Rotating three-dimensional solitons in Bose-Einstein condensates with gravitylike attractive nonlocal interaction

We study formation of rotating three-dimensional high-order solitons (azimuthons) in Bose Einstein condensate with attractive nonlocal nonlinear interaction. In particular, we demonstrate formation of toroidal rotating solitons and investigate their stability. We show that variational methods allow a very good approximation of such solutions and predict accurately the soliton rotation frequency. We also find that these rotating localized structures are very robust and persist even if the initial condensate conditions are rather far from the exact soliton solutions. Furthermore, the presence of repulsive contact interaction does not prevent the existence of those solutions, but allows one to control their rotation. We conjecture that self-trapped azimuthons are generic for condensates with attractive nonlocal interaction.

Figure 1

The left panels show the dependencies of the azimuthon width $\sigma$ on the parameter $\rho =8\pi Na/R_c$ (top) and the chemical potential $E$ on the width $\sigma$ (bottom). Black curves show results from the variational approach; dashed blue curves represent the asymptotic limits for $\sigma \to \infty$ (without repulsive contact interaction). The right panel shows the rotation frequency $\Omega$ as a function of the width $\sigma$. Black dots denote results obtained from numerical simulations of the GPE (5). All plots are for $p=0.7$.

Figure 2

Dynamics of the 3D stable solitons in gravity-like BEC. Iso-surfaces of the normalized density $\left|\psi \right|{}^2$ are depicted for different evolution times; the interior density distribution is represented in grayscales. The initial variational parameters used are $\sigma =1$ and $p=1$ (torus, $\rho =19.4$; iso-density surface at $\left|\psi \right|{}^2=0.039$) for the upper row; $p=0$ (dipole, $\rho =19.7$; iso-density surface at $\left|\psi \right|{}^2=0.071$) for the middle one; and finally, $p=0.7$ (azimuthon, $\rho =19.4$; iso-density surface at $\left|\psi \right|{}^2=0.044$). The sense of the rotation ($\Omega =0.64$) is indicated by the arrow.

Figure 3

Azimuthon rotation frequency $\Omega$ versus modulation parameter $p$ in gravity-like BEC for $\sigma =0.6\approx {\sigma }_s$ ($\rho =48.0\dots 65.4$, left panel) and $\sigma =3$ ($\rho =5.2\dots 5.5$, right panel). Black curves show results from the variational approach; dashed blue curves represent the asymptotic limits for $\sigma \to \infty$ (without repulsive contact interaction). Black dots denote results obtained from numerical simulations of the GPE (5).

Figure 4

The upper row shows iso-density surfaces at $\left|\psi \right|{}^2=0.16$ for the very slow rotating ($\Omega \approx 0$) azimuthon with $p=0.7$ and $\sigma =0.61$ ($\rho =47.5$). The lower row shows a fast counter-rotating ($\Omega =-2.2$) azimuthon with $p=0.7$, $\sigma =0.5$ ($\rho =93.9$), and iso-density surface at $\left|\psi \right|{}^2=0.34$. Same plot style as in Fig. 2.

Figure 5

The upper row shows iso-density surfaces at $\left|\psi \right|{}^2=0.040$ of the time evolution of a sphere ($\rho =18.8$). The imprinted vortex phase is shown in the inset; a stable torus is formed. The lower row shows the formation of a rotating dipole azimuthon by using initially an ellipsoid ($\rho =19.6$; iso-density surface at $\left|\psi \right|{}^2=0.039$). Same plot style as in Fig. 2.