Power-law expansion cosmology in Schr\"odinger-type formulation

We investigate non-linear Schr\"{o}dinger-type formulation of cosmology of which our cosmological system is a general relativistic FRLW universe containing canonical scalar field under arbitrary potential and a barotropic fluid with arbitrary spatial curvatures. We extend the formulation to include phantom field case and we have found that Schr\"{o}dinger wave function in this formulation is generally non-normalizable. Assuming power-law expansion, $a \sim t^q$, we obtain scalar field potential as function of time. The corresponding quantities in Schr\"{o}dinger-type formulation such as Schr\"{o}dinger total energy, Schr\"{o}dinger potential and wave function are also presented.Comment: 9 pages, 5 figures, Revtex 4, accepted by Astroparticle Physics, more Ref. adde

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