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

Abstract

We study the nonlinear Schrödinger equation with logarithmic nonlinearity on a star graph G . At the vertex an interaction occurs described by a boundary condition of delta type with strength α ∈ R . We investigate the orbital stability and the spectral instability of the standing wave solutions e i ω t Φ ( x ) to the equation 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.