Long time stability of Hamiltonian Derivative Nonlinear Schordinger equations without potential

In this paper, we will give an abstract Birkhoff normal form theorem for  unbounded infinite dimensional  Hamiltonian systems. By this result we obtain that  the solution to DNLS equations without potential under periodic boundary condition with typical small enough initial value remains small over a long time interval. The length of the time interval equals to $e^{-|\ln R|^{1+\gamma}}$ with $0<\gamma<1/5$ as the initial value is smaller than $R$.