with the term associated to the interaction between dark solitons, contains a new term introduced by the external trap [@Frantzeskakis2010].

Again, we focus on the symmetric case $\mu_1=\mu_2=\mu$, where the local chemical potential reads $\mu(z)=\mu-V(z)=(g+g_{12})\,n(z)$. In the particular case of motion around the center of the harmonic potential $\mu(z)\rightarrow \mu$, and by following the same procedure as in the untrapped case (with $\hbar/\sqrt{m\mu}$ and $\hbar/\mu$ as space and time units respectively) we can obtain a particle-like evolution for the soliton position $z_0$

![(Color online) Forces (top panel) and total, single-well potential (bottom panel) experienced by each dark soliton forming a dark-dark soliton system with $\mu=1$ and $g_{12}>0$ in the presence of a harmonic trap with $\Omega=0.1$. The dashed (red) line represents the force caused by the trap, whereas the continuous lines represent the effective force due to the interaction with the other dark soliton.[]{data-label="fig:potg12posa001"}](Pot_g12_pos_a_001){width="\linewidth"}

$$\begin{aligned}
\ddot{z}_0=-\frac{\Omega^2}{2 \mu^2} z_0- 
F_{\rm eff}^{(i)} ,
\label{eq:freq_trap}\end{aligned}$$

where the right hand side is the total effective force acting on the soliton $F_{\rm eff}=-dU_{\rm eff}/d\,z_0$, and embraces the superposition of the action of the external trap and the soliton interaction. For small-amplitude oscillations $z_0\approx 0$, we get $$\begin{aligned}
\ddot{z}_0=-\left(\frac{\Omega^2}{2\mu^2}+\frac{8}{15}\frac{g_{12}}{g}
\right)z_0+\frac{64}{63}\frac{g_{12}}{g}z_0^3+O(z_0^5),
\label{eq:cl2}\end{aligned}$$ where the balance between the 