Inverse scattering theory with stochastic initial potentials

Elgin, J.N.

doi:10.1016/0375-9601(85)90549-3

Cited by 56 publications

(19 citation statements)

References 6 publications

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“…As it is well known, one of the important effects leading to a loss of the soliton signal is the so-called Gordon-Haus effect [8,9] that manifests itself as fluctuations of the soliton position Y (6). One can estimate the variance as Y 2 ∼ DT 3 .…”

Section: Modifications Of the Basic Model By In-line Control Elementsmentioning

confidence: 99%

“…For different control schemes, one must add to the right-hand side of (14) the terms that originate from the additional terms in the right-hand sides of Eqs. (8), (9), (11) and (67), we shall call the sum of such terms U (see section below). Now, to derive the equations for α, β, η, y, s k , s * k one should find projections of Eq.…”

Section: Separation Of the Discrete Spectrummentioning

confidence: 99%

“…The fiber dispersion converts frequency variations in the arrival times jitter known as the Gordon-Haus effect [8] (see also [9] where the mathematical theory of the effect has been developed). Contribution of the Gordon-Haus effect to the total bit error rate (BER) is the probability that a soliton (corresponding to elementary "one") will arrive outside the detection window.…”

Section: Introductionmentioning

confidence: 99%

“…That makes it possible to reduce the formally infinite-dimensional problem to the analysis of the finite set of soliton parameters and effectively find the error probability for a single soliton transmission under different control schemes. The probability density function (PDF) is essentially Gaussian for timing jitter [8][9][10][11] in systems without control and may have substantially non-Gaussian tails in systems with in-line filtering and amplitude modulation. Here, we consider a single-soliton propagation, the effects related to soliton interaction are described in [2,3,[12][13][14].…”

Section: Introductionmentioning

confidence: 99%

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Non-Gaussian error probability in optical soliton transmission

Falkovich

Kolokolov

Лебедев

et al. 2004

Physica D: Nonlinear Phenomena

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We find the probability distribution of the fluctuating parameters of a soliton propagating through a medium with additive noise. Our method is a modification of the instanton formalism (method of optimal fluctuation) based on a saddle-point approximation in the path integral. We first solve consistently a fundamental problem of soliton propagation within the framework of noisy nonlinear Schrödinger equation. We then consider model modifications due to in-line (filtering, amplitude and phase modulation) control. It is examined how control elements change the error probability in optical soliton transmission. Even though a weak noise is considered, we are interested here in probabilities of error-causing large fluctuations which are beyond perturbation theory. We describe in detail a new phenomenon of soliton collapse that occurs under the combined action of noise, filtering and amplitude modulation.

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Section: Modifications Of the Basic Model By In-line Control Elementsmentioning

confidence: 99%

Section: Separation Of the Discrete Spectrummentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Non-Gaussian error probability in optical soliton transmission

Falkovich

Kolokolov

Лебедев

et al. 2004

Physica D: Nonlinear Phenomena

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“…This long-ranged triple, and nonrandom character of the interaction, along with the sensitivity of the phenomenon to phases are generic features of any problem explaining the adiabatic evolution of solitons in the presence of induced radiation. However, if spatiotemporal (short-correlated both in t and z) noise is considered, the radiation effect, equivalent to the one considered in the letter, is masked by a jitter in relative soliton positions, δy 2 ~ z 3 [19][20][21], where measures the intensity of the noise. Different solitons jitter independently; i.e., fluctuations of intersoliton separations are described by the same δy.…”

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confidence: 99%

Shedding and interaction of solitons in imperfect medium

et al. 2001

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1 We consider long-range soliton interaction mediated by radiation in nonlinear 1 d system with frozen disorder. The problem is of a great importance for nonlinear fiber optics of the next generation (see, e.g., [1,2]), and it is also of general relevance for any of the traditional fields, like plasma physics, where propagation of solitary waves is possible. Our aim here is to answer the following sets of fundamental questions:What statistics describe the radiation emitted due to disorder by a single soliton or a pattern of solitons? How far do the radiation wings extend from the peak of the soliton(s)? What is the structure of the wings?How strong is the radiation mediating interaction between the solitons? How is the interaction modified if we vary the soliton positions and phases within a pattern of solitons?We focus on the dynamics of wave packets. The universal coarse-grained description of a wave packet envelope is given by the nonlinear Schrödinger equation (NLS) [3][4][5]. We consider the 1 d problem motivated mainly by applications to fiber optics [6] (1)The medium (fiber) is imperfect; i.e., various macroscopic characteristics of the fiber fluctuate in space. Fluctuations of the dispersion coefficient, d , are believed to be one of the major sources of disorder present in real fibers [7]. This disorder is frozen; i.e., d is a random function of z . We assume that d fluctuates on short spatial scales and that the fiber is homogeneous on larger scales. The averaged value of d is a constant, which can be rescaled to unity by changing the units 1 This article was submitted by the authors in English., where 〈ξ〉 = 0. According to the Central Limit Theorem [8], ξ at scales larger than the correlation length can be treated as a homogeneous Gaussian random process with zero mean and described by the quantity D = , which is the noise intensity. The pair correlation function of ξ is (2) We assume that the disorder is weak; i.e., D Ӷ 1.At z = 0, a sequence of well-separated solitons is launched. In an ideal medium ( ξ = 0), each of the solitons is preserved dynamically gaining, according to the exact single-soliton solution of Eq. (1), only a multiplicative phase factor. Because the medium is imperfect, the solitons, perturbed by impurities, shed radiation. The first problem is to describe the radiation. The soliton looses energy shedding radiation. Another problem is to describe the degradation of a single soliton. The tails of different solitons interfere with each other, forming a collective background. This fluctuating background affects all solitons. It results in the emergence of a long-range effective intersoliton interaction, which is the final (but not the least) problem to be addressed. The long-range interaction dominates the direct interaction due to overlapping of soliton tails, as this direct interaction decays exponentially with separation [9,10]. The emergence of the long-range interaction between the imperfect solitons in the pure ( ξ = 0) NLS, mediated by the emitted radiation, was noted in [11]....

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Chaotic Dynamics of Solitons and Breathers

Abdullaev¹,

Darmanyan²

1989

Springer Proceedings in Physics

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Inverse scattering theory with stochastic initial potentials

Cited by 56 publications

References 6 publications

Non-Gaussian error probability in optical soliton transmission

Non-Gaussian error probability in optical soliton transmission

Shedding and interaction of solitons in imperfect medium

Chaotic Dynamics of Solitons and Breathers

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