On the standing waves of the NLS-log equation with point interaction on a star graph

TL;DR

This paper analyzes the stability of standing wave solutions to a nonlinear Schrödinger equation with logarithmic nonlinearity on a star graph, focusing on the effects of point interactions at the vertex.

Contribution

It introduces a novel analysis of orbital and spectral stability for NLS-log equations with point interactions on star graphs using extension theory and perturbation methods.

Findings

Identifies conditions for orbital stability of standing waves.

Determines spectral instability for certain configurations.

Provides a framework for analyzing nonlinear equations on graph structures.

Abstract

We study a nonlinear Schr\"odinger equation with logarithmic nonlinearity on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\in \mathbb{R}$. We investigate orbital stability and spectral instability of the standing wave solutions $e^{i\omega t}\mathbf{\Phi}(x)$ to the equation when the profile $\mathbf\Phi(x)$ has mixed structure (i.e. has bumps and tails). In our approach we essentially use the extension theory of symmetric operators by Krein - von Neumann, and the analytic perturbations theory.