  The paper is devoted to numerical study of stability of nonlinear localized
modes ("gap solitons") for the spatially one-dimensional Gross-Pitaevskii
equation (1D GPE) with periodic potential and repulsive interparticle
interactions. We use the Evans function approach combined with the exterior
algebra formulation in order to detect and describe weak oscillatory
instabilities. We show that the simplest ("fundamental") gap solitons in the
first and in the second spectral gaps can undergo oscillatory instabilities for
certain values of the frequency parameter (i.e., the chemical potential). The
number of unstable eigenvalues and the associated instability rates are
described. Several stable and unstable more complex (non-fundamental) gap
solitons are also discussed. The results obtained from the Evans function
approach are independently confirmed using the direct numerical integration of
the GPE.
