the numerical solution of the excitation spectrum from Eq. for the out-of-phase anomalous mode (dash-dotted lines).[]{data-label="fig:freqg12nega00175"}](Freq_g12_neg_a_00175.pdf){width="\columnwidth"}

As anticipated, in this case the configuration of overlapped solitons at $z_0*=0$ is unstable for $|g_{12}/g|>g_c$. This instability can also be detected by the appearance of an imaginary frequency in the excitation spectrum. Fig. \[fig:freqg12nega00175\] shows our results for the linear excitations of such a stationary state with $\mu=10\, \Omega$ from the solution of the Bogoliubov equations (dash-dotted lines). The frequency of the out-of-phase anomalous mode (see a discussion of this mode in next section) takes real values for $|g_{12}/g|<g_c$ and pure imaginary for $|g_{12}/g|>g_c$. This critical point $g_c$ is associated to the change of the total effective potential, from single-well to double-well.

Along with the change of the total effective potential two new stable fixed points appear in the system. In Fig. \[fig:eqpos2\] we compare these points (red dots), obtained by extracting the mean position of the soliton oscillations around the equilibrium positions, with our analytical approach Eq. (orange line).The latter fails for increasing values of $|g/g_{12}|$. However the direct numerical computation of Eq. (blue line) provides a very good agreement for the regime of interest, and shows that the distance between equilibrium 