Ill-posedness for the Half wave Schrödinger equation

Abstract

We study ill-posedness for the half wave Schrödinger equation introduced by Xu \cite{Xu}. Ill-posedness is obtained in the super-critical or at the critical space. The proof is based on the argument established by Christ, Colliander and Tao \cite{CCT1}. For the critical space, we use the standing wave solution, which was proved the existence by Bahri, Ibrahim and Kikuchi \cite{BIK}.