a bound solitonic state in this case.

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

In the presence of axial harmonic trapping $V(z)=\frac{1}{2}\Omega^2 z^2$, we rely on the Thomas-Fermi (TF) approximation to study the strong (interparticle) interacting regime ($\mu_i>>\Omega$) [@Pitaevskii2003]. There the inhomogeneous ground state densities, $n_i(z)=|\phi_i|^2$, are given by $$\begin{aligned}
n_i(z)=\frac{g\mu_i(z)-g_{12}\mu_j(z)}{g^2-g^2_{12}},
\label{eq:TF}\end{aligned}$$ where $\mu_i(z)=\mu_i-V(z)$ are the local chemical potentials. In the limit of $g_{12}<<g$, we get $$\begin{aligned}
g\,n_i(z)={\mu_i(z)}-\frac{g_{12}}{g}{\mu_j(z)},\end{aligned}$$ and, as a result, the equations of motion for soliton solutions Eq. (\[eq:v\]) become $$\begin{aligned}
&i \frac{\partial v_i}{\partial t}  + 
\frac{1}{2}\frac{\partial^2 v_i}{\partial z^2}- \mu_i(z)  
\left(|v_{i}|^2-1\right)v_i = 
\nonumber \\
&=- \frac{g_{12}}{g}\mu_j(z) \left(|v_{i}|^2-|v_{j}|^2\right)v_i 
-\frac{\partial v_i}{\partial z}\,\frac{d\ln \phi_i}{dz} .\end{aligned}$$ The local chemical potentials and the last term in the right hand side of this equation are the main differences with respect to the untrapped case Eq. (\[eq:v3\]). In spite of these differences, and due to the fact that the background densities change slowly inside the TF regime ($dV/dz\rightarrow 0$), an analogue perturbative approach can still be followed, and the governing equation for the dark soliton $v_i$ is $$\begin{aligned}
i \frac{\partial v_i}{\partial t}  + \frac{1}{2}\frac{\partial^2 v_i}{\partial 
z^2}-\mu_i(z) v_i \left(|v_{i}|^2-1\right)=P(v_i,v_j;V(z)),
\label{eq:v5}\end{aligned}$$ where the perturbation $P(v_i,v_j;V)$ is given by $$\begin{aligned}
&P(v_i,v_j;V)\approx - \frac{g_{12}}{g} \mu_j(z)
\left(|v_{i}|^2-|v_{j}|^2\right)v_i+
\nonumber\\
&\hspace{.5cm} 
+\frac{1}{2\mu(z)}\frac{dV}{dz}\frac{\partial v_i}{\partial 
z},\end{aligned}$$ which, along 