Canonical coordinates with tame estimates for the defocusing NLS equation on the circle

TL;DR

This paper constructs real analytic, canonical coordinates for the defocusing NLS equation on the circle, enabling better analysis in perturbation theory by normalizing the Hamiltonian up to order three.

Contribution

It introduces tame, real analytic canonical coordinates near invariant tori for the defocusing NLS, tailored for perturbation theory applications.

Findings

Coordinates satisfy tame estimates with derivatives

Hamiltonian expressed in normal form up to third order

Coordinates facilitate analysis of perturbations

Abstract

In a case study for integrable PDEs, we construct real analytic, canonical coordinates for the defocusing NLS equation on the circle, specifically taylored towards the needs in perturbation theory. They are defined in neighbourhoods of families of finite dimensional invariant tori and are shown to satisfy together with their derivatives tame estimates. When expressed in these coordinates, the dNLS Hamiltonian is in normal form up to order three.