# Stability and Dynamics of Dark-Bright Soliton Bound States Away from the Integrable Limit
###### Abstract
Furthermore, the key intuition that repulsion mediated by the dark solitons at short distances, and attraction mediated by anti-symmetric bright solitons at longer distances, would be counterbalanced, leading in turn to a bound state formation, has been enriched by taking into account the significant role of the cross-component interaction energy term. The crucial dependence of the latter on the inter-species interaction coefficient has been analyzed. It was shown that anti-symmetric stationary states exist and remain robust for a wide parametric window of inter-species repulsions. Importantly, an exponential instability of the anti-symmetric states was identified upon crossing a critical inter-species repulsion. The latter was found to be associated with a subcritical pitchfork bifurcation, giving rise to asymmetric stationary states with mass imbalanced bright soliton counterparts. In the present work, we extend the aforementioned findings of [50] upon considering significant deviations from the integrable limit. By this, we mean that we always utilize a more realistic selection of the nonlinear inter- and intra-species repulsion coefficients motivated by \({}^{87}\)Rb (see details in Section 3), under which it is never possible that all coefficients are equal (which would represent the integrable limit). In our analysis, we vary the inter-species repulsion towards both the immiscible (i.e., dominated by inter-species repulsion and thus phase separated in the ground state) regime, but also towards the miscible (i.e., dominated by intra-species repulsion) regime. To this end, we investigate the stability and dynamics of the anti-symmetric states, the so-called \"solitonic gluons\" [24], as well as the above-mentioned asymmetric modes covering both the miscible and immiscible parameter regime. In particular, the full excitation spectrum of these symbiotic states is explored in detail by means of a Bogolyubov-de Gennes linearization analysis [1]. It is found that, for the asymmetric modes, the linearization spectrum is strongly affected when crossing the immiscibility-miscibility threshold, which is, in turn, directly connected with a rapid change of their density profiles as observed already in [50]. As a next step, the stability and dynamics of the anti-symmetric DB pairs are exploredpast the immiscibility-miscibility threshold. It is found that, in addition to the destabilization in the immiscible regime reported in [50], a second critical point occurs deep in the miscible regime, rendering the anti-symmetric state unstable once more. This instability scenario is found to be associated with a _supercritical_ pitchfork bifurcation, giving rise to another family of mass imbalanced (asymmetric) symbiotic structures which are found to be stable. As the inter-species repulsion is decreased, the miscible character of this regime alters the bright soliton component, resulting into asymmetric pairs with the bright solitons \"living\" on top of a finite background. Comparing and contrasting the instability mechanisms in the different parameter regimes, the long-time evolution of the asymmetric and anti-symmetric states is performed numerically at different interaction ratios. The presentation is structured as follows. In Section 2, a description of the theoretical model and prior results regarding the existence of the anti-symmetric DB soliton pairs are provided. Furthermore, we briefly comment on the methods to be used for the numerical analysis of our findings. Section 3 contains the results regarding both the stability and the dynamics of asymmetric and anti-symmetric DB soliton pairs beyond the integrable limit. Finally, in Section 4, we summarize our findings and discuss future perspectives.