Extreme Waves

Theories on the bosonic nature of dark matter are a promising alternative to the cold dark matter model. Here we consider a dark matter halo in the state of a Bose-Einstein condensate, subject to the gravitation of a black hole. In the low energy limit, we bring together the general relativity in the Schwarzschild metric and the quantum description of the Bose-Einstein condensate. The model is solvable in the Fermi normal coordinates with the so-called highly nonlocal approximation and describes tidal deformations in the condensate wave function. The black hole deforms the localized condensate until the attraction of the compact object overcomes self-gravitation and destabilizes the solitonic dark matter. Moreover, the model can be implemented as a gravitational analog in the laboratory; the time-dependent potential generated by the galactic black hole can be mimicked by an optical trap acting on a conventional condensate.

Accurate short-term prediction of phase-resolved water wave conditions is crucial for decision-making in ocean engineering. However, the initialization of remote-sensing-based wave prediction models first requires a reconstruction of wave surfaces from sparse measurements like radar. Existing reconstruction methods either rely on computationally intensive optimization procedures or simplistic modeling assumptions that compromise real-time capability or accuracy of the entire prediction process. We therefore address these issues by proposing a novel approach for phase-resolved wave surface reconstruction using neural networks based on the U-Net and Fourier neural operator (FNO) architectures. Our approach utilizes synthetic yet highly realistic training data on uniform one-dimensional grids, that is generated by the high-order spectral method for wave simulation and a geometric radar modeling approach. The investigation reveals that both models deliver accurate wave reconstruction results and show good generalization for different sea states when trained with spatio-temporal radar data containing multiple historic radar snapshots in each input. Notably, the FNO-based network performs better in handling the data structure imposed by wave physics due to its global approach to learn the mapping between input and desired output in Fourier space.

We present hydrodynamics experiments investigating the interaction between two stream flows of soliton gas, known as monochromatic soliton gases, with identical mean amplitudes but opposite mean velocities. Our study involves recording the space-time evolution of these two soliton gases in a 148-meter long water tank, where the dynamics of the soliton gases, at the leading order, is described by the focusing one-dimensional nonlinear Schrodinger equation. To change the interaction strength, we varied the relative initial velocity of the two gases. We measured the changes in macroscopic density and velocity resulting from the interaction. Our experimental results are in good quantitative agreement with the predictions of the spectral kinetic theory of soliton gas. We should note that these results hold true despite the presence of perturbative higher-order effects that break the integrability of the wave dynamics.

The localized running waves in the discrete Josephson transmission lines (JTL), constructed from Josephson junctions (JJ) and capacitors, are analytically studied. The quasicontinuum approximation reduces calculation of the running wave properties to the problem of equilibrium of an elastic rod in the potential field. Making additional approximation, the problem is reduced to the motion of the fictitious Newtonian particle in the potential well. It is shown that there exist running waves in the form of supersonic kinks and solitons and their velocities and profiles are calculated. It is shown that the nonstationary smooth waves which are small perturbations on the homogeneous nonzero background are described by Korteweg–de Vries equation and those on zero background by modified Korteweg–de Vries equation. The effect of dissipation on the running waves in JTL is also studied and it is found that in the presence of the resistors, shunting the JJ and/or in series with the ground capacitors, the only possible stationary running waves are the shock waves, whose profiles are also found. Finally, in the framework of Stocks expansion, the nonlinear dispersion and modulation stability in the discrete JTL are studied.

Recently, nonlinear Fourier analysis based on the Korteweg–De Vries (KdV) equation has been applied to a large number of rogue and non-rogue waves measured at a shallow water site in the southern North sea (Teutsch et al., NHESS, 2023). Strongly outstanding solitons in the nonlinear spectrum have been found to indicate rogue waves with high probability. Under ideal uni-directional propagation according to the KdV equation, the nonlinear soliton spectrum would stay constant and strongly outstanding solitons might provide a useful predictor for certain rogue waves. However, it is not clear how the nonlinear soliton spectrum changes under propagation in the presence of directional spread. In this study, we therefore use a High-Order spectral method (HOS) to simulate the two-dimentional propagation of nonlinear wavefields with varying directional spreads in shallow water. Our goals are a) to investigate the application of the NFT in directional shallow water sea states, and b) to investigate how strongly the nonlinear soliton spectrum changes for different directional spreads and propagation distances.

Previous studies pointed out that the occurrence of the extreme wave height is significantly related to the quasi-resonant four-wave interaction in the modulated waves. From the numerical-experimental study over an uneven bottom, the nonlinear effect caused by the bathymetry change also contributes to the occurrence of extreme events in the unidirectional waves. To comprehensively analyze the two-dimensional wave field, this study develops an evolution model for a directional random wavefield based on the depth-modified Nonlinear Schrödinger equation, which considers the nonlinear resonant interactions, wave refraction and the wave shoaling the shallow water. Through Monte Carlo simulation, we discuss the directional effect on the four-wave interaction in the wave train and the maximum wave height distribution from deep to shallow water with a slow varying slope.

Dissipative quartic solitons In mode-locked lasers have garnered attention for their energy-width scaling which allows the generation of ultrashort pulses with high energies. In this work, we numerically studied dissipative quartic solitons in a distributed model for mode-locked lasers in the presence of both positive and negative fourth-order dispersion (4OD). We explored the impact that laser gain and loss as well as the spectral filtering may have on the energy-width relation of the output pulses, finding that the most energetic and narrowest solutions occur for negative 4OD, with the energy having an inverse cubic dependence with the width in most cases, changing however with the sign of 4OD. Our simulations showed that the spectral filtering has the biggest contribution in the generation of ultra-short highly energetic pulses, with pulses having widths as low as 39 fs and energies as high as 392 nJ .

We present two distinct approaches to estimating the exponent related to the power-law decaying distribution of optical rogue waves observed in supercontinuum generated in a single-mode fiber. That is, we answer the question of how rogue these waves are. The first is a generalization of the well-known Hill estimator, and the second relies on estimating all parameters of a multi-parameter model. We show that the model shows a good correspondence with experimental data, and that the two estimating approaches provide consistent results, which are significantly more accurate than those obtained with earlier estimation methods. Furthermore, alternative visualization through the tail function revealed the presence of pump depletion as well as detector saturation leading to the breakdown of power-law behavior for the largest observations. We characterized this breakdown via a combination of an exponential and a generalized Pareto distribution. Additionally, we have uncovered a weak memory effect in the data, which can be attributed to changes in the refractive index in the single-mode fiber.

We have conducted a theoretical investigation on the modulational instability (MI) phenomenon in birefringent optical media with pure quartic dispersion and weak Kerr nonlocal nonlinearity. Our findings suggest that nonlocality causes instability regions to expand, as confirmed by direct numerical simulations that show the emergence of Akhmediev breathers (ABs) in the total energy context. Moreover, the balanced competition between nonlocality and other nonlinear and dispersive effects provides an opportunity to generate long-lived structures, which enhances our understanding of soliton dynamics in pure-quartic dispersive optical systems and creates new avenues for research in fields related to nonlinear optics and lasers.

Statistical properties of soliton fields during maximal focusing Authors: T.V.Tarasova, A.V.Slunyaev For the efficient description of the soliton gas dynamics kinetic equations were derived [1,2]. They characterize the transport of the soliton spectral density, but due to the violation of the wave linear superposition principle, do not provide information about the wave solution itself (which can be water surface displacement, intensity of electromagnetic fields, etc.). In particular, the questions about the probability distribution for wave amplitudes or about the values of the wave field statistical moments remain unanswered. Multisoliton solutions, which can be formally written in a closed form using the Inverse Scattering Transform or related methods for integrable equations, are very cumbersome, what makes their analytical and even numerical analysis difficult. The direct numerical simulation of evolution equations is commonly used to study the soliton gas evolution, which also becomes complicated in the case of a dense gas (i.e. when many solitons interact simultaneously). The focus of this study is made on the dynamics of soliton interactions governed by the classic Korteweg -- de Vries (KdV) equation. The use of an ultra-high-precision procedure based on the Darboux transformation made it possible to compute the exact $N$-soliton solutions $u_N$ when $N$ is large [3] and to calculate their statistical moments $\mu_n (t) = \int_{-\infty}^{+\infty} {u_N^n(x,t) dx}$, $n \in \mathbb{N}$, with high accuracy. In this work we present a general idea that dense ensembles of KdV-type solitons of the same sign can be considered as strongly-nonlinear / small-dispersion wave states, what allows to express the statistical moments in terms of the spectral parameters of the associated scattering problem analytically [4]. A particular case when dense soliton states can occur is synchronous multisoliton collisions, for which $u_N(-x,-t)=u_N(x,t)$ holds. This symmetry condition should correspond in some sense to the maximal focusing of solitons at $t = 0$ at the point $x = 0$. The soliton amplitudes are set decaying exponentially, so that they form a geometric series with the ratio $d>1$, $A_j = 1/d^{j-1}$, $j=1,...,N$. Time dependences of statistical moments are investigated for many-soliton solutions. It is shown that during the interaction of solitons of the same sign the wave field is effectively smoothed out. When $d$ is sufficiently close to 1, and $N$ is large, the statistical moments remain approximately constant within long time spans, when the solitons are located most densely. This quasi-stationary state is characterized by greatly reduced statistical moments and by the density of solitons close to some critical value. This state may be treated as the small-dispersion limit, what makes it possible to analytically estimate all high-order statistical moments. While the focus of the study is made on the Korteweg--de Vries equation and its modified version, a much broader applicability of the results to equations that support soliton-type solutions is discussed. \textbf{References} \myitem [1] V. E. Zakharov, ``Kinetic equation for solitons,'' JETP 33, 538-541 (1971). \myitem [2] G. A. El and A. M. Kamchatnov, ``Kinetic equation for a dense soliton gas'', Phys. Rev. Lett. 95, 204101 (2005). \myitem [3] T.V. Tarasova, A.V. Slunyaev, Properties of synchronous collisions of solitons in the Korteweg -- de Vries equation. Communications in Nonlinear Science and Numerical Simulation 118, 107048 (2022). doi: 10.1016/j.cnsns.2022.107048 \myitem [4] A.V. Slunyaev, T.V. Tarasova, Statistical properties of extreme soliton collisions. Chaos 32, 101102 (2022). doi: 10.1063/5.0120404

Explicit rational solutions of self-focusing nonlinear Schrödinger equation from first order to fourth order have been studied by [1]. In this study, we numerically solve discrete nonlinear Schrödinger equation under periodic boundary conditions. We have properly chosen first and higher-order solutions as an initial condition. We show a relationship between the increasing order of rational order and the amplitude of rogue waves. Furthermore, we predict that long-living rogue waves occur when higher-order solutions are considered as the initial condition. References: [1] N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, «Rogue waves and rational solutions of the nonlinear Schrödinger equation,» Physical Review E, 80, 026601, 2009.