points increases with $|g_{12}/g|$ instead of being saturated at $2 z_0^*=1.44$.

![(Color online) Fixed points of the effective double well potential given by Eq. (continuous blue line) as a function of $g_{12}/g$. The continuous (orange) line represents the analytical result and the (red) dots are the mean position of the soliton oscillations extracted from the numerical solution of GP equation. []{data-label="fig:eqpos2"}](eq_pos2){width="1.\columnwidth"}

![Spatio-temporal evolution of a dark-dark-soliton density obtained from the numerical solution of the GP Eq. for $g_{12}/g=-0.046$ and harmonic trapping $\Omega=0.1$. Each panel corresponds to a different condensate. The dashed (red) lines represent the position of the equilibrium points given by . []{data-label="fig:doublewell0046"}](insta_outcenter_0046){width="\columnwidth"}

We have numerically solved the GP Eq. (\[eq:ebmr\_dim\]) for the real-time evolution of  dark-dark solitons with values of $g_{12}$ below and above the bifurcation point $g_c$ for the change of stability. First, we have computed a case (see Fig. \[fig:doublewell0046\]) with overlapped solitons situated at $z_0=0$ and $g_{12}/g=-0.046<g_c$, hence unstable according to the linear prediction. The initial stationary state has been perturbed with white-noise of $1\%$ amplitude. As expected the system is unstable, and eventually the solitons separate by moving towards the minima of the effective double well potential Eq. (\[eq:fixed\_double\]). In this case, each dark 