On a generalized derivative nonlinear Schr\"odinger equation

TL;DR

This paper studies a generalized derivative nonlinear Schrödinger equation, proving the existence of wave operators for small asymptotic states and demonstrating scattering behavior for small initial data in $H^2( eal)$, using a method based on an associated system.

Contribution

It introduces new results on wave operator existence and scattering for a generalized derivative NLS, extending previous methods to a broader class of equations.

Findings

Existence of wave operators under smallness conditions.

Scattering of solutions for small initial data in $H^2( eal)$.

Use of an associated system approach from prior work.

Abstract

We consider a generalized derivative nonlinear Schr\''odinger equation. We prove existence of wave operator under an explicit smallness of the given asymptotic states. Our method bases on studying the associated system used in \cite{Tinpaper4}. Moreover, we show that if the initial data is small enough in $H^2(\mathbb{R})$ then the associated solution scatters up to a Gauge transformation.