potential. As concerns the bifurcations of the various branches of solutions, we show that particular states emerge or disappear for certain chemical potential thresholds, which are determined analytically, in some cases, by using a Galerkin-type approach [50]. For each branch, we also study the stability of the pertinent solutions via a Bogoliubov-de Gennes (BdG) analysis, which reveals the role of the anomalous (negative energy) modes in the excitation spectra as concerns the emergence of oscillatory instabilities of dark solitons: it is found that such instabilities occur due to the collision of an anomalous mode with a positive energy mode. Furthermore, based on the weakly-interacting limit of the model, we study analytically small-amplitude oscillations of the one- and multiple-dark-soliton states. Particularly, we obtain equations of motion for the single and multiple solitons from which we determine the characteristic soliton frequencies; the latter, are found to be, in many cases in good agreement with the eigenfrequencies found in the framework of the BdG analysis. The paper is organized as follows. We present our model in section II and provide the theoretical framework of our analysis. In Sec. III, we study the bifurcations of the branches of the solutions of the model and study their stability via the BdG equations. In Sec. IV, we provide some analytical estimates for the soliton frequencies based on the dynamics of solitons and compare our findings to the ones obtained by the BdG analysis. Sec. V is devoted to a study of the soliton dynamics numerically, and shows the manifestation of instabilities when they arise. Finally, in Sec.VI, we present our conclusions. We consider a BEC confined in a highly elongated trap, with longitudinal and transverse confining frequencies (denoted by ω$_{z}$ and ω ⊥ , respectively) such that ω$_{z}$ ≪ ω ⊥ . In such a case, use of the adiabatic approximation, together with a variational approach for determin- ing the local transverse chemical potential, leads to the following 1D GP equation for the longitudinal condensate’s wave function [57, 59]: where ψ ( z, t ) is normalized to the number of atoms, i.e., N = ∫ + ∞ −∞ | ψ | $^{2}$dx , a is the s -wave scattering length, m is the atomic mass, and V$_{ext}$ ( z ) is the longitudinal part of the external i ℏ ∂$_{t}$ψ = [$_{−}$ ℏ 2 2 m ∂ 2 z + V$_{ext}$ ( z ) + ℏ ω ⊥ √ 1 + 4 a | ψ | 2 ] ψ, (1) 4 II. MODEL AND THEORETICAL FRAMEWORK A. The effective 1D mean-field model and BdG analysis