The PT Symmeter

V. Achilleos, P. G.Kevrekidis, D. J. Frantzeskakis, R. Carretero-Gonzalez

We examine a prototypical nonlinear Schrodinger 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.

http://arxiv.org/abs/1208.2445
Pattern Formation and Solitons (nlin.PS); Soft Condensed Matter (cond-mat.soft)

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

We consider the nonlinear analogues of Parity-Time (\(\mathcal{PT}\)) symmetric linear systems exhibiting defocusing nonlinearities. We study the ground state and excited states (dark solitons and vortices) of the system and report the following remarkable features. For relatively weak values of the parameter \(\varepsilon\) controlling the strength of the \(\mathcal{PT}\)-symmetric potential, excited states undergo (analytically tractable) spontaneous symmetry breaking; as \(\varepsilon\) is further increased, the ground state and first excited state, as well as branches of higher multi-soliton (multi-vortex) states, collide in pairs and disappear in blue-sky bifurcations, in a way which is strongly reminiscent of the linear \(\mathcal{PT}\)-phase transition —thus termed the nonlinear \(\mathcal{PT}\)-phase transition. Past this critical point, initialization of, e.g., the former ground state leads to spontaneously emerging “soliton (vortex) sprinklers”.

http://arxiv.org/abs/1202.1310
Pattern Formation and Solitons (nlin.PS); Soft Condensed Matter (cond-mat.soft)