A remark on the Half wave Schr\"odinger equation in the energy space

TL;DR

This paper studies the initial value problem for the half wave Schrödinger equation in the energy space, establishing local well-posedness for certain nonlinearities under specific initial data conditions.

Contribution

It proves local well-posedness of the half wave Schrödinger equation in the energy space for odd power nonlinearities with additional initial data assumptions.

Findings

Established local well-posedness in the energy space

Identified conditions on initial data for well-posedness

Focused on odd power nonlinearities

Abstract

We investigate the Cauchy problem for the half wave Schr\"odinger equation in the energy space. We derive the local well-posedness in the energy space for the odd power type nonlinearities under certain additional assumption for the initial data, namely $\hat{u}_0 \in L^1_{\xi, \eta}(\mathbb{R}^2)$.