Existence and orbital stability of standing-wave solutions of the NLS-log equation on a tadpole graph

TL;DR

This paper investigates the existence and orbital stability of standing wave solutions for the nonlinear logarithmic Schrödinger equation on a tadpole graph, providing the first such analysis on this type of graph.

Contribution

It establishes the existence and stability of standing waves for NLS-log on tadpole graphs, introducing a novel analysis method for this setting.

Findings

Existence of standing wave solutions on tadpole graphs.

Orbital stability of these solutions under certain conditions.

Identification of spectral properties of the linearized operator.

Abstract

This work aims to study some dynamical aspects of the nonlinear logarithmic Schr\"odinger equation (NLS-log) on a tadpole graph, namely, a graph consisting of a circle with a half-line attached at a single vertex. By considering Neumann-Kirchhoff boundary conditions at the junction we show the existence and the orbital stability of standing wave solutions with a profile determined by a positive single-lobe state. Via a splitting-eigenvalue method, we identify the Morse index and the nullity index of a specific linearized operator around a positive singlelobe state. To our knowledge, the results contained in this paper are the first to study the (NLS-log) on tadpole graphs. In particular, our approach has the prospect of being extended to study stability properties of other bound states for the (NLS-log) on a tadpole graph or other non-compact metric graph such as a looping-edge graphs.