Extreme Waves

Optical self-trapped beams induced by self-focusing instabilities have already been studied since the late 1960s. Typically, the self-focusing of a high-power laser beam in nonlinear dielectric medium has been shown to induce the breaking of a cylindrical Gaussian beam, thus giving rise to multiple filaments in space. Recently, multimode optical fibers have been shown to be essential for bridging the gap between nonlinear optics in bulk media and single-mode fibers. The understanding of the transition between the two fields remains complex due to intermodal nonlinear processes and spatiotemporal couplings, e.g., some striking phenomena observed in bulk media with ultrashort pulses have not yet been unveiled in such waveguides. However, exact theoretical developments of nonlinear waves propagating in single-mode fibers may provide an excellent basis through the suitable use of the one-dimensional nonlinear Schrödinger equation. In this contribution, we will theoretically and numerically investigate a particular regime of modulation instability in a multimode optical fiber, for which the nonlinear stage is described by analytical breather solutions with new spatial features. An overview of both the noise-driven and the coherent seeding regimes of this nonlinear optical process will be presented.

We report the real-time full-field measurements of modulation instability under pulsed excitation in the broadband regime. Combining spectral Interferometry with an ultra-broadband phase-stable reference field, we achieve sub-30 fs temporal resolution allowing to resolve localized structures in this particular regime.

We report the observation of different features of Fermi recurrences Pasta Ulam Tsingou in an advanced photonic fiber system. We have developed a nearly lossless fiber optic device in which the longitudinal evolution of the electric field (intensity and phase) can be followed. This allowed us to capture the rich dynamics of this complex nonlinear system, such as the observation of symmetry breaking, the path to thermalization of the Fermi Pasta Ulam Tsingou recurrence process or the direct observation of complex breathers solutions predicted analyticaly . References: 1. Mussot, A. et al. Fibre multi-wave mixing combs reveal the broken symmetry of Fermi–Pasta–Ulam recurrence. Nature Photonics 12, 303–308 (2018). 2. G. Vanderhaegen, et al. “Heterodyne optical time-domain reflectometer combined with active loss compensation: a practical tool for investigating Fermi Pasta Ulam recurrence process and breathers dynamics in optical fibers”, FRONTIERS IN PHYSICS doi: 10.3389/fphy.2021.637812 (2021)

We report the existence of stable, dissipative light bullets in diffractive and dispersive Kerr cavities. These three-dimensional (3D) localized structures consist of either isolated, bounded, or clustered structures forming well-defined 3D patterns. They can be considered stationary states in the reference frame, moving with the group velocity of light inside the cavity. The number of LBs and their 3D distribution are determined by the initial conditions, while their maximum power remains constant for a fixed value of the system parameters. Their bifurcation diagram allows us to explain this phenomenon as a manifestation of the homoclinic snaking for dissipative light bullets. However, as the injected beam intensity increases, the LBs lose their stability, and the cavity field exhibits giant 3D pulses of short duration. The statistical characterization of the pulse amplitude reveals a long-tailed probability distribution, indicating the occurrence of extreme events.

The smooth transition between stable, Talbot-effect-dominated and modulational instable nonlinear optical wave propagation is described as superposition of oscillating, growing and decaying eigenmodes of the common linearized theory of modulation instability. Results explain well the initial stage of experiments on modulation instability and breather excitation in spatial-spatial optical platforms. An increased accuracy of instability gain measurements and a more selective and well-directed breather excitation are demonstrated experimentally. The experiments were performed in a new platform with adjustable effective cubic nonlinearity.

The cubic focusing nonlinear Schroedinger (NLS) equation is the simplest model describing the modulation instability (MI) of 1+1 dimensional quasi monochromatic waves in weakly nonlinear media, the main physical mechanism for the generation of rogue waves (RWs) in nature. An analytic theory of periodic RWs of NLS has been developed, allowing one to solve the periodic Cauchy problem for small initial perturbations of the unstable background solution (the periodic Cauchy problem of the RWs). A perturbation theory has also been constructed, allowing one to study the order one effects of physical perturbations of NLS on the periodic RW dynamics. In both theories the solution is given, to leading order, in terms of elementary functions of the initial data. In this talk, after summarizing the main aspects of the above theories, we present recent analytic results on the theory of RWs in multidimensions, mainly concentrating on the Davey-Stewartson 2 equation, an integrable 2+1 dimensional generalization of NLS. The results illustrated in this talk have been obtained in collaboration with P. Grinevich and F. Coppini.

Semiconductor lasers under continuous-wave optical injection display a rich variety of complex dynamical regimes. In this presentation, I will focus on a particular regime in which the laser output intensity displays ultrahigh pulses, reminiscent of optical rogue waves. I will present experimental and numerical results obtained by using nonlinear time-series analysis and machine learning. I will analyze the effect of a weak modulation of the laser current, and show that it can either trigger or fully suppress extreme pulses. I will also discuss the effectiveness of several popular machine learning methods for predicting the height of the next pulse.

The talk focuses on the numerical simulations of modulation instability and wave turbulence in optical fibers using the generalized nonlinear Schrödinger equation. The simulation method used is the split-step method, which is a popular technique due to its ease of implementation. However, the talk highlights the risk of nonphysical spurious instabilities that can result from a careless application of this standard numerical technique. We discuss the reasons why such instabilities occur and provide solutions to avoid them. The results are important for applications such as supercontinuum generation or collection of statistics of the extreme events, where the optical spectrum is wide and complex, making spurious instabilities difficult to detect.

Amin Chabchoub1,2 • 1Disaster Prevention Research Institute, Kyoto University, Japan • 2School of Civil Engineering, The University of Sydney, Australia Breather solutions of the nonlinear Schrödinger equation, also referred to as solitons on finite and continuous background, have been known since the late 70s. Some of these pulsating localized structures describe the nonlinear stage of the modulation instability in a variety of nonlinear dispersive media and are prized to model and control rogue wave events in such wave systems. The talk will elaborate on the latest groundbreaking breather observations in optics, hydrodynamics, plasma, and Bose-Einstein condensates. A particular focus will be devoted to applications, for instance in marine engineering as well as ocean rogue wave modeling and prediction. Moreover, perspectives on implementations and limitations of breather modeling for extreme sea waves will also be discussed.

Seit Jahrtausenden fährt der Mensch zur See. Und wohl schon seit Anbeginn berichten Seefahrer von gewaltigen und furchteinflößenden Erfahrungen und Begegnungen. Mit Meeresungeheuern, Seeschlangen, Riesenkraken. Vor allem und nachdrücklich durchziehen die Berichte aber immer wieder riesenhaft anmutende Wellen, die sich plötzlich und unerwartet auftürmen und bisweilen ganze Schiffe mit Mann und Maus verschlingen. Seemannsgarn, so dachte man lange an Land. Erst seit wenigen Jahrzehnten ist sich nun aber auch die Wissenschaft bewusst geworden, dass spontan auftretende große Wellen auf dem Meer keine Mär, sondern Realität sind. Vom Wellental zum Wellenberg können 30 bis 40 Meter erreicht werden – Kirchturmhöhe. Wie kann es zu solchen extremen Meereswellen kommen und welche Rolle spielen sie heute in Schiffbau, Schiffahrt und Meerestechnik?

Several concepts for the description of nonlinear pulse propagation in dispersive optical media will be introduced. Among those are envelope models, short-pulse equations, as well as our generalized approach [1,2]. Effects like limitations for ultrashort optical pulses [3], cusp formation [4], and solitons that mimic event horizons for smaller (dispersive) optical waves (DW's) [5] will be presented. Moreover, it will be demonstrated, both numerically and more efficiently by a new analytic theory [5,6], that small optical waves can be used to control such solitons [6,7]. This opens a new concept of all-optical switching, with interesting new applications. In particular, the parasitic Raman effect can be completely compensated by properly chosen DW’s, such that the soliton properties remain unchanged by this support [8]. References [1] S. Amiranashvili, U. Bandelow and A. Mielke, "Calculation of ultrashort pulse propagation based on rational approximations for medium dispersion", Opt. Quantum Electron. 44:241-246 (2012) [2] Sh. Amiranashvili, U. Bandelow, and A. Mielke. Padé approximant for refractive index and nonlocal envelope equations. Opt. Commun., 283:480-485 (2010) [3] S. Amiranashvili, U. Bandelow and N. Akhmediev, "Few-cycle optical solitary waves in nonlinear dispersive media", Phys. Rev. A 87:013805 (2013) [4] Sh. Amiranashvili, U. Bandelow, N. Akhmediev, Spectral properties of limiting solitons in optical fibers. Optics Express, Vol. 22, No. 24, (2014) pp. 30251-30256 [5] S. Pickartz, U. Bandelow, Sh. Amiranashvili, Adiabatic theory of solitons fed by dispersive waves, Phys. Rev. A 94:033811/1-12 (2016) [6] S. Pickartz, U. Bandelow, Sh. Amiranashvili, Efficient all-optical control of solitons Opt. Quantum Electron., Vol. 48, (2016) pp. 503:1-7 [7] S. Pickartz, U. Bandelow, Sh. Amiranashvili, Asymptotically stable compensation of soliton self-frequency shift Opt. Lett., Vol. 42, (2017) pp. 1416-1419 [8] U. Bandelow, Sh. Amiranashvili, S. Pickartz, Stabilization of Optical Pulse Transmission by Exploiting Fiber Nonlinearities, J. of Lightwave Technology, Vol. 38, No. 20, (2020) pp. 5743-5747

We show that nonuniformity can trigger drift instabilities in the evolving amplitude of localised Faraday patterns. We performed an experimental study of Faraday waves on the free surface of a fluid under localised vertical driving. It is shown that above a threshold of instability, the localised and nondrifting drive triggers drift instabilities in the surface patterns. We provide a theoretical explanation of the phenomenon based on a nonuniform version of the pdnlS equation as a simple model for parametric driven patterns. Under the assumption of a vertical drive modulated by a spatially spread Gaussian profile featured by an extrinsic length scale, we derived an evolution equation governing the slow dynamics of the pattern amplitude. The resulting normal form includes two unreported SPM terms that are linked to complex dynamics in the underlying patterns. We show that such terms trigger drift via nonlinear symmetry breaking induced by a nonlinear gradient. Numerical simulations of the pdnlS equation are in agreement with both the theory and experimental data. These results extend our general understanding of nontrivial dynamical effects of nonuniformity in nonlinear extended parametric oscillators and how pattern selection can be dramatically affected by the nonuniform nature of the input of energy.

We present our recent theoretical results on complex nonlinear interactions of coherent solitary wave structures living on an unstable background -- breathers. As a theoretical model, we use the focusing one-dimensional nonlinear Schrödinger equation. First, we discuss the theoretical description and experimental observation of the nonlinear mutual interactions between a pair of co-propagative breathers -- breather molecules [1]. As a general case, we show that the resulting bound state of breathers exhibits molecule like behavior with quasiperiodic oscillatory dynamics. At the same time, for specific commensurate conditions, the molecule oscillations become precisely periodic. Then we derive general analytical expressions describing the space-phase shifts that breathers acquire after collisions with each other [2]. Based on these expressions, we propose a general approach to managing multiple interactions of breathers [3]. The breather management approach allows adjusting the initial positions and phases of more than two moving breathers to observe various desired wave states at controllable moments of wave evolution. As proof of principle, we consider a couple of separated pairs of breathers initially synchronized in small-amplitude patterns. We obtain an explicit expression for the separation interval between the pairs so that the interactions of the breathers from the neighboring patterns lead to the formation of an extreme amplitude wave or recurrence to the initial small-amplitude state. Experiments carried out on a light wave platform with a nearly conservative optical fiber system accurately reproduce the predicted dynamics and prove the viability of our nonlinear wave theory. In addition, we consider generalizations of the scalar breathers theory to the vector two-component NLSE describing polarized light and show examples of resonance vector breathers transformations [4]. [1] G. Xu, A. Gelash, A. Chabchoub, V. Zakharov and B. Kibler, Breather wave molecules. Phys. Rev. Lett., 2019. [2] A.A. Gelash, Formation of rogue waves from a locally perturbed condensate, Phys. Rev. E, 2018. [3] A. Gelash, G. Xu, and B. Kibler, Management of breather interactions. Phys. Rev. Research, 2022. [4] A. Gelash and A. Raskovalov, Vector breathers in the Manakov system. Stud. Appl. Math. 2023.

Understanding spectral downshifting is the heart of ocean wave research. The evolution of ocean waves was described by a fetch law that associates wind-wave growth with the elongation of the mean wavelength. Historically, the mechanism of spectral downshifting is considered explicable by the nonlinear energy transfer among spectral components due to four-wave resonance. The theory, first presented by Hasselmann (1962), is derived from a conservative system. There are alternative mechanisms based on dissipation. Tulin (1996) considered the imbalance of energy and momentum loss due to dissipation leading to an inevitable spectral downshifting. Donelan et al. (2012) considered the redistribution of energy from short to long waves by spilling breakers. These alternative theories both successfully explained the fetch law. Our recent observational and experimental studies showed that spectral downshifting under sea ice cannot be explained by conventional weak nonlinear interaction and linear dissipation. During the 2019 campaign in the Okhotsk Sea, we deployed a wave buoy on ice and concurrently observed waves by a ship-borne stereo imaging system. During a wave event in Feb. 2019, the incoming swell of around 10 s anomalously downshifted to 15~17 s as it propagated through the Marginal Ice Zone for about 90 km. The significant wave height was reduced from 0.34 m to 0.25 m. Since the energy at the low frequency increased, a conventional frequency-dependent dissipation is not at work. Partly motivated by this finding, we have conducted an experiment in the wave-ice tank of the modulated wave train propagating under the brash ice. The tank length of 8 m was too short for a modulated wave train of carrier wavelength of about 0.9 m to see a considerable development of modulation without ice. However, when the same wave system propagated under ice, the peak of the spectrum shifted to the lower sideband. This spectral downshifting cannot be explained by a linear dissipation, as the energies at the lower frequencies increased. These two studies imply that waves under the ice may undergo a dissipation-driven nonlinear downshifting, yet to be proven.

Light flow in nonlinear media can exhibit quantum hydrodynamical features which are profoundly different from those of classical fluids. We show that rather extremes regime of quantum hydrodynamics can be accessed by exploring two paradigmatic problems in fluid dynamics. The first one is the piston problem for light, and its generalization, where the piston acts on a concomitant abrupt change of photon density. Our experiment reveals regimes featuring optical rarefaction (retracting piston) or shock (pushing piston) wave pairs, and most importantly the transition to a peculiar type of flow, occurring above a precise critical piston velocity, where the light shocks are smoothly interconnected by a large contrast, periodic, fully nonlinear wave. The second one is the temporal photonic analogue of the dam-break phenomenon for shallow water by exploiting a fiber optics setup. We clearly observe the decay of the step-like input (photonic dam) into a pair of oppositely propagating rarefaction wave and dispersive shock wave. Our results show that fiber-based optical systems are peerless testbeds to investigate the extension of fluid dynamics problems to superfluid regimes by taking benefit of the analogy between optics and fluid dynamics supported by the universality of the nonlinear Schrodinger equation.

Because time and space play a similar role in wave propagation, wave propagation is affected as well by spatial modulation or by time modulation of the refractive index. We will discuss the role of time modulation and show that sudden changes of the medium properties generate instant wave sources that emerge instantaneously from the entire wavefield and can be used to control wavefield and to revisit the way to create time-reversed waves. Then we will extend the approach to periodic time modulation. The difference between periodic time modulation and periodic spatial modulation will be emphasized and the way an antenna radiates in these two kinds of situation will be discussed. Then the coupling of an antenna with the emitted wavefield in a time modulated medium will be discussed and we show that a kind of wave-particle duality can be observed.

Abstract: The phenomenon of resonant radiation emitted by Peregrine solitons in cubic media has been recently investigated, showing that, unlike bright or dark solitons of the nonlinear Schrödinger equation, the radiation process is affected by the intrinsic local longitudinal variation of the soliton wavenumber [1]. In this workshop, we evince the peculiar resonant radiation emitted by extreme waves in a quadratic nonlinear medium [2,3], described by the second harmonic generation model, in the absence of higher-order dispersions, owing to its second harmonic component. We give criteria for calculating the radiated frequencies based on material's parameters, waves' amplitude, finding excellent agreement with numerical simulations. The results are important in the prediction and characterization of resonant radiation in ultrafast second harmonic generation. References: 1. F. Baronio, S. Chen, S. Trillo, Resonant Radiation from Peregrine solitons, Optics Letters 45, 427 (2020). 2. L. Bu, F. Baronio, S. Chen, and S. Trillo, Resonant radiation emitted by solitary waves in cascading quadratic media, Optics Express 21, 8307-8324 (2023). 3. L. Bu, F. Baronio, S. Chen, and S. Trillo, Quadratic Peregrine solitons resonantly radiating without higher order dispersion, Optics Letters 47, 2370-2373 (2023).

Photonic media undergoing spatiotemporal variations of the refractive index are recently becoming physically realisable due to the development of new materials and experimental techniques. In this talk I will review old known results and the recent progress on the linear and nonlinear properties of these media. Solitons propagating in the momentum-gap, parametric amplification and the analogous of supercontinuum generation will be presented.

Fiber lasers offer an attractive platform to study the nonlinear interaction dynamics that can lead to the build-up of instabilities and extreme events. We will discuss the soliton dynamics in thulium-doped mode-locked fiber lasers in transition regimes between stable fundamentally mode-locked states and multi-pulsing regimes. Using dispersive Fourier transform measurements to capture the pulse-by-pulse dynamics, high intensity fluctuations were analyzed during the build-up of multi-pulsing states in the laser cavity. In a versatile thulium-doped fiber laser capable of emitting both conventional solitons and dissipative solitons, soliton explosions were induced by adjusting an intracavity polarization controller. Incoherent dissipative solitons and their coherence properties were recorded, offering insights into the underlying pulse dynamics of transition states between conventional an dissipative solitons.

During recent years the complexity of non-linear rogue gravity waves on the surface of water has become apparent. A key mechanism is modulation instability and its nonlinear evolution in the form of Akhmediev and Peregrine breathers. We discuss how the findings on the non-linear physics of rogue ocean waves relates to methods for radar based rogue wave prediction in the ocean. A numerical approach based on sparse data assimilation methods combined with the nonlinear higher order spectral method is presented and discussed with respect to its potentials and limitations.

Orazio Descalzi Complex Systems Group, Universidad de los Andes, Santiago, Chile e-mail: odescalzi@miuandes.cl Since Hasegawa & Tappert [1] derived an equation describing the dynamical evolution of the envelope of the transverse electric field for short optical pulses in a dielectric single mode fiber waveguide in the presence of anomalous group velocity dispersion, many studies have led to new terms that have been added to this equation giving account of gain and losses of energy as much as the Raman effect. This generalized equation is the cubic complex Ginzburg-Landau equation with nonlinear gradient terms, which are associated with Raman scattering, delayed nonlinear gain, self-steepening and dispersion of the nonlinear gain. Facao and Carvalho [2-4] found that stable dissipative solitons can arise for the above mentioned equation including moving fixed shaped and oscillating pulses. We have shown [5-6] that a mechanical analog containing contribution from a potential and from a nonlinear viscous term can explain the existence of localized solutions and can be used to set up a feedback mechanism. Recently, a transition to localized spatio-temporal disorder of pulses has been studied in the cubic complex Ginzburg-Landau equation with nonlinear gradient terms [7]. In addition, a scenario with rich dynamics has been found in the collision of these dissipative solitons [8]. Finally, stochastic aspects of the dissipative solitons stabilized by nonlinear gradient terms will be discussed during the talk [9]. References [1] A. Hasegwa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973) [2] M. Facao and M.I. Carvalho, Phys. Rev. E 92, 022922 (2015) [3] M. Facao and M.I. Carvalho, Phys. Rev. E 96, 042220 (2017) [4] M.I. Carvalho and M. Facao, Phys. Rev. E 100, 032222 (2019) [5] O. Descalzi, J. Cisternas, and H.R. Brand, Phys. Rev. E 100, 052218 (2019) [6] O. Descalzi and H.R. Brand, Chaos 30, 043119 (2020) [7] O. Descalzi, C. Cartes, and H.R. Brand, Phys. Rev. E 105, L062201 (2022) [8] O. Descalzi, C. Cartes, and H.R. Brand, Phys. Rev. E 103, 042215 (2021) [9] O. Descalzi, C. Cartes, and H.R. Brand, Phys. Rev. E 103, L050201 (2021)

Two-dimensional soliton equations such as the KP equation and the Davey-Stewartson (DS) system have broad classes of exact solutions. It is important to understand the soliton interaction in the two- dimensional soliton equations because interesting phenomena not seen in one-dimensional cases can be observed. In this talk, I will show some recent results about exact solutions and soliton interactions of DS2 equation.

Rogue waves are known to be much more common on jet currents, Agulhas current is a notorious example [1]. A possible explanation was put forward in [2]: for the waves trapped on a current robust long-lived envelope solitary waves become possible, such wave patterns cannot exist in the absence of the current. The characteristic nonlinear scales in the real ocean suggest that the traditional ray theory cannot describe the evolution of nonlinear trapped waves. In order to fill this gap in the mathematical theory, the asymptotic weakly nonlinear mode approach was developed in [2,3]. The modulational instability and envelope solitons and breathers it generates within the frameworks of the nonlinear Schrodinger equation (NLSE) are recognized prototypes of oceanic rogue waves. Ongoing intense research effort is dedicated to the investigation of complex interactions between envelope solitons and breathers within the integrable NLSE and its non-integrable generalizations, including the strongly nonlinear frameworks. In this work we investigate interactions between envelope solitons of essentially nonlinear trapped waves by means of the direct numerical simulation of the Euler equations. The solitary waves remain localized in both horizontal directions. We demonstrate the high efficiency of the developed analytic nonlinear mode theory for description of the long-lived solitary patterns up to remarkably steep waves, robustness of the solitons in the course of interactions, and the possibility of extreme wave generation as a result of solitons’ collisions. These robust solitary waves obtained from the Euler equations without weak nonlinearity assumptions are viewed as a plausible model of rogue waves on jet currents. References: 1. Mallory J.K Abnormal waves on the South East coast of South Africa. The International hydrographic review, 51, 99-129, 1974. 2. Shrira V.I. and Slunyaev A.V. Nonlinear dynamics of trapped waves on jet currents and rogue waves, Phys. Review E, 89, 041002(R), 2014. 3. Shrira V.I. and Slunyaev A.V. Trapped waves on jet currents: asymptotic modal approach, J. Fluid Mech., 738, 65-104, 2014

Fluid motion under water waves includes an Eulerian return flow in the direction opposite to wave-packet propagation that is of importance for accurately modelling the transport of tracers in the ocean like sediments, plastic pollution, oil, etc. The return flow is related to the derivative in space of the zero-harmonic component of the velocity potential $\phi_0$. It turns out that this contribution is not consistently taken into account in some derivations of the high-order nonlinear Schrödinger equation (HONLS) using the multiscale development at arbitrary depth, which therefore do not correctly reproduce experimental results in the deep-water limit. We show how to formulate a Neumann problem for $\phi_0$ which can be solved at fourth order in steepness for arbitrary depth. The derivative of such term is thus included in the HONLS in both cases of propagation in time and in space, the latter being relevant in water tank experiments. We compare the results of the simulations obtained using our model to those obtained with previously published fourth-order models. While our model is equivalent to alternative formulations in intermediate water, it is more accurate in deep water. Therefore, it provides an accurate description of the evolution of narrow-banded wave packets at fourth-order in steepness relevant for depths ranging from intermediate to deep water.

Modulation instability is a fundamental physical effect that is in the roots of many natural phenomena as well as laboratory observations. These include morphogenesis, pattern formation, rogue waves, supercontinuum generation in optical fibres, etc. MI is tightly related to the Fermi-Pasta-Ulam recurrence. MI is known at least since 1950-ies. Interest to this universal effect increases every year. Despite the relative simplicity of its mathematical treatment, there are still some unresolved issues that require further analysis. One of the features of the MI that has been overlooked in previous years of studies will be considered in this talk.

A comparison of empirical data of waves and turbulence based on common multi-point statistics is presented. The complexity of both systems is described by special scale-dependent Fokker-Planck equations derived from the data. This stochastic description allows to define entropy values for all wave events and wind fluctuations. Negative entropy is associated with extreme events; moreover, the statistics of the entropy values follow the fluctuation theorems. Thus, negative entropy events and positive entropy events balance each other out. Finally, the stochastic approach for these complex systems allows the definition of instantons and the search for the rarest events, even those that are not measured. For waves, we find the "three sisters" as a very rare event.

Tropical cylcones have produced some of the largest waves observed on the ocean surface due to the extremely high winds. The largest waves occur in the right front quadrant of these storms where there are long durations and consistent alignment of wind and wave directions. To accurately predict the magnitude of these extreme waves the wind input and dissipation must be specified, however there is very limited field data in these conditions for obvious reasons. Hence the SUrge-STructure-Atmosphere INteraction (SUSTAIN) facility at the University of Miami was constructed to examine air-sea processes in extreme wind and waves. The 23-m long and 6-m wide wave tank in SUSTAIN is capable of generating winds in excess of 90 m s-1 (at 10-m reference height), which correspond to those found in very strong Category 5 hurricanes. The tank features a multidirectional wave maker to allow generating swell on top of young windsea. The air-sea momentum transfer has been measured in SUSTAIN in winds up to 40 m s-1 using eddy covariance method, and up to 50 m s-1 using a momentum budget method, both established in the existing literature. We observed a modified drag saturation in high winds by analyzing the relationships between air-sea momentum flux and wind waves, and mechanically generated longer waves. We have measured the distribution of the wind-flow separation with phase over the longer waves and the relationship that they have on short wave growth. It was found that the re-attachment point of the seperated wind flow has an important impact on momentum transfer that should be considered in wave modeling. For future work the newly observed relationships will be used to formulate a comprehensive wave growth source function for use in next-generation coupled atmosphere-wave-ocean prediction models. The function will be tested in numerical simulations of realistic major hurricanes.

The nonlinear Schrodinger equation permits rogue wave solutions that can explain sudden extreme formations in nature. We generalize the nonlinear Schrodinger equation to investigate the influence of various higher-order effects on the rogue wave solutions. In the presence of perturbations, the rogue wave solutions undergo various intermediary process of transformation which eventually becomes a collection of solitons. These observations reveal the process that creates a large number of solitons in the process of modulation instability-induced supercontinuum generation in optical fibers.

Light propagation in space in nonlinear media manifests complex dynamics and rogue waves. In this context, I discuss how the engineering of spatial beam propagation in photorefractive crystals enables an effective and versatile platform to study experimentally non-dissipative nonlinear wave dynamics. Starting from the observation of the Fermi-Pasta-Ulam-Tsingou recurrence and the box-problem dynamics for spatial optical waves, I present experimental results on extreme waves in three dimensions and show soliton collision, fusion, and chaos. I show how any ensemble of nonlinear waves can act as analog computing systems by exploiting neuromorphic computing paradigms such as the extreme learning machine and reservoir computing. Finally, I move to optical waves in biological matter and present the observation of optical rogue waves in spheroids of tumor cells, a finding that opens the way for using extreme waves for biomedical applications.

Extreme value stastistics is a well established mathematical theory for the estimation of the frequency and severity of extreme events. We discuss recent progress in applications to long-range correlated time series and to non-stationary time series. We illustrate these concepts using daily precipitation data and water levels of the river Elbe, with the particular focus on whether changes due to climate change can be detected with statistical significance.

Amin Chabchoub1,2 • 1Disaster Prevention Research Institute, Kyoto University, Japan • 2School of Civil Engineering, The University of Sydney, Australia Rogue waves are known to appear in a variety of nonlinear dispersive media. In the ocean, such large-amplitude waves can emerge either in offshore or nearshore zones. The formation of large-amplitude waves in deep-water has been well-documented and intensively studied in the last decades. That said, wave processes near the coasts are known to be dominated by wave reflection, which have a significant influence on the incident waves’ dynamics. This experimental study aims at improving understanding of severe wave focusing and statistics in a counter-propagating wave environment. To address this, several regular and irregular wave trains have been generated and tracked in a water wave tank while varying the beach inclination to allow several reflection conditions. The collected data confirms the variation of the role of nonlinearity in the focusing process when considering such cross-wave conditions. Numerical simulations, based on weakly nonlinear and coupled wave evolution models, are in very good agreement with the laboratory experiments.

Optical rogue waves have been extensively studied in the past two decades. However, observations of multidimensional extreme wave events remain surprisingly scarce. In this talk, we present the experimental demonstration of the spontaneous generation of spatially localized two-dimensional beams in a quadratic nonlinear crystal, which are composed by twin components at the fundamental and the second-harmonic frequencies. These localized spots of light emerge from a wide background beam, and eventually disappear as the laser beam intensity is progressively increased. We also experimentally studied the conditions for the wave-breaking of spatially extended optical beams in the process of second harmonic generation. Whenever the pump energy of the picosecond-long fundamental beam reaches a critical level, the beam shape at the second harmonic in a KTP crystal breaks into small filaments. These filaments exhibit extreme local intensity peaks, and their statistical distribution can be modified by the input energy of the fundamental beam. Moreover, by analyzing similar wave-breaking dynamics in a PPLN crystal in the presence of a higher nonlinear quadratic response, we observe that the spatial beam breaking may even gradually vanish as the laser intensity grows larger, leading to a spatial reshaping into a smooth and wider beam, accompanied by a substantial broadening of its temporal spectrum.

Temporal optical solitons, conservative and dissipative, have been dominated by pulses affected by second order dispersion (group velocity dispersion, GVD) and nonlinearities, in particular the Kerr effect. Nevertheless, the introduction of fourth order dispersion (4OD), with or without GVD, may also permit the existence of solitons. The result is known from the 1990s [1,2] but gained new momentum with the observation of laser pulses in cavities dominated by 4OD, that have been named as pure quartic soliton laser [3]. The topic of quartic solitons has also been recently explored in models for microcavities for the generation of frequency combs [4,5] . The conservative quartic solitons are already reasonably studied, including one hump pulses and one hump with side lobes and presenting energy that scales inversely with the third power of temporal width [6.7]. However, the quartic dissipative solitons, to which type the quartic soliton of lasers and microcavities belong and for which we anticipate a richer behavior, are still less explored. Here we study some of the models associated with mode-locked lasers in the presence of 4OD, searching for regions of existence of plain or oscillatory solitons, investigating their characteristics and novel features. [1] M. Karlsson and A. Höök, Opt. Commun. 104, 303 (1994). [2] N. Akhmediev, A. Buryak, and M. Karlsson, Opt. Commun. 110, 540 (1994) [3] J. Runge, D. Hudson, K. Tam, C. Sterke, and A. Blanco-Redondo, Nat. Photonics 14, 492–497 (2020) [4] H. Taheri and A. B. Matsko, Opt. Lett. 44, 3086 (2019) [5] S. Yao, K. Liu, and C. Yang, Opt. Express 29, 8312 (2021) [6] K. K. K. Tam, T. J. Alexander, A. Blanco-Redondo, and C. M. de Sterke, Opt. Lett. 44, 3306 (2019). [7] K. K. K. Tam, T. J. Alexander, A. Blanco-Redondo, and C. M. de Sterke, Phys. Rev. A 101, 043822 (2020).

When making random sea states, for practical reasons we often remove the high frequency tail which would normally be present in a wave spectrum from numerical simulations or are unable to create such a broad range in the laboratory. The non-linear physics present will quickly reestablish this tail. This talk examines whether suppressing part of the spectrum can lead to a higher number of rogue waves. We investigate this numerically using different models analysing different scenarios. We find that this can have a surprisingly large effect. We attribute this to a lack of equilibrium in the curtailed spectrum although the full physics of what is going on remains unclear.

Nonlinear Fourier transforms (NFTs), which are traditionally also known as scattering transforms, offer unique capabilities for the analysis of nonlinear waves in engineering and the applied sciences. For example, they can extract hidden soliton components within a signal, or indicate whether it is close to being potentially modulationally unstable. The interest in applying NFTs to real-world time series, including rogue waves, has been increasing in the last years. However, studies considering real-world rogue waves so far concentrated on individual events. In contrast, we recently analyzed large datasets of rogue and non-rogue waves measured in the sea. Such analyses require reliable numerical implementations of the NFT, which ideally also indicate numerical problems to the user. We discuss recent progress in this challenging area and highlight open issues.

When shoaling waves approach a plateau atop a shoal, they reach a maximum likelihood of becoming rogue waves. This physical phenomena seems unconnected with engineering properties of wave mechanics at first. However, a closer look shows that the mean water level change due to wave shoaling controls the rate of increase in the rogue wave occurrence, with the former being a function of the slope magnitude. We extend this finding by investigating the mean water level due to interactions with opposing and following currents over a flat bottom in deep water, providing a possible explanation for a quasi-symmetrical amplification as a function of (U/c_{g})^{2}. Finally, we also show how to obtain an effective solution for the nonlinear shoaling coefficient over steep slopes from the mean water level instead of the energy flux conservation.

Wave hydrodynamics at high latitudes in the Arctic and Southern Oceans is affected by the presence of sea ice where feedback mechanisms affect both sea ice and water wave properties. The understanding of wave propagation in these areas is therefore essential to model this key component of the Earth climate system. The most significant physical impact of sea ice is the dampening of the waves during a long-term evolution. The nonlinear Schrödinger equation (NLS) is a fundamental weakly nonlinear model for ocean waves, which describes the wave evolution in the third-order of approximation and the growth-decay cycles of unstable modes, also known as modulational instability (MI). A dissipative NLS (d-NLS) is used to model the evolution of unstable waves with a frequency-dependent dissipation, as quantified and determined from in-situ measurements, to simulate a viscous sea ice layer on the water surface. The numerical simulations show that the focusing recurrence in sea ice is preserved, also in the presence of sea ice dissipation, however, in its phase-shifted form and with the increase of recurrence frequency for higher dissipation. We also show that the frequency-dependent dissipation breaks the symmetry between the first order left and right sideband.

The emergence of extreme water waves statistics in laboratory experiments has been numerously experimentally and theoretically studied in the past years [1, 2, 3]. It has been demonstrated that for unidirectional irregular sea states in deep water, the Gaussian free surface elevation series imposed by the wave-maker are strongly affected by the nonlinear waves propagation , leading to extreme non Gaussian height and crest distributions as the distance from the wave-maker increases. The phenomenon is usually explained by nonlinear interactions among the waves fields components, generating modulational instabilities that affect the waves envelop and increase the number of extreme events. In the present work, we intend to clearly identify those nonlinear waves interactions in a statistically reliable experimental data-set. We rely on 30 realizations of a narrow banded unidirectonal sea state generated in the Ecole Centrale de Nantes 150m-long towing tank. The analysis focus on the high order spectra (bi- and tri- spectra) computed at several positions of the domain. The later quantities provide a frequency map of the 3 and 4 waves interactions occurring in the wave fields. A focus is also done on the groupiness of the free surface elevation that characterize the modulation of the envelop. The results demonstrate a strong link between the extreme crest distributions, the modulations of the envelop and the amount of 4 waves interactions. [1] Onorato, M., Waseda, T., Toffoli, A., Cavaleri, L., Gramstad, O., Janssen, P. A. E. M., ... & Trulsen, K. (2009). Statistical properties of directional ocean waves: the role of the modulational instability in the formation of extreme events. Physical review letters, 102(11), 114502. [2] Tayfun, M. A., & Fedele, F. (2007). Wave-height distributions and nonlinear effects. Ocean engineering, 34(11-12), 1631-1649. [3] Fedele, F. (2015). On the kurtosis of deep-water gravity waves. Journal of Fluid Mechanics, 782, 25-36.

In this work, two new mathematical frameworks are presented for the description of a fluid system containing extremely large waves. The first deals with a mathematical model of Coupled Envelope Evolution Equations newly derived by Li (2023, J. Fluid. Mech., Vol. 960) for the long-term propagation of nonlinear surface gravity waves. This model is as accurate as the High-Order Spectral method and shares a few advantaged features as a Nonlinear Schrödinger (NLS) equation-based model for water waves. It offers a viable path towards a better understanding of the roles of extremely large waves in large-scale oceanic processes. The second framework presents a NLS equation-based model for deepwater waves in the presence of a background flow. The flow is assumed to be horizontally oriented, and the magnitude of its velocity varies slowly in space. A new interpretation of the roles of Stokes drift, an Eulerian return flow, and a background vertically-sheared current have been provided in the modulational instability (MI) of Stokes waves. The results in particular show that a current opposing a long-crested wave group can enhance the oblique MI while suppressing completely the MI which arises from sideband waves in the directions parallel to the wave group. This demonstrates clearly the roles of a background flow on rogue waves, owing to that the MI has been well recognized as a plausible cause to their generation.

Optical dynamic holography, which deals with nonlinear coupling of laser beams in dynamic nonlinear-optical media, currently covers a wide range of advanced applications [1]. It is close related to modern technology for creating and processing optical images, since each laser beam can be considered as an optical wavefront carrying information. At present, the conception of describing dynamic holography as a network for optical information processing and an optical computer is confirmed in studies devoted to holographic artificial intelligence machines (AIM) [2]. The mathematical modeling reduces in the simplest case of two (or four) interacting beams to a nonlinear system for three equations: coupled wave equations, including the coupling function as the amplitude of the dynamic grating, and the evolutionary equation for this function, which contains its temporal relaxation. Furthermore, such a system reduces to a single complex Ginzburg-Landau equation (CGLE) [3-4]. Solutions of the type of longwise dissipative solitons (LDS) are obtained for this equation, which describe the spatial localization for the light intensity envelope in the interference pattern along the longitudinal direction of the medium thickness. Our present work is devoted to the study of non-stationary dynamics of wave-coupling in dynamic holographic systems. Nonlinear-optical properties of liquid crystal cells are used as a real model of a dynamic medium. We defined the dependence of the output intensities of laser beams on the time parameters of the nonlinear system, such as the duration of laser pulses, the frequency in pulse trains, and the recording and relaxation time constants of the dynamic grating arising in the nonlinear medium. We have found that a sharp peak in the output light intensity occurs in a narrow region of the ratio of these time values. This intensity peak can be more increased by driven a phase noise in the input pulses. The novelty of this work lies in new approach to the study of methods and effects of manipulating the parameters of laser pulses during the coherent interaction of waves in nonlinear media. It consists in finding an experimental implementation of such exotic solutions as "extreme events" and "rogue-waves". From the point of view of practical applications, the results obtained have prospects for the development of new effective laser scanning technologies, namely holographic deflectors and modulators based on liquid crystals; as well as for the development of new frequency coding with information security, which include both software based on mathematical algorithms and hardware security devices. [1] J. Sheridan, et.al., Roadmap on holography, J. Opt., 22, 123002, 2020. [2] F. Laporte, et.al., Simulating self-learing in photorefractive optical reservoir computers, Scientific Reports, 11:2701, 2021. [3] S. Bugaychuk, R. Conte, Ginzburg-Landau equation for dynamical four -wave mixing in gain nonlinear media with relaxation, Phys. Rev. E, 80, 066603, 2009. [4] S. Bugaychuk, R. Conte, Nonlinear amplification of coherent waves in media with soliton-type refractive index pattern, Phys. Rev. E, 86, 026603, 2012.

The puzzle of extreme waves attracted the attention of scientists at the beginning of this century [1]. The 2 most important feature of such waves is characterized as follows – The waves that appear from nowhere and disappear without a trace [2]. Thus, according to [2] the properties of these mysterious waves resemble the properties virtual particles. Books were written, to one degree or another, devoted to them [3,4]. To describe them mathematical, equations and solutions were used which described huge waves with continuous and smooth profiles. At the same time, there is experimental and numerical data which shows that sometimes the profiles of these waves can lose continuity, collapse, and the waves themselves are transformed into some clots of matter. Thus, strongly nonlinear, multivalued processes are observed. During these processes, extreme, multivalued waves arise. They may be described by scalar fields and corresponding nonlinear equations. It has wave solutions corresponding to the Leonhard Euler figures [5] at singular points and in their vicinity. Pierre Laplace connected Euler's results with extreme wave processes [6]. However, this relationship was not further explored until the early 21st century. In particular, the similarity was not emphasized between the mathematical description of the Euler figures and extreme waves. Extreme waves can change in time and space so that they break up into waves and particles, or into a chain of isolated particle-waves. The idea is consistently carried out in this report that the nonlinear theory opens up the possibility of describing elementary particles not as harmonic functions but as extreme particle-waves. References. [1] Smith С. B. Extreme Waves. Joseph Henry Press, 2006; [2] Akhmediev, N., Ankiewicz, A., Taki, M. Waves that appear from nowhere and disappear without a trace. Phys. Lett. A, 373, 675-678, 2009; [3] Kharif, C., Pelinovsky, E., Slunyaev, A. Rogue Waves in the Ocean. Springer, 2009; [4] Pelinovsky, E..Kharif, C. Extreme Ocean Waves. Springer, 2015; [5] Euler L., Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, 1744, Leonhardi Euleri Opera Omnia Ser. I, vol. 14. [6] Laplace P. S. Œuvres complètes de Laplace, volume 4. Gauthier-Villars, 1880. Volume 4 of Laplace’s Complete Works.

The stability of surface waves in deep water has traditionally been approached via linearisation, starting with various model equations, such as the NLS equation or (without restriction to narrow bandwidth) the Zakharov equation. Owing to the form of the deep-water dispersion relation, energy exchange among surface water waves - and thus the possibility of instability - occurs when four or more wavenumbers are involved. We shall employ the compact, Hamiltonian description of four-wave interaction due to Krasitskii. In the usual mathematical sense, instability requires starting from a solution to a set of equations, and describes the evolution of perturbations to that solution. A handful of such explicit solutions - the monochromatic (Stokes’) wave and bichromatic wave train - therefore form the backbone of classical instability results. The simplest of these is the Benjamin-Feir (BFI) or modulational instability. In all cases the entire nonlinear dynamics can be described by the level lines of a certain planar Hamiltonian function. This enables the explicit and straightforward computation of steady state solutions, which have been the object of much recent interest. It is found that the classical linear stability criteria are algebraically identical with the criteria for orbital instability of associated nullclines of the system. Moreover, certain heteroclinic orbits can be identified as new discrete-mode breather solutions. The reduction to planar dynamics provides immediate and novel insight into these cases of isolated wave-wave interaction.

Nonlinear wave interactions leading to extreme waves have received a lot of attention in the deep-water depth regime. In contrast, far less attention has been given to waves evolving in intermediate and shallow water depths. This study addresses this shortcoming in two ways. First, an extensive database of high-quality field measurements is analysed in conjunction with experimental simulations to establish the appropriate wave distributions. These relate to long records of random, short-crested seas on flat bed bathymetries. Second, a very extensive laboratory study on uniform sea-bed slopes with varying inclinations is presented. Very fine spatial resolution and a novel sea-state generation methodology is used to investigate the evolution of extreme waves as they propagate shoreward. The results arising in both the flat bed and sloping bed studies are used to study extreme wave statistics and the competing effects of nonlinearity and wave breaking. These are subsequently used to derive a predictive model for crest height statistics.