Intermittency in integrable turbulence

Abstract

We examine the statistical properties of nonlinear random waves that are ruled by the one-dimensional defocusing and integrable nonlinear Schrödinger equation. Using fast detection techniques in an optical fiber experiment, we observe that the probability density function of light fluctuations is characterized by tails that are lower than those predicted by a Gaussian distribution. Moreover, by applying a bandpass frequency optical filter, we reveal 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. These phenomena are very well described by numerical simulations of the one-dimensional nonlinear Schrödinger equation. DOI: 10.1103/PhysRevLett.113.113902 PACS numbers: 42.65.Sf, 05.45.-a The way through which nonlinearity and randomness interplay to influence the propagation of waves is of major interest in various fields of investigation, such as, e.g., hydrodynamics, wave turbulence, optics, or condensedmatter physics. If the spatiotemporal dynamics of the wave system is predominantly influenced by some disorder found inside the nonlinear medium, localized eigenmodes may emerge through the process of Anderson localization [1]. Conversely, if randomness mostly arises from the initial field, linear and nonlinear effects occurring inside the propagation medium interplay to modify the statistical properties of the incoherent wave launched as initial condition. This process, extensively studied during the past years, may result in the formation of extreme or rogue waves [2][3][4][5].In this context, the nonlinear Schrödinger equation (NLSE) plays a special role because it provides a universal model that rules the dynamics of many nonlinear wave systems. In the fields of oceanography and laser filamentation, numerical simulations of the NLSE with random initial conditions have been made in 2 þ 1 dimensions to study the emergence of rogue waves [6][7][8]. In the context of nonlinear fiber optics, the propagation of incoherent waves is often treated in 1 þ 1 dimensions from generalized NLSEs including high-order linear and nonlinear terms [2,[9][10][11]. Wave turbulence (WT) theory provides an appropriate framework for the statistical treatment of the interaction of random nonlinear waves that are described by these nonintegrable equations [12].If third-order nonlinearity and second-order (linear) dispersion dominate other physical effects in a onedimensional medium, wave propagation is ruled by the 1D NLSE, which is an integrable equation. In the focusing regime, the 1D NLSE possesses soliton and breather solutions that have been recently observed in various physical systems [13][14][15]. Considering a random input field, these breather solutions may exist embedded in random waves, thus behaving as prototypes of rogue waves that modify the statistical properties of the random input field [16,17]. In the defocusing regime, the 1D NLSE has dark or gray soliton solutions. These solutions have also bee...