Stability of dark solitons in a bubble Bose-Einstein condensate

TL;DR

This work addresses the stability of dark solitons on the surface of a spherical bubble Bose-Einstein condensate by performing a Bogoliubov–de Gennes analysis for discrete angular modes $m$ and by deriving analytical and numerical results across small and large interaction regimes $ε$. It shows that for $ε o$ small, solitons are stable for all $m$, while for $ε \gtrsim 8.37$ exactly one unstable mode exists for each $m ≥ 2$, with the unstable mode dominating the instability and driving snake-like deformations that generate $m$ vortex dipoles; asymptotically the instability thresholds satisfy $ε_m ≈ 4 m(m-1)$. The authors verify the spectral predictions with time-dependent GPE simulations that exhibit the formation of vortex dipoles and discuss the topological constraint that vortices on a sphere must occur in zero-net-circulation pairs. These results offer concrete guidance for experiments in bubble-trapped BECs and advance understanding of curvature-induced stability in closed two-dimensional quantum fluids.

Abstract

The dynamic stability of dark solitons trapped on the surface of a two-dimensional spherical bubble is investigated. In this spherical geometry of the Bose-Einstein condensate, dark solitons are found to be unstable for the interaction parameter $ε \gtrsim 8.37$, since discrete angular modes drive snake instabilities, with the generation of vortex dipoles. We show analytically and numerically that, for each angular mode $m \ge 2$, there exists exactly one unstable mode whose dominance determines the number m of vortex dipoles. Time-dependent simulations confirm the formation of vortex dipoles.

Figures (4)

• Figure 1: (Color on-line) In (a), we show profiles of dark solitons, $f(\theta)$ (upper panel), with corresponding densities $|f(\theta)|^2$ (lower panel), as function of $\theta$, for three different values of $\mu$, with $\varepsilon$ obtained numerically by using the shooting method. In (b), the chemical potential $\mu$ is presented as a function of $\varepsilon$ for dark solitons. Solid lines correspond to the numerical data, with dashed and circles referring to $\mu=2+\frac{9}{10}\varepsilon$ and $\mu=\frac{\varepsilon}{2}+\sqrt{\varepsilon}+1$, respectively, in agreement with \ref{['dark-soliton']} and \ref{['dark-soliton-large']}.
• Figure 2: (Color on-line) By varying $\varepsilon$, we show for different $m$-angular modes (indicated by the symbols and lines) that the imaginary part of the eigenvalues $\omega$ (positive defined), obtained from \ref{['nls-spectrum']} (solid lines), start increasing from zero when the corresponding lowest eigenvalue $\omega^-_m$ of $L_m^-$ (dashed lines) becomes negative. For a given $\varepsilon$, the dominant unstable mode $m$ is provided by the largest ${\rm Im}(\omega_m)$.
• Figure 3: Dark-soliton dynamics, for given time instants $t$, with dominant instability modes $m=2$ (left panels, $\varepsilon=20$), $m=3$ (center panels, $\varepsilon=50$), and $m=4$ (right panels, $\varepsilon=100$), shown by the respective densities $|\psi|^2$. In 3D graphics, the darker the region, the less dense it is. Panels (d), (h), and (l) show the corresponding $|\psi(\frac{\pi}{2},\phi)|^2$ for the same instants $t$ represented in the upper 3D plots. The dashed lines are for $t=0$; the red dotted-dashed, when snake instabilities occur; and the solid-blue lines, after the breakup in vortex pairs.
• Figure 1 S2: Shooting method for dark solitons. The left boundary condition is $\tilde{f}'(0)=0$. For fixed $\mu$ one shoots $\tilde{f}(0)$ and propagate Eq. \ref{['a2']} till the condition $\tilde{f}(\pi/2)=0$ is satisfied.