Solitons and Their Ghosts in \(\mathcal{P}\mathcal{T}\) -Symmetric Systems with Defocusing Nonlinearities

  • V. Achilleos
  • P. G. Kevrekidis
  • D. J. Frantzeskakis
  • R. Carretero-González
Chapter

Abstract

We examine a prototypical nonlinear Schrödinger model bearing a defocusing nonlinearity and Parity-Time (\(\mathcal{P}\mathcal{T}\)) 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 \(\mathcal{P}\mathcal{T}\)-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 \(\mathcal{P}\mathcal{T}\)-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.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • V. Achilleos
    • 1
  • P. G. Kevrekidis
    • 2
  • D. J. Frantzeskakis
    • 1
  • R. Carretero-González
    • 3
  1. 1.Department of PhysicsUniversity of AthensAthensGreece
  2. 2.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  3. 3.Nonlinear Dynamical Systems Group, Department of Mathematics and Statistics, Computational Science Research CenterSan Diego State UniversitySan DiegoUSA

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