Solitons and their ghosts in PT-symmetric systems with defocusing nonlinearities

TL;DR

This paper investigates the behavior of dark solitons and vortices in PT-symmetric nonlinear Schrödinger systems with defocusing nonlinearities, revealing bifurcations, ghost states, and stability phenomena through numerical analysis.

Contribution

It introduces the concept of ghost states and analyzes bifurcations and stability of solitons and vortices in PT-symmetric systems with defocusing nonlinearities.

Findings

Dark solitons destabilize via symmetry breaking bifurcation.

Ground state and dark soliton vanish at a PT-phase transition.

Ghost states influence system dynamics despite not being exact solutions.

Abstract

We examine a prototypical nonlinear Schr\"odinger model bearing a defocusing nonlinearity and Parity-Time (PT) symmetry. For such a model, the solutions can be identified numerically and characterized in the perturbative limit of small gain/loss. There we find two fundamental phenomena. First, the dark solitons that persist in the presence of the PT-symmetric potential are destabilized via a symmetry breaking (pitchfork) bifurcation. Second, the ground state and the dark soliton die hand-in-hand in a saddle-center bifurcation (a nonlinear analogue of the PT-phase transition) at a second critical value of the gain/loss parameter. The daughter states arising from the pitchfork are identified as "ghost states", which are not exact solutions of the original system, yet they play a critical role in the system's dynamics. A similar phenomenology is also pairwise identified for higher excited states, with e.g. the two-soliton structure bearing similar characteristics to the zero-soliton one, and the three-soliton state having the same pitchfork destabilization mechanism and saddle-center collision (in this case with the two-soliton) as the one-dark soliton. All of the above notions are generalized in two-dimensional settings for vortices, where the topological charge enforces the destabilization of a two-vortex state and the collision of a no-vortex state with a two-vortex one, of a one-vortex state with a three-vortex one, and so on. The dynamical manifestation of the instabilities mentioned above is examined through direct numerical simulations.