Sharp global well-posedness for 1D NLS with derivatives

TL;DR

This paper proves that the one-dimensional derivative nonlinear Schrödinger equation is globally well-posed in the Sobolev space H^s for s ≥ 1/2, using a linear-nonlinear decomposition method to handle the nonlinearity.

Contribution

It establishes the sharp global well-posedness result at the critical regularity s=1/2 for the 1D derivative NLS, improving understanding of its solution behavior.

Findings

Global well-posedness in H^s for s ≥ 1/2

Use of linear-nonlinear decomposition method

Refined almost conservation law at critical regularity

Abstract

We show that the 1d derivative nonlinear Schr\"{o}dinger equation (\ref{equ}) is globally well-posed in $H^s(\mathbb{R})$ for $s\geq 1/2$. We use the linear-nonlinear decomposition method to take advantage of the local smoothing effect of the nonlinearity, which enables us to establish a refined version of the almost conservation law. Note that $H^{1/2}$ is the endpoint that we have uniform continuous for the solution map and hence our result is sharp.