Rogue Waves Research Articles

Pattern dynamics of higher-order rogue waves in the nonlinear Schrödinger–Boussinesq equation

Pattern dynamics of higher-order rogue waves in the nonlinear Schrödinger–Boussinesq equation

An Empirical Model for Skillful Rogue Wave Forecasts

Abstract One of the leading goals of rogue wave research is to develop a robust rogue wave warning system to mitigate the danger they pose. One such system has been developed by the European Centre for Medium-Range Weather Forecasts (ECMWF), called the freak wave warning system (FWWS), based on nonlinear wave effects. The FWWS predicts maximum expected wave envelope height as a risk parameter for forecasts. Recently, a data-driven alternative has been proposed by Häfner et al., which was distilled from a neural network using wave buoy observations. However, it has yet to be evaluated by a spectral wave model for application to operational wave forecasting. The data-driven, learned model emphasizes bandwidth-controlled linear superposition as the predominant mechanism in crest-to-trough rogue wave generation, while nonlinear effects are a secondary term. The present work evaluates the performance of the empirical model using output from an ECMWF global wave hindcast. We find that the prediction models based on bandwidth effects have the highest log likelihood scores, with the empirical model outperforming all other tested models. In contrast, the expected maximum envelope wave height from the FWWS does not predict the occurrence of rogue waves. These results indicate that the empirical model with wave model input is a skillful predictor and should be considered for operational implementation to improve rogue wave forecasting. Significance Statement Rogue waves are unexpectedly large and unpredictable waves. Encounters with rogue waves can result in damage to marine vessels and offshore infrastructures. Research is devoted to developing systems to predict the risk of rogue wave events. A novel predictive model has been shown to perform well in predicting rogue wave occurrences based on wave buoy observations. This model is a symbolic expression distilled from an artificial neural network that incorporates known rogue wave dynamics and that can be evaluated alongside traditional risk estimates with minor adjustments to operational forecasting systems. This study evaluates the newly proposed empirical model in a forecasting setting. Our work demonstrates the efficacy of the empirical model for operational forecasting purposes.

Modulational instability, generation, and evolution of rogue waves in the generalized fractional nonlinear Schrödinger equations with power-law nonlinearity and rational potentials.

In this paper, we investigate several properties of the modulational instability (MI) and rogue waves (RWs) within the framework of the generalized fractional nonlinear Schrödinger (FNLS) equations with rational potentials. We derive the dispersion relation for a continuous wave (CW), elucidating the relationship between the wavenumber and the instability growth rate of the CW solution in the absence of potentials. This relationship is primarily influenced by the power parameter σ, the Lévy index α, and the nonlinear coefficient g. Our theoretical findings are corroborated by numerical simulations, which demonstrate that MI occurs in the focusing context. Furthermore, we study the RW generations in both cubic and quintic FNLS equations with two types of time-dependent rational potentials, which make both cubic and quintic NLS equations support the exact RW solutions. Specifically, we show that the introduction of these two potentials allows for the excitations of controllable RWs in the defocusing regime. When these two potentials become the time-independent cases such that the stable W-shaped solitons with non-zero backgrounds are generated in these cubic and quintic FNLS equations. Moreover, we consider the excitations of higher-order RWs and investigate the conditions necessary for their generations. Our analysis reveals the intricate interplay between the system parameters and the potential configurations, offering insights into the mechanisms that facilitate the emergence of higher-order RWs. Finally, we find the separated controllable multi-RWs in the defocusing cubic FNLS equation with time-dependent multi-potentials. This comprehensive study not only enhances our understanding of MI and RWs in the fractional nonlinear wave systems, but also paves the way for future research in related nonlinear wave phenomena.

Statistics of unidirectional wave groups with and without freak waves observed in the Norwegian Sea

The statistical properties of observed wave groups are essential for designing marine structures. However, the characteristics of group energy, length, and profiles remain unclear. This paper analyzes more than 1 million measured ocean unidirectional wave groups in deep water of the Norwegian Sea during a decade. By classifying wave groups into ordinary and extreme categories based on the presence of a freak wave, it is found that both the distributions of the non-dimensional group energy and group duration follow the Generalized extreme value functions. Moreover, the statistics of wave groups are significantly influenced by the spectral width, with wave steepness having negligible effects. The ratio of the average group duration between extreme and ordinary categories varies slightly from 1.4 to 1.8, although the energy of extreme wave groups can reach 3.0–4.5 times than that of ordinary wave groups. Furthermore, unlike the typical shape of a freak wave with a high wave crest or deep wave trough significantly larger than the surrounding waves, consecutive large waves resembling the “three sisters” are quite common in this location. However, NewWave theory generally underestimates the wave amplitudes surrounding a freak wave, leading to the predicted energy of the most likely extreme wave groups being only about 50–80% of the measured values. Finally, a new modified model is proposed to predict the average shapes of extreme wave groups. After testing numerous wave cases, the model accurately captures the mean morphology of extreme wave groups in the Norwegian Sea.

Families of exact similaritons with flexible modulations in N-coupled homogeneous and inhomogeneous systems

In this paper, a family of exact solutions to N-coupled nonlinear Schrödinger (N-CNLS) equations including bright and dark solitons, Akhmediev breathers, Kuznetsov-Ma breathers, and rogue waves are derived with the aid of a generic nonlinear wave assumption. As an application of the solutions, we present a scheme to realize amplitude coding in an identical waveguide system. Furthermore, by a general self-similar transformation method, exact controllable similaritons of N-coupled inhomogeneous NLS (N-CINLS) equations are constructed under relaxed compatibility conditions. Flexible modulations of similaritons are investigated in a typical inhomogeneous system. Based on the general self-similar transformation, we further study various and novel modulations of composite similariton of 2-CINLS regimes. The various similaritons presented here may be expected to find application in the control and transmission of nonlinear waves in realizable complex systems including those based on optical fibers and Bose-Einstein condensates.

Experimental and numerical study on focused wave generation using different energy modes

This paper reports an experimental and numerical study on the generation of freak waves related to the spatiotemporal focusing mechanism, where multiple water waves converge at a specific point in space and time to produce a large transient wave. A new energy distribution mode of constant-difference wave number (CDK) within the wave group is proposed and applied to both a physical and a numerical flume to generate focused waves, with primary attention given to the water surface elevation, underlying kinematics, and internal spectrum structure during the focused wave generation process. The classical experiment by Baldock et al. [“A laboratory study of nonlinear surface waves on water,” Philos. Trans. R. Soc., Ser. A 354, 649–676 (1996)], which used a constant-difference wave period within the wave group, is chosen for comparison. Spectral analysis shows that the CDK energy distribution mode can easily stimulate multiwave resonance in the focused wave group, where significant energy transfer from the main frequencies to higher harmonics is observed. Compared with the data from Baldock et al. [“A laboratory study of nonlinear surface waves on water,” Philos. Trans. R. Soc., Ser. A 354, 649–676 (1996)], the CDK wave group produces larger focused wave crests with higher propagation speeds and larger velocity gradients near the water surface due to strong multiwave resonance. The effects of the input spectrum, incident amplitude and initial water depth under different energy modes on focused-wave generation are analyzed. All the results indicate that the CDK energy distribution mode can produce larger focused waves compared to other energy distribution modes, which offers a novel approach for generating freak waves in the laboratory.

The Northern Cross Fast Radio Burst project

Context. Fast radio bursts (FRBs) are energetic, millisecond-duration radio pulses observed at extragalactic distances and whose origins are still a subject of heated debate. A fraction of the FRB population have shown repeating bursts, however it’s still unclear whether these represent a distinct class of sources. Aims. We investigated the bursting behaviour of FRB 20220912A, one of the most active repeating FRBs known thus far. In particular, we focused on its burst energy distribution, linked to the source energetics, and its emission spectrum, with the latter directly related to the underlying emission mechanism. Methods. We monitored FRB 20220912A at 408 MHz with the Northern Cross radio telescope and at 1.4 GHz using the 32-m Medicina Grueff radio telescope. Additionally, we conducted 1.2 GHz observations taken with the upgraded Giant Meter Wave Radio Telescope (uGMRT) searching for a persistent radio source coincident with FRB 20220912A, and included high energy observations in the 0.3–10 keV, 0.4–100 MeV and 0.03–30 GeV energy range. Results. We report 16 new bursts from FRB 20220912A at 408 MHz during the period between October 16th 2022 and December 31st 2023. Their cumulative spectral energy distribution follows a power law with slope αE = −1.3 ± 0.2 and we measured a repetition rate of 0.19 ± 0.03 hr−1 for bursts having a fluence of ℱ ≥ 17 Jy ms. Furthermore, we report no detections at 1.4 GHz for ℱ ≥ 20 Jy ms. These non-detections imply an upper limit of β < −2.3, with β being the 408 MHz – 1.4 GHz spectral index of FRB 20220912A. This is inconsistent with positive β values found for the only two known cases in which an FRB has been detected in separate spectral bands. We find that FRB 20220912A shows a decline of four orders of magnitude in its bursting activity at 1.4 GHz over a timescale of one year, while remaining active at 408 MHz. The cumulative spectral energy distribution (SED) shows a flattening for spectral energy Eν ≥ 1031 erg Hz−1, a feature seen thus far in only two hyperactive repeaters. In particular, we highlight a strong similarity between FRB 20220912A and FRB 20201124A, with respect to both the energy and repetition rate ranges. We also find a radio continuum source with 240 ± 36 μJy flux density at 1.2 GHz, centered on the FRB 20220912A coordinates. Finally, we place an upper limit on the γ to radio burst efficiency η to be η < 1.5 × 109 at 99.7% confidence level, in the 0.4–30 MeV energy range. Conclusions. The strong similarity between the cumulative energy distributions of FRB 20220912A and FRB 20201124A indicate that bursts from these sources are generated via similar emission mechanisms. Our upper limit on β suggests that the spectrum of FRB 20220912A is intrinsically narrow-band. The radio continuum source detected at 1.2 GHz is likely due to a star formation environment surrounding the FRB, given the absence of a source compact on millisecond scales brighter than 48 μJy beam−1. Finally, the upper limit on the ratio between the γ and radio burst fluence disfavours a giant flare origin for the radio bursts unlike observed for the Galactic magnetar SGR 1806-20.

Nonlinear tunneling of partially nonlocal dark-dark annular sneaker waves under a parabolic potential

Annular rogue waves have been reported, but little is known about how to excite and control them, particularly their tunneling effect. From the projecting relation between the vector partially nonlocal nonlinear Schrödinger equation possessing different transverse-directional diffractions and the parabolic potential and vector constant-coefficient nonlinear Schrödinger equation, via the Darboux method, partially nonlocal dark-dark annular sneaker wave approximate solutions are found. Two aspects of the study are carried out: (i) four excitations of partially nonlocal dark-dark annular sneaker waves including complete, delayed, valley-maintained and prohibitive excitations and (ii) nonlinear tunneling of partially nonlocal dark-dark annular sneaker waves are studied. The influence of two parameters R and l on dark-dark annular sneaker waves is also analyzed. The partially nonlocal characteristics of dark-dark annular sneaker waves passing through the nonlinear well implies that the layer of annular sneaker wave structures increases in xyz-coordinate when the value of the Hermite parameter adds. These findings will further our understanding of the partially nonlocal nonlinear wave in the disciplines of cold atom and optical communication.

Conversion mechanisms and transformed waves for the (3 + 1)-dimensional nonlinear equation

In this paper, we focus on investigating the (3 + 1)-dimensional nonlinear equation which is used to describe the propagation of waves in the shallow water. The study begins with the application of the Hirota bilinear method to obtain N-soliton solution. Building on this foundation, the research delves into the construction of first-order breather wave by imposing complex conjugate constraints on the parameters of two solitons. Further analysis of the characteristic lines of breathers leads to the derivation of conversion conditions. Under this specific condition, a series of nonlinear transformed waves are presented, including quasi-kink solitons, W-shaped kink solitons, oscillation W-shaped kink solitons, multipeaks solitons, quasi-periodic waves, and line rogue waves. Each of these transformed waves exhibits unique structural and dynamic properties, enriching the understanding of wave behavior in higher-dimensional nonlinear systems. The study also explores the nonlinear superposition mechanism between solitary wave and periodic wave. This mechanism elucidates the formation process of nonlinear waves, explaining how their locality and oscillatory characteristics emerge from the superposition of different wave components. Moreover, the geometric properties of the two characteristic lines of the waves are analyzed to understand the time-varying nature of the transformed waves. This temporal analysis is crucial for predicting the evolution and interaction of these waves over time. Finally, the research extends to the higher-order breather wave and explores the interactions among various waves. These interactions reveal the complex dynamics that may arise in the (3 + 1)-dimensional nonlinear systems and provide deeper insights into the interactions among different wave structures.

Hydrodynamic modulation instability triggered by a two-wave system.

The modulation instability (MI) is responsible for the disintegration of a regular nonlinear wave train and can lead to strong localizations in the form of rogue waves. This mechanism has been studied in a variety of nonlinear dispersive media, such as hydrodynamics, optics, plasma, mechanical systems, electric transmission lines, and Bose-Einstein condensates, while its impact on applied sciences is steadily growing. It is well-known that the classical MI dynamics can be triggered when a pair of small-amplitude sidebands are excited within a particular frequency range around the main peak frequency. That is, a three-wave system, consisting of the carrier wave together with a pair of unstable sidebands, is usually adopted to initiate the wave focusing process in a numerical or laboratory experiment. Breather solutions of the nonlinear Schrödinger equation (NLSE) revealed that MI can generate much more complex localized structures, beyond the three-wave system initialization approach or by means of a continuous spectrum. In this work, we report an experimental study for deep-water surface gravity waves asserting that a MI process can be triggered by a single unstable sideband only, and thus, initialized from a two-wave process when including the contribution of the peak frequency. The experimental data are validated against fully nonlinear hydrodynamic numerical wave tank simulations and show very good agreement. The long-term evolution of such unstable wave trains shows a distinct shift in the recurrent Fermi-Pasta-Ulam-Tsingou focusing cycles, which are captured by the NLSE and fully nonlinear hydrodynamic simulations with some distinctions.

Rogue wave patterns of two-component nonlinear Schrödinger equation coupled to the Boussinesq equation

Rogue wave patterns of two-component nonlinear Schrödinger equation coupled to the Boussinesq equation

Breather–breather interactions, rational and semi-rational solutions of an integrable spin system in optical fibers

This paper investigates an integrable spin system (or Heisenberg ferromagnet-type equation or Nurshuak-IIA equation) which is an integrable generalization of the Heisenberg ferromagnet equation that raises from magneto-optical devices. The N-fold Darboux transformation of the Nurshuak-IIA equation will be constructed. With the selection of several parameters, a variety of the solutions will be produced, including breather–breather interactions, higher-order rogue waves and semi-rational solutions.

Statistical distribution of surface elevation by using quadratic forms of normal random variables

This paper is concerned with the statistical properties of surface elevation, which is crucial for the design and operation of marine and coastal structures and is helpful to the prediction of rogue waves. A semi-analytic quadratic model is proposed to describe the statistical distribution of surface elevation by using the theory of quadratic forms of normal random variables, in which the characteristic function and cumulants are given in analytical form, and the probability density is obtained by the numerical inversion of the characteristic function using the fast Fourier transform algorithm. Compared with Monte Carlo simulations, the quadratic model presents excellent performance in describing the probability density function of surface elevation, especially at the tails, for varying degrees of steepness, bandwidth, and directional spreading in both finite and infinite water depth. The effect of these parameters that characterize sea states on the statistical properties of surface elevation is also investigated. Through a comparative study, the quadratic model is superior to previously existing theoretical models.

Modified three-dimensional nonlinear Schrödinger equation in finite water depth for gravity waves with influence of slowly varying currents

A new modified three-dimensional (3D) nonlinear Schrödinger equation (3DMNLSE) is derived for gravity waves in the presence of wind, dissipation, and two-dimensional slowly varying currents, which include transverse and longitudinal currents in finite water depth. The effect of currents on modulational instability (MI) is investigated. The divergence (convergence) effect of longitudinal favorable (adverse) currents decreases (increases) the MI growth rate and region of instability while suppressing (enhancing) the formation of freak waves. Meanwhile, the transverse currents hardly affect the occurrence of MI. Furthermore, some results from the simulations with the space evolution of 3DMNLSE are presented. The results show the ubiquitous occurrence of freak waves in 3D wave fields under certain sets of initial conditions. We demonstrate that larger waves can be triggered when a weakly modulated wave train enters a region of adverse currents. The maximum amplitude of a freak wave depends on the ratio of the current velocity to the wave phase velocity.

Rogue-wave structures for a generalized (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles

This study explores the behavior of higher-order rogue waves within a (3+1)-dimensional generalized nonlinear wave equation in liquid-containing gas bubbles. It creates the investigated equation’s Hirota D-operator bilinear form. We employ a generalized formula with real parameters to obtain the rogue waves up to the third order using the direct symbolic technique. The analysis reveals that the second and third-order rogue solutions produce two and three-waves, respectively. To gain deeper insights, we use the Cole-Hopf transformation on the transformed variables ξ and η to produce a bilinear equation. Using the system software Mathematica, the dynamic analysis presents the graphics for the obtained solutions in transformed ξ, η, and original spatial-temporal coordinates x, y, z, t. These visualizations reveal rogue waves’ intricate structure and evolution, capturing their localized interactions and significant presence within nonlinear systems. We demonstrate that rogue waves, characterized by their substantial height and sudden appearance, are prevalent in various nonlinear events. The equation examined in this study offers valuable insights into the evolution of longer waves with smaller amplitudes, which is particularly relevant in fields such as fluid dynamics, dispersive media, and plasmas. The implications of this research extend across multiple scientific domains, including fiber optics, oceanography, dusty plasma, and nonlinear systems, where understanding the behavior of rogue waves is crucial for both theoretical and practical applications.

Rogue waves on the periodic background for a higher-order nonlinear Schrödinger–Maxwell–Bloch system

In this paper, we construct the rogue wave solutions on the background of the Jacobian elliptic functions for a higher-order nonlinear Schrödinger–Maxwell–Bloch system. The Jacobian elliptic function traveling wave solutions as the seed solutions are considered. Through the approach of the nonlinearization of the Lax pair and Darboux transformation method, the rogue waves and the line rogue waves on the Jacobian elliptic functions dn and cn background are obtained, respectively.

Analysis of rogue wave in the mid-infrared supercontinuum under femtosecond weak seed pulse conditions based on deep learning

The generation process of rogue wave (RW) is affected by noise, which is an unstable state, and the existence of RW will reduce the stability of mid-infrared supercontinuum. However, the process of studying RW requires a large amount of data simulation and statistics, and traditional methods are time-consuming and inefficient. Therefore, this paper adopts long short-term memory (LSTM) neural network to obtain the spectrum information after transmission for a certain distance according to the waveform information of the incident pulse. The results show that the LSTM neural network structure can train and predict the peak power, time deviation information, time intensity evolution and spectrum evolution of RW after 10 cm propagation with only changing the number of internal units. And it performs well on both large and small data sets.

On examining the predictive capabilities of two variants of the PINN in validating localized wave solutions in the generalized nonlinear Schrödinger equation

We introduce a novel neural network structure called strongly constrained theory-guided neural network (SCTgNN), to investigate the behaviour of the localized solutions of the generalized nonlinear Schrödinger (NLS) equation. This equation comprises four physically significant nonlinear evolution equations, namely, the NLS, Hirota, Lakshmanan–Porsezian–Daniel and fifth-order NLS equations. The generalized NLS equation demonstrates nonlinear effects up to quintic order, indicating rich and complex dynamics in various fields of physics. By combining concepts from the physics-informed neural network and theory-guided neural network (TgNN) models, the SCTgNN aims to enhance our understanding of complex phenomena, particularly within nonlinear systems that defy conventional patterns. To begin, we employ the TgNN method to predict the behaviour of localized waves, including solitons, rogue waves and breathers, within the generalized NLS equation. We then use the SCTgNN to predict the aforementioned localized solutions and calculate the mean square errors in both the SCTgNN and TgNN in predicting these three localized solutions. Our findings reveal that both models excel in understanding complex behaviour and provide predictions across a wide variety of situations.

Rogue waves of the Nizhnik-Novikov-Veselov equation via self-mapping transformation

This paper introduces a new two-dimensional self-mapping transformation applied to the Nizhnik-Novikov-Veselov equation, resulting in the generation of numerous rogue wave solutions. We discover that temporal-localized and spatiotemporal-localized two-dimensional rogue waves respectively. Notably, these rogue waves emerge from a zero background and subsequently exhibit both algebraic and exponential decay patterns. The proposed technique offers a potential tool for constructing rogue-like waves within (2+1)-dimensional nonlinear wave frameworks. The findings presented here serve as a robust mathematical foundation for advancing both theoretical understanding and practical applications of rogue waves.

Retrieval of lump, breather, interactions, and rogue wave solutions to the fractional complex paraxial wave dynamical model with sensitivity analysis

In this research, the Hirota bilinear method and the modified Sardar sub-equation (MSSE) techniques are used to investigate the generation and detection of soliton structures in the fractional complex paraxial wave dynamical (FPWD) model together with Kerr media. By employing the aforementioned techniques, we derive lump and different exact solitary wave solutions for the selected model, which has not been documented in previous literature. We manifested some novel lump soliton solutions, including the homoclinic breather wave, periodic cross rational wave, the M-shaped interaction with rogue and kink waves, the M-shaped rational solution, the M-shaped rational solution with one and two kink waves, and multi-wave solutions. Furthermore, for intellectual curiosity, we also amalgamated the rich spectrum of soliton solutions such as W-shape, periodic, dark, bright, combo, rational, exponential, mixed trigonometric, and hyperbolic soliton wave solutions inherent in the FPWD equation. We also undertake sensitivity analysis to examine the resilience of the selected model in the face of variations in initial circumstances and parameters, which provides insights into the system’s sensitivity to perturbations. Furthermore, we investigate the ramifications of these findings for a variety of physical systems, including optics, fluid dynamics, and plasma physics. These findings are to gain a better knowledge of nonlinear wave phenomena and fresh insights into the dynamics of complex systems by combining the Hirota bilinear technique and the MSSE method.