moving along the z axis have also been found in strictly cylindrical geometry. The dependence of solutions on the form of the function f(q) has been analyzed.
The local induction equation, approximately describing dynamics of a quantized vortex filament in a trapped Bose-Einstein condensate in the Thomas-Fermi regime on a spatially nonuniform density background ρ(r)ρ({\bf r})ρ(r) and taking dimensionless form Rt=ϰb+[∇ln⁡ρ(R)×τ]{\mathbf R}_t=\varkappa {\mathbf b}+[\nabla\lnρ({\mathbf R})\times {\boldsymbol τ}]Rt​=ϰb+[∇lnρ(R)×τ] (where ϰ\varkappaϰ is a local curvature of the filament, b{\mathbf b}b is the unit binormal vector, and τ{\boldsymbol τ}τ is the unit tangent vector), is shown to admit a finite-dimensional reduction if the density profile is an isotropic Gaussian, ρ∝exp⁡(−∣r∣2/2)ρ\propto\exp(-|{\bf r}|^2/2)ρ∝exp(−∣r∣2/2). The reduction corresponds to a geometrically perfect vortex ring centered at position A(t){\bf A}(t)A(t), with orientation and size both determined by a vector B(t){\bf B}(t)B(t). Parameters A{\bf A}A and B{\bf B}B exhibit the same dynamics as velocity and position of a Newtonian particle do in 3D: A˙=B/∣B∣2−B\dot {\bf A}={\bf B}/|{\bf B}|^2-{\bf B}A˙=B/∣B∣2−B, and B˙=A\dot {\bf B}={\bf A}B˙=A.
We derive, from the Gross-Pitaevskii equation, an exact expression for the velocity of any vortex in a Bose-Einstein condensate, in equilibrium or not, in terms of the condensate wave function at the center of the vortex. In general, the vortex velocity is a sum of the local superfluid velocity, plus a correction related to the density gradient near the vortex. A consequence is that in rapidly rotating, harmonically trapped Bose-Einstein condensates, unlike in the usual situation in slowly rotating condensates and in hydrodynamics, vortices do not move with the local fluid velocity. We indicate how Kelvin's conservation of circulation theorem is compatible with the velocity of the vortex center being different from the local fluid velocity. Finally, we derive an exact wave function for a single vortex near the rotation axis in a weakly interacting system, from which we derive the vortex precession rate.
The dynamics of interacting quantum vortices in a quasi-two-dimensional spatially inhomogeneous Bose–Einstein condensate, whose equilibrium density vanishes at two points of the plane with a possible presence of an immobile vortex with a few circulation quanta at each point, has been considered in a hydrodynamic approximation. A special class of density profiles has been chosen, so that it proves possible to calculate analytically the velocity field produced by point vortices. The equations of motion have been given in a noncanonical Hamiltonian form. The theory has been generalized to the case where the condensate forms a curved quasi-two-dimensional shell in the three-dimensional space.
The paper describes several features of vortex filaments in a weakly non-ideal Bose-Einstein condensate (BEC) with a counterflow. The study focuses on a process of reconnection and equations of motion. A primary motivation of this study is a fact that quantum vortex dynamics for superfluid helium was obtained previously at a phenomenological level. But some critical problems can be solved through a study of a corresponding problem in BEC. By considering a vortex filament as a crossing line for surfaces where the real and imaginary parts of a macroscopic wave function ψ ( r , t ) turn into zero, we derived the equations of motion for a vortex filament. It is remarkable that the equation structure coincides completely with a structure of equation for vortex filament dynamics in superfluid helium that has been obtained from other (phenomenological) foundations.