# Stability and Dynamics of Dark-Bright Soliton Bound States Away from the Integrable Limit
## 3 Numerical Results  
Having verified that stationary anti-symmetric pairs of symbiotic matter waves can be found in a wide range of values of the inter-species repulsion coefficient \(g_{12}\), a natural next step is to consider the fate of these solutions under small perturbations, providing information on their stability in the different regions of existence. The latter is explored by using the BdG linearization analysis discussed in Section 2.3. For the presentation of our results, we will distinguish the miscible (\(g_{12}<g_{12_{th}}\)) and immiscible (\(g_{12}>g_{12_{th}}\)) regimes, separated by the miscibility-immiscibility threshold [56]\(g_{12_{th}}=\sqrt{g_{22}}=0.975\) for our choice of \(g_{22}=0.95\). In Figure 2, the BdG spectrum of the anti-symmetric stationary two dark-bright soliton states is shown as a function of \(g_{12}\). In particular, both the real (\(\Re(\omega)\)) and the imaginary parts (\(\Im(\omega)\)) of the eigenfrequencies \(\omega\) are depicted in the top (a) and middle (b) panels, respectively. Panels (c) and (d) depict profiles of the dark and bright wave functions at selected values of \(g_{12}\) in the miscible and immiscible regime, respectively. Two general comments can be made before examining in detail the excitation spectrum. The most significant one is that, within the background spectrum denoted with blue circles, there exist two distinguished modes. The trajectories of the latter are illustrated with red stars. These modes are the so-called anomalous modes since they possess a so-called negative Krein signature \(K\)[57], which, for the two-component system considered herein, is defined as:  
\[K=\omega\int\left(|a|^{2}-|b|^{2}+|c|^{2}-|d|^{2}\right)dx. \tag{12}\]  
The sign of this quantity is a topological property associated with the excited nature of this state, and the eigenvectors of such anomalous modes result in a variation of the solitary waves (as opposed to a variation of the system's background). Furthermore, we note in passing that each continuous symmetry of the system corresponds to a pair of zero modes, \(\omega=0\), in the BdG spectrum. Thus, we expect three pairs of such modes related, respectively, with the conservation of the particle number (or the \(U(1)\) gauge invariance) in each of the two components, and with the translation invariance due to the absence of a confining potential. This is confirmed in the numerical data. Just by inspecting the trajectories of the two anomalous modes that appear in the spectrum, their very different behavior becomes apparent. As it is observed in panel (a) of Figure 2, the higher frequency mode decreases almost monotonically upon increasing \(g_{12}\). We were able not only to identify this mode but also to relate it with an out-of-phase vibration of the bound DB soliton pair around its equilibrium distance. The latter can be done by adding the corresponding BdG eigenvector, which turns out to be localized in the vicinity of the DB pair, to the relevant stationary anti-symmetric state (results not shown here). Next, and also in the same panel of Figure 2, let us closely follow the trajectory of the lower-lying anomalous mode. The trajectory of this mode is more complicated than the former one. In particular, starting from the aforementioned reference point, \(g_{12_{th}}\approx 1\), and increasing the interspecies interactions towards the immiscible regime, i.e., for \(g_{12}>g_{12_{th}}\), it can be seen that there exists a critical point \(g_{12_{\sigma}}^{(1)}=1.18\) where this mode destabilizes. This destabilization corresponds to an eigenvalue zero crossing and signals the instability of the anti-symmetric configuration deep in the immiscible regime. Notice the non-zero imaginary part that appears past this critical point shown in panel (b) of Figure 2. The existence and destabilization of this mode was found to be related with a symmetry breaking of the bright soliton component, being linked, in turn, to other stationary states that are mass imbalanced with respect to their bright soliton counterpart.