$\psi_i$ and $\phi_i$ satisfy the coupled GP equations for the same chemical potential: $$\begin{aligned}
&\mu_i \phi_i = 
\left(-\frac{1}{2}\frac{\partial^2}{\partial z^2}+\frac{1}{2}\Omega^2 z^2+
g \phi_{i}^2 + g_{12} \phi_{j}^2 \right)\phi_{i}
\label{eq:ebmr20}
\\
&i \frac{\partial}{\partial t} \psi_i = 
\left(-\frac{1}{2}\frac{\partial^2}{\partial z^2}+\frac{1}{2}\Omega^2 z^2+
g |\psi_{i}|^2 + g_{12} |\psi_{j}|^2 \right)\psi_{i} \,,
\label{eq:ebmr2}\end{aligned}$$ where $\Omega=\Omega_z/\Omega_\perp$ is the trap aspect ratio, and we have written the equations in dimensionless form by using transverse trap units, $a_\perp=\sqrt{\hbar/m\Omega_\perp}$ as unit length and $t_\perp=\Omega_\perp^{-1}$ as unit time, and for the sake of a simple notation have kept the same letters for the dimensionless couplings ${g}$ and ${g}_{12}$. So, from Eqs. (\[eq:ebmr20\])-(\[eq:ebmr2\]) we can obtain corresponding equations for the soliton wave functions $v_i(z,t)$ $$\begin{aligned}
&\left(i \frac{\partial}{\partial t}  
+ \frac{1}{2}\frac{\partial^2 }{\partial z^2}- g\phi^2_{i} 
\left(|v_{i}|^2-1\right) - g_{12}\phi^2_j\left(|v_{j}|^2-1\right)\right)v_i=
\nonumber\\
&\hspace{.5cm} =-\frac{\partial v_i}{\partial z}\,\frac{\partial \ln 
\phi_i}{\partial z}\,,
\label{eq:v}\end{aligned}$$ where the external trap does not appear explicitly, although its information is encoded in the background wavefunctions $\phi_i$.

Untrapped case
--------------

Here we set $\Omega=0$, hence the ground state solutions to Eq. (\[eq:ebmr20\]) are the constant density $|\phi_i|^2=n_i$ states $$n_i=\frac{g\mu_i-g_{12}\mu_j}{g^2-g^2_{12}}. 
\label{eq:n_i}$$ The equations of motion for soliton states Eq. (\[eq:v\]) can be written as $$\begin{aligned}
\left(i \frac{\partial }{\partial t}  +
\frac{1}{2}\frac{\partial^2 }{\partial z^2}
- g\,n_i\, \left(|v_{i}|^2-1\right)
- g_{12}\,n_j\,  \left(|v_{j}|^2-1\right)\right)v_i=0.
\label{eq:v2}\end{aligned}$$

