Dynamics of the NLS-log equation on a tadpole graph

Abstract: This work aims to study some dynamics issues 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 $\delta$-type boundary conditions at the junction we show the existence and the orbital stability of standing-waves 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 an a priori positive single-lobe state. To our knowledge, the results contained in this paper are the first in studying 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 another non-compact metric graph such as a looping edge graph.