In the present work, we focus on the cases of twosite (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.

No abstract

No abstract

We study the effects of management of the PT-symmetric part of the potential within the setting of Schrödinger 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. I. INTRODUCTIONIt has been about a decade and a half since the radical and highly innovative proposal of C. Bender and his collaborators [1] regarding the potential physical relevance of Hamiltonians respecting Parity (P) and time-reversal (T) symmetries. While earlier work was focused on an implicit postulate of solely self-adjoint Hamiltonian operators, this proposal suggested that these fundamental symmetries may allow for a real operator spectrum within a certain regime of parameters which is regarded as the regime of exact PT-symmetry. On the other hand, beyond a critical parametric strength, the relevant operators may acquire a spectrum encompassing imaginary or even genuinely complex eigenvalues, in which case, we are referring (at the linear level) to the regime of broken PT-phase.These notions were intensely studied at the quantum mechanical level, chiefly as theoretical constructs. Yet, it was the fundamental realization that optics can enable such "open" systems featuring gain and loss, both at the theoretical [2-5] and even at the experimental [6,7] level, that propelled this activity into a significant array of new directions, including the possibility of the interplay of nonlinearity with PT-symmetry. In this optical context, the well-known connection of the Maxwell equations with the Schrödinger equation was utilized, and Hamiltonians of the form H = −(1/2)∆ + V (x) were considered at the linear level with the PT-symmetry necessitating that the potential satisfies the condition V (x) = V ⋆ (−x). Yet another physical context where such systems have been experimentally "engineered" recently is that of electronic circuits; see the work of [8] and also the review of [9]. In parallel to the recent experimental developments, numerous theoretical groups have explored various features of both linear PT-symmetric potentials [10-36] and even of nonlinear ones such where a PT-symmetric type of gain/loss pattern appears in the nonlinear term [37][38][39][40].Our aim in the present work is to combine this highly active research theme of PT-symmetry with another topic of considerable recent interest in the physics of optical and also atomic systems, namely that of "ma...