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

Abstract

We study the nonlinear Schrödinger equation (NLS) on a star graph $\mathcal{G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $α\in \mathbb{R}$. We investigate an orbital instability of the standing waves $e^{iωt}\mathbfΦ(x)$ of NLS-$δ$ equation with attractive power nonlinearity on $\mathcal{G}$ when the profile $Φ(x)$ 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-$δ$ equation with repulsive nonlinearity.