condensates their frequency coincide with the oscillation frequency of the solitons in the trap [@Busch2000]. There are also two different anomalous modes for non null $g_{12}$, an in-phase one with constant energy, and an out-of-phase mode whose energy increases with $g_{12}$. The in-phase anomalous mode is associated with the small amplitude, abreast oscillations of the solitons in the trap, hence it is the same as in decoupled condensates: $\omega_0=\Omega/\sqrt{2}$. However, the out-of-phase anomalous mode is associated with the relative motion of the solitons. This mode, given by the analytical expression Eq. , is depicted (orange line) in the upper panel of Fig. \[fig:spectrumrealimag\], in good agreement with the numerical results for small values of $g_{12}/g$.

The lower panel of Fig. \[fig:spectrumrealimag\] present our numerical results for the imaginary part of the spectrum. As previously commented in the untrapped case, these instabilities are characterized by the collision of the out-of-phase anomalous mode with the spin mode associated with the background. These collisions produce Hamiltonian-Hopf bifurcations where a complex frequency quartet appears in the excitation spectrum. It is interesting to see that the out-of-phase anomalous mode only collides with odd spin modes. Regions of stability and instability alternates up to a value 