Logarithmic NLS equation on star graphs: existence and stability of standing waves

TL;DR

This paper studies the logarithmic Schrödinger equation on star graphs, establishing existence, stability, and characterization of standing wave solutions through variational and compactness methods.

Contribution

It introduces a novel analysis of the logarithmic NLS on star graphs, proving existence, stability, and ground state properties for the first time.

Findings

Existence of multiple standing wave solutions.

Ground states are minimizers of the action on the Nehari manifold.

Ground states are orbitally stable.

Abstract

In this paper we consider the logarithmic Schr\"{o}dinger equation on a star graph. By using a compactness method, we construct a unique global solution of the associated Cauchy problem in a suitable functional framework. Then we show the existence of several families of standing waves. We also prove the existence of ground states as minimizers of the action on the Nehari manifold. Finally, we show that the ground states are orbitally stable via a variational approach.