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

The dynamics of a vortex filament in a Bose–Einstein condensate whose equilibrium density in the reference frame rotating at the angular velocity $\mathbf{\Omega}$ is Gaussian with the quadratic form $\mathbf{r}\cdot\hat D\mathbf{r}$ 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 $\mathbf{R}(\beta,t)=\beta \mathbf{M}(t) +\mathbf{N}(t)$ of a straight vortex, where $\beta$ is the longitudinal parameter and is the time. The vortex slips over the surface of an ellipsoid, which follows from the conservation laws $\mathbf{N}\cdot \hat D \mathbf{N}=C_1$ and $\mathbf{M}\cdot \hat D \mathbf{N}=C_0=0$. The equation of the evolution of the tangential vector $M(t)$ appears to be closed and has integrals of motion $\mathbf{M}\cdot \hat D \mathbf{M}=C_2$ and $(|\mathbf{M}| -\mathbf{M}\cdot\hat G\mathbf{\Omega})=C$, with the matrix $\hat G=2(\hat I \,\mathrm{Tr}\, \hat D -\hat D)^{-1}$. Crossing of the respective isosurfaces specifies trajectories in the phase space.