Stability of dark solitons in a Bose-Einstein condensate trapped in an   optical lattice
  We investigate the stability of dark solitons (DSs) in an effectively one-dimensional Bose-Einstein condensate in the presence of the magnetic parabolic trap and an optical lattice (OL). The analysis is based on both the full Gross-Pitaevskii equation and its tight-binding approximation counterpart (discrete nonlinear Schr{\"o}dinger equation). We find that DSs are subject to weak instabilities with an onset of instability mainly governed by the period and amplitude of the OL. The instability, if present, sets in at large times and it is characterized by quasi-periodic oscillations of the DS about the minimum of the parabolic trap. 
P.G. Kevrekidis, R. Carretero-Gonz\'alez, G. Theocharis, D.J.   Frantzeskakis, and B.A. Malomed