# Stability and Dynamics of Dark-Bright Soliton Bound States Away from the Integrable Limit
## 3 Numerical Results
As before, the upper row of panels (a)-(d) shows the spatio-temporal evolution of the dark soliton counterpart, while panels (e)-(h) depict the corresponding  
Figure 5: (a) bifurcation diagram obtained by measuring the relative bright imbalance \(\Delta N_{b}\) as a function of the inter-species interaction coefficient \(g_{12}\). The stable anti-symmetric and asymmetric branches are denoted with solid blue lines, while the unstable anti-symmetric one is shown with a dashed green line. Notice that the asymmetric branches exist before the critical point, \(g_{12,}^{(2)}=0.81\), but are stable verifying the supercritical nature of the bifurcation. (b–d) BdG spectra at three different values of \(g_{12}\) showcasing the stability of the asymmetric states. In all cases, the anomalous modes are illustrated with red stars. The associated stationary DB profiles are depicted in panels (e–g). (h–m) spatio-temporal evolution of the stationary asymmetric states shown in panels (e–g) showing the dynamical stability of these symbiotic structures. Panels (h–j) ((k–m)) correspond to the dark (bright) soliton component. bright component. From the stability analysis presented above, it is expected that for all values of the interspecies interaction coefficient \(g_{12_{cr}}^{(2)}<g_{12}<g_{12_{cr}}^{(1)}\), the anti-symmetric two dark-bright soliton states exist as stable configurations, and as such should be robust throughout the propagation. This latter result is confirmed in panels (c) and (g) of Figure 6, for \(g_{12}=0.82\), which is slightly above the lower critical point. However, and as anticipated, a very different picture is found below the critical point \(g_{12_{cr}}^{(2)}\) depicted in panels (b) and (f), (a) and (e), respectively. Starting with the former, we observe that slightly below \(g_{12_{cr}}^{(2)}\), the initial stationary state quickly decays and in (b) and (f) we observe periodic tunneling of the bright component between the two dark solitons, while the dark solitons are only relatively weakly affected here. It is worth mentioning that similar tunneling dynamics have been identified and interpreted in terms of a bosonic Josephson junction model in [58], but with the crucial difference that the soliton pair was further supported by the restoring force of a harmonic trap in that work, while, in our present setup, there is no external potential that would keep the dark solitons in place. Remarkably, in this regime, despite the mass exchange, the bound soliton pair does not disintegrate. Further decreasing \(g_{12}\) in panels (a) and (e), the effects of the instability are more drastic. The time scale of the bright component oscillations decreases and the dark solitons vibrate more strongly. After some time of almost periodic oscillations, a more irregular type of motion sets in, with both dark solitons (one filled by most of the bright component, the other almost empty) eventually moving towards positive \(x\) while separating and recolliding in the process, suggesting still a kind of effective attractive interaction between them. This decay mechanism deep in the miscible regime is to be contrasted to the unstable dynamics in the immiscible regime above \(g_{12_{cr}}^{(1)}\). In panels (d) and (h), we show the time evolution of the anti-symmetric two-DB soliton state at \(g_{12}=1.4\). While initially almost no dynamics are visible in the densities, especially no oscillations within the bright component, on intermediate time scales, a strong asymmetry in the bright filling builds up (see again [50] for a more detailed discussion) and subsequently the filled and the empty dark soliton split, showing no sign of effective attraction. Notice that the above described decaying mechanism is rather similar to the one observed for the unstable asymmetric states for \(g_{12}>1\) (see panels (d) and (h) of Figure 4).