Quasi-periodic Solutions of a Derivative Nonlinear Schr\"odinger Equation

TL;DR

This paper proves the existence of a family of real analytic quasi-periodic solutions with two Diophantine frequencies for a 1D derivative nonlinear Schrödinger equation using Birkhoff normal form and KAM techniques.

Contribution

It establishes the existence of quasi-periodic solutions for a derivative nonlinear Schrödinger equation with a novel application of Birkhoff normal form and KAM theory.

Findings

Existence of quasi-periodic solutions with two Diophantine frequencies.

Application of partial Birkhoff normal form and KAM method.

Analytic solutions on a periodic domain.

Abstract

This paper is concerned with a one dimensional (1D) derivative nonlinear Schr\"odinger equation with periodic boundary conditions \begin{equation*} \mi u_t+u_{xx}+\mi |u|^2u_x=0, \ \ x\in \mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z}. \end{equation*} We show that above equation admits a family of real analytic quasi-periodic solutions with two Diophantine frequencies. The proof is based on a partial Birkhoff normal form and KAM method.