Schrödinger Soliton from Lorentzian Manifolds

Details Schrödinger Soliton from Lorentzian Manifolds Schrödinger Soliton from Lorentzian Manifolds

2009-10-09

arxiv.org/abs/0910.1759v3

Mathematics

Differential Geometry

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

Scientific paper

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.

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