Showing papers on "Rogue wave published in 2010"
TL;DR: The Peregrine soliton was observed experimentally for the first time by using femtosecond pulses in an optical fiber as mentioned in this paper, which gave some insight into freak waves that can appear out of nowhere before simply disappearing.
Abstract: The Peregrine soliton — a wave localized in both space and time — is now observed experimentally for the first time by using femtosecond pulses in an optical fibre. The results give some insight into freak waves that can appear out of nowhere before simply disappearing.
1,158 citations
TL;DR: The Darboux transformation technique is modified to show how to construct the hierarchy of rational solutions of the Hirota equation, a modified nonlinear Schrödinger equation that takes into account higher-order dispersion and time-delay corrections to the cubic nonlinearity.
Abstract: The Hirota equation is a modified nonlinear Schrodinger equation (NLSE) that takes into account higher-order dispersion and time-delay corrections to the cubic nonlinearity In describing wave propagation in the ocean and optical fibers, it can be viewed as an approximation which is more accurate than the NLSE We have modified the Darboux transformation technique to show how to construct the hierarchy of rational solutions of the Hirota equation We present explicit forms for the two lower-order solutions Each one is a regular (nonsingular) rational solution with a single maximum that can describe a rogue wave in this model Numerical simulations reveal the appearance of these solutions in a chaotic field generated from a perturbed continuous wave solution
392 citations
TL;DR: In this article, the authors analytically give the financial rogue waves in the nonlinear option pricing model due to Ivancevic, which is nonlinear wave alternative of the Black-Scholes model.
Abstract: We analytically give the financial rogue waves in the nonlinear option pricing model due to Ivancevic, which is nonlinear wave alternative of the Black—Scholes model. These rogue wave solutions may he used to describe the possible physical mechanisms for rogue wave phenomenon in financial markets and related fields.
301 citations
TL;DR: Evidence of strong four-wave coupling in nonlinear waves (high tricoherence) is presented, which points to modulation instability as the main mechanism in rogue waves.
Abstract: We report the first observation of extreme wave events (rogue waves) in parametrically driven capillary waves. Rogue waves are observed above a certain threshold in forcing. Above this threshold, frequency spectra broaden and develop exponential tails. For the first time we present evidence of strong four-wave coupling in nonlinear waves (high tricoherence), which points to modulation instability as the main mechanism in rogue waves. The generation of rogue waves is identified as the onset of a distinct tail in the probability density function of the wave heights. Their probability is higher than expected from the measured wave background.
295 citations
TL;DR: In this article, the authors discuss multi-Peregrine breather solutions to the nonlinear Schrodinger equations which are relevant in the description of rogue waves in hydrodynamics or in nonlinear optics.
Abstract: We discuss multi-Peregrine breather solutions to the nonlinear Schrodinger equations which are relevant in the description of rogue waves in hydrodynamics or in nonlinear optics. We also describe some basic properties of multi-positon and positon-soliton solutions to the Korteweg-de Vries equations and speculate about their possible links with freak waves.
256 citations
TL;DR: In this paper, the formation of breathers as prototypes of freak waves is studied within the framework of the classic "focussing" nonlinear Schrodinger (NLS) equation.
Abstract: The formation of breathers as prototypes of freak waves is studied within the framework of the classic ‘focussing’ nonlinear Schrodinger (NLS) equation. The analysis is confined to evolution of localised initial perturbations upon an otherwise uniform wave train. For a breather to emerge out of an initial hump, a certain integral over the hump, which we refer to as the “area”, should exceed a certain critical value. It is shown that the breathers produced by the critical and slightly supercritical initial perturbations are described by the Peregrine soliton which represents a spatially localised breather with only one oscillation in time and thus captures the main feature of freak waves: a propensity to appear out of nowhere and disappear without trace. The maximal amplitude of the Peregrine soliton equals exactly three times the amplitude of the unperturbed uniform wave train. It is found that, independently of the proximity to criticality, all small-amplitude supercritical humps generate the Peregrine solitons to leading order. Since the criticality condition requires the spatial scale of the initially small perturbation to be very large (inversely proportional to the square root of the smallness of the hump magnitude), this allows one to predict a priori whether a freak wave could develop judging just by the presence/absence of the corresponding scales in the initial conditions. If a freak wave does develop, it will be most likely the Peregrine soliton with the peak amplitude close to three times the background level. Hence, within the framework of the one-dimensional NLS equation the Peregrine soliton describes the most likely freak-wave patterns. The prospects of applying the findings to real-world freak waves are also discussed.
241 citations
TL;DR: The appearance of rogue waves is well known in oceanographic, optics, and cold-matter systems as mentioned in this paper, and there is even a possibility for the existence of atmospheric rogue waves.
Abstract: The appearance of rogue waves is well known in oceanographics, optics, and cold matter systems. Here we show a possibility for the existence of atmospheric rogue waves.
227 citations
TL;DR: In this paper, the authors present a special issue "discussions and debates on the subject of "rogue waves" where the authors have the chance to discuss the basic concepts of an emerging topic in science.
Abstract: This short editorial contains the introductory remarks for the special issue “discussions and debates” on the subject of “rogue waves”. This issue is the first of its kind, in the sense that the authors have the chance to discuss the basic concepts of an emerging topic in science. Based on these discussions, an attempt to give a “definition” of a rogue wave is made.
219 citations
TL;DR: Microwave transport experiments have been performed in a quasi-two-dimensional resonator with randomly distributed conical scatterers with branching structures similar to those observed in stationary imaging of electron flow, which confirm that caustics in the ray dynamics are responsible for these structures.
Abstract: Microwave transport experiments have been performed in a quasi-two-dimensional resonator with randomly distributed conical scatterers. At high frequencies, the flow shows branching structures similar to those observed in stationary imaging of electron flow. Semiclassical simulations confirm that caustics in the ray dynamics are responsible for these structures. At lower frequencies, large deviations from Rayleigh's law for the wave height distribution are observed, which can only partially be described by existing multiple-scattering theories. In particular, there are "hot spots" with intensities far beyond those expected in a random wave field. The results are analogous to flow patterns observed in the ocean in the presence of spatially varying currents or depth variations in the sea floor, where branches and hot spots lead to an enhanced frequency of freak or rogue wave formation.
188 citations
TL;DR: In this paper, numerically rogue waves in two-component Bose-Einstein condensates were studied and it was shown that rogue wave solutions exist only for certain combinations of the nonlinear coefficients describing two-body interactions.
Abstract: We study numerically rogue waves in the two-component Bose-Einstein condensates which are described by the coupled set of two Gross-Pitaevskii equations with variable scattering lengths. We show that rogue wave solutions exist only for certain combinations of the nonlinear coefficients describing two-body interactions. We present the solutions for the combinations of these coefficients that admit the existence of rogue waves.
179 citations
TL;DR: In this paper, the authors reported analytical nonautonomous rogons for the inhomogeneous nonlinear Schrodinger equation with variable coefficients in terms of rational-like functions by using the similarity transformation and direct ansatz, which can be used to describe the possible formation mechanisms for optical, oceanic and matter rogue wave phenomenon in optical fibres, the deep ocean, and Bose-Einstein condensates, respectively.
TL;DR: In this article, numerical simulations of nonlinear waves have been compared in order to assess the ability of numerical models to describe the evolution of weakly nonlinear wave and predict the probability of occurrence of extreme waves within a variety of random directional wave fields.
Abstract: Nonlinear modulational instability of wavepackets is one of the mechanisms responsible for the formation of large-amplitude water waves. Here, mechanically generated waves in a three-dimensional basin and numerical simulations of nonlinear waves have been compared in order to assess the ability of numerical models to describe the evolution of weakly nonlinear waves and predict the probability of occurrence of extreme waves within a variety of random directional wave fields. Numerical simulations have been performed following two different approaches: numerical integration of a modified nonlinear Schrodinger equation and numerical integration of the potential Euler equations based on a higher-order spectral method. Whereas the first makes a narrow-banded approximation (both in frequency and direction), the latter is free from bandwidth constraints. Both models assume weakly nonlinear waves. On the whole, it has been found that the statistical properties of numerically simulated wave fields are in good quantitative agreement with laboratory observations. Moreover, this study shows that the modified nonlinear Schrodinger equation can also provide consistent results outside its narrow-banded domain of validity.
TL;DR: In this article, the authors considered two wave modes existing in plasmas, the ion-sound wave and the Alfven wave, and the results of numerical solution of the Gardner equation with the modulationally unstable initial condition are presented.
Abstract: Generation of large-amplitude short-lived wave groups from small-amplitude initial perturbations in plasmas is discussed. Two particular wave modes existing in plasmas are considered. The first one is the ion-sound wave. In a plasmas with negative ions it is described by the Gardner equation when the negative ion concentration is close to critical. The results of numerical solution of the Gardner equation with the modulationally unstable initial condition are presented. These results clearly show the possibility of generation of freak ion-acoustic waves due to the modulational instability. The second wave mode is the Alfven wave. When this wave propagates at a small angle with respect to the equilibrium magnetic field, and its wave length is comparable with the ion inertia length, it is described by the DNLS equation. Studying the evolution of an initial perturbation using the linearized DNLS equation shows that the generation of freak Alfven waves is possible due to linear dispersive focusing. The numerical solution of the DNLS equation reveals that the nonlinear dispersive focusing can also produce freak Alfven waves.
TL;DR: The focusing nonlinear Schrodinger equation as mentioned in this paper describes generic nonlinear phenomena, including waves in the deep ocean and light pulses in optical fibres, and supports a hierarchy of recently discovered rational solutions.
Abstract: The focusing nonlinear Schrodinger equation, which describes generic nonlinear phenomena, including waves in the deep ocean and light pulses in optical fibres, supports a whole hierarchy of recently discovered rational solutions. We present recurrence relations for the hierarchy, the pattern of zeros for each solution and a set of integral relations which characterizes them.
TL;DR: A new model of pump noise in supercontinuum and rogue wave generation is presented and it is found that for four-wave mixing (FWM) a narrow spectral line width initially leads to a build-up of FWM from quantum noise, whereas a broad spectral linewidth initially leading to a gradual broadening of the pump spectrum.
Abstract: A new model of pump noise in supercontinuum and rogue wave generation is presented. Simulations are compared with experiments and show that the new model provides significantly better agreement than the currently ubiquitously used one-photon-per-mode model. The new model also allows for a study of the influence of the pump spectral line width on the spectral broadening mechanisms. Specifically, it is found that for four-wave mixing (FWM) a narrow spectral line width (≲ 0.1 nm) initially leads to a build-up of FWM from quantum noise, whereas a broad spectral line width (≳ 1 nm) initially leads to a gradual broadening of the pump spectrum. Since the new model provides better agreement with experiments and is still simple to implement, it is particularly important that it is used for future studies of the statistical properties of nonlinear spectral broadening, such as the formation of rogue waves.
TL;DR: Using symmetry analysis, a higher-dimensional similarity transformation is presented reducing the (3+1) -dimensional inhomogeneous nonlinear Schrödinger (NLS) equation with variable coefficients and parabolic potential to the (1-dimensional NLS equation with constant coefficients, to obtain rogue wavelike solutions localized in three dimensions.
Abstract: Using symmetry analysis we systematically present a higher-dimensional similarity transformation reducing the (3+1)-dimensional inhomogeneous nonlinear Schrodinger (NLS) equation with variable coefficients and parabolic potential to the (1+1)-dimensional NLS equation with constant coefficients. This transformation allows us to relate certain class of localized exact solutions of the (3+1)-dimensional case to the variety of solutions of integrable NLS equation of the (1+1)-dimensional case. As an example, we illustrated our technique using two lowest-order rational solutions of the NLS equation as seeding functions to obtain rogue wavelike solutions localized in three dimensions that have complicated evolution in time including interactions between two time-dependent rogue wave solutions. The obtained three-dimensional rogue wavelike solutions may raise the possibility of relative experiments and potential applications in nonlinear optics and Bose-Einstein condensates.
TL;DR: In this paper, the authors discuss optical rogue wave generation in terms of collisions and turbulence processes, and show that the rogue wave can emerge from either third-order dispersion or Raman scattering independently.
TL;DR: In this article, the authors present a statistical analysis of a large sample of individual wave steepness data collected from measurements of the surface elevation in laboratory facilities and the open sea under a variety of sea state conditions.
Abstract: The breaking of waves is an important mechanism for a number of physical, chemical and biological processes in the ocean. Intuitively, waves break when they become too steep. Unfortunately, a general consensus on the ultimate shape of waves has not been achieved yet due to the complexity of the breaking mechanism which still remains the least understood of all processes affecting waves. To estimate the limiting shape of ocean waves, here we present a statistical analysis of a large sample of individual wave steepness. Data were collected from measurements of the surface elevation in laboratory facilities and the open sea under a variety of sea state conditions. Observations reveal that waves are able to reach steeper profiles than the Stokes' limit for stationary waves. Due to the large number of records this finding is statistically robust. Copyright © 2010 by the American Geophysical Union.
Landau Institute for Theoretical Physics1, Ohio State University2, University of Sheffield3, University of Franche-Comté4, Loughborough University5, Lancaster University6, University of Turin7, Tallinn University of Technology8, Australian National University9, University of California, Los Angeles10, University of Göttingen11, École normale supérieure de Cachan12
TL;DR: In this paper, the discussion inputs by the contributors of the special issue on the subject of rogue waves were discussed, and the authors provided a discussion input for each of the contributors.
Abstract: This paper contains the discussion inputs by the contributors of the special issue on the subject of rogue waves.
TL;DR: In this article, the authors provide some general physical insights into the emergence of rogue wave events from optical turbulence by analyzing the long term evolution of the field, and identify three turbulent regimes that lead to specific rogue wave event: (i) persistent and coherent rogue quasi-solitons, (ii) intermittent-like rogue quasisolitons that appear and disappear erratically, and (iii) sporadic rogue waves events that emerge from turbulent fluctuations as bursts of light or intense flashes.
TL;DR: In this article, high intensity second sound (temperature-entropy) waves within a resonant cavity have been observed in superfluid helium, with a constant oscillatory driving force at the resonant frequency, and there are fluxes of energy towards both high and low frequencies.
Abstract: Rogue waves have been observed in superfluid helium. The experimental system consists of high intensity second sound (temperature-entropy) waves within a resonant cavity. Under steady state conditions, with a constant oscillatory driving force at the resonant frequency, the waves are turbulent and there are fluxes of energy towards both high and low frequencies. Rogue waves appear under the nonequilibrium conditions that prevail shortly after the drive has been switched on, prior to establishment of the steady state. The experiment is described briefly, relevant results are presented and discussed theoretically in terms of nonlinear wave interactions, and possible connections to rogue waves on the ocean are considered.
TL;DR: In this article, a third generation model, SWAN, was employed for wave simulation and the results were compared with the recorded wave data, which revealed that the calibration of the wave model for high waves led to the overestimation of low waves.
TL;DR: In this paper, the modulational instability in crossing seas was considered as a potential mechanism for the formation of freak waves, and the problem was discussed in terms of a system of two coupled Nonlinear Schroedinger equations.
Abstract: We consider the modulational instability in crossing seas as a potential mechanism for the formation of freak waves. The problem is discussed in terms of a system of two coupled Nonlinear Schroedinger equations. The asymptotic validity of such system is discussed. For some specific angles between the two wave trains, the equations reduce to an integrable system. A stability analysis of these equations is discussed. Furthermore, we present an analytical study of the maximum amplification factor for an unstable plane wave solution. Results indicate that angles between 10∘ and 30∘ are the most probable for establishing a freak wave sea. We show that the theoretical expectations are consistent with numerical simulations of the Euler equations.
TL;DR: In this article, the authors consider how long wavelength spectral filtering influences the characteristics of the statistical distribution of peak power, and contrast the statistics of the spectrally filtered supercontinuum generation with the statistics for both the peak power of the most red-shifted soliton in the SC and the maximum peak power across the full temporal field with no spectral selection.
Abstract: Numerical simulations are used to discuss various aspects of “optical rogue wave” statistics observed in noise-driven fiber supercontinuum generation associated with highly incoherent spectra. In particular, we consider how long wavelength spectral filtering influences the characteristics of the statistical distribution of peak power, and we contrast the statistics of the spectrally filtered SC with the statistics of both the peak power of the most red-shifted soliton in the SC and the maximum peak power across the full temporal field with no spectral selection. For the latter case, we show that the unfiltered statistical distribution can still exhibit a long-tail, but the extreme-events in this case correspond to collisions between solitons of different frequencies. These results confirm the importance of collision dynamics in supercontinuum generation. We also show that the collision-induced events satisfy an extended hydrodynamic definition of “rogue wave” characteristics.
TL;DR: In this article, the authors used the Volume Of Fluid (VOF) method to simulate extreme wave generation using wave focusing in a 2D numerical model and compared the results with theoretical solutions and experimental data.
Abstract: Numerical simulations of extreme wave generation are carried out by using the Volume Of Fluid (VOF) method. Extreme waves are generated based on wave focusing in a 2-D numerical model. To validate the capability of the VOF-based model described in this article, the propagation of regular waves is computed and compared with the theoretical results. By adjusting the phases of wave components, extreme waves are formed at given time and given position in the computation. The numerical results are compared with theoretical solutions and experimental data. It is concluded that the present model based on the VOF technique can provide acceptably accurate numerical results to serve practical purposes.
TL;DR: In this paper, the spectrum width plays an important role in the evolution of the random wavefield and strongly affects the variation of the wave spectrum as well as the parameters that characterize the deviation of wavefield statistics from that corresponding to the Gaussian distribution.
Abstract: [1] Results of extensive experiments on propagation of unidirectional nonlinear random waves in a large wave tank are presented. The nonlinearity of the wavefield determined by the characteristic wave amplitude and the dominant wave length was retained constant in various series of experimental runs. In each experimental series, initial spectra of different shape and/or width were considered. Every series contained sufficient number of independent realizations to ensure reliable statistics. Evolution of various statistical parameters along the tank was investigated. It is demonstrated that the spectrum width plays an important role in the evolution of the random wavefield and strongly affects the variation of the wave spectrum as well as of parameters that characterize the deviation of the wavefield statistics from that corresponding to the Gaussian distribution. In particular, in a random wavefield that initially contains independent free harmonics within a narrow spectrum, extremely steep waves appear more often in the process of evolutions than predicted by a Rayleigh distribution, while for wider initial wave spectra the probability of those waves decreases sharply and is well below the Rayleigh values.
TL;DR: In this article, the authors examine the occurrence of rogue internal waves in the Gardner equation, which is an extended version of the Korteweg-de Vries equation with quadratic and cubic nonlinearity.
Abstract: Rogue waves can be categorized as unexpectedly large waves, which are temporally and spatially localized. They have recently received much attention in the water wave context, and also been found in nonlinear optical fibers. In this paper, we examine the issue of whether rogue internal waves can be found in the ocean. Because large-amplitude internal waves are commonly observed in the coastal ocean, and are often modeled by weakly nonlinear long wave equations of the Korteweg-de Vries type, we focus our attention on this shallow-water context. Specifically, we examine the occurrence of rogue waves in the Gardner equation, which is an extended version of the Korteweg-de Vries equation with quadratic and cubic nonlinearity, and is commonly used for the modelling of internal solitary waves in the ocean. Importantly, we choose that version of the Gardner equation for which the coefficient of the cubic nonlinear term and the coefficient of the linear dispersive term have the same sign, as this allows for modulational instability. From numerical simulations of the evolution of a modulated narrow-band initial wave field, we identify several scenarios where rogue waves occur.
TL;DR: In this article, a stochastic model of sea storms for describing long-term statistics of extreme wave events is presented, which generalizes Boccotti's equivalent triangular storm model by describing an actual storm history in the form of a generic power law.
Abstract: A stochastic model of sea storms for describing long-term statistics of extreme wave events is presented. The formulation generalizes Boccotti’s equivalent triangular storm model by describing an actual storm history in the form of a generic power law. The latter permits the derivation of analytical solutions for the return periods of extreme wave events and associated statistical properties. Lastly, the relative validity of the new model and its predictions is assessed by analyzing wave measurements retrieved from two NOAA National Oceanographic Data Center (NODC) buoys in the Atlantic and Pacific Oceans.
TL;DR: In this paper, the quasi arbitrary Lagrangian-Eulerian finite element method (QALE-FEM) is combined with a commercial software (StarCD) to simulate the interaction between freak waves and winds with an improved computational efficiency.
Abstract: This paper presents a newly developed approach for the numerical modelling of wind effects on the generation and dynamics of freak waves. In this approach, the quasi arbitrary Lagrangian-Eulerian finite element method (QALE-FEM) developed by the authors of this paper is combined with a commercial software (StarCD). The former is based on the fully nonlinear potential model, in which the wind-excited pressure is modelled using a modified Jeffreys’ model (C. Kharif, et al. J. Fluid Mech. 594:209-247,2008). The latter has a volume of fluid (VOF) solver which can handle violent air-wave interaction problems. The combination can simulate the interaction between freak waves and winds with an improved computational efficiency. The numerical approach is validated by comparing its predictions with experimental data. Satisfactory agreements are achieved. Detailed numerical investigations of the interaction between winds and 2D freak waves are carried out, which not only explore different air flow states but also reveal the wind effects on the change of freak wave profiles. Both breaking and non-breaking freak waves are considered.
TL;DR: In this paper, the authors presented a statistical analysis of freak waves measured during the 203 h of observation on sea surface elevation at a location in the coastal zone of the Baltic Sea (2.7 m depth) during June-July 2008.
Abstract: . We present a statistical analysis of freak waves1 measured during the 203 h of observation on sea surface elevation at a location in the coastal zone of the Baltic Sea (2.7 m depth) during June–July 2008. The dataset contains 97 freak waves occurring in both calm and stormy weather conditions. All of the freak waves are solitary waves, 63% of them having positive shape, 17.5% negative shape and 19.5% sign-variable shape. It is suggested that the freak waves can be divided into two groups. Those of the first group, which includes 92% of the freak waves, have an amplification factor (ratio of freak wave height to significant wave height) which does not vary from significant wave height and has values largely within the range of 2.0 to 2.4; while for the second group, which contain the most extreme freak waves, amplification factors depend strongly on significant wave height and can reach 3.1. Analysis based on the Generalised Pareto distribution is used to describe the waves of the first group and lends weight to the identification of the two groups. It is suggested that the probable mechanism of the generation of freak waves in the second group is dispersive focussing. The time-frequency spectra of the freak waves are studied and dispersive tracks, which can be interpreted as dispersive focussing, are demonstrated. 1 taken to be waves whose height is 2 or more times greater than the significant wave height