In this paper, we prove that there exists some small such that the derivative nonlinear Schrödinger equation (DNLS) is globally well-posed in the energy space, provided that the initial data satisfies . This result shows us that there are no blow-up solutions whose masses slightly exceed , 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 , we show the blow-up for the solution with negative energy.
"Global well-posedness for the nonlinear Schrödinger equation with derivative in energy space." Anal. PDE 6 (8) 1989 - 2002, 2013. https://doi.org/10.2140/apde.2013.6.1989