Stability of gap soliton complexes in the nonlinear Schrödinger equation with periodic potential and repulsive nonlinearity

The work is devoted to numerical investigation of stability of stationary localized modes (“gap solitons”) for the one-dimentional nonlinear Schrödinger equation (NLSE) with periodic potential and repulsive nonlinearity. Two classes of the modes are considered: a bound state of a pair of in-phase and out-of-phase fundamental gap solitons (FGSs) from the first bandgap separated by various numbers of empty potential wells. Using the standard framework of linear stability analysis, we computed the linear spectra for the gap solitons by means of the Fourier collocation method and the Evans function method. We found that the gap solitons of the first and second classes are exponentially unstable for odd and even numbers of separating periods of the potential, respectively. The real parts of unstable eigenvalues in corresponding spectra decay exponentially with the distance between FGSs. On the contrary, we observed that the modes of the first and second classes are either linearly stable or exhibit weak oscillatory instabilities if the number of empty potential wells separating FGSs is even and odd, respectively. In both cases, the oscillatory instabilities arise in some vicinity of upper bandgap edge. In order to check the linear stability results, we fulfilled numerical simulations for the time-dependent NLSE by means of a finite-difference scheme. As a result, all the considered exponentially unstable solutions have been deformed to long-lived pulsating formations whereas stable solutions conserved their shapes for a long time.