Multi-instability of gap solitons and dynamics of nonlinear excitations in the array of optical fibers

The nonlinear localized excitations and gap multi-instability phenomena in the array of optical fibers are investigated analytically by means of the method of multiple scales combined with the averaging method. The perturbative analysis in nonlinear discrete systems leads to a system of four coupled nonlinear oscillated equations without second members. The method of multiple scales is used to show the different possibilities of instability. The averaging method is considered to show how the angular frequency received the perturbation with the evolution of time.