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Results concerning the existence and spectral stability and instability ofmultiple periodic wave solutions for the nonlinear Schr\"odinger system with\textit{dnoidal} and \textit{cnoidal} profile will be determined in thismanuscript. The spectral analysis for the corresponding linearized operator isestablished by using the comparison theorem and tools of Floquet theory. Themain results are determined by applying the spectral stability theory in\cite{KapitulaKevrekidisSandstedeI} and \cite{KapitulaKevrekidisSandstedeII}via Krein signature.

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