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2011, Physical Review Letters
We show that rogue waves can be triggered naturally when a stable wave train enters a region of an opposing current flow. We demonstrate that the maximum amplitude of the rogue wave depends on the ratio between the current velocity U 0 and the wave group velocity c g. We also reveal that an opposing current can force the development of rogue waves in random wave fields, resulting in a substantial change of the statistical properties of the surface elevation. The present results can be directly adopted in any field of physics in which the focusing nonlinear Schrödinger equation with nonconstant coefficient is applicable. In particular, nonlinear optics laboratory experiments are natural candidates for verifying experimentally our results.
Physics Reports, 2013
Rogue waves is the name given by oceanographers to isolated large amplitude waves, that occur more frequently than expected for normal, Gaussian distributed, statistical events. Rogue waves are ubiquitous in nature and appear in a variety of different contexts. Besides water waves, they have been recently reported in liquid Helium, in nonlinear optics, microwave cavities, etc. The first part of the review is dedicated to rogue waves in the oceans and to their laboratory counterpart with experiments performed in water basins. Most of the work and interpretation of the experimental results will be based on the nonlinear Schrödinger equation, an universal model, that rules the dynamics of weakly nonlinear, narrow band surface gravity waves. Then, we present examples of rogue waves occurring in different physical contexts and we discuss the related anomalous statistics of the wave amplitude, which deviates from the Gaussian behavior that were expected for random waves. The third part of the review is dedicated to optical rogue waves, with examples taken from the supercontinuum generation in photonic crystal fibers, laser fiber systems and two-dimensional spatiotemporal systems. In particular, the extreme waves observed in a two-dimensional spatially extended optical cavity allow us to introduce a description based on two essential conditions for the generation of rogue waves: nonlinear coupling and nonlocal coupling. The first requirement is needed in order to introduce an elementary size, such as that of the solitons or breathers, whereas the second requirement implies inhomogeneity, a mechanism needed to produce the events of mutual collisions and mutual amplification between the elementary solitons or wavepackets. The concepts of ''granularity'' and ''inhomogeneity'' as joint generators of optical rogue waves are introduced on the basis of a linear experiment. By extending these concepts to other systems, rogue waves can be classified as phenomena occurring in the presence of many uncorrelated ''grains'' of activity inhomogeneously distributed in large spatial domains, the ''grains'' being of linear or nonlinear origin, as in the case of wavepackets or solitons.
Journal of Fluid Mechanics, 2015
Interaction with an opposing current amplifies wave modulation and accelerates nonlinear wave focusing in regular wavepackets. This results in large-amplitude waves, usually known as rogue waves, even if the wave conditions are less prone to extremes. Laboratory experiments in three independent facilities are presented here to assess the role of opposing currents in changing the statistical properties of unidirectional and directional mechanically generated random wavefields. The results demonstrate in a consistent and robust manner that opposing currents induce a sharp and rapid transition from weakly to strongly non-Gaussian properties. This is associated with a substantial increase in the probability of occurrence of rogue waves for unidirectional and directional sea states, for which the occurrence of extreme and rogue waves is normally the least expected.
Physical Review E, 2013
We show experimentally that a stable wave propagating into a region characterized by an opposite current may become modulationally unstable. Experiments have been performed in two independent wave tank facilities; both of them are equipped with a wavemaker and a pump for generating a current propagating in the opposite direction with respect to the waves. The experimental results support a recent conjecture based on a current-modified nonlinear Schrödinger equation which establishes that rogue waves can be triggered by a nonhomogeneous current characterized by a negative horizontal velocity gradient. PACS number(s): 05.45.Yv, Ocean waves are characterized by a statistically small steepness and often (but not always; see, for example, [1]) a weakly nonlinear approach is sufficient to capture some of the intriguing aspects hidden in the fully nonlinear primitive equations. This weakly nonlinear approach is also shared by other fields of physics such as nonlinear optics [2] and plasma physics where small parameters can be individuated and asymptotic expansions can be used to simplify the original equations. If the considered physical process is not only weakly nonlinear but also narrow banded then the lion's share is played by the nonlinear Schrödinger equation (NLS). Being an exactly integrable equation via the inverse scattering transform [4], bizarre analytical solutions have been found in the past: besides traveling waves, breathers or multibreather solutions have been found and observed in hydrodynamics , nonlinear optics , and plasma experiments. Starting from , such solutions have been considered as prototypes of rogue waves. The early stages of the so-called Akhmediev breather solution [6] describes the exponential growth of slightly perturbed plane waves, i.e., it corresponds to the classical modulational instability process . For water waves in infinite water depth, the instability is active when εN 1/ √ 2, where ε = k 0 a 0 is the initial steepness of the plane wave, k 0 the wave number, a 0 its amplitude, and N = ω 0 / the number of waves under the modulation; ω 0 is the angular frequency corresponding to the carrier wave number k 0 and the angular frequency of the modulation. The whole picture is by now pretty well understood and relies on the fact that the medium in which waves propagate is homogeneous. In terms of the NLS equation this means that the coefficients of the dispersive and nonlinear terms do not depend on the spatial coordinates. Much more complicated and intriguing is the case in which the medium changes its properties along the direction of propagation of the waves. This situation is much more difficult to treat in terms of simplified models because it turns out that in general the resulting modified NLS does not share the property of integrability as the standard NLS, and analytical breather solutions can be found only in special cases (see some examples in ).
European Journal of Mechanics - B/Fluids, 2003
A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin-Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrödinger equation, the Davey -Stewartson system, the Korteweg -de Vries equation, the Kadomtsev -Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.
Optical and Quantum Electronics, 2018
In this article, the first and second-order rogue wave solutions are obtained which are localized in both space and time that appear from nowhere and disappear without a trace. The coupled NLSEs with time-dependent coefficients are considered that describe the effects of ultrashort optical pulse propagation in nonlinear optics and quantum physics. The similarity transformation is used to investigate these rational-like (rogue wave) solutions. Moreover, the 3D graphical representations and contour plots have depicted with different parameters of gravity field and external magnetic field.
Nonlinear instability and refraction by ocean currents are both important mechanisms that go beyond the Rayleigh approximation and may be responsible for the formation of freak waves. In this paper, we quantitatively study nonlinear effects on the evolution of surface gravity waves on the ocean, to explore systematically the effects of various input parameters on the probability of freak wave formation. The fourth-order current-modified nonlinear Schrödinger equation (CNLS 4 ) is employed to describe the wave evolution. By solving CNLS 4 numerically, we are able to obtain quantitative predictions for the wave height distribution as a function of key environmental conditions such as average steepness, angular spread, and frequency spread of the local sea state. Additionally, we explore the spatial dependence of the wave height distribution, associated with the buildup of nonlinear development.
Coastal Engineering Proceedings, 2014
The occurrence probabilities of Rogue Waves in different nonlinear states are investigated based on high-order spectral method (HOSM), which is a direct phase-resolved numerical method. The focus is given to the occurrence probability of Rogue Waves in the nonlinear evolution stage where the Benjamin-Feir Instability is not the dominate mechanism due to the quartet resonance interactions for wave fields with a broad range in frequencies. The initial wave trains are generated from Stokes waves and two sidebands. Based on the simulation, we find that the Kurtosis evolves distinctly at three nonlinear stages and shows a weak relation to the probabilities of the Rouge Waves. We also introduce a simple Entropy formula, turning out to close a stable value.
Annals of Physics, 2012
We present, analytically, self-similar rogue wave solutions (rational solutions) of the inhomogeneous nonlinear Schrödinger equation (NLSE) via a similarity transformation connected with the standard NLSE. Then we discuss the propagation behaviors of controllable rogue waves under dispersion and nonlinearity management. In an exponentially dispersion-decreasing fiber, the postponement, annihilation and sustainment of self-similar rogue waves are modulated by the exponential parameter σ. Finally, we investigate the nonlinear tunneling effect for self-similar rogue waves. Results show that rogue waves can tunnel through the nonlinear barrier or well with increasing, unchanged or decreasing amplitudes via the modulation of the ratio of the amplitudes of rogue waves to the barrier or well height.
Nonlinearity, 2012
We investigate rogue waves in deep water in the framework of the nonlinear Schrödinger (NLS) and Dysthe equations. Amongst the homoclinic orbits of unstable NLS Stokes waves, we seek good candidates to model actual rogue waves. In this paper we propose two selection criteria: stability under perturbations of initial data, and persistence under perturbations of the NLS model. We find that requiring stability selects homoclinic orbits of maximal dimension. Persistence under (a particular) perturbation selects a homoclinic orbit of maximal dimension all of whose spatial modes are coalesced. These results suggest that more realistic sea states, described by JONSWAP power spectra, may be analyzed in terms of proximity to NLS homoclinic data. In fact, using the NLS spectral theory, we find that rogue wave events in random oceanic sea states are well predicted by proximity to homoclinic data of the NLS equation.
Nonlinearity
We propose a novel, analytically tractable, scenario of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schrödinger (NLS) equation with the initial condition in the form of a rectangular barrier (a "box"). We use the Whitham modulation theory combined with the nonlinear steepest descent for the semi-classical inverse scattering transform, to describe the evolution and interaction of two counter-propagating nonlinear wave trains-the dispersive dam break flows-generated in the NLS box problem. We show that the interaction dynamics results in the emergence of modulated largeamplitude quasi-periodic breather lattices whose amplitude profiles are closely approximated by the Akhmediev and Peregrine breathers within certain spacetime domain. Our semi-classical analytical results are shown to be in excellent agreement with the results of direct numerical simulations of the small-dispersion focusing NLS equation.
2020
Summary. We discuss physical and statistical properties of rogue wave generation in deep water from the perspective of the focusing Nonlinear Schrödinger equation and some of its higher order generalizations. Numerical investigations and analytical arguments based on the inverse spectral theory of the underlying integrable model, perturbation analysis, and statistical methods provide a coherent picture of rogue waves associated with nonlinear focusing events. Homoclinic orbits of unstable solutions of the underlying integrable model are certainly candidates for extreme waves, however for more realistic models such as the modified Dysthe equation two novel features emerge: (a) a chaotic sea state appears to be an important mechanism for both generation and increased likelihood of rogue waves; (b) the extreme waves intermittently emerging from the chaotic background can be correlated with the homoclinic orbits characterized by maximal coalescence of their spatial modes.
Nature, 2007
Recent observations show that the probability of encountering an extremely large rogue wave in the open ocean is much larger than expected from ordinary wave-amplitude statistics 1-3 . Although considerable effort has been directed towards understanding the physics behind these mysterious and potentially destructive events, the complete picture remains uncertain. Furthermore, rogue waves have not yet been observed in other physical systems. Here, we introduce the concept of optical rogue waves, a counterpart of the infamous rare water waves. Using a new real-time detection technique, we study a system that exposes extremely steep, large waves as rare outcomes from an almost identically prepared initial population of waves. Specifically, we report the observation of rogue waves in an optical system, based on a microstructured optical fibre, near the threshold of soliton-fission supercontinuum generation 4,5 -a noise-sensitive 5-7 nonlinear process in which extremely broadband radiation is generated from a narrowband input 8 . We model the generation of these rogue waves using the generalized nonlinear Schrödinger equation 9 and demonstrate that they arise infrequently from initially smooth pulses owing to power transfer seeded by a small noise perturbation.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science
Rogue waves appearing on deep water or in optical fibres are often modelled by certain breather solutions of the focusing nonlinear Schrödinger (fNLS) equation which are referred to as solitons on finite background (SFBs). A more general modelling of rogue waves can be achieved via the consideration of multiphase, or finite-band, fNLS solutions of whom the standard SFBs and the structures forming due to their collisions represent particular, degenerate, cases. A generalized rogue wave notion then naturally enters as a large-amplitude localized coherent structure occurring within a finite-band fNLS solution. In this paper, we use the winding of real tori to show the mechanism of the appearance of such generalized rogue waves and derive an analytical criterion distinguishing finite-band potentials of the fNLS equation that exhibit generalized rogue waves.
2015
We investigate the generation mechanisms and evolution dynamics of rogue (or freak) waves using direct phase-resolved nonlinear wave-field simulations. The focus is on the understanding of the effects of nonlinear broadband wave-wave interactions on rogue wave development and characteristics. Large-scale long-time simulations of nonlinear long-crested wave-fields with various wave spectrum parameters are obtained. Based on these simulations, we find that nonlinear self-focusing of wave groups and nonlinear wave group interactions play a critical role in the formation and development of rogue waves in deep seas. Occurrence of rogue waves is closely correlated to kurtosis of the wave-field, with larger kurtosis associated with rogue waves of higher height. Moreover, occurrence of rogue waves (especially of large height) is usually correlated with broadband wave spectra. Background The occurrence of rogue/freak waves in deep seas has been observed (e.g. Haver 2000). Due to exceptionall...
Fluids, 2019
The formation of rogue oceanic waves may be the result of different causes. Various factors (winds, currents, dispersive focussing, depth, nonlinear focussing and instability) make this subject intriguing, and yet its understanding is quite relevant to practical issues. Here, we deal only with the nonlinear character of this dynamics, which has been recognised as the main ingredient to rogue wave formation. In this perspective, the formation of rogue waves requires a non-vanishing and unstable background such as a nonlinear regular wave train with attractive self-interaction. The simplest, best known model of such dynamics is the universal nonlinear Schrödinger equation. This has proven to serve as a good approximation in various contexts and over a broad range of experimental settings. This model aims to give the slow evolution of the envelope of one monochromatic wave due to nonlinearity. Here, we naturally consider the same problem for the envelopes of two weakly resonant monochr...
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2013
Rogue waves observed in the ocean and elsewhere are often modelled by certain solutions of the nonlinear Schrodinger equation, describing the modulational instability of a plane wave and the subsequent development of multi-phase nonlinear wavetrains. In this paper, we describe how integrability and application of the inverse scattering transform can be used to construct a class of explicit asymptotic solutions that describe this process. We discuss the universal mechanism of the onset of multiphase nonlinear waves (rogue waves) through the sequence of successive multi-breather wavetrains. Some applications to ocean waves and laboratory experiments are presented.
2012
We construct and discuss a semirational, multiparametric vector solution of coupled nonlinear Schrödinger equations (Manakov system). This family of solutions includes known vector Peregrine solutions, bright-and dark-rogue solutions, and novel vector unusual freak waves. The vector rogue waves could be of great interest in a variety of complex systems, from optics and fluid dynamics to Bose-Einstein condensates and finance.
2009
Freak waves, or rogue waves, are one of the fascinating manifestations of the strength of nature. These devastating "walls of water" appear from nowhere, are short-lived and extremely rare. Despite the large amount of research activities on this subject, neither the minimum ingredients required for their generation nor the mechanisms explaining their formation have been given. Today, it is possible to reproduce such kind of waves in optical fibre systems. In this context, we demonstrate theoretically and numerically that convective instability is the basic ingredient for the formation of rogue waves. This explains why rogues waves are extremely sensitive to noisy environments.
Physical Review E, 2018
Rogue waves are extreme and rare fluctuations of the wave field that have been discussed in many physical systems. Their presence substantially influences the statistical properties of an incoherent wave field. Their understanding is fundamental for the design of ships and offshore platforms. Except for very particular meteorological conditions, waves in the ocean are characterised by the so-called JONSWAP (Joint North Sea Wave Project) spectrum. Here we compare two unique experimental results: the first one has been performed in a 270-meter wave tank and the other in optical fibers. In both cases, waves characterised by a JONSWAP spectrum and random Fourier phases have been launched at the input of the experimental device. The quantitative comparison, based on an appropriate scaling of the two experiments, shows a very good agreement between the statistics in hydrodynamics and optics. Spontaneous emergence of heavy tails in the probability density function of the wave amplitude is observed in both systems. The results demonstrate the universal features of rogue waves and provide a fundamental and explicit bridge between two important fields of research. Numerical simulations are also compared with experimental results.
Physical Review Letters, 2011
Experimental observations of rare giant pulses or rogue waves were done in the output intensity of an optically injected semiconductor laser. The long-tailed probability distribution function of the pulse amplitude displays clear non-Gaussian features that confirm the rogue wave character of the intensity pulsations. Simulations of a simple rate equation model show good qualitative agreement with the experiments and provide a framework for understanding the observed extreme amplitude events as the result of a deterministic nonlinear process.
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