FIG. 4. (Color online) BdG spectra for four different values of the optical lattice strength for the symmetric dark soliton state (with the soliton located at the trap center): from left to right, V$_{0}$ = 0, V$_{0}$ = 8 . 3 × 10 − 14 eV, V$_{0}$ = 1 . 65 × 10 − 13 eV and V$_{0}$ = 4 . 96 × 10 − 13 eV. The solid lines correspond to the predictions of Eq. (21). Symbol ⋇ (blue) denotes a positive (or zero for a vanishing eigenfrequency), symbol × (green) a negative and symbol + (red) a vanishing Krein sign mode (for non-zero eigenfrequencies). chemical potential. The figure shows the constant eigenfrequency of the dipolar mode [see straight (blue) line] at the value of the trap frequency ω$_{z}$ and the anomalous mode bifurcating from the dipolar mode in the linear limit. As mentioned above, the eigenfrequency of the anomalous mode reaches asymptotically the value ω$_{ds}$ = 30 . 36 Hz. The second panel of Fig. 4 shows the BdG spectrum for V$_{0}$ = 8 . 3 × 10 − 14 eV. All eigen- frequencies are real, so the state is still stable. The dashed line is the prediction by Eq. (21) using ω$_{ds}$ = ω$_{z}$$_{/}$√$_{2, while the solid line is the prediction with}$ ω$_{ds}$ = 30 . 36 Hz. The dipole mode remains approximately constant for such a small optical lattice strength. However, the degeneracy of the anomalous mode and the dipole mode in the linear limit disappears. The third panel shows the spectra for V$_{0}$ = 1 . 65 × 10 − 13 eV. In this case, the anomalous mode becomes purely imaginary for increasing chemical potential, reflecting the fact that the state becomes unstable. Notice that the numerical result and the analytical prediction, ω$_{ODE}$ = i √ | ω 2 ds − k $^{2}$V$_{0}$ | , are in a good agreement for large values of the chemical potential, where the TF limit is reached and Eq. (19) is valid. Since V$_{0}$ > V$_{0}$$_{,cr}$ , this state is expected to be unstable in the TF limit. The fourth panel shows the spectra for V$_{0}$ = 4 . 96 × 10 − 13 eV. In this case, the anomalous mode becomes purely imaginary already at small chemical potentials. Furthermore, the 20  0  20  40  60  80 r  (2 Hz)  0  20  40  60  2.4  2.8  3.2 i  (2 Hz)  (10$^{-12}$ eV)  0  20  40  60  80 r  (2 Hz)  0  20  2  2.4  2.8  3.2 i  (2 Hz)  0  20  40  60  80  2  2.4  2.8  3.2 r  (2 Hz)  (10$^{-12}$ eV)  0  20  40  60  80  2  2.4  2.8  3.2 r  (2 Hz)  (10$^{-12}$ eV)