PT-symmetric oligomers: Analytical solutions, linear stability, and nonlinear dynamics

Abstract

In the present work we focus on the case of (few-site) configurations respecting the parity-time (PT ) symmetry, i.e., with a spatially odd gain-loss profile. We examine the case of such "oligomers" with not only two sites, as in earlier works, but also the cases of three and four 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 -symmetry breaking point in the case of the trimer, and symmetry-breaking bifurcations, as well as more complex, even asymmetric solutions in the case of the quadrimer with nontrivial properties in their linear stability and in their nonlinear dynamics. The linearization around the obtained solutions and their dynamical evolution, when unstable, are discussed.

Nonlinear switching and solitons in PT‐symmetric photonic systems

“…(19) shows that there are up to five (at least one) branches of stationary solutions of Eqs. (18) depending on the particular values of b and k. The dependence of these branches on gain/loss parameter γ for particular case b = k = 1 has been discussed in detail in [58,68]. Three different branches were found: first branch is mostly unstable (except small region that splits unstable region into two domains), while the second one is chiefly stable.…”

“…The nonlinear effects in PT-symmetric systems can be utilized for an efficient control of light including all-optical low-threshold switching and unidirectional invisibility [24,56,57]. The possibility to engineer PT-symmetric oligomers, which may include nonlinearity, triggers a broad variety of studies on both the few-site systems and entire PT-symmetric lattices, including onedimensional PT-symmetric dimer [35,58], trimer [58,59], quadrimer [58,60], 2D PT-symmetric plaquettes [60,61], PT-symmetric finite/infinite chains [62][63][64][65], necklaces [66] and multicore fibers [67].…”

“…PT-symmetric trimer, first discussed in [58] and revisited in [34,68] PT-symmetric interposition of these waveguides with nonzero PT-symmetry breaking threshold: equal gain and loss parameters, i.e. …”

Interaction of sine-Gordon kinks and breathers with a parity-time-symmetric defect

“…These include, among others, unconventional beam refraction [13], Bragg scattering [14], symmetry-breaking transitions [4] and associated ghost states [15][16][17][18], a loss-induced optical transparency [5], conical diffraction [19], a new type of Fano resonance [20], chaos [21], nonlocal boundary effects [22], optical switches [23] and diodes [24,25], phase sensitivity of light dynamics [26][27][28], and the possibility of linear and nonlinear wave amplification and filtering [29][30][31]. Unexpected instabilities were also * saadatmand.d@gmail.com † dmitriev.sergey.v@gmail.com ‡ borisovdi@yandex.ru § kevrekid@math.umass.edu identified at the level of PT -symmetric lattices, and nonlinear modes were identified in few-site oligomers, as well as in full lattice settings both in one dimension [32][33][34][35][36][37] and even in two dimensions [38]. Extensions of PT -symmetric considerations in the setting of active media (of unequal gain and loss) have also recently been proposed [39,40].…”

A PT -Symmetric Dual-Core System with the Sine-Gordon Nonlinearity and Derivative Coupling

“…Equation (24) gives rise to the commonly-known vast set of single-and multi-soliton solutions [13,14], which generates the respective two-component solutions via Equation (23). The analysis of the stability of this solution is a subject for a separate work.…”