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

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 $\rho({\bf r})$ and taking dimensionless form ${\mathbf R}_t=\varkappa {\mathbf b}+[\nabla\ln\rho({\mathbf R})\times {\boldsymbol \tau}]$ (where $\varkappa$ is a local curvature of the filament, ${\mathbf b}$ is the unit binormal vector, and ${\boldsymbol \tau}$ is the unit tangent vector), is shown to admit a finite-dimensional reduction if the density profile is an isotropic Gaussian, $\rho\propto\exp(-|{\bf r}|^2/2)$. 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 $\dot {\bf B}={\bf A}$.