An Analysis on Different Algorithms and Methods of Schrodinger–Poisson–Slater Equation: Investigating the Existence and Computation of Solitary Waves and Eigenpairs in a Schrodinger-Poisson System

Abstract

We study the existence of radially symmetric solitarywaves for a non-linear Schrodinger-Poisson system. In contrast to all previousresults, we consider the presence of a positive potential, of interest inphysical applications. We present a new implementation of the two-grid methodfor computing extremum eigenpairs of self-adjoint partial differentialoperators with periodic boundary conditions. A novel two-grid centereddifference method is proposed for the numerical solutions of the nonlinearSchrödinger–Poisson (SP) eigenvalue problem. We solve the Poisson equation toobtain the nonlinear potential for the nonlinear Schrödinger eigenvalueproblem.