Long Time Stability of Hamiltonian Derivative Nonlinear Schrödinger Equations Without Potential

In this paper, we prove an abstract Birkhoff normal form theorem for some unbounded infinite dimensional Hamiltonian systems. Based on this result we obtain that the solution to Derivative Nonlinear Schrödinger equations under periodic boundary condition with typical small enough initial value remains small in the Sobolev norm \( H^{\textbf{s}}(\mathbb {T})\) over a long time interval. The length of the time interval is equal to \(e^{|\ln R|^{1+\gamma }}\) with \(0<\gamma <1/5\) as the initial value is smaller than \(R\ll 1\).