On the spectral stability of periodic waves of the coupled Schrödinger equations

•We consider the coupled nonlinear Schrödinger equations.•We study the existence and the stability of the periodic standing waves.•We show that these waves are spectrally stable for certain parameter values.•We use Floquet theory and do some numerical computations.

In this present work we consider the periodic standing wave solutions for the coupled nonlinear Schrödinger equations. We restrict our problem to two cases and construct the solutions explicitly. Then we make the stability analysis for each case. We show that the relevant stationary solutions are spectrally stable for certain parameters. For those parameters, we also present our numerical results in the spectral analysis.