# Stability and Dynamics of Dark-Bright Soliton Bound States Away from the Integrable Limit
## 3 Numerical Results
The identification of these asymmetric states which exist below \(g_{12_{\sigma}}^{(1)}\) and collide with the anti-symmetric branch at \(g_{12_{\sigma}}^{(1)}\) in a subcritical pitchfork bifurcation was established in [50]. It is worth mentioning, at this point, that such asymmetric states were also analytically obtained in [31] in the integrable limit of the theory. However, to the best of our knowledge, the stability of such states has not been addressed. The following paragraphs will be devoted to a discussion of these asymmetric two-DB solutions, before we then return to an analysis of the BdG spectrum of the anti-symmetric pair, Figure 2, at small values of \(g_{12}\). As mentioned above, the asymmetric and the anti-symmetric two-DB states coincide at the bifurcation point \(g_{12_{\sigma}}^{(1)}\). Below this critical \(g_{12}\), the anti-symmetric state is stable and there are two symmetry-broken asymmetric solutions which are unstable (as is characteristic of a subcritical pitchfork bifurcation). Indeed, these asymmetric branches can themselves be continued towards much smaller values of \(g_{12}\) and exist both in the immiscible and the miscible regime (see the profiles shown in  
Figure 2: Full BdG spectrum of anti-symmetric two-DB soliton states, upon varying the inter-species interaction coefficient \(g_{12}\) within the interval \(0.75\leq g_{12}\leq 1.5\). (**a**) real part, \(\Re(\omega)\), of the eigenfrequencies \(\omega\) as a function of \(g_{12}\). The corresponding imaginary part, \(\Im(\omega)\), is shown in panel (**b**). Upon increasing \(g_{12}\), there exist two critical values, \(g_{12_{\sigma}}^{(2)}=0.81\), and \(g_{12_{\sigma}}^{(1)}=1.18\), indicated with light blue and gray boxes, respectively, below and above which the anti-symmetric branch destabilizes. The trajectories of the two anomalous modes (see text) appearing in the spectrum are shown with red stars. (**c,d**) profiles of the anti-symmetric states for \(g_{12}=0.75\), and \(g_{12}=1.4\), i.e., deep in the miscible and immiscible regime, respectively. Figure 3c,d). In both panels, dashed black lines denote the wavefunction of the dark soliton component, while solid green lines depict the corresponding bright soliton counterpart. To gain further insight regarding the nature of the instability of these asymmetric states, their full BdG spectrum is illustrated in Figure 3a,b. Once more, both the real \(\Re(\omega)\) and the imaginary \(\Im(\omega)\) parts of the eigenfrequencies \(\omega\) are shown as a function of the interspecies interaction. As expected from the above discussion, only one of the anomalous modes appears in the excitation spectrum of these mass imbalanced states and is depicted with stars in red. Replacing the second anomalous mode, there is now throughout a purely imaginary frequency signaling the instability of the asymmetric branch. Remarkably, as the immiscibility-miscibility threshold is crossed, the growth rate of the instability is drastically decreased, rendering these states only weakly unstable for \(g_{12}<g_{12_{ab}}\). This latter observation is in close contact with the change in the spatial character of these asymmetric states, and also in line with our previous findings [50]. Namely, as the interspecies interactions decrease towards the miscible region, the asymmetric states change gradually from only weakly asymmetric (i.e., weakly mass imbalanced with respect to their bright amplitudes), to maximally asymmetric (i.e., almost purely dark/dark-bright bound states). The difference is clearly seen in the profiles provided in panels (c) and (d) of Figure 3. The existence of such maximally asymmetric bound states can be intuitively understood as follows. We have found in [50] that the effective interaction between dark solitons (\(E_{dd}\)) is repulsive while the interaction between OP bright solitons (\(E_{bb}\)) with fixed \(g_{22}=0.95\) and \(\mu=2/3\) is attractive.