the system (i.e., the effect of transverse directions). The analysis of these equations of motion led to the determination of the characteristic soliton frequencies, which were then compared to the eigenfrequencies found in the framework of the BdG analysis; the agreement between the two was generally found to be very good. Perhaps even more importantly, the simple physical (i.e., particle) picture we adopted allowed us to reveal the role of the optical lattice strength (i.e., the height of the barrier in the double-well setting) in the stability of the dark solitons, as well as its role on the coupling between solitons. Particularly, it was found that sufficiently strong barriers lead to instability of symmetric and asymmetric soliton states, while it results in an effective decoupling of multi-solitons: the coupling between neighboring solitons as described by an effective repulsive potential, becomes negligible for sufficiently strong barriers. We should also note that generally the states possessing a dark soliton at the center of the trap were found to be most strongly unstable, due to the emergence of a progressively larger (the larger the nonlinearity) imaginary eigenfrequency. Finally, we have performed systematic numerical simulations, based on direct numerical integration of the quasi-one-dimensional model, to investigate the dynamics of dark solitons in the double-well setup. We studied the manifestation of instabilities, when present, and found in all cases, a very good agreement between the predictions based on the BdG analysis and the numerical results. Instabilities have been observed to manifest themselves through the exponential divergence of the soliton from its initial location (when associated with an imaginary pair of eigenfrequencies) or the oscillatory growth (when associated with a complex quartet of eigenfrequencies). An interesting direction for a future study would be a similar analysis in a higher- dimensional setting: this would refer to vortices confined in double-well potentials (or more complex “energy surfaces” such as quadruple-well potentials [71]) in higher-dimensional BECs. Such studies are presently in progress. The work of D.J.F. was partially supported by the Special Account for Research Grants of the University of Athens. P.G.K. gratefully acknowledges support from NSF-DMS-0349023 (CAREER) and NSF-DMS-0806762, as well as from the Alexander von Humboldt Founda- ACKNOWLEDGMENTS. 31