Asymptotic Stability of Solitons to Nonlinear Schrodinger Equations on Star Graphs

TL;DR

This paper proves the asymptotic stability of solitons in nonlinear Schrödinger equations on star graphs, advancing understanding of their long-term behavior with new dispersive estimates and scattering theory techniques.

Contribution

It introduces a novel approach to proving stability of solitons on star graphs using dispersive estimates and scattering theory, partially resolving an open problem.

Findings

Established dispersive estimates for the linearized operator

Proved asymptotic stability of solitons on star graphs

Applied Born's series and scattering theory techniques

Abstract

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.