On a class of solutions to the generalized derivative Schr\"odinger equations

TL;DR

This paper investigates the initial value problem for a class of generalized derivative Schrödinger equations, establishing local well-posedness for small initial data using advanced analytical techniques.

Contribution

It introduces new well-posedness results for generalized derivative Schrödinger equations with small data in weighted Sobolev spaces, extending previous methods.

Findings

Proves local well-posedness for small initial data

Utilizes Kato smoothing effects in the analysis

Extends existing results to a broader class of equations

Abstract

In this work we shall consider the initial value problem associated to the generalized derivative Schr\"odinger equations \begin{equation*} \p_tu=i\p_x^2u + \mu\,|u|^{\a}\p_xu, \hskip10pt x,t\in\R, \hskip5pt 0<\a \le 1\;\, {\rm and}\;\, |\mu|=1, \end{equation*} and \begin{equation*} \p_tu=i\p_x^2u + \mu\,\p_x\big(|u|^{\a}u\big), \hskip10pt x,t\in\R, \hskip5pt 0<\a \le 1\;\, {\rm and}\;\, |\mu|=1. \end{equation*} Following the argument introduced by Cazenave and Naumkin \cite{Cazenave} we shall establish the local well-posedness for a class of small data in an appropriate weighted Sobolev space. The other main tools in the proof include the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schr\"odinger equation established by Kenig-Ponce-Vega in \cite{KPV1}.