by the repulsive interparticle interactions $g_{12}>0$, we excite the relative motion of the solitons by imposing the initial ansatz $$\begin{aligned}
\begin{aligned}
&\psi_1=\sqrt{n}\tanh\left(z-z_0\right)
\\
&\psi_2=\sqrt{n}\tanh\left(z+z_0\right),
\end{aligned}\end{aligned}$$ and we fix $\mu=1$. Figures \[fig:osci003\]–\[fig:osci01\] show the comparison between the subsequent motion of the solitons from the numerical solution of GP Eq. , and the analytical prediction by Eq. (\[eq:freq\_rel\]) fitted to $y(t)=z_0\cos{wt}$. As can be seen, it provides a reasonable good estimate for small $g_{12}$ (Fig. \[fig:osci003\]) but fails for larger $g_{12}$ (Fig. \[fig:osci01\]) or also long times.

In order to obtain a more quantitative comparison we have run different real time evolutions for varying $g_{12}$ and equal chemical potential $\mu=1$. By tracking the position of the solitons (at minimum density), we have computed their characteristic frequency from a Fourier analysis in time. The numerical results are presented in Fig. \[fig:fourier\], and show a very good agreement with our analytical prediction for small values of $g_{12}/g$.

Trapped case
------------

### Attractive interaction between condensates: ${g_{12}<0}$

![(Color online) Frequency of the dark soliton relative motion in a harmonic trap with $\Omega=0.1$. The real (top panel) and imaginary (bottom) parts of this frequency are shown according to our analytical prediction Eq. (continuous orange line) and 