Nonlinear stationary states in PT-symmetric lattices

Kevrekidis, P. G.; Pelinovsky, Dmitry E.; Tyugin, Dmitry

doi:10.48550/arxiv.1303.3298

Cited by 4 publications

(9 citation statements)

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“…Notice that expansions (9) suggest that the nonlinear modes w bifurcating from the PT -invariant eigenvector w will also be PT invariant: PT w = w. This fact is confirmed in numerous previous studies, see e.g. [4,6,7,9,11,13,27], where a large library of results dedicated to nonlinear modes bifurcating from the non-degenerate (simple) linear eigenstates and supported by the nonlinearities of the N L wPT type can be found. Existence of the families of nonlinear modes was also established through the analytical continuation from the the anticontinuum limit [11] (the technique well known in the theory of conservative lattices [37]).…”

Section: Bifurcations Of the Families Of Nonlinear Modes Simple Eigen...supporting

confidence: 80%

“…We start by considering families of nonlinear modes bifurcating from a linear eigenstate of the PT -symmetric Hamiltonian H. Let b be a simple real eigenvalue and w is the corresponding eigenvector solving linear problem (6). According to the discussion in Sec.…”

Section: Bifurcations Of the Families Of Nonlinear Modes Simple Eigen...mentioning

confidence: 99%

“…In particular, exact solutions were reported for the nonlinear PTsymmetric dimer (a system of two coupled nonlinear oscillators) in [3]. Families of nonlinear discrete modes received particular emphasis in [4,5,6,7], while branches of the solutions were obtained for finite lattices like PT -symmetric trimers and oligomers [5,8,9], as well as for infinite PT -symmetric chains [6,7,10,11].…”

Section: Introductionmentioning

confidence: 99%

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

Zezyulin¹

2013

J. Phys. A: Math. Theor.

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We consider finite-dimensional nonlinear systems with linear part described by a parity-time (PT -) symmetric operator. We investigate bifurcations of stationary nonlinear modes from the eigenstates of the linear operator and consider a class of PT -symmetric nonlinearities allowing for existence of the families of nonlinear modes. We pay particular attention to the situations when the underlying linear PT -symmetric operator is characterized by the presence of degenerate eigenvalues or exceptional-point singularity. In each of the cases we construct formal expansions for small-amplitude nonlinear modes. We also report a class of nonlinearities allowing for the system to admit one or several integrals of motion, which turn out to be determined by the pseudo-Hermiticity of the nonlinear operator.

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Section: Bifurcations Of the Families Of Nonlinear Modes Simple Eigen...supporting

confidence: 80%

Section: Bifurcations Of the Families Of Nonlinear Modes Simple Eigen...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Zezyulin¹

2013

J. Phys. A: Math. Theor.

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“…On the contrary, we want to highlight that this is different than the "worst case scenario" of the Schrödinger dimer of the RWA. As illustrated in [23], in the latter at worst an exponential (indefinite) growth of the amplitude may arise (but no finite time blow up). Our numerical computations indicate that the switching behavior reported above is only possible provided that the growth rate (i.e.…”

Section: A Soft Potentialmentioning

confidence: 99%

“…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 [9,11,23,24]. A direct comparison of the RWA-derived Schrödinger dimer reveals natural similarities, but also significant differences between the two models.…”

Section: Introductionmentioning

confidence: 99%

PT-symmetric dimer of coupled nonlinear oscillators

et al. 2013

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We provide a systematic analysis of a prototypical nonlinear oscillator system respecting PT symmetry i.e., one of them has gain and the other an equal and opposite amount of loss. Starting from the linear limit of the system, we extend considerations to the nonlinear case for both soft and hard cubic nonlinearities identifying symmetric and antisymmetric breather solutions, as well as symmetry-breaking variants thereof. We propose a reduction of the system to a Schrödinger-type PT -symmetric dimer, whose detailed earlier understanding can explain many of the phenomena observed herein, including the PT phase transition. Nevertheless, there are also significant parametric as well as phenomenological potential differences between the two models and we discuss where these arise and where they are most pronounced. Finally, we also provide examples of the evolution dynamics of the different states in their regimes of instability.

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PT-symmetry management in oligomer systems

Horne¹,

Cuevas²,

Kevrekidis³

et al. 2013

J. Phys. A: Math. Theor.

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We study the effects of management of the PT-symmetric part of the potential within the setting of Schrödinger dimer and trimer oligomer systems. This is done by rapidly modulating in time the gain/loss profile. This gives rise to a number of interesting properties of the system, which are explored at the level of an averaged equation approach. Remarkably, this rapid modulation provides for a controllable expansion of the region of exact PT-symmetry, depending on the strength and frequency of the imposed modulation. The resulting averaged models are analyzed theoretically and their exact stationary solutions are translated into time-periodic solutions through the averaging reduction. These are, in turn, compared with the exact periodic solutions of the full non-autonomous PT-symmetry managed problem and very good agreement is found between the two. I. INTRODUCTIONIt has been about a decade and a half since the radical and highly innovative proposal of C. Bender and his collaborators [1] regarding the potential physical relevance of Hamiltonians respecting Parity (P) and time-reversal (T) symmetries. While earlier work was focused on an implicit postulate of solely self-adjoint Hamiltonian operators, this proposal suggested that these fundamental symmetries may allow for a real operator spectrum within a certain regime of parameters which is regarded as the regime of exact PT-symmetry. On the other hand, beyond a critical parametric strength, the relevant operators may acquire a spectrum encompassing imaginary or even genuinely complex eigenvalues, in which case, we are referring (at the linear level) to the regime of broken PT-phase.These notions were intensely studied at the quantum mechanical level, chiefly as theoretical constructs. Yet, it was the fundamental realization that optics can enable such "open" systems featuring gain and loss, both at the theoretical [2-5] and even at the experimental [6,7] level, that propelled this activity into a significant array of new directions, including the possibility of the interplay of nonlinearity with PT-symmetry. In this optical context, the well-known connection of the Maxwell equations with the Schrödinger equation was utilized, and Hamiltonians of the form H = −(1/2)∆ + V (x) were considered at the linear level with the PT-symmetry necessitating that the potential satisfies the condition V (x) = V ⋆ (−x). Yet another physical context where such systems have been experimentally "engineered" recently is that of electronic circuits; see the work of [8] and also the review of [9]. In parallel to the recent experimental developments, numerous theoretical groups have explored various features of both linear PT-symmetric potentials [10-36] and even of nonlinear ones such where a PT-symmetric type of gain/loss pattern appears in the nonlinear term [37][38][39][40].Our aim in the present work is to combine this highly active research theme of PT-symmetry with another topic of considerable recent interest in the physics of optical and also atomic systems, namely that of "ma...

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Nonlinear stationary states in PT-symmetric lattices

Cited by 4 publications

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

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

PT-symmetric dimer of coupled nonlinear oscillators

PT-symmetry management in oligomer systems

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