Global Existence for the Derivative Nonlinear Schr\"{o}dinger Equation with Arbitrary Spectral Singularities

TL;DR

This paper proves global well-posedness for the derivative nonlinear Schrödinger equation in a specific function space, leveraging integrability and spectral analysis to handle initial data with spectral singularities.

Contribution

It establishes global existence results for DNLS without spectral restrictions, extending previous work by incorporating spectral singularities analysis.

Findings

Global well-posedness in $H^{2,2}(\

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Abstract

We show that the derivative nonlinear Schr\"odinger (DNLS) equation is globally well-posed in the weighted Sobolev space $H^{2,2}(\mathbb{R})$. Our result exploits the complete integrability of DNLS and removes certain spectral conditions on the initial data, thanks to Xin Zhou's analysis on spectral singularities in the context of inverse scattering.