have derived analytical expressions describing their relative motion both in harmonic traps and untrapped systems. Contrary to the case of solitons in scalar condensates, our theoretical model predicts that the interaction between dark solitons excited in different condensates is attractive (repulsive) for repulsive (attractive) interparticle interactions $g_{12}$. In harmonically trapped systems, the scenario is specially interesting for negative $g_{12}$, where the effective potential felt by the solitons modifies its shape as a function of $g_{12}$ (through a pitchfork bifurcation) from a single-well potential to a double-well potential, then allowing for stationary states made of solitons located at different positions.

The theoretical analytical predictions have been shown to be in good agreement with the numerical solutions of the Gross Pitaevskii equation for the real time evolution, and with the Bogoliubov equations for the linear excitations of the dark-dark solitons. In particular, we have demonstrated that the resonance of two out-of-phase modes, the anomalous one giving the frequency of the relative motion between solitons, and the lowest energy mode associated to the spin density excitation of the background, give rise to instabilities (Hopf bifurcations) that produce the decay of dark-dark solitons. This fact translate in finite systems, either harmonically trapped condensates or 