Breathers in Hamiltonian                                        PT                                 -Symmetric Chains of Coupled Pendula under a Resonant Periodic Force

Chernyavsky, Alexander; Pelinovsky, Dmitry E.

doi:10.3390/sym8070059

Cited by 13 publications

(38 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The work that followed focussed on chains of  -symmetric couplers [6,[32][33][34]. The corresponding theoretical studies progressed from a single Schrödinger dimer [17][18][19][20] and oligomer [28,[35][36][37][38], to  -symmetric dimer arrays [39][40][41][42][43][44]. The subsequent analyses included the effects of diffraction of spatial beams and dispersion of temporal pulses, that is, included an additional spatial or temporal dimension [40,[45][46][47][48][49].…”

Section: Introductionmentioning

confidence: 99%

Solitons in ${\mathscr{P}}{\mathscr{T}}$-symmetric ladders of optical waveguides

2017

View full text Add to dashboard Cite

We consider a  -symmetric ladder-shaped optical array consisting of a chain of waveguides with gain coupled to a parallel chain of waveguides with loss. All waveguides have the focusing Kerr nonlinearity. The array supports two co-existing solitons, an in-phase and an antiphase one, and each of these can be centred either on a lattice site or midway between two neighbouring sites. We show that both bond-centred (i.e. intersite) solitons are unstable regardless of their amplitudes and parameters of the chain. The site-centred in-phase soliton is stable when its amplitude lies below a threshold that depends on the coupling and gain-loss coefficient. The threshold is lowest when the gain-to-gain and loss-to-loss coupling constant in each chain is close to the interchain gain-to-loss coupling coefficient. The antiphase site-centred soliton in the strongly-coupled chain or in a chain close to the  -symmetry breaking point, is stable when its amplitude lies above a critical value and unstable otherwise. The instability growth rate of solitons with small amplitude is exponentially small in this parameter regime; hence the small-amplitude solitons, though unstable, have exponentially long lifetimes. On the other hand, the antiphase soliton in the weakly or moderately coupled chain and away from the  -symmetry breaking point, is unstable when its amplitude falls in one or two finite bands. All amplitudes outside those bands are stable.

show abstract

Section: Introductionmentioning

confidence: 99%

Solitons in ${\mathscr{P}}{\mathscr{T}}$-symmetric ladders of optical waveguides

2017

View full text Add to dashboard Cite

show abstract

“…where (A n , B n ) are amplitudes for nearly harmonic oscillations and (X n , Y n ) are remainder terms. Rigorous justification of the asymptotic expansions (3) in a similar context has been developed in [12], see also [2,6]. From the conditions that the remainder terms (X n , Y n ) remain bounded as the system evolves, it can be shown by straightforward computations that the amplitudes (A n , B n ) satisfy the discrete nonlinear Schrödinger (dNLS) equations in the following form:…”

Section: Derivation Of the Pt -Symmetric Dnls Modelmentioning

confidence: 97%

“…Such systems can be formulated mathematically by using the concept of parity (P) and time-reversal (T ) symmetries, which was used first to characterize the non-Hermitian Hamiltonians [3] and has now been widely observed in many physical experiments [4,17]. The presence of Hamiltonian formulation for a class of PT -symmetric dNLS equations allows to employ methods of Hamiltonian dynamics to characterize stability and long-time dynamics of breathers in the parametrically driven chains of coupled oscillators [6,7].…”

Section: Introductionmentioning

confidence: 99%

Coupled pendula chains under parametricPT-symmetric driving force

et al. 2017

View full text Add to dashboard Cite

We consider a chain of coupled pendula pairs, where each pendulum is connected to the nearest neighbors in the longitudinal and transverse directions. The common strings in each pair are modulated periodically by an external force. In the limit of small coupling and near the 1 : 2 parametric resonance, we derive a novel system of coupled PT -symmetric discrete nonlinear Schrödinger equation, which has Hamiltonian symmetry but has no gauge symmetry. By using the conserved energy, we find the parameter range for the linear and nonlinear stability of the zero equilibrium. Numerical experiments illustrate how destabilization of the zero equilibrium takes place when the stability constraints are not satisfied. The central pendulum excites nearest pendula and this process continues until a dynamical equilibrium is reached where each pendulum in the chain oscillates at a finite amplitude.

show abstract

“…More specifically, the Hamiltonian structure has been revealed for PT -symmetric coupled oscillators [6], for some completely integrable dimer models [7,8,9,10,11,12], and for chains of PT -symmetric pendula [13]. We also mention that systems which do not posses PT symmetry but display characteristics of conservative and dissipative ones, are also known; they are described by time-reversible Hamiltonians [14].…”

Section: Introductionmentioning

confidence: 98%

Solitons in a Hamiltonian $\newcommand{\PT}{\mathcal{ PT}}{\boldsymbol \PT}$ -symmetric coupler

Zezyulin¹

2017

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Abstract. We introduce a nonlinear parity-time-symmetric dispersive coupler which admits Hamiltonian and Lagrangian formulations. We show that, in spite of the gain and dissipation, the model has several conservation laws. The system also supports a variety of exact solutions. We focus on exact bright solitons and demonstrate numerically that they are dynamically stable in a wide parameter range and undergo elastic interactions, thus manifesting nearly-integrable dynamics. Physical applications of the introduced model in the theory of Bose-Einstein condensates in nonlinear lattices are discussed.

show abstract

Breathers in Hamiltonian PT -Symmetric Chains of Coupled Pendula under a Resonant Periodic Force

Cited by 13 publications

References 35 publications

Solitons in ${\mathscr{P}}{\mathscr{T}}$-symmetric ladders of optical waveguides

Solitons in ${\mathscr{P}}{\mathscr{T}}$-symmetric ladders of optical waveguides

Coupled pendula chains under parametricPT-symmetric driving force

Solitons in a Hamiltonian $\newcommand{\PT}{\mathcal{ PT}}{\boldsymbol \PT}$ -symmetric coupler

Contact Info

Product

Resources

About