A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation

Abstract

We give a sufficient condition for global existence of the solutions to a generalized derivative nonlinear Schrödinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear Schrödinger equation (DNLS). For (DNLS), Wu proved that the solution with the initial data $u_0$ is global if $\left\Vert u_0 \right\Vert_{L^2}^2<4π$ by the sharp Gagliardo--Nirenberg inequality in the paper "Global well-posedness on the derivative nonlinear Schrödinger equation", Analysis & PDE 8 (2015), no. 5, 1101--1112. The variational argument gives us another proof of the global existence for (DNLS). Moreover, by the variational argument, we can show that the solution to (DNLS) is global if the initial data $u_0$ satisfies that $\left\Vert u_0 \right\Vert_{L^2}^2=4π$ and the momentum $P(u_0)$ is negative.