Orbital instability of standing waves for NLS equation on star graphs

Abstract:

We consider a nonlinear Schrödinger (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 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 Schrödinger 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.