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The work is devoted to numerical investigation of stability of stationarylocalized modes ("gap solitons") for the one-dimentional nonlinearSchr\"odinger equation (NLSE) with periodic potential and repulsivenonlinearity. Two classes of the modes are considered: a bound state of a pairof in-phase and out-of-phase fundamental gap solitons (FGSs) from the firstbandgap separated by various number of empty potential wells. Using thestandard framework of linear stability analysis, we computed the linear spectrafor the gap solitons by means of the Fourier collocation method and the Evansfunction method. We found that the gap solitons of the first and second classesare exponentially unstable for odd and even numbers of separating periods ofthe potential, respectively. The real parts of unstable eigenvalues incorresponding spectra decay with the distance between FGSs exponentially. Onthe contrary, we observed that the modes of the first and second classes areeither linearly stable or exhibit weak oscillatory instabilities if the numberof empty potential wells separating FGSs is even and odd, respectively. In bothcases, the oscillatory instabilities arise in some vicinity of upper bandgapedge. In order to check the linear stability results, we fulfilled numericalsimulations for the time-dependent NLSE by means of a finite-difference scheme.As a result, all the considered exponentially unstable solutions have beendeformed to long-lived pulsating formations whereas stable solutions conservedtheir shapes for a long time.

Stability of gap soliton complexes in the nonlinear Schr\"odinger equation with periodic potential and repulsive nonlinearity