# Stability and Dynamics of Dark-Bright Soliton Bound States Away from the Integrable Limit
## 3 Numerical Results
Departing from \(g_{12}\approx 1\) towards the miscible regime, and in particular by following once again the lower-lying anomalous BdG mode, we observe that as \(g_{12}\) decreases a second critical point that exists deep within the miscible side. The eigenvalue zero crossing occurs at \(g_{12\alpha}^{(2)}=0.81\), and is indicated by the solid light blue box. We note here that this latter critical point was not discussed in our previous work [50], where only larger values of \(g_{12}\) were studied. The destabilization of the aforementioned mode suggests the existence of a second pitchfork bifurcation. This is, once again, related with a symmetry breaking of the bright soliton component, resulting in mass imbalanced (i.e., asymmetric) two DB states. Indeed, we were able to identify these _new_ pairs. In contrast to the previous instability  
Figure 4: Space-time evolution of the unstable asymmetric two-DB soliton state for different values of the inter-species interaction coefficient \(g_{12}\). From left to right, the interaction is increased from \(g_{12}=0.75\) to \(g_{12}=1.1\), while panels (**a–d**) ((**e–h**)) correspond to the densities \(|\psi_{d}|^{2}\) (\(|\psi_{b}|^{2}\)) of the dark (bright) component. scenario, these mass imbalanced states exist past the critical point but are _stable_, i.e., the pitchfork deep in the miscible domain is found to be _supercritical_. The corresponding bifurcation diagram is shown in panel (a) of Figure 5. In order to obtain this diagram, we measure the relative bright imbalance defined as \(\Delta N_{b}\equiv\left(\int_{-\infty}^{0}|u_{b}|^{2}dx-\int_{0}^{\infty}|u_{b} |^{2}dx\right)/\int_{-\infty}^{\infty}|u_{b}|^{2}dx\) upon varying \(g_{12}\). Notice that four branches are identified, i.e., three stable ones consisting of two asymmetric and an anti-symmetric branch all denoted by solid blue lines, and one unstable anti-symmetric branch illustrated with a dashed green line. To further demonstrate the stability of the new asymmetric symbiotic pairs in panels (b)-(d), the BdG spectra are shown for different values of the inter-species interactions below the associated critical point \(g_{12\sigma}^{(2)}=0.81\). Two anomalous modes appear in the linearization spectra of these asymmetric states illustrated with red stars. The respective stationary wave profiles are depicted in panels (e)-(g), where the dark (bright) soliton wavefunction is shown with a dashed black (solid green) line. We observe, that upon decreasing \(g_{12}\), the asymmetry between the bright solitons increases, as was also the case for the respective asymmetric but unstable states found in the immiscible regime. Furthermore, for \(g_{12}<0.7\), a background gradually builds up for the bright solitons as is evident in the stationary state shown in panel (e) of Figure 5, revealing the miscible character of the regime supporting these states. Our detailed BdG analysis indicates that, for all values within the miscible regime that we have checked, the asymmetric states exist as stable configurations and should remain dynamically robust. This result is verified and highlighted in panels (h)-(m) of Figure 5, where we use as initial condition the stationary asymmetric states depicted in panels (e)-(g). We note that, in all three cases, panels (h)-(j) ((k)-(m)) show the evolution of the dark (bright) soliton component. Having studied both the static and the dynamical properties of the new asymmetric structures that bifurcate from the OP dark-bright states, we now turn our attention to the dynamics of the anti-symmetric DB waveforms. In particular, we explore the long time evolution of the OP DB states so as to reveal the decay mechanisms that such a pair suffers from. The dynamics at different values of \(g_{12}\) are summarized in Figure 6, all initiated at equilibrium.