# An unstable three dimensional KAM torus for the quintic NLS

## TL;DR

This paper constructs a three-dimensional invariant torus for the quintic nonlinear Schrödinger equation on the circle, demonstrating its linear instability, while showing two-dimensional tori are stable, using Birkhoff and KAM techniques.

## Contribution

It introduces the existence of a linearly unstable three-dimensional KAM torus for the quintic NLS, contrasting with the stability of two-dimensional tori, via a novel application of Birkhoff and KAM methods.

## Key findings

- Three-dimensional KAM torus is linearly unstable.
- Two-dimensional tori are always linearly stable.
- Application of Birkhoff and KAM techniques to NLS.

## Abstract

We consider the quintic nonlinear Schr{\"o}dinger on the circle. By applying a Birkhoff procedure and a KAM theorem, we exihibit a three dimension invariant torus that is linearly unstable. In comparison, we also prove that two dimensional tori are always linearly stable.