Oscillatory instabilities of gap solitons in a repulsive Bose–Einstein condensate

Highlights: • We study numerically the oscillatory instabilities of fundamental gap solitons. • We use the Evans function combined with the exterior algebra formulation. • The case of repulsive interactions is considered. • The oscillatory instability found for fundamental gap solitons in the first and second gaps. • The results are confirmed by direct simulations of the dynamics. 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 gap 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.