model
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The dynamics of a BEC at zero temperature can be accurately described within the mean field approach in terms of a wave-function $\psi(\mathbf{r},t)$. In a one dimensional setting, the wave functions of two trapped, density-coupled BECs are governed by corresponding Gross-Pitaevskii equations $$\begin{aligned}
\begin{aligned}
&i\hbar \frac{\partial}{\partial t} \psi_1 = 
\left(\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial z^2}+\frac{m\Omega_z^2 
z^2}{2}+ g |\psi_{1}|^2 + g_{12} |\psi_{2}|^2 \right)\psi_{1}
\\
&i\hbar \frac{\partial}{\partial t} \psi_2 = 
\left(\frac{-\hbar^2}{2m}\frac{\partial^2}{\partial z^2}+\frac{m\Omega_z^2 
z^2}{2}+ g |\psi_{2}|^2 + g_{12} |\psi_{1}|^2 \right)\psi_{2}\, ,
\end{aligned}
\label{eq:ebmr_dim}\end{aligned}$$ where $\Omega_z$ is the angular frequency of the axial harmonic trapping, $g=2\hbar \Omega_\perp a$ is the reduced 1D strength of the interaction between particles of the same condensate, which is proportional to the energy of the transverse trapping $\hbar \Omega_\perp$ and to the scattering length $a$, and $g_{12}=2\hbar \Omega_\perp 
a_{12}$ is the strength of the density coupling between condensates, proportional to the scattering length between particles of different condensates $a_{12}$.

A stationary soliton solution to Eq. (\[eq:ebmr\_dim\]) $\psi_i(z,t)$, with $i=1,2$, can be written as the product of a time-independent (and real function) background component $\phi_i$ and the soliton excitation $v_i$, so that $$\begin{aligned}
\psi_i(z,t)=\phi_i(z) v_i(z,t) e^{-i \mu_i t/\hbar},
\label{DS}\end{aligned}$$ where $\mu_i$ is the chemical potential. It is important to remark that both 