Breathers in Hamiltonian ${\cal PT}$-symmetric chains of coupled pendula under a resonant periodic force

Long-time stability of breathers in Hamiltonian ${ \mathcal P }{ \mathcal T }$-symmetric lattices

“…Each pair of coupled pendula (u n , v n ) are hung on a common string with a periodically varying tension coefficient propositional to γ. When the frequency of the periodic force is in 1 : 2 resonance with the frequency of pendula detuned by Ω, the system of Newton's equations of motion has been shown in [6] to reduce asymptotically to the system of amplitude equations (1). Similar systems of amplitude equations were derived previously in a number of physically relevant applications [2,4,5].…”

“…This technique was introduced for the PT -symmetric systems in [12,14] and was applied to the system of amplitude equations (1) in [6]. Here we recall the main facts about these breathers obtained in [6]. The values ±E 0 with E 0 := Ω 2 − γ 2 correspond to bifurcation of the small-amplitude solutions.…”

“…Remark 1. As shown in [6], the zero equilibrium of the PT -symmetric dNLS equation ( 1) is linearly stable if |γ| < γ 0 , where the PT phase transition threshold γ 0 is given by…”

“…Existence and spectral stability of breathers can be characterized in the limit of small coupling constant ǫ, when breathers bifurcate from solutions of the dimer equation arising at a single site, say the central site at n = 0. This technique was introduced for the PT -symmetric systems in [12,14] and was applied to the system of amplitude equations (1) in [6]. Here we recall the main facts about these breathers obtained in [6].…”

“…The small-amplitude solutions are connected with the large-amplitude solutions for Ω > γ > 0, whereas the branches of small-amplitude and large-amplitude solutions are disconnected for Ω < −γ < 0. Every time-periodic solution supported at the central dimer for ǫ = 0 is continued uniquely and smoothly with respect to the small coupling parameter ǫ by the implicit function arguments [6]. The resulting breather is symmetric about the central site and PT -symmetric so that…”