Global $H^2$-solutions for the generalized derivative NLS on $\mathbb{T}$

TL;DR

This paper establishes the global existence of $H^2$ solutions for the generalized derivative nonlinear Schrödinger equation on the 1D torus, solving an open problem in the field.

Contribution

It introduces a novel energy construction and analysis method to prove global solutions for a challenging nonlinear PDE on the torus.

Findings

Proves global existence of $H^2$ solutions.

Develops new energy estimates to handle nonlinear terms.

Addresses an open problem from 2015.

Abstract

We prove global existence of $H^2$ solutions to the Cauchy problem for the generalized derivative nonlinear Schr\"{o}dinger equation on the 1-d torus. This answers an open problem posed by Ambrose and Simpson (2015). The key is the extraction of the terms that cause the problem in energy estimates and the construction of suitable energies so as to cancel the problematic terms out by effectively using integration by parts and the equation.