Schr\"odinger Soliton from Lorentzian Manifolds

TL;DR

This paper introduces Schr"odinger solitons as special solutions to the Schr"odinger flow from Riemannian or Lorentzian manifolds into K"ahler manifolds, establishing well-posedness and existence results, including for the hyperbolic Ishimori system.

Contribution

It defines Schr"odinger solitons in the context of Lorentzian manifolds and proves their existence and well-posedness, extending the theory to wave maps with potential.

Findings

Established local well-posedness of the Schr"odinger flow system.

Proved global existence in 1+1 dimensions.

Demonstrated existence of Schr"odinger solitons for the hyperbolic Ishimori system.

Abstract

In this paper, we introduce a new notion named as Schr\"odinger soliton. So-called Schr\"odinger solitons are defined as a class of special solutions to the Schr\"odinger flow equation from a Riemannian manifold or a Lorentzian manifold $M$ into a K\"ahler manifold $N$. If the target manifold $N$ admits a Killing potential, then the Schr\"odinger soliton is just a harmonic map with potential from $M$ into $N$. Especially, if the domain manifold is a Lorentzian manifold, the Schr\"odinger soliton is a wave map with potential into $N$. Then we apply the geometric energy method to this wave map system, and obtain the local well-posedness of the corresponding Cauchy problem as well as global existence in 1+1 dimension. As an application, we obtain the existence of Schr\"odinger soliton of the hyperbolic Ishimori system.