KAM for the derivative nonlinear Schrödinger equation with periodic boundary conditions

This paper is concerned with the derivative nonlinear Schrödinger equation with periodic boundary conditionsiut+uxx+i(f(|u|2)u)x=0,x∈T:=R/2πZ, where f   is real analytic in some neighborhood of the origin in CC, f(0)=0f(0)=0, and f′(0)≠0f′(0)≠0. 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.