VI. CONCLUSIONS In this work, we have performed a detailed study of the statics and dynamics of matter- wave dark solitons in a cigar-shaped Bose-Einstein condensate (BEC) confined in a double- well potential. The latter, was assumed to be formed as a combination of a usual harmonic trap and a periodic (optical lattice) potential. For our analysis, we have adopted and used an effectively 1D mean-field model, namely a Gross-Pitaevskii (GP)-like equation with a non-cubic nonlinearity, which has been used successfully in other works to describe dark solitons in BECs in the dimensionality crossover regime between 1D and 3D. As a first step in our analysis, we studied the existence and stability of both the one- and multiple-dark soliton states. In particular, starting from the respective linear problem, we have used the continuation with respect to the chemical potential (number of atoms) to reveal the different branches of purely nonlinear solutions, including the ground state, as well as excited states, in the form of single or multiple dark solitons. We have shown that the presence of the optical lattice is crucial for the existence of nonlinear states that do not have a linear counterpart: these are actually asymmetric dark soliton states, with solitons located in one well (rather than in both wells) of the double-well potential, and do not exist in the non-interacting — linear — limit. We have systematically studied the bifurcations of the various branches of solutions and showed how each particular state emerges or disappears for certain chemical potential thresholds; the latter were determined analytically, in some cases, by using a Galerkin-type approach, in very good agreement with the numerical results. For each branch, we have also studied the stability of the pertinent solutions via a Bogoliubov-de Gennes (BdG) analysis. This way, we have discussed the role of the anomalous (negative energy) modes in the excitation spectra and revealed two types of instabilities of dark solitons, corresponding to either purely imaginary or genuinely complex eigenfrequencies; the latter were found to occur due to the collision of an anomalous mode with a positive energy mode and are associated with the so-called oscillatory instabilities of dark solitons. We have also adopted a simple analytical model to study small-amplitude oscillations of the one- and multiple-dark-soliton states. Particularly, equations of motion for the single and multiple solitons were presented in the weakly-interacting limit (where the model becomes the usual cubic 1D GP equation), and modified them to take into regard the dimensionality of 30