Soliton gas: Theory, numerics, and experiments.

The concept of soliton gas was introduced in 1971 by 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 (rarefied) soliton gas, solitons with random amplitude and phase parameters are almost nonoverlapping. More recently, the concept has been extended to dense gases in which solitons strongly and continuously interact.

Interaction of soliton gases in deep-water surface gravity waves.

Soliton gases represent large random soliton ensembles in physical systems that display integrable dynamics at leading order. We report hydrodynamic experiments in which we investigate the interaction between two beams or jets of soliton gases having nearly identical amplitudes but opposite velocities of the same magnitude. The space-time evolution of the two interacting soliton gas jets is recorded in a 140-m-long water tank where the dynamics is described at leading order by the focusing one-dimensional nonlinear Schrödinger equation.

Topological Properties of Floquet Winding Bands in a Photonic Lattice.

The engineering of synthetic materials characterized by more than one class of topological invariants is one of the current challenges of solid-state based and synthetic materials. Using a synthetic photonic lattice implemented in a two-coupled ring system we engineer an anomalous Floquet metal that is gapless in the bulk and shows simultaneously two different topological properties. On the one hand, this synthetic lattice presents bands characterized by a winding number.

Spatiotemporal observation of higher-order modulation instability in a recirculating fiber loop.

We experimentally investigate higher-order seeded modulation instability in an optical fiber experiment. The recirculating loop configuration with round trip losses compensation enables the observation in single-shot of the spatiotemporal evolution of an initially modulated continuous field revealing intricate yet deterministic dynamics. By tuning the modulation period, a continuous transition between perfectly coherent and purely noise-driven dynamics is observed that we characterize by means of a statistical study.

Nonlinear dispersion relation in integrable turbulence.

We investigate numerically and experimentally the concept of nonlinear dispersion relation (NDR) in the context of partially coherent waves propagating in a one-dimensional water tank. The nonlinear random waves have a narrow-bandwidth Fourier spectrum and are described at leading order by the one-dimensional nonlinear Schrödinger equation. The problem is considered in the framework of integrable turbulence in which solitons play a key role.

Observation of a giant nonlinear wave-packet on the surface of the ocean.

In many physical systems such as ocean waves, nonlinear optics, plasma physics etc., extreme events and rare fluctuations of a wave field have been widely observed and discussed. In the field of oceanography and naval architecture, their understanding is fundamental for a correct design of platforms and ships, and for performing safe operations at sea.

Solitonic model of the condensate.

We consider a spatially extended box-shaped wave field that consists of a plane wave (the condensate) in the middle and equals zero at the edges, in the framework of the focusing one-dimensional nonlinear Schrodinger equation. Within the inverse scattering transform theory, the scattering data for this wave field is presented by the continuous spectrum of the nonlinear radiation and the soliton eigenvalues together with their norming constants; the number of solitons N is proportional to the box width. We remove the continuous spectrum from the scattering data and find analytically the specific corrections to the soliton norming constants that arise due to the removal procedure.

Numerical spectral synthesis of breather gas for the focusing nonlinear Schrödinger equation.

We numerically realize a breather gas for the focusing nonlinear Schrödinger equation. This is done by building a random ensemble of N∼50 breathers via the Darboux transform recursive scheme in high-precision arithmetics. Three types of breather gases are synthesized according to the three prototypical spectral configurations corresponding the Akhmediev, Kuznetsov-Ma, and Peregrine breathers as elementary quasiparticles of the respective gases.

Nonlinear Spectral Synthesis of Soliton Gas in Deep-Water Surface Gravity Waves.

Soliton gases represent large random soliton ensembles in physical systems that exhibit integrable dynamics at the leading order. Despite significant theoretical developments and observational evidence of ubiquity of soliton gases in fluids and optical media, their controlled experimental realization has been missing. We report a controlled synthesis of a dense soliton gas in deep-water surface gravity waves using the tools of nonlinear spectral theory [inverse scattering transform (IST)] for the one-dimensional focusing nonlinear Schrödinger equation.

Single-shot observation of breathers from noise-induced modulation instability using heterodyne temporal imaging.

We report phase and amplitude measurements of large coherent structures originating from the noise-induced modulation instability in optical fibers. By using a specifically designed time-lens system (SEAHORSE) in which aberrations are compensated, the complex field is recorded in single-shot over long durations of 200 ps with sub-picosecond resolution. Signatures of Akhmediev breather-like patterns are identified in the ultrafast temporal dynamics in very good agreement with numerical predictions based on the nonlinear Schrödinger equation.

Bound State Soliton Gas Dynamics Underlying the Spontaneous Modulational Instability.

We investigate the fundamental phenomenon of the spontaneous, noise-induced modulational instability (MI) of a plane wave. The statistical properties of the noise-induced MI, observed previously in numerical simulations and in experiments, have not been explained theoretically. In this Letter, using the inverse scattering transform (IST) formalism, we propose a theoretical model of the asymptotic stage of the noise-induced MI based on N-soliton solutions of the focusing one-dimensional nonlinear Schrödinger equation.

Early stage of integrable turbulence in the one-dimensional nonlinear Schrödinger equation: A semiclassical approach to statistics.

We examine statistical properties of integrable turbulence in the defocusing and focusing regimes of one-dimensional small-dispersion nonlinear Schrödinger equation (1D-NLSE). Specifically, we study the 1D-NLSE evolution of partially coherent waves having Gaussian statistics at time t=0. Using short time asymptotic expansions and taking advantage of the scale separation in the semiclassical regime we obtain a simple explicit formula describing an early stage of the evolution of the fourth moment of the random wave field amplitude, a quantitative measure of the "tailedness" of the probability density function.

Statistical Properties of the Nonlinear Stage of Modulation Instability in Fiber Optics.

We present an optical fiber experiment in which we examine the space-time evolution of a modulationally unstable plane wave initially perturbed by a small noise. Using a recirculating fiber loop as an experimental platform, we report the single-shot observation of the noise-driven development of breather structures from the early stage to the long-term evolution of modulation instability. Performing single-point statistical analysis of optical power recorded in the experiments, we observe decaying oscillations of the second-order moment together with the exponential distribution in the long-term evolution, as predicted by Agafontsev and Zakharov [Nonlinearity 28, 2791 (2015).

Nonlinear Evolution of the Locally Induced Modulational Instability in Fiber Optics.

We report an optical fiber experiment in which we study the nonlinear stage of modulational instability of a plane wave in the presence of a localized perturbation. Using a recirculating fiber loop as the experimental platform, we show that the initial perturbation evolves into an expanding nonlinear oscillatory structure exhibiting some universal characteristics that agree with theoretical predictions based on integrability properties of the focusing nonlinear Schrödinger equation. Our experimental results demonstrate the persistence of the universal evolution scenario, even in the presence of small dissipation and noise in an experimental system that is not rigorously of an integrable nature.

Nonlinear spectral analysis of Peregrine solitons observed in optics and in hydrodynamic experiments.

The data recorded in optical fiber and in hydrodynamic experiments reported the pioneering observation of nonlinear waves with spatiotemporal localization similar to the Peregrine soliton are examined by using nonlinear spectral analysis. Our approach is based on the integrable nature of the one-dimensional focusing nonlinear Schrödinger equation (1D-NLSE) that governs at leading order the propagation of the optical and hydrodynamic waves in the two experiments. Nonlinear spectral analysis provides certain spectral portraits of the analyzed structures that are composed of bands lying in the complex plane.

Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation.

We report experimental confirmation of the universal emergence of the Peregrine soliton predicted to occur during pulse propagation in the semiclassical limit of the focusing nonlinear Schrödinger equation. Using an optical fiber based system, measurements of temporal focusing of high power pulses reveal both intensity and phase signatures of the Peregrine soliton during the initial nonlinear evolution stage. Experimental and numerical results are in very good agreement, and show that the universal mechanism that yields the Peregrine soliton structure is highly robust and can be observed over a broad range of parameters.

Optical Random Riemann Waves in Integrable Turbulence.

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, prebreaking stage being described by a system of interacting random Riemann waves.

Single-shot observation of optical rogue waves in integrable turbulence using time microscopy.

Optical fibres are favourable tabletop laboratories to investigate both coherent and incoherent nonlinear waves. In particular, exact solutions of the one-dimensional nonlinear Schrödinger equation such as fundamental solitons or solitons on finite background can be generated by launching periodic, specifically designed coherent waves in optical fibres. It is an open fundamental question to know whether these coherent structures can emerge from the nonlinear propagation of random waves.

Inverse scattering transform analysis of rogue waves using local periodization procedure.

The nonlinear Schrödinger equation (NLSE) stands out as the dispersive nonlinear partial differential equation that plays a prominent role in the modeling and understanding of the wave phenomena relevant to many fields of nonlinear physics. The question of random input problems in the one-dimensional and integrable NLSE enters within the framework of integrable turbulence, and the specific question of the formation of rogue waves (RWs) has been recently extensively studied in this context. The determination of exact analytic solutions of the focusing 1D-NLSE prototyping RW events of statistical relevance is now considered as the problem of central importance.

A workflow to process 3D+time microscopy images of developing organisms and reconstruct their cell lineage.

The quantitative and systematic analysis of embryonic cell dynamics from in vivo 3D+time image data sets is a major challenge at the forefront of developmental biology. Despite recent breakthroughs in the microscopy imaging of living systems, producing an accurate cell lineage tree for any developing organism remains a difficult task. We present here the BioEmergences workflow integrating all reconstruction steps from image acquisition and processing to the interactive visualization of reconstructed data.

Statistics of a turbulent Raman fiber laser.

We report the experimental study of statistical properties of partially coherent waves emitted by a Raman fiber laser operating in the normal dispersion regime. Using an asynchronous optical sampling technique, we accurately measure the probability density function of the optical power of the Stokes wave that exhibits strong and fast fluctuations. As predicted from numerical simulations presented by Randoux et al.

Optical rogue waves in integrable turbulence.

We report optical experiments allowing us to investigate integrable turbulence in the focusing regime of the one-dimensional nonlinear Schrödinger equation (1D NLSE). In analogy with broad spectrum excitation of a one-dimensional water tank, we launch random initial waves in a single mode optical fiber. Using an original optical sampling setup, we measure precisely the probability density function of optical power of the partially coherent waves rapidly fluctuating with time.

Intermittency in integrable turbulence.

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.

Transient buildup of the optical power spectrum in Raman fiber lasers.

We implement an experimental technique enabling to study the transient buildup of the optical power spectrum in a Raman fiber laser. We investigate the way through which the laser optical power spectrum broadens before reaching its shape at steady-state.

Experimental evidence of extreme value statistics in Raman fiber lasers.

We present an experiment in which the intracavity broadened spectrum of a Raman fiber laser is sliced by using a narrow-bandwidth optical filter. High-contrast fast fluctuations of the Stokes power are observed at the output of the optical filter. The statistics of the power fluctuations strongly change with the central wavelength of the filter, and rare extreme events are found to be generated in the far wings of the spectrum.