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This paper is concerned with the derivative nonlinear Schrodinger equationwith periodic boundary conditions  $$\mathbf{i}u_t+u_{xx}+\mathbf{i}\Big(f(x,u,\bar{u})\Big)_x=0,\quadx\in\mathbb{T}:=\mathbb{R}/2\pi\mathbb{Z},$$ where $f$ is an analytic functionof the form $$f(x,u,\bar{u})=\mu|u|^2u+f_{\geq4}(x,u,\bar{u}),\quad0\neq\mu\in\mathbb{R},$$ and $f_{\geq4}(x,u,\bar{u})$ denotes terms of order atleast four in $u,\bar{u}$. We show the above equation possesses Cantor familiesof smooth quasi-periodic solutions of small amplitude. The proof is based on aninfinite dimensional KAM theorem for unbounded perturbation vector fields.

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