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 Schrdinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear Schrdinger equation (DNLS). For (DNLS), Wu (2015) proved that the solution with the initial data u 0 is global if u 0 2 L 2 < 4 by the sharp Gagliardo-Nirenberg inequality. 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 u 0 2 L 2 = 4 and the momentum P(u 0 ) is negative.