Local well-posedness for hyperbolic-elliptic Ishimori equation

TL;DR

This paper proves local well-posedness for the hyperbolic-elliptic Ishimori equation with small initial data in Sobolev spaces, using advanced perturbative methods adapted from prior work.

Contribution

It extends the well-posedness theory for the Ishimori equation by applying and adapting methods of Ionescu and Kenig to a hyperbolic-elliptic setting.

Findings

Local well-posedness established for s > 3/2

Methods of Ionescu and Kenig successfully adapted to this problem

Perturbative approach effective for small data

Abstract

In this paper we consider the hyperbolic-elliptic Ishimori initial-value problem. We prove that such system is locally well-posed for small data in $H^{s}$ level space, for $s> 3/2$. The new ingredient is that we develop the methods of Ionescu and Kenig \cite{IK} and \cite{IK2} to approach the problem in a perturbative way.