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

Abstract

In this paper, we prove that there exists some small ${\epsilon }_{\ast }>0$ such that the derivative nonlinear Schrödinger equation (DNLS) is globally well-posed in the energy space, provided that the initial data $u_0\in H^1\left(ℝ\right)$ satisfies $\parallel u_0{\parallel }_{L^2}<\sqrt{2\pi }+{\epsilon }_{\ast }$. 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 the nonlinear Schrödinger equation with critical nonlinearity. The technique used is a variational argument together with the momentum conservation law. Further, for the DNLS on the half-line ${ℝ}^{+}$, we show the blow-up for the solution with negative energy.