and down panel correspond to component 1 and 2 respectively. The continuous (colour) lines represent the evolution of the dark solitons positions given by Eq. (\[eq:freq\_rel\]).[]{data-label="fig:osci003"}](untrap_osci_0015){width="\columnwidth"}

![Same as Fig. \[fig:osci003\] but with $g_{12}=0.1$. []{data-label="fig:osci01"}](untrap_osci_01){width="\columnwidth"}

![(Color online) Soliton relative motion frequency as a function of the interparticle strength ratio in the absence of external trap. The continuous (blue) line represents our theoretical prediction Eq. (\[eq:freq\_rel\]) and the (orange) dots are values extracted by means of Fourier analysis from the numerical solution of the time dependent GP equation.[]{data-label="fig:fourier"}](Fourier2){width="\columnwidth"}

The excitation frequencies of the stationary states can be readily extracted by solving the Bogoliubov equations Eq. (\[eq:BdG\]). In the Appendix we analytically show that there are two zero modes of excitation at $g_{12}/g=0$, and $1$, and hence that the system of overlapped dark solitons is expected to be unstable in between. However such analysis assume an infinite system, and the situation is quite different in systems of finite size, where ranges of dynamical stability can be found. Fig. \[fig:bog\_notrap\] shows two examples of this phenomenon, where the Bogoliubov modes are computed for overlapped solitons with the same chemical potential $\mu=0.5$ in 1D rings of different sizes $L=20,\; 40$. The instability emerge from a 