Orbital stability of periodic waves and black solitons in the cubic defocusing NLS equation

Thierry Gallay, Dmitry Pelinovsky

Published 2014-09-23Version 1

Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown to be linearly stable and the Floquet-Bloch spectrum of the linearized operator has been explicitly computed in [Bottman, Deconinck, and Nivala, 2011]. We combine here the first four conserved quantities of the NLS equation to give a direct proof that small amplitude periodic waves are orbitally stable with respect to subharmonic perturbations, with period equal to an integer multiple of the period of the wave. We also show that the black soliton is an unconditional minimizer of a fourth-order energy functional, which gives a simple proof of orbital stability with respect to perturbations in $H^2(\mathbb{R})$.