## Nonlinear stationary states in PT-symmetric lattices

Panayotis G. Kevrekidis, Dmitry E. Pelinovsky, Dmitry Y.Tyugin

In the present work we examine both the linear and nonlinear properties of two related PT-symmetric systems of the discrete nonlinear Schrodinger (dNLS) type. First, we examine the parameter range for which the finite PT-dNLS chains have real eigenvalues and PT-symmetric linear eigenstates. We develop a systematic way of analyzing the nonlinear stationary states with the implicit function theorem at an analogue of the anti-continuum limit for the dNLS equation. Secondly, we consider the case when a finite PT-dNLS chain is embedded as a defect in the infinite dNLS lattice. We show that the stability intervals of the infinite PT-dNLS lattice are wider than in the case of a finite PT-dNLS chain. We also prove existence of localized stationary states (discrete solitons) in the analogue of the anti-continuum limit for the dNLS equation.
Numerical computations illustrate the existence of nonlinear stationary states, as well as the stability and saddle-center bifurcations of discrete solitons.

http://arxiv.org/abs/1303.3298
Pattern Formation and Solitons (nlin.PS); Dynamical Systems (math.DS)

## Parity-time symmetric coupler with birefringent arms

K. Li, D. A. Zezyulin, V. V. Konotop, P. G. Kevrekidis

In this work, we propose a PT-symmetric coupler whose arms are birefringent waveguides as a realistic physical model which leads to a so-called quadrimer i.e., a four complex field setting. We seek stationary solutions of the resulting linear and nonlinear model, identifying its linear point of PT symmetry breaking and examining the corresponding nonlinear solutions that persist up to this point, as well as, so-called, ghost states that bifurcate from them. We obtain the relevant symmetry breaking bifurcations and numerically follow the associated dynamics which give rise to growth/decay even within the PT-symmetric phase. Our obtained stationary nonlinear solutions are found to terminate in saddle-center bifurcations which are analogous to the linear PT-phase transition.

http://arxiv.org/abs/1212.1676
Quantum Physics (quant-ph)

## PT-symmetric lattices with spatially extended gain/loss are generically unstable

D.E. Pelinovsky, P.G. Kevrekidis, D.J. Frantzeskakis

We illustrate, through a series of prototypical examples, that linear parity-time (PT) symmetric lattices with extended gain/loss profiles are generically unstable, for any non-zero value of the gain/loss coefficient. Our examples include a parabolic real potential with a linear imaginary part and the cases of no real and constant or linear imaginary potentials. On the other hand, this instability can be avoided and the spectrum can be real for localized or compact PT-symmetric potentials. The linear lattices are analyzed through discrete Fourier transform techniques complemented by numerical computations.

http://arxiv.org/abs/1211.5815
Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

## Linear and Nonlinear PT-symmetric Oligomers: A Dynamical Systems Analysis

M. Duanmu, K. Li, R. L. Horne, P. G. Kevrekidis, N. Whitaker

In the present work we focus on the cases of two-site (dimer) and three-site (trimer) configurations, i.e. oligomers, respecting the parity-time (PT) symmetry, i.e., with a spatially odd gain-loss profile. We examine different types of solutions of such configurations with linear and nonlinear gain/loss profiles. Solutions beyond the linear PT-symmetry critical point as well as solutions with asymmetric linearization eigenvalues are found in both the nonlinear dimer and trimer. The latter feature is absent in linear PT-symmetric trimers, while both of them are absent in linear PT symmetric dimers. Furthermore, nonlinear gain/loss terms enable the existence of both symmetric and asymmetric solution profiles (and of bifurcations between them), while only symmetric solutions are present in the linear PT-symmetric dimers and trimers. The linear stability analysis around the obtained solutions is discussed and their dynamical evolution is explored by means of direct numerical simulations. Finally, a brief discussion is also given of recent progress in the context of PT-symmetric quadrimers.

http://arxiv.org/abs/1210.3871
Quantum Physics (quant-ph)

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

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)

## PT-symmetric Double Well Potentials Revisited: Bifurcations, Stability and Dynamics

A.S. Rodrigues, K. Li, V. Achilleos, P.G. Kevrekidis, D.J. Frantzeskakis, Carl M. Bender

In this work we analyze PT-symmetric double-well potentials based on a two-mode picture. We reduce the problem into a PT-symmetric dimer and illustrate that the latter has effectively two fundamental bifurcations, a pitchfork (symmetry-breaking bifurcation) and a saddle-center one, which is the nonlinear analog of the PT-phase-transition. It is shown that the symmetry breaking leads to ghost states (amounting to growth or decay); although these states are not true solutions of the original continuum problem, the system’s dynamics closely follows them, at least in its metastable evolution. Past the second bifurcation, there are no longer states of the original continuum system. Nevertheless, the solutions can be analytically continued to yield a new pair of branches, which is also identified and dynamically examined. Our explicit analytical results for the dimer are directly compared to the full continuum problem, yielding a good agreement.

http://arxiv.org/abs/1207.1066
Pattern Formation and Solitons (nlin.PS); Quantum Gases (cond-mat.quant-gas); Mathematical Physics (math-ph)

## Nonlinear PT-symmetric plaquettes

Kai Li, P. G. Kevrekidis, Boris A. Malomed, Uwe Guenther

We introduce four basic two-dimensional (2D) plaquette configurations with onsite cubic nonlinearities, which may be used as building blocks for 2D PT -symmetric lattices. For each configuration, we develop a dynamical model and examine its PT symmetry. The corresponding nonlinear modes are analyzed starting from the Hamiltonian limit, with zero value of the gain-loss coefficient. Once the relevant waveforms have been identified (chiefly, in an analytical form), their stability is examined by means of linearization in the vicinity of stationary points. This reveals diverse and, occasionally, fairly complex bifurcations. The evolution of unstable modes is explored by means of direct simulations. In particular, stable localized modes are found in these systems, although the majority of identified solutions is unstable.

http://arxiv.org/abs/1204.5530
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th)

## Dark solitons and vortices in PT-symmetric nonlinear media: from spontaneous symmetry breaking to nonlinear PT phase transitions

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)

## PT-Symmetric Oligomers: Analytical Solutions, Linear Stability and Nonlinear Dynamics

K. Li, P.G. Kevrekidis

In the present work we focus on the case of (few-site) configurations respecting the PT-symmetry. We examine the case of such “oligomers” with not only 2-sites, as in earlier works, but also the cases of 3- and 4-sites. While in the former case of recent experimental interest, the picture of existing stationary solutions and their stability is fairly straightforward, the latter cases reveal a considerable additional complexity of solutions, including ones that exist past the linear PT-breaking point in the case of the trimer, and more complex, even asymmetric solutions in the case of the quadrimer with nontrivial spectral and dynamical properties. Both the linear stability and the nonlinear dynamical properties of the obtained solutions are discussed.

http://arxiv.org/abs/1102.0809
Pattern Formation and Solitons (nlin.PS); Quantum Gases (cond-mat.quant-gas)