We study the statics and dynamics of dark solitons in a cigar-shaped
Bose-Einstein condensate confined in a double-well potential. Using a
mean-field model with a non-cubic nonlinearity, appropriate to describe the
dimensionality crossover regime from one to three dimensional, we obtain
branches of solutions in the form of single- and multiple-dark soliton states,
and study their bifurcations and stability. It is demonstrated that there exist
dark soliton states which do not have a linear counterpart and we highlight the
role of anomalous modes in the excitation spectra. Particularly, we show that
anomalous mode eigenfrequencies are closely connected to the characteristic
soliton frequencies as found from the solitons' equations of motion, and how
anomalous modes are related to the emergence of instabilities. We also analyze
in detail the role of the height of the barrier in the double well setting,
which may lead to instabilities or decouple multiple dark soliton states.Comment: 35 pages, 12 figure
