two harmonic force terms (inside the parenthesis) leads to different scenarios depending on the sign of the coupling interaction $g_{12}$.

![(Color online) Same as Fig. \[fig:potg12posa001\] for $g_{12}<0$. Three cases are depicted for different $g_{12}/g$ values, below (blue), at (orange) and above (green) the threshold $g_c$ indicating the change in the shape of the total effective potential from single well $g_{12}/g< c_c$ to double well $g_{12}/g>g_c$ (see text). []{data-label="fig:potg12nega001"}](Pot_g12_neg_a_001){width="\columnwidth"}

### Repulsive coupling: ${g_{12}>0}$

Figure \[fig:potg12posa001\] shows the effective force and effective potential defined in Eq. (\[eq:freq\_trap\]) for varying values of $g_{12}/g$ at a given trapping $\Omega=0.1$. As can be seen, the different parameters do not produce qualitative changes in the potential, which presents a single minimum capable to bound the coupled solitons. As in the untrapped case, our theory predicts that the dark solitons will perform small-amplitude oscillations around $z_0=0$ with angular frequency (in transverse oscillator units) $$w=\left(\frac{\Omega^2}{2}+\frac{8}{15}\frac{g_{12}}{g} 
\mu^2\right)^\frac { 1 } { 2 }.
\label{eq:fre}$$

As we will see later, our numerical results demonstrate that the minimum supports a bound state that exists and is stable for small $g_{12}/g$. However, for increasing values of this parameter both stable and unstable cases can be found depending on the particular valued 