* /weɪv to͞o ˈsikˌstēn/

## Book of abstracts

## Vorticity effect on the generation of rogue waves in shallow water

## Modeling dynamics and light localization in novel photonic structures

## Extreme solitons in optical fibers

## Extreme events induced by spatiotemporal chaos in an extended microcavity laser

Extreme events such as rogue waves in optics and fluids are often associated with the merging dynamics of coherent structures. We present experimental and numerical results on the physics of extreme events appearance in a spatially extended semiconductor microcavity laser with intracavity saturable absorber [1]. This system can display deterministic irregular dynamics only thanks to spatial coupling through diffraction of light. We have identified parameter regions where extreme events are encountered and established the origin of this dynamics in the emergence of spatiotemporal chaos, by studying the proportion of extreme events and the Kaplan-Yorke dimension of the attractor.

Examining the power spectrum of the dynamics give some hints on the possible dynamical scenarios explaining the variation of the proportion of extreme events found.

**References**

[1] F. Selmi, S. Coulibaly, Z. Loghmari, I. Sagnes, G. Beaudoin, M. G. Clerc, S. Barbay, Phys. Rev. Lett. 116, 013901 (2016).

## Optical Nondiffractive and Nondispersive Wave Packets of Hydrodynamic Origin

## Modelling wave transformation over coastal profiles with a nonlinear and dispersive potential model, and application to freak waves in variable water depth

With the objective of modeling coastal wave dynamics taking into account nonlinear and dispersive effects in variable bottom conditions, an accurate nonlinear potential flow model is developed. The model is based on the time evolution of two surface quantities: the free surface position and the free surface velocity potential (Zakharov, 1968). Following Tian and Sato (2008) a spectral approach is used to resolve vertically the velocity potential in the whole domain, by decomposing the potential using the orthogonal basis of Chebyshev polynomials.

The mathematical theory and numerical development are described, and the model is then validated with the application of several 1DH test cases including (i) the propagation of nonlinear regular wave over a submerged bar (experiments by Dingemans (1994)), (ii) the transformation of nonlinear irregular waves over a barred beach (experiments by Becq-Girard et al. (1999)), (iii) the generation of nonlinear harmonics on a submerged, with comparisons to small-scale experiments currently performed at ESPCI (Paris). All test cases results agree well with experimental data, confirming the model's ability to simulate accurately nonlinear and dispersive waves.

The model is then applied to study the formation of freak waves in coastal waters, with attention paid to the effect of the bottom slope on the statistics of these extreme waves. The simulation results are compared with laboratory experiments performed in a wave flume by Kashima et al. (2103). In particular, the reproduction of the most extreme waves in the experimental records is studied. The distribution of extreme wave heights is compared to existing formulas, such as Mori & Janssen (2006). The evolution of statistical parameters from deep to shallow water conditions is also analyzed, in particular the high-order statistical moments: skewness and kurtosis.

**References**

Benoit M., Raoult C., Yates M.L. (2014) Fully nonlinear and dispersive modelling of surf zone waves: non-breaking tests. In Coastal Engineering Proceedings, (ICCE'2014), 15-20 July 2014, Seoul (Korea), 1(34), waves.15, DOI: 10.9753/icce.v34.waves.15.

Raoult C., Benoit M., Yates M.L. (2016) Validation of a fully nonlinear and dispersive wave model with laboratory non-breaking experiments. Coastal Engineering. DOI : 10.1016/j.coastaleng.2016.04.003

Yates M.L., Benoit M. (2015) Accuracy and efficiency of two numerical methods of solving the potential flow problem for highly nonlinear and dispersive water waves. International Journal for Numerical Methods in Fluids. Vol. 77(10), pp 616-640. DOI: 10.1002/fld.3992

## Rogue Waves within the Framework of Integrable Systems

## Identifying spatio-temporal and spectral dynamics of mode-locked fibre lasers

## Some dynamical aspects of rogue water waves

## Background-enhanced dispersive shock generation in optical fibers

## Scale-symmetry in soliton solutions and a possible explanation of rogue waves

*(cancelled)*

## Nonlinear propagation of partially coherent waves : an experimental comparison between optics and hydrodynamics

We present experiments in which nonlinear random waves propagate in systems that are described at leading order by the focusing one-dimensional nonlinear Schrodinger equation. We compare the propagation of waves in a water tank and light propagation inside a single mode optical fiber.

We have performed an optical fiber experiment similar in its principle to a hydrodynamical experiment realized in 2004 [1]. In this experiment, a random complex field having a gaussian statistics at initial stage propagates inside the one-dimensional water tank. The spectrum of the fluctuations taken as initial condition is the so-called JONSWAP spectrum that has been measured in Oceanography experiments [2].

In our work, we specifically focus on the study of the statistical changes experienced by the nonlinear waves in the two propagation media. Our Optical and Hydrodynamical experiments are realized with reduced identical parameters. Using an original optical sampling method, we were able to measure the evolution of the statistics of power fluctuations with a temporal resolution of 250fs [3]. We compare these results to the ones obtained earlier in the Hydrodynamical experiment.

In the two experiments, we observe the emergence of large amplitude events resulting in heavy- tailed deviations from gaussian statistics. The results are quantitatively comparable. Two differences are noticeable : firstly the transcient regime is slower in water waves than in optical waves. Secondly, at long distance of propagation, optical waves exhibit a stationnary state whereas the kurtosis of water waves decreases.

We also perform numerical simulations of the nonlinear Schrodinger equation that describes the propagation. We explain some of the differences by the role of high order term and by the probable appearance of wave breaking phenomena after some propagation distance.

**References**

[1] M. Onorato, A.R. Osborne, and M Serio, Observation of strongly non-Gaussian statistics for random sea surface gravity waves in wave flume experiments, Phys. Rev. E, 70, 067302(4), (2004).

[2] G.J. Komen, L. Caveleri, M. Donelan, K. Hasselman, S. Hasselman, and P.A.E.M. Janssen, Dynamics and Modelling of Oceans Waves Cambridge University Press,(1994).

[3] P. Walczak, S. Randoux and P. Suret, Optical Rogue Waves in Integrable Turbulence Phys. Rev. Lett.,114, 143903(5)(2015).

## Fluids of Light

## Observation of resonant interactions among gravity surface waves

Wave resonant interactions constitute an efficient mechanism to exchange energy between nonlinear waves. Although these interactions are the fundamental ingredient of wave turbulence, only few experiments exist in the case of gravity surface waves.

I will talk about experiments on resonant interactions among gravity surface waves in a large basin. Two crossing sinusoidal waves interact nonlinearly to give birth to a resonant wave whose properties are fully characterized, and compared to the weakly nonlinear wave interaction theory. I will also discuss results of experiments for stronger nonlinear waves or for off-resonance conditions.

This work has been funded by ANR Turbulon, and performed in collaboration with F. Bonnefoy (EC Nantes), G. Michel, B. Semin (ENS Paris), S. Aumaître, T. Humbert, (CEA Saclay), M. Berhanu and F. Haudin (MSC Paris)

*(cancelled)*

## Real-time capture of rogue waves emerging from spontaneous modulation instability

Modulation instability (MI) is one of the most fundamental processes of nonlinear science [1]. In nonlinear fiber optics, MI develops when a weak perturbation on top of a continuous wave field experiences exponential amplification and evolves into strongly-localized breather structures with large intensity. Although MI has been studied for decades, there continues to be intense interest in understanding its dynamics because when triggered from noise, it generates high amplitude and statistically-rare “rogue waves” of importance in hydrodynamics and optics [2]. Due to experimental limitations, however, the ultrafast instability dynamics of noise-driven spontaneous MI in optics have never been directly observed experimentally. Using a time-lens magnifier system, we report for the first time the measurement of both the statistics and intensity profiles of transient breather structures emerging from spontaneous modulation instability. Our measurements are in excellent agreement with numerical simulations. The technique has tremendous potential as a laboratory tool in the study of nonlinear instabilities.

**References**

[1] V. E. Zakharov and L. A. Ostrovsky, Phys. D 238, 540-548 (2009)

[2] J. M. Dudley, F. Dias, M. Erkintalo, G. Genty, Nature Photon. 8, 755–764 (2014)

## Extreme events and phase dynamics in a forced semiconductor ring laser

The study of extreme events in optical systems has attracted lots of attention essentially due to their formal similarity to emblematic oceanic rogue waves [1]. Due to this origin, optical fibers have long been the system of choice for their analysis [2]. However, the understanding of extreme events is challenging and potentially useful in many nonlinear systems beyond oceanography and optical experiments may shed light on many generic phenomena leading to the appearance of rogue events. This is especially true when single shot spatially and temporally resolved measurements can be performed together with the accumulation of very large data sets. Such measurements are of course notoriously difficult to achieve in nonlinear fiber optics due to the involved time scales which often oblige the experimentalist to make use of spectral measurement techniques.

In this contribution, we analyse the nucleation of extreme events in a fast (nanosecond time scale) spatially extended oscillatory medium with coherent forcing. In particular, we focus on the predictability of individual events and their analysis in terms of the (spatio-temporally resolved) phase dynamics of the system.

The experimental system is a strongly multimode ring semiconductor laser [3] consisting of a millimetric semiconductor active medium enclosed in a 1-m long optical cavity under coherent external forcing. While similar experiments have often been performed in single mode lasers (described by lowdimensional dynamical systems [4]) the description of this particular experiment requires the use of partial differential equations taking into account the propagation of the electric field. In the single-mode case, the many accessible dynamical regimes are well documented both experimentally and theoretically and include frequency and phase locking, bistability, excitability and chaos. In the multimode case (and especially with the geometrical characteristics of this particular experiment), the system is much less documented in particular due to the stiffness of the model equations and its very large number of degrees of freedom.

We analyze our results thanks to a set of partial differential equations describing the field and semiconductor medium evolution [3]. In some cases the full set of PDEs can be reduced to a single soliton forming equation, namely a modified Ginzburg- Landau equation with nearly resonant forcing [5,6], providing both a simple numerical system and a connection to a much broader class of systems. Indeed, this model markedly differs from the nonlinear Schrödinger equation not only due to the presence of dissipation, but also due to its oscillatory nature. Here, we analyze the impact of phase dynamics in the formation of coherent structures in this system, from phase solitons to extreme events. In phase-unstable regimes, where usually strong multimode competition occurs, we attempt the prediction of extreme events based on local phase space reconstruction.

**References**

[1] Solli, D. R., Ropers, C., Koonath, P., & Jalali, B. (2007). Optical rogue waves.Nature, 450 (7172), 1054-1057.

[2] Dudley, John M., et al. "Instabilities, breathers and rogue waves in optics."Nature Photonics 8.10 (2014): 755-764.

[3] Gustave, François, et al. "Dissipative phase solitons in semiconductor lasers." Physical Review Letters 115.4 (2015): 043902.

[4] Bonatto, Cristian, et al. "Deterministic optical rogue waves." Physical review letters 107.5 (2011): 053901.

[5] Chaté, Hugues, Arkady Pikovsky, and Oliver Rudzick. "Forcing oscillatory media: phase kinks vs. synchronization." Physica D: Nonlinear Phenomena131.1 (1999): 17-30.

[6] Gibson, Christopher J., Yao, Alison M. and Oppo, Gian-Luca, Optical Rogue Waves in Vortex Turbulence, Phys. Rev. Lett. 116, 043903 (2016).

## New experiments on modulation instability and extreme waves in scalar/vector and conservative/dissipative nonlinear fiber systems

Since the seminal works by A. Hasegawa and co-workers in the 1980s, the modulation instability phenomenon has been widely studied and used in optical fibers, in particular for generating high-repetition-rate soliton trains and for parametric amplification of weak signals. Modulation instability is also known as a general precursor of highly localized wave structures through amplification of perturbations. We review here our recent experiments performed in conservative physical systems based on nonlinear fiber optics [4], that evidence a large class of exact pulsating solutions of the nonlinear Schrödinger equation, called breathers and rogue wave solutions. Based on the coherent seeding of the modulation instability process in scalar and vector fiber systems, we confirm the existence of super-regular breathers and dark vector rogue waves. These results have shed new lights on extreme nonlinear dynamics and related analogies between optics and hydrodynamics.

Moreover, we present our recent investigations into the modulation instability process in dissipative systems based on a nonlinear fiber ring cavity [5]. In this special case, the cavity boundary conditions are known to play, in addition to the fiber nonlinearity and dispersion, a crucial role in the propagation dynamics. We show that such systems exhibit complex spatiotemporal dynamics that lead to the emergence of extreme events for very simple parameters.

**References**

[1] B. Frisquet et al., Phys. Rev. A 89, 023821 (2014)

[2] B. Kibler et al., Phys. Rev. X 5, 041026 (2015)

[3] B. Frisquet et al., Phys. Rev. A 92, 053854 (2015)

[4] B. Frisquet et al., Sci. Rep. 6, 20785 (2016)

[5] A. Bendahmane et al., SPIE Photonics Europe, 9894-10 (2016)

[6] A. Bendahmane et al., in preparation (2016)

## Numerical study of the stability of the Peregrine solution

## Superfluid turbulence and Kelvin wave cascade: coexistence of weak-wave turbulence and strong hydrodynamical turbulence

## Microwave realization of Particle-like Scattering States

In my talk I will present an experimental realization of particle like scattering states [1], i.e. collimated waves supported only on a single classical trajectory or ray. First I will introduce the concept of these states and how they can be obtained using only information of the transmission matrix, more precisely the Wigner-Smith time-delay operator. To finally implement them experimentally we need wave-front shaping techniques, which we realize using a set of antennas excited via IQ-modulators. Finally the experimentally obtained particle like scattering states will be presented and discussed.

[1] Generating Particle-like Scattering States in Wave Transport, Stefan Rotter, Philipp Ambichl, and Florian Libisch, Phys. Rev. Lett. 106, 120602 (2011)

## Emergent spacetime geometries and nonlinear acoustic waves in photon-fluids

The propagation of finite-amplitude waves in defocusing media is investigated in the framework of the hydrodynamic description of light as a photon fluid. We show that in the appropriate limits, the wave dynamics can described in purely geometrical terms, where an effective curved spacetime determining the propagation of the waves is generated by the waves themselves. The spacetime geometry emerges naturally as a result of the interaction between a density wave and the self-induced background flow. Remarkably, a similar situation occurs in General Relativity: mass distributions evolve in a curved spacetime metric that is modified by mass itself.

In this context, we offer an alternative view of some nonlinear effects in acoustic waves propagation. In particular, as observed in real fluids, different points of the wave profile propagate at different velocities leading to the self-steepening of the wave front and the subsequent formation of a shock. We show that the self-steepening effect can be interpreted as a kind of gravitational influence on the wave by its own effective metric, while the wave-breaking process is associated to the formation of a curvature singularity.

## Two-dimensional Rogue Waves Induced by Nonlinear Optical Vortices

The first demonstration of optical rogue waves in a microstructured optical fibre [1] triggered a major scientific interest in the understanding of the physical mechanisms behind the origin and development of optical rogue waves, the probability of observing them, and the type of optical systems capable of generating these extreme events [2]. For nonlinear optical cavities, experimental evidence of deterministic rogue waves was first provided in lasers with injected signals [3]. The dynamics of many of these systems, however, are confined to one spatial dimension or less.

In this talk we describe a mechanism for the generation of fully two-dimensional spatio- temporal rogue waves in the presence of turbulence of interacting optical vortices. We consider the dynamics in the transverse plane of the complex Ginzburg-Landau (CGL) and complex Swift-Hohenberg equations in the presence of an external forcing. Without spatio- temporal coupling, chaos, turbulence and, consequently, rogue waves are forbidden. With spatio-temporal coupling and below the locking threshold, we demonstrate phase instabilities leading to amplitude instabilities and then to regimes of defect-mediated turbulence with interacting optical vortices [4]. Depending on the density of the moving vortices, short distance interactions lead to sudden, rare, large and randomly positioned peaks of the light intensity. The statistics of these events is highly non-Gaussian and the probability density function is well described by the Weibull distribution [4]. The small aspect ratio, the full 2D character and the quick dynamics represent the major advantages of transverse optical devices for studying the generation and control of rare events with applications, by universality, in hydrodynamics and oceanography. We note that the CGL model equations have a broad range of applications in optics, ranging from broad area lasers to optical parametric oscillators, and to polaritons in semiconductor microcavities.

**References:**

[1] D. R. Solli, C. Ropers, P. Koonath, and B. Jalai, “Optical rogue waves,” Nature 450, 1054 (2007)

[2] J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nature Photonics 8, 755 (2014)

[3] C. Bonatto et al., “Deterministic Optical Rogue Waves,” Phys. Rev. Lett. 107, 053901 (2011)

[4] C. Gibson, A. M. Yao and G.-L. Oppo, “Optical rogue waves in vortex turbulence,” Phys. Rev. Lett. 116, 043903 (2016)

## Dispersive Alfvén wave phase mixing and focusing in a collisionless plasma

## Adiabatic theory of champion solitons

We consider the interaction of an optical soliton with a low-intensity dispersive wave copropagating in a nonlinear optical fiber. In specially prepared cases the dispersive wave is reflected at the soliton barrier, whilst the soliton acquires a permanent shift in frequency and may experience switching to a new state with a considerable increase of peak power.

We present an adiabatic theory that quantifies the soliton behaviour and suggests optimal initial parameter values for the dispersive wave. Our approach resides in a revised soliton perturbation theory combined with quantum mechanical scattering theory for the dispersive wave.

## Optical wave turbulence and rogue waves

The wave turbulence theory provides a unified nonequilibrium thermodynamic formulation of random nonlinear optical waves [1]. The talk will review different formalisms: the wave turbulence kinetic equation describing wave condensation and thermalization, as well as its breakdown, the long-range Vlasov equation in analogy with gravitation, the weak Langmuir turbulence kinetic equation and its description of spectral shock and collapse singularities. Depending on the regime and the considered formalism, extreme events manifest themselves under different forms. We will discuss in particular whether extreme events can be interpreted as the natural large fluctuations of the phase-transition to soliton condensation.

**References**

[1] Picozzi, Garnier, Hansson, Suret, Randoux, Millot, Christodoulides, Optical wave turbulence, Phys. Reports 542, 1 (2014).

## Spatial rogue waves in photorefractive ferroelectrics

We discuss experimental observations of extreme optical events in photorefractive ferroelectric crystals demonstrating a new optical system to study abnormal wave phenomena [1]. Spatial rogue waves emerge as the highly-nonlinear beam propagation is affected by out-of equilibrium properties and large stochastic fluctuations in media response. Ongoing investigations to inspect the role of optical turbulence and spatial incoherence in their generation are also reported.

**References**

[1] D.Pierangeli, F.Di Mei, C.Conti, A.J.Agranat and E.DelRe, Spatial Rogue Waves in Photorefractive Ferroelectrics, Phys. Rev. Lett. 115, 093901 (2015).

## On the origin of heavy tail statistics in equations of the nonlinear Schrödinger type: an exact relation

We study the formation of extreme events in incoherent systems described by the nonlinear Schrödinger type of equations. We derive an identity that relates the evolution of the kurtosis (a measure of the relevance of the tails in the probability density function) of the wave amplitude to the rate of change of the width of the Fourier spectrum of the wave field. The result is exact for all dispersive systems characterised by a nonlinear term of the form of the one contained in the nonlinear Schrödinger equation. Numerical simulations are also performed to confirm our findings. Our work sheds some light on the origin of rogue waves in incoherent dispersive nonlinear media ruled by local cubic nonlinearity. (joint work with Miguel Onorato, Gennady El, Stéphane Randoux, and Pierre Suret)

## Observation of star-shaped surface gravity waves

## Identification of rogue waves from scattering transform analysis of periodized waveforms

The nonlinear Schrödinger equation (NLSE) stands out as the dispersive nonlinear partial differential equation that plays prominent role in the modeling and understanding of features 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 [Agafontsev:15,Randoux:14,Walczak:15], and the specific question of formation of rogue waves has been recently extensively studied in this context [Toenger:15]. The determination of exact analytic solutions of the focusing 1D-NLSE prototyping rogue waves of statistical relevance is now considered as the problem of central importance. Here we address this question from the perspective of the inverse scattering transform (IST) method that relies on the integrable nature of the wave equation. We develop a conceptually new approach to the rogue wave identification in which appropriate, locally coherent structures are specifically isolated from a globally incoherent wave train to be subsequently analyzed by implementing a numerical IST procedure relying on a spatial periodization of the object under consideration. Using this approach we extend the existing classifications of the prototypes of statistically relevant rogue waves from standard solitons on finite background (SFBs) and their collisions to more general nonlinear modes characterized by the *finite-band* spectra.

**References**

[Agafontsev:15] Agafontsev, D.S. and Zakharov, V.E. Integrable turbulence and formation of rogue waves. Nonlinearity 28, 2791 (2015).

[Randoux:14] Randoux, S., Walczak, P., Onorato, M. and Suret, P. Intermittency in integrable turbulence. Phys. Rev. Lett. 113, 113902 (2014).

[Walczak:15] Walczak, P., Randoux, S. and Suret, P. Optical rogue waves in integrable turbulence. Phys. Rev. Lett. 114, 143903 (2015).

[Toenger:15] Toenger, S. *et al.* Emergent rogue wave structures and statistics in spontaneous modulation instability. Scientific Reports 5 (2015).

## Waves on jet currents: novel mechanisms of freak wave formation

We develop a new paradigm of how to describe linear and weakly nonlinear dynamics of waves on jet currents with a particular emphasis on new mechanisms of freak wave formation.

From numerous seamen accounts and insurer records it has been known for long time that rogue waves events are quite frequent on certain currents, e.g. on notorious in this respect Agulhas current. The theoretical explanation of this fact is still lacking, which is due to our overall poor understanding of wave nonlinear evolution on currents. We address this challenge by developing a new systematic asymptotic theory of waves dynamics on jet currents which does not rely on the WKB approximation. The solutions for jet currents with arbitrary lateral profiles are found by first solving a linear 2D boundary-value problem in terms of an asymptotic series in natural small parameters. In essence, we employ approximate separation of variables, which is shown to be always justified for the oceanic conditions.

There are three types of waves on the current: passing through, reflected and trapped. The key role in the new approach is played by the trapped modes. The trapped modes themselves (rather than comprising them harmonic components) participate in the nonlinear interactions. A general weakly nonlinear theory of trapped mode evolution is being developed. In particular, the corresponding interaction coefficients have been derived. The properties of resonant interactions are qualitatively different from those between the waves in the absence of a current. In particular, three-wave interactions are always allowed in deep water and may play an important role in wave field evolution. There are three main advantages of the developed approach: (i) it is systematic and can identify and address the situations where the commonly adopted paradigm is not applicable; (ii) the current could be almost arbitrary, i.e. weak/strong, or smooth/with sharp edges; (iii) the initially 3-D problem is reduced to solving 1D evolution equations with the lateral and vertical dependence being prescribed by the corresponding modal structure. The modes can participate in both three and four- interactions. From the perspective of rogue wave occurrence we have identified several new mechanisms which have no analogues in the absence of currents.

(i)Acting as a waveguide the current makes wave dynamics essentially one-dimensional, which dramatically enhances the likelihood of freak waves.

(ii) In contrast to the case of no currents robust envelope solitons were found both analytically (in the weakly nonlinear setting) and numerically in the full Euler equations.

(iii) If we allow for weak longitudinal non-uniformity of the currents then for trapped waves blockage becomes possible for dominant wave scales.

## Phase space unraveling for estimating the risk of ocean rogue wave formation

## Optical Rogue Waves in integrable turbulence : direct observation by using Time Microscopy and study of the statistical properties

We report optical experiments and numerical simulations devoted to the investigation of integrable turbulence in the focusing regime of 1D-NLSE [1, 2]. We present in particular the first real time observation of rogue waves (RWs) generated in optical fibers by using a specially-designed Time Microscope (TM) ultrafast acquisition system [2]. RWs with time scale of the order of 300 fs are found to emerge from the propagation of partially coherent waves having initial time scale of 5ps.

Using our TM and another optical sampling setup, we measure precisely the probability density function (PDF) of optical power of the nonlinear random waves rapidly fluctuating with time. The PDF of optical power is found to evolve from the exponential distribution to a strong heavy-tailed distribution. The exponential distribution of the power corresponds to a Gaussian statistics for the field. Our experiments thus reveal the occurrence of extreme events (RWs) in integrable turbulence with a probability much higher than predicted by the normal law. Finally we will emphasize the crucial question of the dependence of the stationary statistics on the initial statistics [5, 6].

**References:**

[1] P. Walczak et al., Phys. Rev. Lett. 114, 143903 (2015)

[2] P. Suret et al., arXiv:1603.01477 (2016)

[3] J. M. Dudley et al., Nat. Photon. 8, 755 (2014)

[4] S. Randoux, et al., arXiv preprint arXiv:1512.04707.

[5] D.S. Agafontsev and V E Zakharov Nonlinearity, 28, 8 (2015)

[6] J. M. Soto-Crespo et al., Phys. Rev. Lett. 116 (2016) 103901.

## Rogue waves formation from Eckhaus modulation instabilities in optical fiber cavities

## Spatio-temporal extreme events in a laser with saturable absorber

Rogue waves in optics have been recently attracting great attention, for the possibility of studying these events in a well controlled environment. Here we show numerical results about extreme events in the (2D) intensity profile of a broad-area semiconductor laser with saturable absorber.

This system can emit a turbulent state, where the light intensity oscillates aperiodically in space and time. We developed a numerical method for the detection of spatio-temporal maxima of intensity in this turbulent regime, and we compare their distribution with heavy-tail distributions of generalised extreme value theory. We can demonstrate the existence of extreme events according to both the two most common definitions of rogue waves threshold. We believe that our system, being intrinsically two-dimensional, could give some precious insights on the focusing mechanisms at the origin of rogue waves in oceans, which could be absent in one-dimensional systems such as fibers.

## Extreme Events in Laser Systems

## Observing Zabusky and Kruskal scenario: breaking of harmonic waves

We give a fresh look at the famous problem investigated numerically by Zabusky and Kruskal in 1965 [1], *i.e.* the breaking of a harmonic wave in the weakly dispersing case and the associated Fermi-Pasta-Ulam recurrence. We report experimental results obtained for surface gravity waves in a shallow water tank [2]. We show that the results are accurately interpreted in terms of analytical formulas developed in the context of a WKB approach for the scattering problem associated with the the Korteweg-De Vries equation [3]. A parallel with experiments in optics described in terms of nonlinear Schrodinger equation is also proposed [4].

**References**

[1] N. J. Zabusky and M. D. Kruskal, Phys. Rev. Lett. 15, 240 (1965).

[2] S. Trillo, G. Deng, G. Biondini, M. Klein, G. F. Clauss, A. Chabchoub, M. Onorato, Hydrodynamical realisation and theoretical description of the Zabusky-Kruskal numerical experiment , submitted to Phys. Rev. Lett.

[3] G.. Deng, G. Biondini, and S. Trillo, Physica D, in press, doi:10.1016/j.physd.2016.03.003

[4] J. Fatome, C. Finot, G. Millot, A. Armaroli, and S. Trillo, Phys. Rev. X 4, 021022 (2014).

## Rogue waves and spatiotemporal wave coupling phenomena in multimode fibers

Optical fibers provide a cost effective and readily accessible test bed to probe complex nonlinear wave coupling phenomena. In this talk, we describe intriguing recent theoretical developments and experimental studies involving nonlinear multimode optical fibers [1]. Exotic behaviors are observed, ranging from analog gravity coupled transient black holes in birefringent fibers [2], to the emission of arrays of multi-octave spanning powerful sidebands, and spatial beam collapse in highly multimode fibers [3-5]. All these experiments share the useful property that are carried out using relatively short spans of standard and cheap telecom optical fibers.

**References**

[1] Picozzi Antonio, Millot Guy, Wabnitz Stefan (2015). Nonlinear optics: Nonlinear virtues of multimode fibre. NATURE PHOTONICS, vol. 9, p. 289-291

[2] Frisquet Benoit, Kibler Bertrand, Morin Philippe, Baronio Fabio, Conforti Matteo, Millot Guy, Wabnitz Stefan (2016). Optical Dark Rogue Wave. SCIENTIFIC REPORTS, vol. 6,

[3] Wright Logan G., Wabnitz Stefan, Christodoulides Demetrios N., Wise Frank W. (2015). Ultrabroadband Dispersive Radiation by Spatiotemporal Oscillation of Multimode Waves. PHYSICAL REVIEW LETTERS, vol. 115

[4] K. Krupa et al., “Observation of geometric parametric instability induced by the periodic spatial self-imaging of multimode waves”, http://arxiv.org/abs/1602.04991, 2016 - accepted for publication in Phys. Rev. Lett.

[5] K. Krupa et al., “Spatial beam self-cleaning in multimode fiber”, http://arxiv.org/abs/1603.02972, 2016.

## Nonlinear propagation of partially coherent waves in optical fibers

We present optical fiber experiments in which we study the nonlinear propagation of partially coherent waves. We focus our attention on systems of random waves that are well described by the integrable one dimensional nonlinear Schrödinger equation. We investigate both experimentally and numerically the statistical evolution of power fluctuations of random waves in the normal and anomalous dispersion regime of the optical fiber.

Using fast detection techniques, our experiments show that nonlinear propagation strongly influences the statistics of the partially coherent waves by producing deviations from the normal distribution. In particular, in anomalous dispersion regime, the probability of emergence of extreme events is significantly enhanced whereas it is strongly reduced in the normal dispersion regime.

Moreover, these experimental results are reproduced in a quantitive way by our numerical simulations of the one dimensional nonlinear Schrödinger equation. Futhermore, we observe the emergence of coherent structures embedded in the random fluctuations which are similar to solitons on finite background in the case of the anomalous regime of the optical fiber.