# The orbital stability of elliptic solutions of the Focusing Nonlinear Schr\"{o}dinger Equation

## TL;DR

This paper investigates the spectral and orbital stability of elliptic solutions to the focusing nonlinear Schrödinger equation, demonstrating that smaller amplitude solutions are stable under various perturbations through spectral analysis and Lyapunov functionals.

## Contribution

It establishes the spectral and orbital stability of elliptic solutions of the focusing NLS using integrability and conserved quantities, extending stability results to larger classes of perturbations.

## Key findings

- Smaller amplitude elliptic solutions are spectrally stable.
- Spectrally stable solutions are also orbitally stable.
- Construction of a Lyapunov functional confirms orbital stability.

## Abstract

We examine the stability of the elliptic solutions of the focusing nonlinear Schr\"odinger equation (NLS) with respect to subharmonic perturbations. Using the integrability of NLS, we discuss the spectral stability of the elliptic solutions, establishing that solutions of smaller amplitude are stable with respect to larger classes of perturbations. We show that spectrally stable solutions are orbitally stable by constructing a Lyapunov functional using higher-order conserved quantities of NLS.