Nonlinear Stationary States in PT-Symmetric Lattices

Abstract

In the present work we examine both the linear and nonlinear properties of two related PT-symmetric systems of the discrete nonlinear Schrödinger (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.

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].…”

PT-symmetric dimer of coupled nonlinear oscillators

“…As illustrated in Ref. [28], in the latter, at worst an exponential (indefinite) growth of the amplitude may arise (but no finitetime blowup). Our numerical computations indicate that the switching behavior reported above is only possible provided that the growth rate (i.e., the FE) of the periodic solution is small enough.…”

“…To provide an analytical insight into the above results, we use the rotating-wave approximation (RWA), which approximates the system by the corresponding nonlinear Schrödinger-type PT -symmetric dimer for which everything can be solved analytically, including the stationary states, the symmetry breaking bifurcations, and even the full dynamics [9,11,28,29]. A direct comparison of the RWA-derived Schrödinger dimer reveals natural similarities but also significant differences between the two models.…”

PT-symmetric coupler withχ(2)nonlinearity

“…Panel (e) shows the amplitudes of the two waveguides of the magenta star family which are oscillating quasi-periodically in a similar way at E = 1.4. Panel (f) and (h) illustrate the instability of cyan squares and black hexagrams where the amplitudes of both harmonics of the first waveguide grow exponentially at about t = 50 while the amplitudes of the second waveguide do not appear to grow indefinitely (but contrary to the cubic case, they are also not observed to systematically decay [27]). The stable dynamics of the orange diamonds family at E = −1.5 is also plotted in panel (g).…”

“…Among these, a PT -symmetric quadrimer model naturally appears in the description of light propagation in a birefringent coupler [24]. Discrete solitons in different types of infinite PT -symmetric waveguide arrays were studied numerically [25] and analytical proofs for their existence have been proposed using the anticontnuum limit [26] and via analysis of the modes bifurcating from the linear limit [27]. Solitons in a necklace of coupled dispersive waveguides were reported in [28].…”