  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.
