Tame majorant analyticity for the Birkhoff map of the defocusing Nonlinear Schr\"odinger equation on the circle

TL;DR

This paper constructs a tame majorant analytic Birkhoff map for the defocusing NLS on the circle, providing a new tame version of the Kuksin-Perelman theorem to analyze the equation's integrability.

Contribution

It introduces a novel tame majorant analytic Birkhoff map for the defocusing NLS and develops a new tame Kuksin-Perelman theorem, advancing the understanding of infinite-dimensional integrable systems.

Findings

Birkhoff map is tame majorant analytic near the origin.

The new tame Kuksin-Perelman theorem is established.

Provides a framework for analyzing the NLS on the circle.

Abstract

For the defocusing Nonlinear Schr\"odinger equation on the circle, we construct a Birkhoff map $\Phi$ which is tame majorant analytic in a neighborhood of the origin. Roughly speaking, majorant analytic means that replacing the coefficients of the Taylor expansion of $\Phi$ by their absolute values gives rise to a series (the majorant map) which is uniformly and absolutely convergent, at least in a small neighborhood. Tame majorant analytic means that the majorant map of $\Phi$ fulfills tame estimates. The proof is based on a new tame version of the Kuksin-Perelman theorem, which is an infinite dimensional Vey type theorem.