Orbital instability of standing waves for NLS equation on Star Graphs

TL;DR

This paper investigates the stability of standing waves in a nonlinear Schrödinger equation on star graphs with delta interactions, revealing instability for all non-symmetric and symmetric waves under various interaction strengths.

Contribution

It extends Sturm theory to star graph Schrödinger operators and provides explicit Morse index counts, demonstrating instability of all standing waves for both positive and negative delta interaction strengths.

Findings

All nonsymmetric standing waves are orbitally unstable for negative delta interaction.

All standing waves are orbitally unstable for positive delta interaction.

Explicit Morse index counts for each standing wave are provided.

Abstract

We consider a nonlinear Schr\"{o}dinger (NLS) equation with any positive power nonlinearity on a star graph $\Gamma$ ($N$ half-lines glued at the common vertex) with a $\delta$ interaction at the vertex. The strength of the interaction is defined by a fixed value $\alpha \in \mathbb{R}$. In the recent works of Adami {\it et al.}, it was shown that for $\alpha \neq 0$ the NLS equation on $\Gamma$ admits the unique symmetric (with respect to permutation of edges) standing wave and that all other possible standing waves are nonsymmetric. Also, it was proved for $\alpha<0$ that, in the NLS equation with a subcritical power-type nonlinearity, the unique symmetric standing wave is orbitally stable. In this paper, we analyze stability of standing waves for both $\alpha<0$ and $\alpha>0$. By extending the Sturm theory to Schr\"{o}dinger operators on the star graph, we give the explicit count of the Morse and degeneracy indices for each standing wave. For $\alpha<0$, we prove that all nonsymmetric standing waves in the NLS equation with any positive power nonlinearity are orbitally unstable. For $\alpha>0$, we prove the orbital instability of all standing waves.