Solitons and Their Ghosts in \(\mathcal{P}\mathcal{T}\) -Symmetric Systems with Defocusing Nonlinearities
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 sm
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Abstract 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 zerosoliton 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 onedark soliton. All of the above notions are generalized in two-dimensional settings for vortices, where the topological charge enforces the destabilization of a twovortex state and the collision of a no-vortex state with a two-vortex one, of a
V. Achilleos D.J. Frantzeskakis Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 157 84, Greece e-mail: [email protected]; [email protected] P.G. Kevrekidis Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USA e-mail: [email protected] R. Carretero-González () Nonlinear Dynamical Systems Group, Department of Mathematics and Statistics, Computational Science Research Center, San Diego State University, San Diego, CA 92182-7720, USA e-mail: [email protected]; http://nlds.sdsu.edu/ R. Carretero-González et al. (eds.), Localized Excitations in Nonlinear Complex Systems, Nonlinear Systems and Complexity 7, DOI 10.1007/978-3-319-02057-0__1, © Springer International Publishing Switzerland 2014
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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.
1 Introduction Over the past decade, and since they were originally proposed by Bender and coworkers [1,2], systems characterized by PT -symmetric Hamiltonians have become a subject of intense research efforts. The interest in these systems arises from their fundamental property to exhibit real spectra, while non-Hermitian, thus providing an intriguing alternative to standard Hermitian quantum mechanics. In the case of a standard Schrödinger-type Hamiltonian, with a generally complex potential U , the PT symmetry dictates that the potential satisfies the condition U.x/ D U .x/, where ./ stands for complex conjugation. Optical systems appear to be ideal setti
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