New Explicit and Approximate Solutions of the Newton-Schrödinger System

In this paper, we consider the Newton-Schrödinger system ∇2Ψ=γΦ+a(x)Ψ,∇2Φ=∣Ψ∣2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{rcl} \nabla ^{2} \Psi & =& \left( \gamma \Phi +a(x)\right) \Psi ,\\ \nabla ^{2} \Phi & =& \mid \Psi \mid ^{2}, \end{array} \right. \end{aligned}$$\end{document}which arises in certain quantum transport and chemistry problems. Explicit analytic solutions, which contain an auxiliary parameter, are obtained. An existence and uniqueness theorem to this nonlinear system subject to the boundary conditions is proved. Also, we introduce approximate solutions to the modified Newton-Schrödinger system in the case of spherically-symmetric stationary and time-independence by the Adomian decomposition method.