the range $g_{12}/g\in[0,1]$, for $\mu=10\, \Omega$, within the Thomas Fermi regime of the axial harmonic oscillator. At $g_{12}=0$, corresponding to uncoupled solitons, the linear modes have double degeneracy, and the lowest energy excitations are the anomalous mode, with frequency $\omega_0=\Omega/\sqrt{2}$, and the hydro-dynamical excitations $\omega_n=\sqrt{{n(n+1)}/{2}}\,\Omega$, with $n=1,2,...$ [@Stringari1996; @Busch2000; @Kevrekidis2015; @Frantzeskakis2010]. The degeneracy is broken for non null $g_{12}$ and gives rise to two branches of in-phase and out-of-phase modes. In the hydro-dynamical case, the in-phase modes are associated to excitations in the total density of the background, and remain constant for varying $g_{12}$. On the other hand, the out-of-phase or spin modes account for variations in the difference of the background densities [@Abad2013], and decrease their energy for increasing values of $g_{12}$.

![Real time evolution of three dark-dark solitons with varying coupling out of and inside the first instability region of Fig. \[fig:spectrumrealimag\], showing corresponding stable and unstable dynamics. []{data-label="fig:instab_trap"}](abc_evolution.pdf "fig:"){width="\columnwidth"}\
![Real time evolution of three dark-dark solitons with varying coupling out of and inside the first instability region of Fig. \[fig:spectrumrealimag\], showing corresponding stable and unstable dynamics. []{data-label="fig:instab_trap"}](imaginary_freq_mu_1.pdf "fig:"){width="0.9\columnwidth"}

The anomalous modes are characterized by the negative value of the quantity $norm \times 
energy$ [@MacKay1987], and in scalar 