  In this work we present a systematic theoretical analysis regarding
dark-bright solitons and their interactions, motivated by recent advances in
atomic two-component repulsively interacting Bose-Einstein condensates. In
particular, we study analytically via a two-soliton ansatz adopted within a
variational formulation the interaction between two dark-bright solitons in a
homogeneous environment beyond the integrable regime, by considering general
inter/intra-atomic interaction coefficients. We retrieve the possibility of a
fixed point in the case where the bright solitons are out of phase. As the
inter-component interaction is increased, we also identify an exponential
instability of the two-soliton state, associated with a subcritical pitchfork
bifurcation. The latter gives rise to an asymmetric partition of the bright
soliton mass and dynamically leads to spontaneous splitting of the bound pair.
In the case of the in-phase bright solitons, we explain via parsing the
analytical approximations and monitoring the direct dynamics why no such pair
is identified, despite its prediction by the variational analysis.
