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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 into a K\"ahler manifold . If the target manifold admits a
Killing potential, then the Schr\"odinger soliton is just a harmonic map with
potential from into . Especially, if the domain manifold is a Lorentzian
manifold, the Schr\"odinger soliton is a wave map with potential into . 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.Comment: 22 pages, with lower regularity of the initial data required in the
revised version
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