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

Abstract

The dynamics of a vortex filament in a trapped Bose-Einstein condensate is considered when the equilibrium density of the condensate, in rotating with angular velocity $\mathbf{\Omega }$ coordinate system, is Gaussian with a quadratic form $\mathbf{r}\cdot \hat{D}\mathbf{r}$. It is shown that equation of motion of the filament in the local induction approximation admits a class of exact solutions in the form of a straight moving vortex, $\mathbf{R}(\beta ,t)=\beta \mathbf{M}(t)+\mathbf{N}(t)$, where $\beta$ is a longitudinal parameter, and $t$ is the time. The vortex is in touch with an ellipsoid, as it follows from the conservation laws $\mathbf{N}\cdot \hat{D}\mathbf{N}=C_1$ and $\mathbf{M}\cdot \hat{D}\mathbf{N}=C_0=0$. Equation of motion for the tangent vector $\mathbf{M}(t)$ turns out to be closed, and it has the integrals $\mathbf{M}\cdot \hat{D}\mathbf{M}=C_2$, $(|\mathbf{M}|-\mathbf{M}\cdot \hat{G}\mathbf{\Omega })=C$, where the matrix $\hat G=2(\hat I \mbox{Tr\,} \hat D -\hat D)^{-1}$. Intersection of the corresponding level surfaces determines trajectories in the phase space.

4.5 pages, 10 figures, submitted to JETP Letters

References in corpus (2)

• Bending-wave Instability of a Vortex Ring in a Trapped Bose-Einstein Condensate
• Hamilton's equations of motion of a vortex filament in the rotating Bose-Einstein condensate and their "soliton" solutions

Cited by in corpus (1)

• Parametric instability of oscillations of a vortex ring in a $z$-periodic Bose-Einstein condensate and the recurrence to starting state