has been anticipated, it presents a minimum at $z_0=0$, allowing for the existence of a bound state made of overlapped solitons, and tends exponentially to zero for finite values of the inter-soliton distance $2z_0$ (see Fig. \[fig:pert\]). For small separation $z_0\rightarrow 0$, the Taylor expansion reads $$\begin{aligned}
U_{\rm eff}^{(i)}(z_0)\sim 
\frac{g_{12}}{g}\left(-\frac{1}{6}+\frac{4}{15} z^2_0 \right) +O[z^4_0]\,,
$$

![Top panel: Effective force experienced by a dark soliton forming a dark-dark soliton system for $g_{12}/g=0.01$ in the absence of trapping. Bottom panel: Soliton interaction potential Eq. (\[eq:int\_pot\]) from which the above effective force is derived. The solitons oscillate in a relative motion around the minimum at $z_0=0$.[]{data-label="fig:pert"}](Pot_g12_pos_a_001b){width="\columnwidth"}

and the equation of relative motion Eq. (\[eq:rel\]) simplifies to that of a harmonic oscillator, $\ddot z_0+w_r^2 \,z_0=0$, with angular frequency (for generic chemical potential $\mu$) $$w_r=\mu\sqrt{\frac{8}{15}\frac{g_{12}}{g}}.
 \label{eq:freq_rel}$$ This is the frequency of small oscillations of the two solitons around their center of mass, moving at constant velocity $v_{CM}=(\sin 
\varphi_1+\sin \varphi_2)/2$. It is also interesting to note that this model predicts an instability (imaginary frequency) when $g_{12}<0$, due to the fact that the effective potential $U_{\rm eff}^{(i)}$ presents a maximum for attractive interparticle interactions between particles of different condensates, and as a consequence, it prevents the existence of 