An example of stable excited state on nonlinear Schrödinger equation with nonlocal nonlinearity

Abstract

In this article, we consider the nonlinear Schrödinger equation with nonlocal nonlinearity, which is a generalized model of the Schrödinger--Poisson system (Schrödinger--Newton equations) in low dimensions. We prove global well-posedness in a wider space than in previous results and show the stability of standing waves including excited states. It turns out that an example of stable excited states with high Morse index is contained. Several examples of traveling-wave-type solutions are also given.