Stable and unstable vortex knots in a trapped Bose-Einstein condensate

The dynamics of a quantum vortex torus knot ${\cal T}_{P,Q}$ and similar knots in an atomic Bose-Einstein condensate at zero temperature in the Thomas-Fermi regime has been considered in the hydrodynamic approximation. The condensate has a spatially nonuniform equilibrium density profile $\rho(z,r)$ due to an external axisymmetric potential. It is assumed that $z_*=0$, $r_*=1$ is a maximum point for function $r\rho(z,r)$, with $\delta (r\rho)\approx-(\alpha-\epsilon) z^2/2 -(\alpha+\epsilon) (\delta r)^2/2$ at small $z$ and $\delta r$. Configuration of knot in the cylindrical coordinates is specified by a complex $2\pi P$-periodic function $A(\varphi,t)=Z(\varphi,t)+i [R(\varphi,t)-1]$. In the case $|A|\ll 1$ the system is described by relatively simple approximate equations for re-scaled functions $W_n(\varphi)\propto A(2\pi n+\varphi)$, where $n=0,\dots,P-1$, and $iW_{n,t}=-(W_{n,\varphi\varphi}+\alpha W_n -\epsilon W_n^*)/2-\sum_{j\neq n}1/(W_n^*-W_j^*)$. At $\epsilon=0$, numerical examples of stable solutions as $W_n=\theta_n(\varphi-\gamma t)\exp(-i\omega t)$ with non-trivial topology have been found for $P=3$. Besides that, dynamics of various non-stationary knots with $P=3$ was simulated, and in some cases a tendency towards a finite-time singularity has been detected. For $P=2$ at small $\epsilon\neq 0$, rotating around $z$ axis configurations of the form $(W_0-W_1)\approx B_0\exp(i\zeta)+\epsilon C(B_0,\alpha)\exp(-i\zeta) + \epsilon D(B_0,\alpha)\exp(3i\zeta)$ have been investigated, where $B_0>0$ is an arbitrary constant, $\zeta=k_0\varphi -\Omega_0 t+\zeta_0$, $k_0=Q/2$, $\Omega_0=(k_0^2-\alpha)/2-2/B_0^2$. In the parameter space $(\alpha, B_0)$, wide stability regions for such solutions have been found. In unstable bands, a recurrence of the vortex knot to a weakly excited state has been noted to be possible.