Global solutions for nonlinear Schrödinger equations in de Sitter spacetime

Abstract

We consider the nonlinear Schrödinger 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 Schrödinger equation in de Sitter spacetime is derived as their nonrelativistic limits. Furthermore we consider the Cauchy problem of the Schrödinger equations in Sobolev spaces. Some effects of spatial variation on the problem are remarked.