A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation · arXivDesk
1610.00267Oct 2, 2016To appear in Analysis & PDE. We changed the title. Namely, this paper is a revised version of "Global Well-Posedness on a generalized derivative nonlinear Schrödinger equation"
A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation
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 u0 is global if ∥u0∥L22<4π
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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