T. Gallay and D.E. Pelinovsky

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

Journal of Differential Equations 258 (2015), 3607-3638

Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown in \cite{Decon} to be linearly stable and the Floquet--Bloch spectrum of the linearized operator has been explicitly computed. We combine here the first four conserved quantities of the NLS equation to give a direct proof that cnoidal periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. Our result is not restricted to the periodic waves of small amplitudes.

Nonlinear Schrodinger equation; periodic waves; orbital stability; conserved quantities; Floquet-Bloch spectrum.