Orbital instability of standing waves for NLS equation on Star Graphs

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.