Modified Scattering and Asymptotic Completeness for the Derivative Nonlinear Schr\"odinger equation

TL;DR

This paper establishes modified scattering and asymptotic completeness for the derivative nonlinear Schrödinger equation, marking the first such result in a quasilinear context using advanced analytical techniques.

Contribution

It introduces a novel combination of wave packet testing, bootstrap, and vector field methods to prove asymptotic completeness in a quasilinear setting.

Findings

Proves modified scattering for the derivative NLS.

Establishes asymptotic completeness in a quasilinear framework.

First such result for this class of equations.

Abstract

We prove a modified scattering and asymptotic completeness for the derivative nonlinear Schr\"odinger equation. This is the first result proving asymptotic completeness in a quasilinear setting. Our approach combines the method of testing by wave packets, introduced by Ifrim and Tataru, a bootstrap argument, and the Klainerman Sobolev vector field method.