Nonlinear Evolution of the Locally Induced Modulational Instability in Fiber Optics

Solitons on a finite a background, also called breathers, are solutions of the focusing nonlinear Schrödinger equation, which play a pivotal role in the description of rogue waves and modulation instability. The breather family includes Akhmediev breathers (AB), Kuznetsov-Ma (KM), and Peregrine solitons (PS), which have been successfully exploited to describe several physical effects. These families of solutions are actually only particular cases of a more general three-parameter class of solutions originally derived by Akhmediev, Eleonskii and Kulagin [Theor. Math. Phys. 72, 809-818 (1987)]. Having more parameters to vary, this significantly wider family has a potential to describe many more physical effects of practical interest than its subsets mentioned above. The complexity of this class of solutions prevented researchers to study them deeply. In this paper, we overcome this difficulty and report several new effects that follow from more detailed analysis. Namely, we present the doubly periodic solutions and their Fourier expansions. In particular, we outline some striking properties of these solutions. Among the new effects, we mention (a) regular and shifted recurrence, (b) period doubling and (c) amplification of small periodic perturbations with frequencies outside the conventional modulation instability gain band.

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“…. Formula (20) for the phase is the corrected version of the one presented earlier in [45,48]. The function Q(t, z) has the following expression:…”

Section: The Family Of A-type Solutionsmentioning

confidence: 99%

Doubly periodic solutions of the focusing nonlinear Schrödinger equation: Recurrence, period doubling, and amplification outside the conventional modulation-instability band

et al. 2020

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“…Moreover, the transient regime is characterized by the decaying oscillatory evolution of the statistical moments [19]. To the best of our knowledge, up to now, this remarkable transient regime has not been observed in experiments.In this Letter, we investigate the evolution of a modulationally unstable plane wave by using a recirculating fiber loop [37,38]. Our setup enables the single-shot recording both in space and time of the nonlinear dynamics associated with the noise-driven MI.…”

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confidence: 99%

“…The experimental data analysis also reveal the decaying oscillations of the statistical moments predicted in [19]. Moreover, we demonstrate numerically and experimentally that the autocorrelation of optical power g (2) (τ ) exhibits a clear oscillatory signature that allows to differentiate the wavefield in the nonlinear stage of the MI from random Gaussian fields made up of linear superposition of waves.Our experimental setup is very similar to the one used in [38] and is schematically shown in Fig. 1.…”

mentioning

confidence: 99%

“…In this Letter, we investigate the evolution of a modulationally unstable plane wave by using a recirculating fiber loop [37,38]. Our setup enables the single-shot recording both in space and time of the nonlinear dynamics associated with the noise-driven MI.…”

mentioning

confidence: 99%

“…a ring cavity made up of 10 km single mode fiber (SMF) closed on itself by a 90/10 fiber coupler (FC). A square-shape light pulse having a temporal width around 200 ns is first prepared by using a single frequency laser operating at 1550 nm, an Erbium-doped fiber laser (EDFA) and an acousto-optical modulator (AOM) (see [38] and Supplementary Material for details). Secondly, this long square-pulse is launched into the FC which is arranged in such a way that 90% of the power is recirculated inside the fiber loop.…”

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confidence: 99%

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Statistical Properties of the Nonlinear Stage of Modulation Instability in Fiber Optics

et al. 2019

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We present an optical fiber experiment in which we examine the space-time evolution of a modulationally unstable plane wave initially perturbed by a small noise. Using a recirculating fiber loop as experimental platform, we report the single-shot observation of the noise-driven development of breather structures from the early stage to the long-term evolution of modulation instability. Performing single-point statistical analysis of optical power recorded in the experiments, we observe decaying oscillations of the second-order moment together with the exponential distribution in the long term evolution, as predicted in [D. S. Agafontsev and V. E. Zakharov, Nonlinearity 28, 2791 (2015)]. Finally, we demonstrate experimentally and numerically that the autocorrelation of the optical power g (2) (τ ) exhibits some unique oscillatory features typifying the nonlinear stage of the noise-driven modulation instability and of integrable turbulence.Modulational Instability (MI), also known as Benjamin-Feir Instability, leads to the spontaneous breakup of plane waves into train of pulses. MI has been extensively studied in the last five decades, see e.g. [1,2], and is often seen as the exponential amplification of a weak modulation of a monochromatic carrier wave. This fundamental phenomenon arises in physical systems such as deep water waves [3], nonlinear optical waves [4], Bose-Einstein condensates and matter waves [5], which are described at leading order by the focusing one-dimensional nonlinear Schrödinger equationThe 1DNLSE is integrable and its exact solutions can be found by using the so-called "inverse scattering transform" (IST) [6,7]. For the MI driven by a sinusoidal perturbation of a constant background, the long-term spatio-temporal dynamics is described by breather solutions of the focusing 1DNLSE, such as the Akhmediev Breather (AB) [8][9][10][11][12][13][14][15]. However, natural perturbations are generally not sinusoidal modulations but rather represent a random process [16][17][18][19][20][21][22].Despite the numerous studies devoted to the subject of MI in optical fibers [4,23], it is only recently that single-shot observations of the noise driven MI have been reported. Shot-to-shot fluctuations of the optical spectra have been measured in experiments with light pulses in [20], while breather structures spontaneously emerging from noise superimposed to a continuous wave (CW) field have been reported in [22]. However, in these pioneering studies the spatio-temporal dynamics and the evolution of statistical properties of the light field along the fiber could not be recorded, since the observation was made only at the output of the fiber.From the theoretical point of view, the question of noise-driven MI enters whithin the fundamental framework of integrable turbulence, which was first introduced by V. E. Zakharov in [24] and is now a subject of extensive theoretical [19,[24][25][26][27][28][29][30] and experimental [31][32][33][34][35][36] research.In the long term evolution, the integrable turbulence is ...

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Rational solutions of multi‐component nonlinear Schrödinger equation and complex modified KdV equation

Wang

Erdélyi

2022

Math Methods in App Sciences

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In this paper, the crucial condition to achieve rational solutions of the multi‐component nonlinear Schrödinger equation is proposed by introducing two nilpotent Lax matrices. Taking the series multisections of the vector eigenfunction as a set of fundamental eigenfunctions, an explicit formula of the nth‐order rational solution is obtained by the degenerate Darboux transformation, which is used to generate some new patterns of rogue waves. A conjecture about the degree of the nth‐order rogue waves is summarized. This conjecture also holds for rogue waves of the multi‐component complex modified Korteweg–de Vries equation. Finally, the semi‐rational solutions of the Manakov system are discussed.

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Nonlinear Evolution of the Locally Induced Modulational Instability in Fiber Optics

Cited by 88 publications

References 60 publications

Doubly periodic solutions of the focusing nonlinear Schrödinger equation: Recurrence, period doubling, and amplification outside the conventional modulation-instability band

Doubly periodic solutions of the focusing nonlinear Schrödinger equation: Recurrence, period doubling, and amplification outside the conventional modulation-instability band

Statistical Properties of the Nonlinear Stage of Modulation Instability in Fiber Optics

Rational solutions of multi‐component nonlinear Schrödinger equation and complex modified KdV equation

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