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Periodic waves of the one-dimensional cubic defocusing NLS equation areconsidered. Using tools from integrability theory, these waves have been shownin [Bottman, Deconinck, and Nivala, 2011] to be linearly stable and theFloquet-Bloch spectrum of the linearized operator has been explicitly computed.We combine here the first four conserved quantities of the NLS equation to givea direct proof that cnoidal periodic waves are orbitally stable with respect tosubharmonic perturbations, with period equal to an integer multiple of theperiod of the wave. Our result is not restricted to the periodic waves of smallamplitudes.

Orbital stability in the cubic defocusing NLS equation: I. Cnoidal periodic waves