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The stability of the stationary periodic solutions of the integrable (onedimensional, cubic) defocusing nonlinear Schrödinger (NLS) equation is reasonably well understood, especially for solutions of small amplitude. In this paper, we exploit the integrability of the NLS equation to establish the spectral stability of all such stationary solutions, this time by explicitly computing the spectrum and the corresponding eigenfunctions associated with their linear stability problem. An additionaldoi:10.1088/1751-8113/44/28/285201 fatcat:vtgdj5d4i5ckbn3sokazypqe6a