J. Cuevas-Maraver, A. Khare, P.G. Kevrekidis, H. Xu, A. Saxena

In the present work, we explore the case of a general PT-symmetric dimer in the context of two both linearly and nonlinearly coupled cubic oscillators. To obtain an analytical handle on the system, we first explore the rotating wave approximation converting it into a discrete nonlinear Schrodinger type dimer. In the latter context, the stationary solutions and their stability are identified numerically but also wherever possible analytically. Solutions stemming from both symmetric and anti-symmetric special limits are identified. A number of special cases are explored regarding the ratio of coefficients of nonlinearity between oscillators over the intrinsic one of each oscillator. Finally, the considerations are extended to the original oscillator model, where periodic orbits and their stability are obtained. When the solutions are found to be unstable their dynamics is monitored by means of direct numerical simulations.

http://arxiv.org/abs/1409.7218

Pattern Formation and Solitons (nlin.PS); Chaotic Dynamics (nlin.CD); Exactly Solvable and Integrable Systems (nlin.SI)

N. Lu, J. Cuevas-Maraver, P.G. Kevrekidis

In this work, we explore a prototypical example of a genuine continuum breather (i.e., not a standing wave) and the conditions under which it can persist in a \(\mathcal{P T}\)-symmetric medium. As our model of interest, we will explore the sine-Gordon equation in the presence of a \(\mathcal{P T}\)- symmetric perturbation. Our main finding is that the breather of the sine-Gordon model will only persist at the interface between gain and loss that \(\mathcal{P T}\)-symmetry imposes but will not be preserved if centered at the lossy or at the gain side. The latter dynamics is found to be interesting in its own right giving rise to kink-antikink pairs on the gain side and complete decay of the breather on the lossy side. Lastly, the stability of the breathers centered at the interface is studied. As may be anticipated on the basis of their “delicate” existence properties such breathers are found to be destabilized through a Hopf bifurcation in the corresponding Floquet analysis.

http://arxiv.org/abs/1406.3082

Pattern Formation and Solitons (nlin.PS)

A. Demirkaya, M. Stanislavova, A. Stefanov, T. Kapitula, P.G. Kevrekidis

In the present work we consider the introduction of PT-symmetric terms in the context of classical Klein-Gordon field theories. We explore the implication of such terms on the spectral stability of coherent structures, namely kinks. We find that the conclusion critically depends on the location of the kink center relative to the center of the PT-symmetric term. The main result is that if these two points coincide, the kink’s spectrum remains on the imaginary axis and the wave is spectrally stable. If the kink is centered on the “lossy side” of the medium, then it becomes stabilized. On the other hand, if it becomes centered on the “gain side” of the medium, then it is destabilized. The consequences of these two possibilities on the linearization (point and essential) spectrum are discussed in some detail.

http://arxiv.org/abs/1402.2942

Pattern Formation and Solitons (nlin.PS); Mathematical Physics (math-ph)

K. Li, P. G. Kevrekidis, B. A. Malomed

The subject of the work are pairs of linearly coupled PT-symmetric dimers. Two different settings are introduced, namely, straight-coupled dimers, where each gain site is linearly coupled to one gain and one loss site, and cross-coupled dimers, with each gain site coupled to two lossy ones. The latter pair with equal coupling coefficients represents a “PT-hypersymmetric” quadrimer. We find symmetric and antisymmetric solutions in these systems, chiefly in an analytical form, and explore the existence, stability and dynamical behavior of such solutions by means of numerical methods. We thus identify bifurcations occurring in the systems, including spontaneous symmetry breaking and saddle-center bifurcations. Simulations demonstrate that evolution of unstable branches typically leads to blowup. However, in some cases unstable modes rearrange into stable ones.

http://arxiv.org/abs/1312.3376

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

H.Xu, P.G.Kevrekidis, Q.Zhou, D.J.Frantzeskakis, V.Achilleos, R.Carretero-Gonzalez

We study the nonlinear Schro¨dinger equation with a PT-symmetric potential. Using a hydrodynamic formulation and connecting the phase gradient to the field amplitude, allows for a reduction of the model to a Duffing or a generalized Duffing equation. This way, we can obtain exact soliton solutions existing in the presence of suitable PT-symmetric potentials, and study their stability and dynamics. We report interesting new features, including oscillatory instabilities of solitons and (nonlinear) PT-symmetry breaking transitions, for focusing and defocusing nonlinearities.

http://arxiv.org/abs/1310.7635

Pattern Formation and Solitons (nlin.PS)

H.Xu, P.G.Kevrekidis, Q.Zhou, D.J.Frantzeskakis, V.Achilleos, R.Carretero-Gonzalez

We study the nonlinear Schrodinger equation with a PT-symmetric potential. Using a hydrodynamic formulation and connecting the phase gradient to the field amplitude, allows for a reduction of the model to a Duffing or a generalized Duffing equation. This way, we can obtain exact soliton solutions existing in the presence of suitable PT-symmetric potentials, and study their stability and dynamics. We report interesting new features, including oscillatory instabilities of solitons and (nonlinear) PT-symmetry breaking transitions, for focusing and defocusing nonlinearities.

http://arxiv.org/abs/1310.7635

Pattern Formation and Solitons (nlin.PS)

R.L. Horne, J. Cuevas, P.G. Kevrekidis, N. Whitaker, F.Kh. Abdullaev, D.J. Frantzeskakis

We study the effects of management of the PT-symmetric part of the potential within the setting of Schrodinger dimer and trimer oligomer systems. This is done by rapidly modulating in time the gain/loss profile. This gives rise to a number of interesting properties of the system, which are explored at the level of an averaged equation approach. Remarkably, this rapid modulation provides for a controllable expansion of the region of exact PT-symmetry, depending on the strength and frequency of the imposed modulation. The resulting averaged models are analyzed theoretically and their exact stationary solutions are translated into time-periodic solutions through the averaging reduction. These are, in turn, compared with the exact periodic solutions of the full non-autonomous PT-symmetry managed problem and very good agreement is found between the two.

http://arxiv.org/abs/1308.3738

Pattern Formation and Solitons (nlin.PS)

J. Cuevas, P.G. Kevrekidis, A. Saxena, A. Khare

We provide a systematic analysis of a prototypical nonlinear oscillator system respecting PT-symmetry i.e., one of them has gain and the other an equal and opposite amount of loss. Starting from the linear limit of the system, we extend considerations to the nonlinear case for both soft and hard cubic nonlinearities identifying symmetric and anti-symmetric breather solutions, as well as symmetry breaking variants thereof. We propose a reduction of the system to a Schr\”odinger type PT-symmetric dimer, whose detailed earlier understanding can explain many of the phenomena observed herein, including the PT phase transition. Nevertheless, there are also significant parametric as well as phenomenological potential differences between the two models and we discuss where these arise and where they are most pronounced. Finally, we also provide examples of the evolution dynamics of the different states in their regimes of instability.

http://arxiv.org/abs/1307.6047

Pattern Formation and Solitons (nlin.PS)

K. Li, P. G. Kevrekidis, D. J. Frantzeskakis, C. E. Ruter, D. Kip

In this paper, we revisit one of the prototypical PT-symmetric oligomers, namely the trimer. We find all the relevant branches of “regular” solutions and analyze the bifurcations and instabilities thereof. Our work generalizes the formulation that was proposed recently in the case of dimers for the so-called “ghost states” of trimers, which we also identify and connect to symmetry-breaking bifurcations from the regular states. We also examine the dynamics of unstable trimers, as well as those of the ghost states in the parametric regime where the latter are found to exist. Finally, we present the current state of the art for optical experiments in PT-symmetric trimers, as well as experimental results in a gain-loss-gain three channel waveguide structure.

http://arxiv.org/abs/1306.2255

Quantum Physics (quant-ph)

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)