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

This paper is concerned with the derivative nonlinear Schrodinger equation with periodic boundary conditions $$\mathbf{i}{u}_t+u_{xx}+\mathbf{i}(f(x,u,\bar{u}){)}_x=0,x\in \mathbb{T}:=\mathbb{R}\mathrm{/}2\pi \mathbb{Z},$$ where $f$ is an analytic function of the form $$f(x,u,\bar{u})=\mu \mathrm{\mid }u{\mathrm{\mid }}^{2}u+f_{\ge 4}(x,u,\bar{u}),0\mathrm{\ne }\mu \in \mathbb{R},$$ and $f_{\ge 4}(x,u,\bar{u})$ 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.