Nonlinear Schr\"odinger equations with trapping potentials in higher dimensions

TL;DR

This paper investigates spherically symmetric solutions of nonlinear Schrödinger equations with trapping potentials in higher dimensions, focusing on existence, uniqueness, stability, and dynamical properties, especially for Schrödinger-Newton-Hooke equations.

Contribution

It introduces a method to analyze higher-dimensional nonlinear Schrödinger equations using classical ODE and dynamical systems techniques, overcoming variational method limitations.

Findings

Existence and uniqueness of stationary solutions

Analysis of their frequency and stability

Dynamical properties of the resonant approximation

Abstract

From the mathematical side, nonlinear Schr\"odinger equations are usually investigated via variational methods, that cease to work in higher dimensions. This thesis tries to overcome this problem by focusing on spherically symmetric solutions. Then, one can use classical methods coming from the fields of ordinary differential equations and dynamical systems. The results presented here include existence and uniqueness of the stationary solutions, their frequency, and stability. The dynamical properties of the resonant approximation are also explored. The main focus is given to the Schr\"odinger-Newton-Hooke equations that is shown to be a nonrelativistic limit of perturbations of the anti-de Sitter spacetime.