for all choices of (Ω , V$_{0}$, k ). This equilibrium corresponds to the symmetric one-soliton state, namely a stationary black soliton located at the center of the potential. In the absence of the optical lattice (i.e., for V$_{0}$ = 0) this is the only existing fixed point, i.e., in the case of a purely harmonic trap, only a symmetric one-soliton state can exist, which originates from the first excited state of the linear problem. On the other hand, when the optical lattice is present (i.e., for V$_{0}$ = 0), this situation changes and more fixed points arise when the slope of the curve f$_{2}$ ( z$_{o}$ ) at the origin becomes larger than the one of the straight line f$_{1}$ ( z$_{o}$ ), namely when k $^{2}$V$_{0}$ /ω 2 ds > 1, two new fixed points appear through a supercritical pitchfork, with the newly emerging states corresponding to the bifurcating asymmetric one-soliton states. Therefore, the bifurcating state appears only for a sufficiently strong optical lattice, such that V$_{0}$ > V$_{0}$$_{,cr}$ ≡ ω 2 $_{ds}$/k$^{2}$. Note that for even larger values of the parameter k $^{2}$V$_{0}$ /ω 2 ds more fixed points appear, corresponding to solitons located in further (i.e., more remote from the trap center) wells of the optical lattice; for this reason, these states are not considered herein. Let us now investigate the spectral stability of the fixed points employing the linearization technique (see, e.g., Ref. [70]). Particularly, we first find the corresponding Jacobian matrix (evaluated at the fixed points), which is given by: When evaluated at the trivial fixed point ( z$_{o}$, p$_{o}$ ) = (0 , 0), the corresponding eigenvalue problem for the Jacobian, det( J − λ I ) = 0, leads to eigenvalues λ 2 = − ω 2 $_{ds}$+ k $^{2}$V$_{0}$ . It is clear that in the case of k $^{2}$V$_{0}$ /ω 2 ds < 1, the eigenvalues are imaginary (note that, accordingly, the eigenfrequencies are real since ω = iλ ) and, thus, the fixed point (0 , 0) is a center. In this case, if the dark soliton is weakly displaced from the center of the trap, it performs harmonic oscillations around the center with an oscillation frequency given by ω$_{osc}$ = √ ω 2 ds − k $^{2}$V$_{0}$ . On the other hand, if k 2 V 0 ω 2 ds > 1 then the eigenvalues become real (and, accordingly, the eigenfrequencies are imaginary), hence (0 , 0) becomes a saddle point, while the respective imaginary eigenfrequency is given by ω$_{osc}$ = i √ | ω 2 ds − k $^{2}$V$_{0}$ | . For the newly bifurcating (stable, since V$_{0}$ > ω 2 $_{ds}$/k$^{2}$) fixed points one can obtain the following eigenfrequency J =   0 1 − ω 2 ds + k $^{2}$V$_{0}$ cos(2 kz$_{0}$ ) 0   . ω$_{osc}$ = √ ω 2 ds − k $^{2}$V$_{0}$ cos(2 kz$_{0}$ ) . (21) 17