Stationary modes and integrals of motion in nonlinear lattices with a ${\mathcal {PT}}$-symmetric linear part

Zezyulin, Dmitry A.

doi:10.1088/1751-8113/46/41/415301

Cited by 26 publications

(39 citation statements)

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“…The dimer (10) does share this property; hence the existence of our bilinear conserved quantity Z = 2(ψ 1 ψ * 2 + ψ * 1 ψ 2 ) follows from the theory of Ref. [40]. However the second integral of motion, Eq.…”

Section: B Concluding Remarksmentioning

confidence: 92%

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An exactly solvable $\mathcal {PT}$-symmetric dimer from a Hamiltonian system of nonlinear oscillators with gain and loss

Barashenkov

Gianfreda

2014

J. Phys. A: Math. Theor.

102

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We show that a pair of coupled nonlinear oscillators, of which one oscillator has positive and the other one negative damping of equal rate, can form a Hamiltonian system. Small-amplitude oscillations in this system are governed by a PT -symmetric nonlinear Schrödinger dimer with linear and cubic coupling. The dimer also represents a Hamiltonian system and is found to be exactly solvable in elementary functions. We show that the nonlinearity softens the PT -symmetry breaking transition in the nonlinearly-coupled dimer: stable periodic and quasiperiodic states with large enough amplitudes persist for an arbitrarily large value of the gain-loss coefficient.

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Section: B Concluding Remarksmentioning

confidence: 92%

“…The first one is due to Zezyulin and Konotop [40] who classified PT -symmetric N -component oligomers with a general cubic nonlinearity. These authors prove that the nonlinearity matrix being pseudo-Hermitian with respect to the inversion P is sufficient for the existence of an integral of motion bilinear in ψ and ψ * .…”

Section: B Concluding Remarksmentioning

confidence: 99%

An exactly solvable $\mathcal {PT}$-symmetric dimer from a Hamiltonian system of nonlinear oscillators with gain and loss

Barashenkov

Gianfreda

2014

J. Phys. A: Math. Theor.

102

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“…In Ref. [79], effects of nonlinearities of different types on the stationary and dynamical properties of nonlinear modes in a PT-symmetric finite-dimensional chain have been studied. In particular, an important class of nonlinearities which allow the nonlinear system to admit at least one integral of motion has been identified.…”

Section: Multicore Fibersmentioning

confidence: 99%

Nonlinear switching and solitons in PT‐symmetric photonic systems

Suchkov

Sukhorukov

Huang

et al. 2016

Laser & Photonics Reviews

352

263

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One of the challenges of the modern photonics is to develop all-optical devices enabling increased speed and energy efficiency for transmitting and processing information on an optical chip. It is believed that the recently suggested ParityTime (PT) symmetric photonic systems with alternating regions of gain and loss can bring novel functionalities. In such systems, losses are as important as gain and, depending on the structural parameters, gain compensates losses. Generally, PT systems demonstrate nontrivial non-conservative wave interactions and phase transitions, which can be employed for signal filtering and switching, opening new prospects for active control of light. In this review, we discuss a broad range of problems involving nonlinear PT-symmetric photonic systems with an intensitydependent refractive index. Nonlinearity in such PT symmetric systems provides a basis for many effects such as the formation of localized modes, nonlinearly-induced PT-symmetry breaking, and all-optical switching. Nonlinear PT-symmetric systems can serve as powerful building blocks for the development of novel photonic devices targeting an active light control.

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“…For example, the general system contains in itself the nonlinear Schrödinger-like ladder system with background-controlled intersite resonant coupling [47] enriched by the additional satellite fields. The question whether or not such a reduction is capable to extend our notions about integrability of PT -symmetric models [21,33,52] applicable to arrays of nonlinear optical waveguides with gains and losses requires a separate consideration.…”

Section: Resultsmentioning

confidence: 99%

Four-Wave Semidiscrete Nonlinear Integrable System with 𝒫𝒯-Symmetry

Vakhnenko

2021

JNMP

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The new type of third-order spectral operator suitable to generate new multifield semidiscrete nonlinear systems with two coupling parameters in the framework of zero-curvature equation is proposed. The evolution operator corresponding to the first integrable system in an infinite hierarchy is explicitly recovered and the general form of first integrable system is isolated. The generalized procedure for the direct recursive development of infinite hierarchy of local conservation laws is presented and several lowest local conservation laws and local conserved densities are found. The reduction to the real field amplitudes in general system with unfixed sampling functions is made and the symmetric parametrization of field amplitudes allowing to exclude the redundant field function and to resolve the problem of sampling fixation for the particular realization of reduced integrable system is considered. This parametrization gives rise to the four field nonlinear model which in certain intervals of adjustable coupling parameter could serve as a semidiscrete analogy to the beam-plasma interaction system.

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Stationary modes and integrals of motion in nonlinear lattices with a ${\mathcal {PT}}$-symmetric linear part

Cited by 26 publications

References 50 publications

An exactly solvable $\mathcal {PT}$-symmetric dimer from a Hamiltonian system of nonlinear oscillators with gain and loss

An exactly solvable $\mathcal {PT}$-symmetric dimer from a Hamiltonian system of nonlinear oscillators with gain and loss

Nonlinear switching and solitons in PT‐symmetric photonic systems

Four-Wave Semidiscrete Nonlinear Integrable System with 𝒫𝒯-Symmetry

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