Asymptotic behavior in time of solution for the cubic nonlinear Schrödinger equation on the tadpole graph

The purpose of this paper is to study large time behavior of solution to the cubic nonlinear Schrödinger equation on the tadpole graph which is a ring attached to a semi-infinite line subject to the Kirchhoff conditions at the junction. Note that the cubic nonlinearity belongs borderline between short and long range scatterings on the whole line. We show that if the initial data has some symmetry on the graph which excludes the standing wave solutions, then the asymptotic behavior of solution to this equation is given by the solution to linear equation with logarithmic phase correction by the nonlinear effect.