Symmetry-breaking bifurcations and ghost states in the fractional nonlinear Schr\"{o}dinger equation with a PT-symmetric potential

TL;DR

This paper investigates symmetry-breaking bifurcations and ghost states in a fractional Schrödinger equation with PT-symmetric potential, revealing novel bifurcation phenomena and stability properties relevant to optical systems.

Contribution

It uncovers the existence of ghost states and their relation to symmetry-breaking bifurcations in fractional nonlinear Schrödinger equations with PT-symmetry, a novel insight in the field.

Findings

Bifurcation points destabilize solitons.

Inverse bifurcation restores stability in CQ nonlinearity.

Ghost states with complex propagation constants are created by bifurcation.

Abstract

We report symmetry-breaking and restoring bifurcations of solitons in a fractional Schr\"{o}dinger equation with the cubic or cubic-quintic (CQ) nonlinearity and a parity-time (PT)-symmetric potential, which may be realized in optical cavities. Solitons are destabilized at the bifurcation point, and, in the case of the CQ nonlinearity, the stability is restored by an inverse bifurcation. Two mutually-conjugate branches of ghost states (GSs), with complex propagation constants, are created by the bifurcation, solely in the case of the fractional diffraction. While GSs are not true solutions, direct simulations confirm that their shapes and results of their stability analysis provide a blueprint for the evolution of genuine localized modes in the system.