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Integrable Turbulence with Nonlinear Random Optical Waves

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Rogue and Shock Waves in Nonlinear Dispersive Media

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Abstract

The field of integrable turbulence deals with the general question of statistical changes that are experienced by ensembles of nonlinear random waves propagating in systems ruled by integrable equations. In this chapter, we specifically focus on the one-dimensional nonlinear Schrödinger equation that describes quantitatively very well experiments performed with single mode fibers and optical waves randomly fluctuating in time. As a result of nonlinear propagation, the power spectrum of the random wave broadens and takes exponential wings both in focusing and in defocusing propagation regimes. Heavy-tailed deviations from gaussian statistics are observed in focusing regime while low-tailed deviations from gaussian statistics are observed in defocusing regime. After some transient evolution, the wave system is found to exhibit a statistically stationary state in which neither the probability density function of the wave field nor the spectrum change with the evolution variable. Separating fluctuations of small scale from fluctuations of large scale both in focusing and defocusing regime, we evidence the phenomenon of intermittency; i.e., small scales are characterized by large heavy-tailed deviations from Gaussian statistics, while the large ones are almost Gaussian.

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Notes

  1. 1.

    Kolmogorov-Zakharov cascade may appear when the spectrum (i.e. scales) of nonlinear random waves can be divided into three parts: a pumping spectral range (with an external source of energy, often at large scales), a spectral range with dissipation (often at small scales) and an intermediate inertial range with no dissipation [1, 2]. On the other hand, thermalization emerges in some systems without dissipation and pumping.

  2. 2.

    In the process of wave thermalization, the “equipartition of energy” corresponds to the equipartition of linear kinetic energy (linear part of the Hamiltonian).

  3. 3.

    The resonances in wave mixing are determined by double equalities involving frequencies and wave vectors of Fourier components. In the case of one dimensional four-wave mixing, these equalities are k 1 + k 2 = k 3 + k 4 and ω 1(k 1) +ω 2(k 2) = ω 3(k 3) +ω 4(k 4). ω(k) is the linear dispersion relation. For Eq. (1) where ω(k) = k 2, it is straightforward to show that the two resonant conditions imply k 1 = k 3 or k 1 = k 4 which is a trivial interaction (no energy is exchanged among different Fourier components). In the case of the integrable 1D-NLSE, the changes of the spectrum and of the statistical properties are thus only induced by non resonant terms.

  4. 4.

    We use in this chapter the usual and natural variables (t, x). Note that in single optical fiber experiments, the time t has to be replaced by the evolution variable z that represents the propagation distance along the fiber. The variable of x “becomes” the physical time t. Time and space are thus exchanged and in this case the 1D-NLSE takes the form of Eq. (5).

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Acknowledgements

This work has been partially supported by Ministry of Higher Education and Research, Nord-Pas de Calais Regional Council and European Regional Development Fund (ERDF) through the Contrat de Projets Etat-Région (CPER) 2007–2013, as well as by the Agence Nationale de la Recherche through the LABEX CEMPI project (ANR-11-LABX-0007) and the OPTIROC project (ANR-12-BS04-0011 OPTIROC). The authors acknowledge to T. Grava (Trieste, Italy), G. El (Loughborough University) and M. Onorato (University of Torino) for fruitful discussions. The authors acknowledge P. Walczak for his major contribution to the work presented in this chapter.

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  1. Laboratoire de Physique des Lasers, Atomes et Molecules, UMR-CNRS 8523, Université de Lille, Lille, France

    Stéphane Randoux & Pierre Suret

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  1. Stéphane Randoux
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Correspondence to Stéphane Randoux .

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  1. Dipartimento di Fisica, Università di Torino, Torino, Italy

    Miguel Onorato

  2. Institut Non Linéaire de Nice, Centre National de la Recherche Scientifique, CNRS, Université de Nice-Sophia Antipolis, Valbonne, France

    Stefania Resitori

  3. Dipartimento di Ingegneria dell' Informazione, Università di Brescia, Brescia, Italy

    Fabio Baronio

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Randoux, S., Suret, P. (2016). Integrable Turbulence with Nonlinear Random Optical Waves. In: Onorato, M., Resitori, S., Baronio, F. (eds) Rogue and Shock Waves in Nonlinear Dispersive Media. Lecture Notes in Physics, vol 926. Springer, Cham. https://doi.org/10.1007/978-3-319-39214-1_9

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