8 research outputs found

    Extreme waves statistics for Ablowitz-Ladik system

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    We examine statistics of waves for the problem of modulation instability development in the framework of discrete integrable Ablowitz-Ladik (AL) system. Modulation instability depends on one free parameter h that has the meaning of the coupling between the nodes on the lattice. For strong coupling h<<1 the probability density functions (PDFs) for waves amplitudes coincide with that for the continuous classical Nonlinear Schrodinger (NLS) equation; the PDFs for both systems are very close to Rayleigh ones. When the coupling is weak h~1, there appear highly localized waves with very large amplitudes, that drastically change the PDFs to significantly non-Rayleigh ones, with so-called "fat tails" when the probability of a large wave occurrence is by several orders of magnitude higher than that predicted by the linear theory. Evolution of amplitudes for such rogue waves with time is similar to that of the Peregrine solution for the classical NLS equation.Comment: 5 pages, 3 figure

    Bound state soliton gas dynamics underlying the spontaneous modulational instability

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    We investigate theoretically the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The long-term statistical properties of the noise-induced MI have been previously observed in experiments and in simulations but have not been explained so far. In the framework of inverse scattering transform (IST), we propose a model of the asymptotic stage of the noise-induced MI based on N-soliton solutions (N-SS) of the integrable focusing one-dimensional nonlinear Schr√∂dinger equation (1D-NLSE). These N-SS are bound states of strongly interacting solitons having a specific distribution of the IST eigenvalues together with random phases. We use a special approach to construct ensembles of multi-soliton solutions with statistically large number of solitons N‚ąľ100. Our investigation demonstrates complete agreement in spectral (Fourier) and statistical properties between the long-term evolution of the condensate perturbed by noise and the constructed multi-soliton bound states. Our results can be generalised to a broad class of integrable turbulence problems in the cases when the wave field dynamics is strongly nonlinear and driven by solitons

    Soliton Gas: Theory, Numerics and Experiments

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    The concept of soliton gas was introduced in 1971 by V. Zakharov as an infinite collection of weakly interacting solitons in the framework of Korteweg-de Vries (KdV) equation. In this theoretical construction of a diluted soliton gas, solitons with random parameters are almost non-overlapping. More recently, the concept has been extended to dense gases in which solitons strongly and continuously interact. The notion of soliton gas is inherently associated with integrable wave systems described by nonlinear partial differential equations like the KdV equation or the one-dimensional nonlinear Schr\"odinger equation that can be solved using the inverse scattering transform. Over the last few years, the field of soliton gases has received a rapidly growing interest from both the theoretical and experimental points of view. In particular, it has been realized that the soliton gas dynamics underlies some fundamental nonlinear wave phenomena such as spontaneous modulation instability and the formation of rogue waves. The recently discovered deep connections of soliton gas theory with generalized hydrodynamics have broadened the field and opened new fundamental questions related to the soliton gas statistics and thermodynamics. We review the main recent theoretical and experimental results in the field of soliton gas. The key conceptual tools of the field, such as the inverse scattering transform, the thermodynamic limit of finite-gap potentials and the Generalized Gibbs Ensembles are introduced and various open questions and future challenges are discussed.Comment: 35 pages, 8 figure

    Extreme events in optics: Challenges of the MANUREVA project

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    International audienceIn this contribution we describe and discuss a series of challenges and questions relating to understanding extreme wave phenomena in optics. Many aspects of these questions are being studied in the framework of the MANUREVA project: a multidisciplinary consortium aiming to carry out mathematical, numerical and experimental studies in this field. The central motivation of this work is the 2007 results from optical physics [D. Solli et al., Nature 450, 1054 (2007)] that showed how a fibre-optical system can generate large amplitude extreme wave events with similar statistical properties to the infamous hydrodynamic rogue waves on the surface of the ocean. We review our recent work in this area, and discuss how this observation may open the possibility for an optical system to be used to directly study both the dynamics and statistics of extreme-value processes, a potential advance comparable to the introduction of optical systems to study chaos in the 1970s
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