Stability transitions of NLS action ground-states on metric graphs

TL;DR

This paper investigates the stability of action ground-states for the nonlinear Schrödinger equation on specific metric graphs, revealing a novel stability transition phenomenon near the $L^2$-critical exponent, supported by theoretical analysis and numerical simulations.

Contribution

It introduces the concept of stability transitions near the $L^2$-critical exponent for NLS on metric graphs, a new dynamical feature not previously documented.

Findings

Stability transitions occur as frequency varies, switching from stable to unstable and back.

Theoretical analysis confirms stability behavior in low/high frequency and nonlinear regimes.

Numerical simulations support the theoretical stability transition results.

Abstract

We study the orbital stability of action ground-states of the nonlinear Schr\"odinger equation over two particular cases of metric graphs, the $\mathcal{T}$ and the tadpole graphs. We show the existence of stability transitions near the $L^2$-critical exponent, a new dynamical feature of the nonlinear Schr\"odinger equation. More precisely, as the frequency $\lambda$ increases, the action ground-state transitions from stable to unstable and then back to stable (or vice-versa). This result is complemented with the stability analysis of ground-states in the asymptotic cases of low/high frequency and weak/strong nonlinear interaction. Finally, we present a numerical simulation of the stability of action ground-states depending on the nonlinearity and the frequency parameter, which validates the aforementioned theoretical results.