We present experimental evidence of the universal emergence of the Peregrine soliton predicted in the semi-classical (zero-dispersion) limit of the focusing nonlinear Schrödinger equation [Comm. Pure Appl. Math. 66, 678 (2012)]. Experiments studying higher-order soliton propagation in optical fiber use an optical sampling oscilloscope and frequency-resolved optical gating to characterise intensity and phase around the first point of soliton compression and the results show that the properties of the compressed pulse and background pedestal can be interpreted in terms of the Peregrine soliton. Experimental and numerical results reveal that the universal mechanism under study is highly robust and can be observed over a broad range of parameters, and experiments are in very good agreement with numerical simulations.
We examine integrable turbulence (IT) in the framework of the defocusing cubic one-dimensional nonlinear Schrödinger equation. This is done theoretically and experimentally, by realizing an optical fiber experiment in which the defocusing Kerr nonlinearity strongly dominates linear dispersive effects. Using a dispersive-hydrodynamic approach, we show that the development of IT can be divided into two distinct stages, the initial, pre-breaking stage being described by a system of interacting random Riemann waves. We explain the low-tailed statistics of the wave intensity in IT and show that the Riemann invariants of the asymptotic nonlinear geometric optics system represent the observable quantities that provide new insight into statistical features of the initial stage of the IT development by exhibiting stationary probability density functions.Propagation of nonlinear random waves has recently received much attention in many areas of modern physics such as nonlinear statistical optics [1][2][3][4], hydrodynamics [5], mechanics [6], and cold-atom physics [7]. In all these areas a broad class of wave phenomena is modelled by integrable nonlinear partial differential equations (PDEs). Although the fundamental role of integrable PDEs has been established since the pioneering work of Fermi, Pasta and Ulam in the 1950s [8] the significance of random input problems for such systems was realized only recently, leading to the concept of integrable turbulence (IT) [9][10][11][12][13][14][15][16][17]. In this context, the one-dimensional nonlinear Schrödinger equation (1D-NLSE) plays a prominent role because it describes at leading order wave phenomena relevant to many fields of nonlinear physics.It is now well established from experiments and numerical simulations that heavy-tailed (resp. low-tailed) deviations from gaussian statistics occur in integrable wave systems ruled by the focusing (resp. defocusing) 1D-NLSE [11][12][13]15]. The heavy-tailed deviations from gaussian statistics have their origin in the random formation of bright coherent structures having properties of localization in space and time similar to rogue waves [12,13,18]. On the other hand, the low-tailed deviations are due to random generation of dispersive shock waves (DSWs) and dark solitons [11,15]. One of the key features of IT is the establishment, at long evolution time, of a state in which the statistical properties of the wave system remain stationary. Due to integrable nature of the system, the long-time statistics depends on the statistics of the input random process (cf. [10][11][12]15]). So far, there has been no satisfactory theoretical framework developed for the description of statistical features of IT due to high complexity of nonlinear wave interactions occurring over the course of its development.In this Letter, we examine IT in optical systems described by the defocusing 1D NLSE from the perspective * Electronic address: stephane.randoux@univ-lille1.fr of dispersive hydrodynamics [19], a semi-classical theory of nonlinear dispersiv...
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