Birkhoff Normal Form for the Derivative Nonlinear Schr\"{o}dinger Equation

TL;DR

This paper derives a complete Birkhoff normal form of order six for the derivative nonlinear Schr"{o}dinger equation with periodic boundary conditions, enabling proof of long-term stability for small solutions.

Contribution

It provides the first complete Birkhoff normal form of order six for this equation, advancing the understanding of its long-term dynamics.

Findings

Complete Birkhoff normal form of order six obtained

Long time stability for small amplitude solutions proved

Enhances analytical tools for derivative nonlinear Schr"{o}dinger equations

Abstract

This paper is concerned with the derivative nonlinear Schr\"{o}dinger equation with periodic boundary conditions. We obtain complete Birkhoff normal form of order six. As an application, the long time stability for solutions of small amplitude is proved.