Abstract
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.
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