Stability and long time evolution of the periodic solutions to the two coupled nonlinear Schrödinger equations

The system of two coupled nonlinear Schrödinger equations has wide applications in physics. In the past, the main attention has been their solitary waves. Here we turn our attention to their periodic wave solutions. In this paper, the stability of the periodic solutions is studied analytically and the criteria for the stability are obtained. The long time evolution of the solutions to the coupled system is studied numerically for the unstable case emphasizing wave–wave interactions in nonlinear optics. Different kinds of evolution are observed depending on the coefficients of the system and the parameters of the unperturbed waves and perturbation. For certain ranges of parameters, the evolution appears to be periodic, while for some other ranges of parameters, solitary wave or solitary wave pairs can be excited among the irregular background although often the evolution is completely chaotic.