occupying the two minima of the effective potential at $\mu=10 \, \Omega$ and $g_{12}/g=-0.1<g_c$.

For small distances $\epsilon(t)$ around the fixed points $z_{0_\pm}^*$, from the substitution of $z_0(t)=z_0^*+\epsilon(t)$ in Eq. (\[eq:cl2\]), the solitons oscillate according to $$\ddot{\epsilon}=-\left( w_1(z_{0_\pm}^*)^2 +w^2 \right) \epsilon \equiv 
-w_{\rm eff}^2 \epsilon ,
\label{eq:epsilon}$$ up to linear terms in the perturbation $\epsilon(t)$, where $w$ is the angular frequency given by , and $ w_1(z_0)^2=-2 f(z_0)  
\left(  \mbox{sech}(z_0)- {5}/{\sinh(2z_0)} + 3\right)-$ $\left( \left(1+4e^{4z_0} +e^{ 8z_0}\right)+
\left(-12e^{8z_0}+\left(16e^{4z_0}+8e^{8z_0}\right) z_0 \right) \right)$ $ 64e^{6z_0} / \left(e^{4z_0}-1\right)^5 $.

![(Color online) Bogoliubov modes in 1D rings of different size (untrapped case). The upper panel depicts the real part of the frequencies and the lower panel the imaginary parts. The solid and dashed lines correspond to Eqs. (\[eq:freq\_rel\]) and (\[eq:spin\]), respectively, the crossing of which indicates the appearance of instabilities.[]{data-label="fig:bog_notrap"}](modes_mu05_Z10.pdf){width="\columnwidth"}

Numerical results
=================

In what follows, in order to test our analytical predictions on the dark-dark-soliton dynamics, we first numerically solve the GP equation to obtain these stationary states for varying chemical potentials and interactions strengths. Afterwards, the soliton stability is monitored both in the nonlinear regime (by simulating the real time evolution with the GP equation) and by linear analysis around the equilibrium 