dipole mode eigenfrequency is now significantly modified, upon the reported interval of changes of the chemical potential. The analytical prediction agrees again well with the numerical BdG result in the TF limit. FIG. 5. (Color online) BdG spectra for V$_{0}$ = 1 . 65 × 10 − 13 eV (left panel) and V$_{0}$ = 8 . 3 × 10 − 13 eV (right panel) for the asymmetric dark soliton state, bifurcating from the symmetric one. The notation of eigenstates is the same as in Fig. 4. B. Two dark-soliton states Next, we consider the continuation with increasing chemical potential of the asymmetric states bifurcating from the symmetric one when the latter becomes unstable. Fig. 5 shows the BdG spectra along the asymmetric one-soliton state for V$_{0}$ = 1 . 65 × 10 − 13 eV and V$_{0}$ = 8 . 3 × 10 − 13 eV. The analytical predictions of Eq. (21) with ω$_{ds}$ = ω$_{z}$$_{/}$√$_{2 and}$ ω$_{ds}$ = 30 . 36Hz, respectively plotted with dashed and solid lines are in good agreement with the respective anomalous mode eigenfrequencies. In the case of V$_{0}$ = 8 . 3 × 10 − 13 eV, the difference between the result of Eq. (21) using ω$_{ds}$ = ω$_{z}$$_{/}$√$_{2 and}$ ω$_{ds}$ = 30 . 36 Hz is small, showing that the dynamics is mostly governed by the potential term stemming from the optical lattice. We now consider the case of the two-soliton state, for which we need to incorporate the interaction potential in order to describe their dynamics. Particularly, according to the theoretical framework of Sec. II.B, the effective potential describing the dynamics of two solitons located at z$_{1}$ and z$_{2}$ has the following form: V$_{eff}$ ( z$_{1}$, z$_{2}$ ) = ω 2 $_{ds}$( z 2 1 + z 2 $_{2}$) + V$_{0}$ 2 (cos(2 kz$_{1}$ ) + cos(2 kz$_{2}$ )) + 2 n$_{0}$ sinh $^{2}$( √ n$_{0}$ ( z$_{2}$ − z$_{1}$ )) . (23) 21  0  50  100  150  2.4  2.8  3.2 r  (2 Hz)  (10$^{-12}$ eV)  0  50  100  150 r  (2 Hz)  0  5  2.4  2.8  3.2 i  (2 Hz)  (10$^{-12}$ eV)