Asymptotic Stability of Solitons to Nonlinear Schrodinger Equations on Star Graphs

In this paper, we prove the asymptotic stability of nonlinear Schrodiger equations on star graphs, which partially solves an open problem in D. Noja \cite{DN}. The essential ingredient of our proof is the dispersive estimate for the linearized operator around the soliton with Kirchhoff boundary condition. In order to obtain the dispersive estimates, we use the Born's series technique and scattering theory for the linearized operator.