On the orbital instability of excited states for the NLS equation with the $\delta$-interaction on a star graph

TL;DR

This paper analyzes the stability and instability of standing waves in a nonlinear Schrödinger equation with delta interaction on a star graph, revealing conditions for orbital instability and stability using operator extension theory.

Contribution

It introduces a novel approach using extension theory and perturbation analysis to study orbital stability of NLS on star graphs, avoiding traditional variational methods.

Findings

Orbital instability for standing waves with mixed structure under attractive nonlinearity.

Orbital stability of the unique standing wave with repulsive nonlinearity.

Application of Krein-von Neumann extension theory in stability analysis.

Abstract

We study the nonlinear Schr\"odinger equation (NLS) 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 an orbital instability of the standing waves $e^{i\omega t}\mathbf{\Phi}(x)$ of NLS-$\delta$ equation with attractive power nonlinearity on $\mathcal{G}$ when the profile $\Phi(x)$ 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, avoiding the variational techniques standard in the stability study. We also prove orbital stability of the unique standing wave solution of NLS-$\delta$ equation with repulsive nonlinearity.