
Helmut Brand

09:00  09:15

Sander Wahls
(Karlsruhe Institute of Technology)
Realworld nonlinear Fourier analysis: Numerical Pitfalls
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 realworld time series, including rogue waves, has been increasing in the last years. However, studies considering realworld rogue waves so far concentrated on individual events. In contrast, we recently analyzed large datasets of rogue and nonrogue 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.

09:15  09:30

Saulo Mendes
(University of Geneva)
Energetic implications of mean water level for rogue wave occurrence
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 quasisymmetrical 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.

09:30  09:45

Alberto Alberello
(University of East Anglia)
The dynamics of unstable waves in sea ice
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 longterm evolution. The nonlinear Schrödinger equation (NLS) is a fundamental weakly nonlinear model for ocean waves, which describes the wave evolution in the thirdorder of approximation and the growthdecay cycles of unstable modes, also known as modulational instability (MI).
A dissipative NLS (dNLS) is used to model the evolution of unstable waves with a frequencydependent dissipation, as quantified and determined from insitu 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 phaseshifted form and with the increase of recurrence frequency for higher dissipation.
We also show that the frequencydependent dissipation breaks the symmetry between the first order left and right sideband.

09:45  10:00

Maxime Canard
(Ecole Centrale Nantes)
Characterization of nonlinear interactions leading to extreme waves statistics: study of a unidirectional experimental sea state
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 wavemaker are strongly affected by the nonlinear waves propagation , leading to extreme non Gaussian height and crest distributions as the distance from the wavemaker 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 dataset. We rely on 30 realizations of a narrow banded unidirectonal sea state generated in the Ecole Centrale de Nantes 150mlong 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). Waveheight distributions and nonlinear effects. Ocean engineering, 34(1112), 16311649.
[3] Fedele, F. (2015). On the kurtosis of deepwater gravity waves. Journal of Fluid Mechanics, 782, 2536.

10:00  10:30

Yan Li
(University of Bergen)
Mathematical framework for extremely large waves and their interplay with a background flow
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 longterm propagation of nonlinear surface gravity waves. This model is as accurate as the HighOrder Spectral method and shares a few advantaged features as a Nonlinear Schrödinger (NLS) equationbased model for water waves. It offers a viable path towards a better understanding of the roles of extremely large waves in largescale oceanic processes. The second framework presents a NLS equationbased 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 verticallysheared current have been provided in the modulational instability (MI) of Stokes waves. The results in particular show that a current opposing a longcrested 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.

10:30  11:00

Coffee break

11:00  11:15

Svitlana Bugaychuk
(National Academy of Science Ukraine)
Extreme phenomena of coupled light pulses arisen in dynamic holographic systems
Optical dynamic holography, which deals with nonlinear coupling of laser beams in dynamic nonlinearoptical 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 GinzburgLandau equation (CGLE) [34].
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 nonstationary dynamics of wavecoupling in dynamic holographic systems. Nonlinearoptical 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 "roguewaves". 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 selflearing in photorefractive optical reservoir computers, Scientific Reports, 11:2701, 2021.
[3] S. Bugaychuk, R. Conte, GinzburgLandau 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 solitontype refractive index pattern, Phys. Rev. E, 86, 026603, 2012.

11:15  11:30

Shamil Galiev
(University of Auckland)
Extreme Multivalued Waves in Scalar Fields
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 particlewaves. 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 particlewaves.
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, 675678, 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. GauthierVillars, 1880. Volume 4 of Laplace’s Complete Works.

11:30  11:45

Raphael Stuhlmeier
(University of Plymouth)
A discrete Hamiltonian perspective on some classical instabilities of surface water waves
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 deepwater 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 fourwave 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 BenjaminFeir (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 discretemode breather solutions. The reduction to planar dynamics provides immediate and novel insight into these cases of isolated wavewave interaction.

11:45  12:15

Ioannis Karmpadakis
(Imperial College London)
Extreme wave statistics in intermediate and shallow water depths
Nonlinear wave interactions leading to extreme waves have received a lot of attention in the deepwater 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 highquality field measurements is analysed in conjunction with experimental simulations to establish the appropriate wave distributions. These relate to long records of random, shortcrested seas on flat bed bathymetries. Second, a very extensive laboratory study on uniform seabed slopes with varying inclinations is presented. Very fine spatial resolution and a novel seastate 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.

12:15  12:30

Concluding Remarks

12:30  13:30

Lunch

13:30  14:00

Wrap up discussion and departure
