# Stability and Dynamics of Dark-Bright Soliton Bound States Away from the Integrable Limit
## 4 Discussion  
In the present contribution, we investigated in detail the stability and dynamics of bound pairs of dark-bright symbiotic solitons, which arise as nonlinear matter wave excitations in mixtures of Bose-Einstein condensates featuring inter-atomic repulsion. In particular, we explored the scenario of differently weighted inter- and intra-species interaction coefficients, breaking the integrability of  
Figure 6: Space-time evolution of the anti-symmetric stationary two-DB 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 \(g_{12}=0.82>g_{12_{cr}}^{(2)}\) and then to \(g_{12}=1.4>g_{12_{cr}}^{(1)}\). Panels (**a–d**) ((**e–h**)) correspond to the densities \(|\psi_{d}|^{2}\) (\(|\psi_{b}|^{2}\)) of the dark (bright) component. the relevant nonlinear Schrodinger model. It was argued by means of a recently proposed variational approach [50] and shown numerically that, upon departing from the integrable limit, bound states of such symbiotic entities exist for anti-symmetric bright soliton counterparts, the so-called solitonic gluons. These anti-symmetric states were found to be robust within a bounded interval of the inter-species repulsion coefficient \(g_{12}\), limited by critical points both in the miscible and in the immiscible regime of the model, associated with a supercritical and a subcritical pitchfork bifurcation, respectively. Below and above these boundaries, i.e., deep in the miscible and the immiscible regime, respectively, the anti-symmetric pair becomes unstable. Long-time propagation revealed differences, but also common characteristics of the decay mechanisms in the two domains of instability. Specifically, a striking common feature is the relevance of bright mass transfer between the two dark-bright solitons. In particular, in the miscible domain, we identified new stationary asymmetric states that bifurcate from the anti-symmetric ones in a supercritical pitchfork. The stability of the new asymmetric states was also dynamically confirmed. Moreover, it was shown that a further decrease of the interspecies repulsion results in asymmetric states with the bright solitons living on top of a finite BEC background, highlighting in this way the miscible character of such bound pairs. In contrast to the above picture, upon entering the immiscible regime, we had found in our recent work [50] that the destabilization of the anti-symmetric dark-bright soliton pair is caused by a subcritical pitchfork bifurcation involving an unstable stationary, mass-imbalanced dark-bright soliton pair mode. In the present work, we further explored the range of existence and stability of this asymmetric branch, demonstrating its overall instability and its substantial deformation upon entering the miscible regime. There are many directions that are worth considering in the future along the lines of this work. In particular, by fixing the intra-species interactions \(g_{22}=0.95\) in this work, we have not addressed the fate of the dark-bright soliton pair states when actually approaching the integrable limit, which would require \(g_{12}=g_{22}=1\). In this respect, it would be particularly interesting to see if the asymmetric state fully stabilizes upon restoring integrability or maintains its weakly unstable nature. Towards this direction, and since in the integrable limit exact solutions are available [31; 59], one could link the anti-symmetric and asymmetric states obtained here with the exact families of dark-bright soliton solutions known in the integrable case. Establishing a possible connection of this type would also open a new direction of exploration and understanding of the symbiotic soliton pairs, since, in such a case, one could depart once more from the integrable limit, but having at hand exact analytical expressions for the two-soliton problem rather than the approximate ones constructed only from the single soliton solution, as used in our present work.