of the coupling with $g_{12}/g$ close to 1, but interestingly inside the immiscible regime. The higher the chemical potential, the closer is this value to $g_{12}/g=1$, according with the Bogoliubov analysis for the untrapped case. Such a value can be well approximated within the Thomas-Fermi regime by Eq. (\[eq:spin\]) evaluated at maximum density $n=\mu/(g+g_{12})$. Again the crossing of this analytical frequency with the function for small oscillations Eq. (\[eq:fre\]) provides a good estimate for the beginning of the first instability region in the range $g_{12}/g\in[0,1]$. It is worth noticing that such instability is not captured by Eq. alone, due to the fact that for its derivation the soliton motion was assumed to be decoupled from the background.

Figure \[fig:instab\_trap\] shows examples of stable and unstable dark-dark solitons near the first instability region of Fig. \[fig:spectrumrealimag\]. The evolution of the nonlinear systems develops according the linear stability analysis and demonstrate the existence of dynamically stable coupled solitons, cases (a) and (c), that could be experimentally realized.

Conclusions
===========

The dynamics of dark-dark soliton states in two density coupled BECs has been studied within GP theory. By performing a perturbation analysis in the parameter $g_{12}/g$ for solitons with equal chemical potential, we 