Positive- and negative-mass solitons in Bose-Einstein condensates with optical lattices

We study the dynamics of solitons in Bose-Einstein condensates (BECs) loaded into an optical lattice (OL), which is combined with an external parabolic potential. Chiefly, the one-dimensional (1D) case is considered. First, we demonstrate analytically that, in the case of the repulsive BEC, where the soliton is of the gap type, its effective mass is negative. In accordance with this, we demonstrate that such a soliton cannot be held by the usual parabolic trap, but it can be captured (performing harmonic oscillations) by an anti-trapping inverted parabolic potential. We also study the motion of the soliton in a long system, concluding that, in the cases of both the positive and negative mass, it moves freely, provided that its amplitude is below a certain critical value; above it, the soliton's velocity decreases due to the interaction with the OL. Transition between the two regimes proceeds through slow erratic motion of the soliton. Extension of the analysis for the 2D case is briefly outlined; in particular, novel results are existence of stable higher-order lattice vortices, with the vorticity S≥2, and quadrupoles.