soliton has enough energy to pass through the energy barrier created at the trap center $z_0=0$, so that collisions between them are observed to cause a shift in their trajectories. If instead the two dark solitons are initially situated at different locations, close to the positions of the fixed points Eq. (\[eq:fixed\_double\]) (see Fig. \[fig:osci\_outcenter\_00125\]), the solitons oscillate symmetrically around such points. Their time evolution can be fitted by $x(t)=z_0^*+\left(-z_0^*+z_0 \right)\cos(w_{\rm eff}t)$, where $w_{\rm eff}$ is given by , and provides a good approximation to the real time evolution obtained from the GP equation.

![Same as Fig. \[fig:doublewell0046\] with $g_{12}/g=-0.0125$. In this spatio-temporal evolution each dark soliton is situated off center ($z_0=\pm0.3$, near the equilibrium points given by ). The continuous line is fitted according to our analytical results (see text).[]{data-label="fig:osci_outcenter_00125"}](osci_outcenter_00125198){width="\columnwidth"}

### Repulsive interaction between condensates: ${g_{12}>0}$

![(Color online) Numerical results from the Bogoliubov spectrum of a dark-dark soliton with $\mu=10 \,\Omega$ in a trap. The real (top panel) and imaginary (bottom panel) parts of the excitation frequencies are shown against the interaction strength ratio $g_{12}/g$. The continuous (orange) line represents the analytical values from Eq. .[]{data-label="fig:spectrumrealimag"}](spectrum_trap01_mu1.pdf){width="1.0\columnwidth"}

Figure \[fig:spectrumrealimag\] represents the excitation spectrum of overlapped dark-dark solitons at $z_0^*=0$, in 