Ill-posedness for the Half wave Schr\"odinger equation

TL;DR

This paper investigates the ill-posedness of the half wave Schrödinger equation in certain critical and super-critical function spaces, using standing wave solutions and established analytical methods.

Contribution

It demonstrates ill-posedness in critical and super-critical spaces for the half wave Schrödinger equation, extending previous analytical frameworks.

Findings

Ill-posedness in super-critical space

Ill-posedness at the critical space

Use of standing wave solutions for proof

Abstract

We study ill-posedness for the half wave Schr\"odinger 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}.