Failure of scattering to standing waves for a Schr\"{o}dinger equation with long-range nonlinearity on star graph

TL;DR

This paper proves that solutions to a Schrödinger equation with long-range nonlinearity on star graphs do not tend to standing waves over time, extending previous Euclidean space results to graph structures.

Contribution

It demonstrates non-scattering of solutions for a broad class of boundary conditions on star graphs, generalizing prior Euclidean space analyses to graph settings.

Findings

Non-trivial solutions do not scatter to standing waves.

Results hold for various boundary conditions including Kirchhoff, Dirichlet, δ, and δ'.

Abstract

We consider the Schr\"{o}dinger equation with power type long-range nonlinearity on star graph. Under a general boundary condition at the vertex, including Kirchhoff, Dirichlet, $\delta$, or $\delta'$ boundary condition, we show that the non-trivial global solution does not scatter to standing waves. Our proof is based on the argument by Murphy and Nakanishi, who treated the long-range nonlinear Schr\"{o}dinger equation with a general potential in the Euclidean space, in order to consider general boundary conditions.