Asymptotic behavior for the long-range nonlinear Schr\"odinger equation on star graph with the Kirchhoff boundary condition

TL;DR

This paper studies the long-term behavior of solutions to the nonlinear Schrödinger equation on star graphs, demonstrating modified scattering in some cases and failure of scattering in others, extending known Euclidean space results.

Contribution

It extends the analysis of nonlinear Schrödinger equations to star graphs with Kirchhoff boundary conditions, establishing new scattering and non-scattering results for long-range nonlinearities.

Findings

Proves modified scattering for the initial and final state problems.

Shows failure of scattering for certain power-type long-range nonlinearities.

Extends Euclidean space NLS results to star graph geometries.

Abstract

We consider the cubic nonlinear Schr\"{o}dinger 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 Schr\"{o}dinger equation with power-type long-range nonlinearities. These results are extension of the results for NLS on the one dimensional Euclidean space.