# Stability and Dynamics of Dark-Bright Soliton Bound States Away from the Integrable Limit
###### Abstract  
The existence, stability, and dynamics of bound pairs of symbiotic matter waves in the form of dark-bright soliton pairs in two-component mixtures of atomic Bose-Einstein condensates is investigated. Motivated by the tunability of the atomic interactions in recent experiments, we explore in detail the impact that changes in the interaction strengths have on these bound pairs by considering significant deviations from the integrable limit. It is found that dark-bright soliton pairs exist as stable configurations in a wide parametric window spanning both the miscible and the immiscible regime of interactions. Outside this parameter interval, two unstable regions are identified and are associated with a supercritical and a subcritical pitchfork bifurcation, respectively. Dynamical manifestation of these instabilities gives rise to a redistribution of the bright density between the dark solitons, and also to symmetry-broken stationary states that are mass imbalanced (asymmetric) with respect to their bright soliton counterpart. The long-time dynamics of both the stable and the unstable balanced and imbalanced dark-bright soliton pairs is analyzed. B 2017 7 388 2017 201  
the second component of a binary BEC featuring intra- and inter-species repulsion. Since bright solitons are not self-sustained structures in repulsive (self-defocusing) media, DB solitons are often called symbiotic, that is, the dark soliton can be thought of as acting as an effective potential well trapping the bright soliton [18; 19; 20; 21; 22]. Such symbiotic entities were first observed and theoretically studied in the context of nonlinear optics [23; 24; 25; 26; 27; 28; 29; 30; 31; 32]. However, their experimental realization in the atomic realm [33] opened a new and highly controllable direction towards a deeper understanding of the dynamics and interactions of these states both with each other as well as with external traps [34; 35; 36; 37; 38]. Additionally, current state-of-the art experiments offer the possibility of manipulating in a controllable fashion the nonlinear interactions via the well-established technique of Feshbach resonances [39; 40; 41; 42; 43; 44; 45]. This motivated recent theoretical activities where the static and dynamical properties of dark-bright symbiotic matter waves have been investigated on the level of mean field theory. Mathematically, tuning the interactions corresponds to deviating from the integrable limit [46; 47; 48] of the relevant nonlinear Schrodinger system, where nonintegrability is introduced when considering arbitrary nonlinearity (i.e., interaction) coefficients. The latter nonintegrable setting forms also the main focus of the present effort. In this context, despite the nonintegrability, analytical expressions of specific single-DB soliton states and lattices thereof have been obtained in [49]. Adding a parabolic trapping potential, it was revealed how the effective restoring force acting on the DB soliton depends on the inter-atomic interactions [49], verifying that the particle-like nature [34] of the symbiotic soliton is preserved. In the same spirit, the dependence of the binding energy of a DB soliton on the inter-species interactions was found analytically in [22], where, moreover, a proper bound on a phase imprinted in the bright soliton constituent was obtained (i.e., considering also moving single DB states), above which a breaking of the symbiotic entity was observed. In our previous work [50], the interactions between DB matter waves and the consequent formation of bound states for out-of-phase (anti-symmetric) bright soliton components has been studied. Based on a two-soliton ansatz of the hyperbolic type [37; 49], the full analytical expressions for the interaction energies between two DB solitons were obtained for arbitrary nonlinear coefficients, and in the absence of a confining potential.