A normal form of the non-linear Schr\"odinger equation

TL;DR

This paper studies the normal form transformations of the resonant non-linear Schrödinger equation on a torus, aiming to facilitate the analysis of quasi-periodic solutions in nonlinear wave dynamics.

Contribution

It introduces a normal form framework for the resonant NLS on tori, providing a new approach to analyze quasi-periodic solutions in nonlinear Schrödinger equations.

Findings

Normal form transformation simplifies the resonant NLS.

Facilitates the study of quasi-periodic solutions.

Provides a new analytical tool for nonlinear wave equations.

Abstract

We discuss normal forms of the completely resonant non-linear Schr\"odinger equation on a torus $\T^n$, with particular applications to quasi periodic solutions.