We study the nonlinear Schr\"odinger equation (NLS) on a star graph $\mathcal{ G}$. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\in \mathbb{ R}$. We investigate an orbital instability of the standing waves $e^{ i\omega t}\mathbf{ \Phi}(x)$ of NLS-$\delta$ equation with attractive power nonlinearity on $\mathcal{ G}$ when the profile $\Phi(x)$ 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-$\delta$ equation with repulsive nonlinearity.
Categories: math.SP math.AP