A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schr\"{o}dinger equation

TL;DR

This paper establishes a new sufficient condition for the global existence of solutions to a generalized derivative nonlinear Schrödinger equation using a variational approach, extending previous results and providing alternative proofs.

Contribution

It introduces a variational method to determine global existence criteria for gDNLS, applicable also to the cubic DNLS, and offers new conditions involving initial data norms and momentum.

Findings

Proves global existence under a new variational condition.

Provides an alternative proof for the cubic DNLS case.

Identifies initial data conditions ensuring global solutions.

Abstract

We give a sufficient condition for global existence of the solutions to a generalized derivative nonlinear Schr\"{o}dinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear Schr\"{o}dinger equation (DNLS). For (DNLS), Wu proved that the solution with the initial data $u_0$ is global if $\left\Vert u_0 \right\Vert_{L^2}^2<4\pi$ by the sharp Gagliardo--Nirenberg inequality in the paper "Global well-posedness on the derivative nonlinear Schr\"odinger 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 $\left\Vert u_0 \right\Vert_{L^2}^2=4\pi$ and the momentum $P(u_0)$ is negative.