Failure of scattering to solitary waves for long-range nonlinear Schr\"odinger equations

TL;DR

This paper proves that solutions to certain long-range nonlinear Schrödinger equations with external potentials cannot scatter to solitary or localized waves, extending known non-existence results for scattering states.

Contribution

It establishes the non-existence of scattering to solitary waves in long-range nonlinear Schrödinger equations with external potentials, broadening previous non-scattering results.

Findings

Solutions do not scatter to solitary waves for long-range nonlinearities.

Results apply to power-type and Hartree nonlinearities.

Extends non-existence of scattering states to more general settings.

Abstract

We consider nonlinear Schr\"odinger equations with either power-type or Hartree nonlinearity in the presence of an external potential. We show that for long-range nonlinearities, solutions cannot exhibit scattering to solitary waves or more general localized waves. This extends the well-known results concerning non-existence of non-trivial scattering states for long-range nonlinearities.