2012/08/12 by V. Achilleos, P. G. Kevrekidis, D. J. Frantzeskakis +2
Physics and Astronomy · #Bifurcation #Collision #Ground state #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Phenomenology (philosophy) #Pitchfork bifurcation #Quantum Mechanics and Non-Hermitian Physics #Soliton #Symmetry breaking #cond-mat.soft #nlin.PS
paper · pdf · doi:10.1007/978-3-319-02057-0_1
published as Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity 7, (2014) 3-42 · 17 pages, 16 figures
arxiv created 2012/08/12 · openalex publication_date 2013/10/21 · arxiv updated 2014/12/03 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We examine a prototypical nonlinear Schrödinger 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.