Gap solitons and their linear stability in one-dimensional periodic media

An analytical theory utilizing exponential asymptotics is presented for one-dimensional gap solitons that bifurcate from edges of Bloch bands in the presence of a general periodic potential. It is shown that two soliton families bifurcate out from every Bloch-band edge under self-focusing or self-defocusing nonlinearity, and an asymptotic expression for the eigenvalues associated with the linear stability of these solitons is derived. The locations of these solitons relative to the underlying potential are determined from a certain recurrence relation, that contains information beyond all orders of the usual perturbation expansion in powers of the soliton amplitude. Moreover, this same recurrence relation decides which of the two soliton families is unstable. The analytical predictions for the stability eigenvalues are in excellent agreement with numerical results.