Normal Form for the Schr\"odinger equation with analytic non--linearities

TL;DR

This paper develops a normal form theory for the resonant nonlinear Schrödinger equation on a torus, emphasizing geometric and combinatorial methods, with future work planned on applications to quasi-periodic solutions.

Contribution

It introduces a new normal form framework for the resonant NLS on a torus, highlighting geometric and combinatorial structures.

Findings

Normal form construction for resonant NLS

Geometric and combinatorial analysis methods

Foundation for future quasi-periodic solutions study

Abstract

In this paper we discuss a class of normal forms of the completely resonant non--linear Schr\"odinger equation on a torus. We stress the geometric and combinatorial constructions arising from this study. Further analytic considerations and applications to quasi--periodic solutions will appear in a forthcoming article. This paper replaces a previous version correcting some mistakes.