Schrödinger Soliton from Lorentzian Manifolds

Abstract

In this paper, we introduce a new notion named as Schrödinger soliton. So-called Schrödinger solitons are defined as a class of special solutions to the Schrödinger flow equation from a Riemannian manifold or a Lorentzian manifold $M$ into a Kähler manifold $N$. If the target manifold $N$ admits a Killing potential, then the Schrödinger soliton is just a harmonic map with potential from $M$ into $N$. Especially, if the domain manifold is a Lorentzian manifold, the Schrödinger 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 Schrödinger soliton of the hyperbolic Ishimori system.

22 pages, with lower regularity of the initial data required in the revised version.