Modulation instability and periodic solutions of the nonlinear Schr�dinger equation

Akhmediev, Nail; Korneev, V. I.

doi:10.1007/bf01037866

Cited by 790 publications

(611 citation statements)

References 5 publications

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“…Figure 1(b) represents the evolution of the spectrum versus the fiber length. An immediate observation is that the spectrum is almost symmetric at each side of the pump wavelength, as expected from the theory based on the NLSE equation [9]. This observation confirms that we have achieved the configuration where the TOD does not significantly affect the process.…”

Section: Experimental Setup and Methodssupporting

confidence: 74%

“…In this configuration, the role played by higher-order linear terms is negligible, so the dynamics is well reproduced by the pure nonlinear Schrödinger equation (NLSE). As a consequence, analytic solutions known as Akhmediev breathers (ABs) [9] accurately describe the recurrence process. The latter can be considered as the extension of the modulation-instability process [10] beyond the linear approximation that captures the full nonlinear dynamics of the system.…”

Section: Introductionmentioning

confidence: 99%

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Fermi-Pasta-Ulam Recurrence in Nonlinear Fiber Optics: The Role of Reversible and Irreversible Losses

et al. 2014

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The discovery of the Fermi-Pasta-Ulam (FPU) recurrence phenomenon in the 1950 s was a major step in science that later led to the discovery of solitons in nonlinear physics. More recently, it was shown that optical fibers can serve as a medium for observing the FPU phenomenon. In the present work, we have found experimentally and numerically that in the low-dispersion region of an optical fiber, the recurrence is strongly influenced by the third-order-dispersion (TOD) term. Namely, the presence of TOD leads to several disappearances and recoveries of the FPU recurrence when the central frequency of the pump wave is varied. The effect is highly nontrivial and can be explained in terms of reversible and irreversible losses caused by Cherenkov radiations interacting with a multiplicity of modes sharing the optical energy in the process of its partition.

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Section: Experimental Setup and Methodssupporting

confidence: 74%

Section: Introductionmentioning

confidence: 99%

Fermi-Pasta-Ulam Recurrence in Nonlinear Fiber Optics: The Role of Reversible and Irreversible Losses

et al. 2014

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“…[3][4][5] In response to this difficulty, Akhmediev and co-workers developed another approach to the problem. 12,13 They derived exact analytical spatiotemporal periodic solutions to the NLS equation (in terms of Jacobi elliptic functions) that represent the spatial evolution of initially continuous (or periodically modulated) waves. As a consequence, their approach inherently accounts for an arbitrary number of interacting sideband modes.…”

Section: -10mentioning

confidence: 99%

Experimental study of the reversible behavior of modulational instability in optical fibers

2002

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We report what is to our knowledge the first clear-cut experimental evidence of the reversibility of modulational instability in dispersive Kerr media. It was possible to perform this experiment with standard telecommunication fiber because we used a specially designed 550-ps square-pulse laser source based on the twowavelength configuration of a nonlinear optical loop mirror. Our observations demonstrate that reversibility is due to well-balanced and synchronous energy transfer among a significant number of spectral wave components. These results provide what we believe is the first evidence, in the field of nonlinear optics, of the universal Fermi-Pasta-Ulam recurrence phenomenon that has been predicted for a large number of conservative nonlinear systems, including those described by a nonlinear Schrödinger equation that is relevant to the context of the present study.

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“…In reality, MI exhibits much richer dynamics when one goes beyond the simple linear stability analysis. To address this problem, some exact pulsating solutions (also called breathers) describing the full MI dynamics were derived during the 1970s and 1980s, but only for periodic initial modulations [17][18][19][20][21][22]. These cases provided, as a first step, a powerful framework for interpretation of a specific range of MI-related dynamics that play a fundamental role in the theory of freak waves [23].…”

Section: Introductionmentioning

confidence: 99%

Superregular Breathers in Optics and Hydrodynamics: Omnipresent Modulation Instability beyond Simple Periodicity

et al. 2015

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Since the 1960s, the Benjamin-Feir (or modulation) instability (MI) has been considered as the selfmodulation of the continuous "envelope waves" with respect to small periodic perturbations that precedes the emergence of highly localized wave structures. Nowadays, the universal nature of MI is established through numerous observations in physics. However, even now, 50 years later, more practical but complex forms of this old physical phenomenon at the frontier of nonlinear wave theory have still not been revealed (i.e., when perturbations beyond simple harmonic are involved). Here, we report the evidence of the broadest class of creation and annihilation dynamics of MI, also called superregular breathers. Observations are done in two different branches of wave physics, namely, in optics and hydrodynamics. Based on the common framework of the nonlinear Schrödinger equation, this multidisciplinary approach proves universality and reversibility of nonlinear wave formations from localized perturbations for drastically different spatial and temporal scales.

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Modulation instability and periodic solutions of the nonlinear Schr�dinger equation

Cited by 790 publications

References 5 publications

Fermi-Pasta-Ulam Recurrence in Nonlinear Fiber Optics: The Role of Reversible and Irreversible Losses

Fermi-Pasta-Ulam Recurrence in Nonlinear Fiber Optics: The Role of Reversible and Irreversible Losses

Experimental study of the reversible behavior of modulational instability in optical fibers

Superregular Breathers in Optics and Hydrodynamics: Omnipresent Modulation Instability beyond Simple Periodicity

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