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

TL;DR

This paper proves the orbital stability of cnoidal periodic waves in the cubic defocusing NLS equation using conserved quantities, extending stability results beyond small amplitude waves through a direct proof.

Contribution

It provides a direct proof of orbital stability for cnoidal waves in the defocusing NLS, utilizing conserved quantities and applicable to waves of arbitrary amplitude.

Findings

Cnoidal waves are orbitally stable under subharmonic perturbations.

Stability proof is valid for waves of any amplitude, not just small ones.

Explicit connection between conserved quantities and stability established.

Abstract

Periodic waves of the one-dimensional cubic defocusing NLS equation are considered. Using tools from integrability theory, these waves have been shown in [Bottman, Deconinck, and Nivala, 2011] 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.