where ∗ denotes complex conjugate. This way, we derive from Eq. (3) the following BdG equations: where µ is the chemical potential, and the functions f and g are given by f = g $_{+}$√$_{1 + 4}$$_{n}$$_{0}$$_{,}$ and g = 2 ψ 2 $_{$_{0}$/}$√$_{1 + 4}$$_{n}$$_{0}$ (with n$_{0}$ ≡ | ψ$_{0}$ | $^{2}$). Then, solving Eqs. (5)-(6), we determine the eigenfrequencies ω ≡ ω$_{r}$ + iω$_{i}$ and the amplitudes u and υ of the normal modes of the system. Note that if ω is an eigenfrequency of the Bogoliubov spectrum, so are − ω , ω ∗ and − ω ∗ (due to the Hamiltonian nature of the system), and consequently the occurrence of a complex eigenfrequency always leads to a dynamic instability. Thus, a linearly stable configuration is tantamount to ω$_{i}$ = 0, i.e., all eigenfrequencies being real. An important quantity resulting from the BdG analysis is the amount of energy carried by the normal mode with eigenfrequency ω , namely The sign of this quantity, known as Krein sign [60], is a topological property of each eigen- mode. Importantly, if the normal mode eigenfrequencies with opposite energy (Krein) signs become resonant then, in most cases, there appear complex frequencies in the excitation spectrum, i.e., a dynamical instability occurs [60]. In order to further elaborate on such a possibility, it is relevant to note that modes with complex or imaginary frequencies carry zero energy (due to the condition ( ω − ω $^{∗}$) ∫ dz ( | u | 2 − | υ | $^{2}$) = 0, which must hold for each BdG mode), while anomalous modes have negative energy (see, e.g., Sec. 5.6 of Ref. [1]). The presence of anomalous modes in the excitation spectrum is a direct signature of an energetic instability, which is particularly relevant to the case of dark solitons [21]; such excited states of the system are discussed below. Exact analytical dark soliton solutions of NLS equations with a generalized defocusing nonlinearity, such as Eq. (3), are not available. In fact, in such cases, dark solitons may only be found in an implicit form (via a phase-plane analysis) or in an approximate form (via the 6 B. Dark soliton states [ ˆ H − µ + f ] u + gυ = ωu, (5) [ ˆ H − µ + f ] υ + gu = − ωυ, (6) E = ∫ dz ( | u | 2 − | υ | $^{2}$) ω. (7)