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

TL;DR

This paper derives exact solutions for vortex filament dynamics in a Bose-Einstein condensate with Gaussian density, revealing integrable motion characterized by conserved quantities and phase space trajectories.

Contribution

It presents new exact solutions for vortex motion in a Gaussian density profile, expanding understanding of vortex dynamics in Bose-Einstein condensates.

Findings

Exact solutions describe straight moving vortices in Gaussian condensates.

Conservation laws define the vortex's geometric and dynamic properties.

Phase space trajectories are determined by intersecting level surfaces of conserved quantities.

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 ${\bf\Omega}$ coordinate system, is Gaussian with a quadratic form ${\bf r}\cdot\hat D{\bf r}$. 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, ${\bf R}(\beta,t)=\beta {\bf M}(t) +{\bf N}(t)$, 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 ${\bf N}\cdot \hat D {\bf N}=C_1$ and ${\bf M}\cdot \hat D {\bf N}=C_0=0$. Equation of motion for the tangent vector ${\bf M}(t)$ turns out to be closed, and it has the integrals ${\bf M}\cdot \hat D {\bf M}=C_2$, $(|{\bf M}| -{\bf M}\cdot\hat G{\bf \Omega})=C$, 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.