function of the optical lattice strength V$_{0}$ . The dash-dotted (green) line denotes the non-zero fixed point calculated from Eq. (20) with ω$_{ds}$ = ω$_{z}$$_{/}$√$_{2, while the solid (black) line denotes}$ the non-zero fixed point calculated from Eq. (20) with ω$_{ds}$ = 30 . 36 Hz. Furthermore, the circles denote the position of the node of the nonlinear stationary state — i.e., the position of the dark soliton — obtained by numerical integration of Eq. (1) with µ = 3 . 8 × 10 − 12 eV (in the TF limit). Clearly, at V$_{0}$$_{,cr}$ ≃ 9 . 6 × 10 − 14 eV, the asymmetric one-soliton branch bifurcates through a supercritical pitchfork bifurcation; notice the excellent agreement between the two approaches. The bottom left and right panels of Fig. 3 show the BdG spectra of the symmetric and asymmetric one soliton state, respectively, as a function of the optical lattice strength (for a fixed chemical potential, µ = 3 . 8 × 10 − 12 eV). For sufficiently small lattice strength, V$_{0}$ < V$_{0}$$_{,cr}$ , the symmetric state is stable, while when the lattice strength V$_{0}$ is increased, the magnitude of the instability grows and the prediction of Eq. (21) deviates from the results obtained from the integration of Eq. (3). To explain this discrepancy, it is relevant to recall that the result of the perturbation theory of Sec. II.B is valid only if all perturbation terms are small and of the same order. Nevertheless, for the symmetric one soliton state — with the soliton located at zero — the magnitude of the perturbation term corresponding to the harmonic trap (optical lattice) has its smallest (largest) value; thus, for large V$_{0}$ , these perturbations cannot be of the same order. As concerns the asymmetric state, the prediction of Eq. (21) is in fairly good agreement to the results obtained from the numerical integration of Eq. (3). However, Eq. (21) cannot predict oscillatory instabilities (associated with the existence of complex eigenfrequencies), which occur when the anomalous mode collides with a mode of the background condensate. This can readily be understood by the fact that the derivation of the equation of motion of the dark soliton relies on the assumption that the soliton is decoupled from the background (see, e.g., Refs. [25] and [3] for details), which is not the case for oscillatory instabilities. Next, let us study the continuation of both symmetric and asymmetric one-soliton states with increasing chemical potential for various values of the optical lattice strength. First, we consider the symmetric one-soliton state and in Fig. 4 we show the BdG spectra along the corresponding branch. The first panel shows the case of a pure harmonic trap, i.e., V$_{0}$ = 0, and the dashed line is the prediction from Eq. (21) using ω$_{ds}$ = ω$_{z}$$_{/}$√$_{2. It is clear}$ that the eigenfrequencies are real, indicating the stability of this state for all values of the 19