Asymptotic behavior for the long-range nonlinear Schrödinger equation on the star graph with the Kirchhoff boundary condition

Abstract

We consider the cubic nonlinear Schrödinger equation on the star graph with the Kirchhoff boundary condition. We prove modified scattering for the final state problem and the initial value problem. Moreover, we also consider the failure of scattering for the Schrödinger equation with power-type long-range nonlinearities. These results are extensions of the results for the nonlinear Schrödinger equation on the one-dimensional Euclidean space.