Schr\"odinger Soliton from Lorentzian Manifolds

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.