Spectral stability of multiple periodic waves for the Schrödinger system with cubic nonlinearity

Results concerning the existence and spectral stability/instability of multiple periodic standing wave solutions for a cubic nonlinear Schrdinger system will be shown in this manuscript.Our approach considers periodic perturbations that have the same period of the standing wave solution.To obtain the quantity and multiplicity of non-positive eigenvalues for the corresponding linearized operator, we use the comparison theorem and tools of Floquet theory.The results are then obtained by applying the spectral stability theory via Krein signature as established in [20] and [21].