  The statics, stability and dynamical properties of dark-bright soliton pairs
are investigated motivated by applications in a homogeneous system of
two-component repulsively interacting Bose-Einstein condensate. One of the
intra-species interaction coefficients is used as the relevant parameter
controlling the deviation from the integrable Manakov limit. Two different
families of stationary states are identified consisting of dark-bright solitons
that are either antisymmetric (out-of-phase) or asymmetric (mass imbalanced)
with respect to their bright soliton. Both of the above dark-bright
configurations coexist at the integrable limit of equal intra- and
inter-species repulsions and are degenerate in that limit. However, they are
found to bifurcate from it in a transcritical bifurcation. The latter
interchanges the stability properties of the bound dark-bright pairs rendering
the antisymmetric states unstable and the asymmetric ones stable past the
associated critical point (and vice versa before it). Finally, on the dynamical
side, it is found that large kinetic energies and thus rapid soliton collisions
are essentially unaffected by the intra-species variation, while cases
involving near equilibrium states or breathing dynamics are significantly
modified under such a variation.
