any axial trap, termed untrapped, and the case with a harmonic trap in the axial direction. Within the Hamiltonian approach of the perturbation theory for solitons [@Uzunov1993; @Kivshar1994; @Frantzeskakis2002], we obtain analytical expressions for the adiabatic evolution of the dark-dark soliton. For the untrapped case we find that when the interaction between components is repulsive the dark-dark soliton can be seen as a bound state of two dark solitons performing a relative harmonic motion. For attractive interactions such a bound state can not exist because the dark solitons repel each other. An equivalent study is also performed for the confined case, where the competition between soliton interactions and harmonic trapping is found to produce sizable changes on the dynamics, leading for example to bound states even with attractive interactions. Our analytical results are tested with different numerical techniques: direct simulations of the Gross-Pitaevskii equations, Fourier analysis to extract characteristic frequencies of the motion, and numerical solution of the Bogoliubov equations for the linear excitations of stationary states. In what follows, we first introduce the theoretical model in section II, and next we show our numerical results in section III. To sum up, we present our conclusions in section IV.

Theoretical 