Variational and stability properties of constant solutions to the NLS equation on compact metric graphs

TL;DR

This paper investigates the stability and variational characteristics of constant solutions to the nonlinear Schrödinger equation on compact metric graphs, emphasizing how graph topology and metric influence these properties across different regimes.

Contribution

It provides a comprehensive analysis of the stability and variational nature of constant solutions on compact graphs, highlighting the impact of graph topology and metric features.

Findings

Constant solutions exist for all masses due to compactness.

Stability and minimality depend on the graph's topology and metric properties.

Differences between subcritical and critical regimes are characterized.

Abstract

We consider the nonlinear Schr\"odinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the graph is compact, for every value of the mass there is a constant solution. Our scope is to analyze (in dependence of the mass) the variational properties of this solution, as a critical point of the energy functional: local and global minimality, and (orbital) stability. We consider both the subcritical regime and the critical one, in which the features of the graph become relevant. We describe how the above properties change according to the topology and the metric properties of the graph.