On the non-existence of local Birkhoff coordinates for the focusing NLS equation

TL;DR

This paper demonstrates that for certain potentials, the focusing nonlinear Schrödinger equation cannot be expressed in local Birkhoff coordinates, highlighting limitations in the integrable structure near those potentials.

Contribution

It provides the first proof of non-existence of local Birkhoff coordinates for the focusing NLS near specific potentials.

Findings

Existence of potentials where local Birkhoff coordinates do not exist

Construction of a local normal form of the linearized equation at such potentials

Implication that the integrable structure is not always locally realizable

Abstract

We prove that there exist potentials so that near them the focusing non-linear Schr\"odinger equation does not admit local Birkhoff coordinates. The proof is based on the construction of a local normal form of the linearization of the equation at such potentials.