On the standing waves of the NLS-log equation with point interaction on a star graph

Abstract

We study a nonlinear Schrödinger equation with logarithmic nonlinearity 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 orbital stability and spectral instability of the standing wave solutions $e^{i\omega t}\mathbf{\Phi }(x)$ to the equation when the profile $\mathbf{\Phi }(x)$ 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.

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