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 G. At the vertex an interaction occurs described by a boundary condition of delta type with strength α ∈ ℝ. We investigate the orbital instability of the standing waves eiωtΦ(x) of the NLS-δ equation with attractive power nonlinearity on 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 the orbital stability of the unique standing wave solution to the NLS-δ equation with repulsive nonlinearity.