Dynamical properties, modulation instability analysis and chaotic behaviors to the nonlinear coupled Schrödinger equation in fiber Bragg gratings

The nonlinear coupled Schrödinger equation in fiber Bragg gratings is studied in this paper. The existence of soliton solutions and periodic solutions are proved by qualitative analysis, and exact solutions are given, as well as the parameter condition of each solution is described. Then the modulation instability (MI) analysis is carried out and the linear stability criterion is given. In particular, external perturbation terms are introduced to prove that the equation exists chaotic behaviors.