Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space

Abstract: In this paper, we prove that there exists some small $\varepsilon_>0$, such that the derivative nonlinear Schr\"{o}dinger equation (DNLS) is global well-posedness in the energy space, provided that the initial data $u_0\in H^1(\mathbb{R})$ satisfies $|u_0|{L^2}<\sqrt{2\pi}+\varepsilon$. This result shows us that there are no blow up solutions whose masses slightly exceed $2\pi$, even if their energies are negative. This phenomenon is much different from the behavior of nonlinear Schr\"odinger equation with critical nonlinearity. The technique is a variational argument together with the momentum conservation law. Further, for the DNLS on half-line $\mathbb{R}^+$, we show the blow-up for the solution with negative energy.