of the chemical potential of the system.

### Attractive coupling: ${g_{12}<0}$

The phenomenology is richer for attractive interactions between particles of different condensates. In contrast to the untrapped case, the presence of the harmonic potential allows for the generation of local minima in the total effective potential that can support stationary states made of two solitons. As shown in Fig. \[fig:potg12nega001\], by increasing the parameter $|g_{12}/g|$, the total effective potential modifies from a single well (an also a unique fixed point $z_0^*=0$ in the equation of motion) to a double well potential (with three fixed points). Specifically, the system shows a pitchfork bifurcation at $g_c=|g_{12}/g|={15}(\Omega/\mu)^2/16$, hence for $|g_{12}/g|>g_c$ the fixed point at $z_0^*=0$ loses its stability and two new off-center, stable fixed points appear. From Eq. (\[eq:cl2\]) we get their position at $$z_{0_\pm}^{*} = \pm \frac{1}{\sqrt{\mu}}\sqrt{\frac{21}{40}-
\frac{63}{128} \left|\frac{g}{g_{12}}\right| 
\left(\frac{\Omega}{\mu}\right)^2 }.
\label{eq:fixed_double}$$

![(Color online) Stationary state made of two solitons situated at the fixed points $z_{0_\pm}^*$ of the effective potential (see text) for $\mu=10\,\Omega$ and $g_{12}/g=-0.1$.[]{data-label="fig:separated"}](separated_solitons){width="\columnwidth"}

As can be noted, the separation between minima increases with $\mu/\, \Omega$ for given $g_{12}/g$, and saturates at a distance of $2z_0^*=1.44$. Fig. \[fig:separated\] shows an example of a stationary state with two separated solitons 