Global solutions for nonlinear Schrodinger equations in de Sitter spacetime

We consider the nonlinear Schrodinger equations in the de Sitter spacetime. We first show some derivation of the Einstein equation and several models of the universe for general dimensions and complex metrics. Then some models of the uniform and isotropic universe are considered based on the Einstein equation. After deriveing nonlinear Klein-Gordon equations in de Sitter spacetime, and the nonlinear Schrodinger equation in de Sitter spacetime is derived as their non-relativistic limits. Furthermore we consider the Cauchy problem of the Schrodinger equations in Sobolev spaces. Some effects of spatial variation on the problem are remarked.