On a class of solutions to the generalized derivative Schr\"odinger 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}.
Keywords
Cite
@article{arxiv.1712.00663,
title = {On a class of solutions to the generalized derivative Schr\"odinger equations},
author = {Felipe Linares and Gustavo Ponce and Gleison N. Santos},
journal= {arXiv preprint arXiv:1712.00663},
year = {2017}
}