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

Results concerning the existence and spectral stability and instability of multiple periodic wave solutions for the nonlinear Schrödinger system with $\mathit{\text{dnoidal}}$ and $\mathit{\text{cnoidal}}$ profile will be determined in this manuscript. The spectral analysis for the corresponding linearized operator is established by using the comparison theorem and tools of Floquet theory. The main results are determined by applying the spectral stability theory in \cite{KapitulaKevrekidisSandstedeI} and \cite{KapitulaKevrekidisSandstedeII} via Krein signature.