As we said before, we are going to deal with the coupling between condensates in a perturbative way. Then, assuming that $g_{12}<<g$, the background densities Eq. (\[eq:n\_i\]) become $n_i\approx \mu_i/g-(g_{12}/g)\,\mu_j/g$, and neglecting terms proportional to $g_{12}^2$ in Eq. (\[eq:v2\]) we get $$\begin{aligned}
\begin{aligned}
&i \frac{\partial v_i}{\partial t}  + 
\frac{1}{2}\frac{\partial^2 v_i}{\partial z^2}- \mu_i  
\left(|v_{i}|^2-1\right)v_i = 
\\
&=- \frac{g_{12}}{g}\mu_j 
\left(|v_{i}|^2-|v_{j}|^2\right)v_i\equiv \frac{g_{12}}{g}\mu_j 
P(v_i,v_j).
\end{aligned}
\label{eq:v3}\end{aligned}$$ On the left hand side of this equality we have the well known non-linear Schrödinger equation for the wave function $v_i$, whereas on the right hand side we get a perturbative term in the parameter $g_{12}/g$. In order to obtain analytical expressions for the interactions between solitons, from now on we will focus on the case of equal backgrounds $\phi_1=\phi_2=\phi$, and then $\mu_1=\mu_2=\mu=(g+g_{12})n$. So that, by dividing the whole equation by $\mu$, we set new units for space $\hbar/\sqrt{m\mu}$ and time $\hbar/\mu$. In these units we introduce the general dark soliton solution $$\begin{aligned}
\begin{aligned}
&v_i(z,t)=\cos\varphi_i \tanh \zeta_i + i \sin\varphi_i,
\end{aligned}
\label{soliton}\end{aligned}$$ where $\zeta_i=\cos\varphi_i(z-t\sin\varphi_i )$, and $\varphi_i$ parametrizes the soliton darkness, that is $|v_i|^2=(1-\cos^2\varphi_i \,  
\mbox{sech}^2 \, \zeta_i)$, and the soliton velocity $\dot z_0=\sin \varphi_i$, that is the time derivative of the soliton position $z_0$, taken at minimum 