Invariant Cantor manifolds of quasi-periodic solutions for the derivative nonlinear Schrodinger equation

TL;DR

This paper proves the existence of Cantor families of small amplitude quasi-periodic solutions for a derivative nonlinear Schrödinger equation with periodic boundary conditions using an infinite-dimensional KAM theorem.

Contribution

It establishes the existence of invariant Cantor manifolds of quasi-periodic solutions for the derivative NLS, extending KAM theory to unbounded perturbations.

Findings

Existence of Cantor families of solutions proven

Solutions are smooth and of small amplitude

Application of infinite-dimensional KAM theorem to PDEs

Abstract

This paper is concerned with the derivative nonlinear Schrodinger equation with periodic boundary conditions $$\mathbf{i}u_t+u_{xx}+\mathbf{i}\Big(f(x,u,\bar{u})\Big)_x=0,\quad x\in\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z},$$ where $f$ is an analytic function of the form $$f(x,u,\bar{u})=\mu|u|^2u+f_{\geq4}(x,u,\bar{u}),\quad 0\neq\mu\in\mathbb{R},$$ and $f_{\geq4}(x,u,\bar{u})$ denotes terms of order at least four in $u,\bar{u}$. We show the above equation possesses Cantor families of smooth quasi-periodic solutions of small amplitude. The proof is based on an infinite dimensional KAM theorem for unbounded perturbation vector fields.