Global Well-Posesedness for the Derivative Nonlinear Schrodinger Equation

TL;DR

This paper establishes global well-posedness for the Derivative Nonlinear Schrödinger equation in a broad class of initial conditions within weighted Sobolev spaces, including large data, and sets the stage for future soliton resolution results.

Contribution

It proves global well-posedness for DNLS with initial data in an open dense subset of weighted Sobolev spaces, including large $L^2$-norm data, excluding spectral singularities.

Findings

The set of initial data supporting well-posedness is open and dense.

Global solutions exist for arbitrarily large $L^2$-norm initial data.

The results facilitate future analysis of soliton resolution and stability.

Abstract

We study the Derivative Nonlinear Schr\"odinger (DNLS). equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities corresponding to algebraic solitons). We show that the set of such initial data is open and dense in a weighted Sobolev space, and includes data of arbitrarily large $L^2$-norm. We prove global well-posedness on this open and dense set. In a subsequent paper, we will use these results and a steepest descent analysis to prove the soliton resolution conjecture for the DNLS equation with the initial data considered here and asymptotic stability of $N-$soliton solutions.