Wave amplification in the framework of forced nonlinear Schrödinger equation: The rogue wave context
TL;DR: In this paper, it is shown that the adiabatically slow pumping (the time scale of forcing is much longer than the nonlinear time scale) results in selective enhancement of the solitary part of the wave ensemble.
About: This article is published in Physica D: Nonlinear Phenomena.The article was published on 2015-05-15 and is currently open access. It has received 41 citations till now. The article focuses on the topics: Nonlinear Schrödinger equation & Rogue wave.
Citations
More filters
Australian National University1, University of Burgundy2, University of Brescia3, Texas A&M University at Qatar4, Xi'an Jiaotong University5, Spanish National Research Council6, École Polytechnique Fédérale de Lausanne7, Adolfo Ibáñez University8, Université libre de Bruxelles9, Vrije Universiteit Brussel10, university of lille11, Tampere University of Technology12, University of Franche-Comté13, Leibniz University of Hanover14, Polytechnic University of Catalonia15, University of Auckland16, University of Nice Sophia Antipolis17, University of Florence18
TL;DR: The concept of optical rogue wave was introduced by Solli et al. as discussed by the authors, who defined it as "an optical pulse whose amplitude or intensity is much higher than that of the surrounding pulses".
Abstract: The pioneering paper 'Optical rogue waves' by Solli et al (2007 Nature 450 1054) started the new subfield in optics. This work launched a great deal of activity on this novel subject. As a result, the initial concept has expanded and has been enriched by new ideas. Various approaches have been suggested since then. A fresh look at the older results and new discoveries has been undertaken, stimulated by the concept of 'optical rogue waves'. Presently, there may not by a unique view on how this new scientific term should be used and developed. There is nothing surprising when the opinion of the experts diverge in any new field of research. After all, rogue waves may appear for a multiplicity of reasons and not necessarily only in optical fibers and not only in the process of supercontinuum generation. We know by now that rogue waves may be generated by lasers, appear in wide aperture cavities, in plasmas and in a variety of other optical systems. Theorists, in turn, have suggested many other situations when rogue waves may be observed. The strict definition of a rogue wave is still an open question. For example, it has been suggested that it is defined as 'an optical pulse whose amplitude or intensity is much higher than that of the surrounding pulses'. This definition (as suggested by a peer reviewer) is clear at the intuitive level and can be easily extended to the case of spatial beams although additional clarifications are still needed. An extended definition has been presented earlier by N Akhmediev and E Pelinovsky (2010 Eur. Phys. J. Spec. Top. 185 1-4). Discussions along these lines are always useful and all new approaches stimulate research and encourage discoveries of new phenomena. Despite the potentially existing disagreements, the scientific terms 'optical rogue waves' and 'extreme events' do exist. Therefore coordination of our efforts in either unifying the concept or in introducing alternative definitions must be continued. From this point of view, a number of the scientists who work in this area of research have come together to present their research in a single review article that will greatly benefit all interested parties of this research direction. Whether the authors of this 'roadmap' have similar views or different from the original concept, the potential reader of the review will enrich their knowledge by encountering most of the existing views on the subject. Previously, a special issue on optical rogue waves (2013 J. Opt. 15 060201) was successful in achieving this goal but over two years have passed and more material has been published in this quickly emerging subject. Thus, it is time for a roadmap that may stimulate and encourage further research.
243 citations
TL;DR: In this paper, a three-wave truncation of a damped/forced high-order nonlinear Schrodinger equation for deep-water gravity waves under the effect of wind and viscosity was studied.
Abstract: We study a three-wave truncation of a recently proposed damped/forced high-order nonlinear Schrodinger equation for deep-water gravity waves under the effect of wind and viscosity. The evolution of the norm (wave-action) and spectral mean of the full model are well captured by the reduced dynamics. Three regimes are found for the wind-viscosity balance: we classify them according to the attractor in the phase-plane of the truncated system and to the shift of the spectral mean. A downshift can coexist with both net forcing and damping, i.e., attraction to period-1 or period-2 solutions. Upshift is associated with stronger winds, i.e., to a net forcing where the attractor is always a period-1 solution. The applicability of our classification to experiments in long wave-tanks is verified.
17 citations
TL;DR: In this paper, the evolution of unidirectional nonlinear sea surface waves is calculated numerically by means of solution of the Euler equations, and wave dynamics correspond to quasi-equilibrium states characterized by JONSWAP spectra.
Abstract: The evolution of unidirectional nonlinear sea surface waves is calculated numerically by means of solution of the Euler equations. The wave dynamics corresponds to quasi-equilibrium states characterized by JONSWAP spectra. The spatiotemporal data are collected and processed providing information about the wave height probability and typical appearance of abnormally high waves (rogue waves). The waves are considered at different water depths ranging from deep to relatively shallow cases (k
p
h > 0.8, where k
p is the peak wavenumber, and h is the local depth). The asymmetry between front and rear rogue wave slopes is identified; it becomes apparent for sufficiently high waves in rough sea states at all considered depths k
p
h ≥ 1.2. The lifetimes of rogue events may reach up to 30–60 wave periods depending on the water depth. The maximum observed wave has a height of about three significant wave heights. A few randomly chosen in situ time series from the Baltic Sea are in agreement with the general picture of the numerical simulations.
16 citations
Cites background from "Wave amplification in the framework..."
...Thus, extreme wave statistics due to the modulational instability takes place in transient states; the effects of modulational instability are much less pronounced in stationary sea conditions (see discussion in Slunyaev and Sergeeva 2011; Slunyaev et al. 2015)....
[...]
...The nonlinear mechanism of adiabatic wave enhancement due to decreasing water depth is discussed in Slunyaev et al. (2015)....
[...]
TL;DR: In this paper, a method for the analysis of groups of unidirectional waves on the surface of deep water, which is based on spectral data of the scattering problem in the approximation of a nonlinear Schrodinger equation, is proposed.
Abstract: We propose a method for the analysis of groups of unidirectional waves on the surface of deep water, which is based on spectral data of the scattering problem in the approximation of a nonlinear Schrodinger equation. The main attention is paid to the robustness and accuracy of the numerically obtained spectral data. Various methods of choosing the wave number of the carrier wave, which rely on the analysis of the local Fourier transform and the zero-crossing wave analysis, are considered. The most robust wave numbers have been chosen on the basis of two model examples. A method for improving the accuracy of the soliton amplitude prediction, which uses the “feedback” in solving the associated scattering problem, is proposed. In the wave steepness range from 0.15 to 0.30, the accuracy of determining the amplitude of the soliton group by this technique lies in a range of 10%.
13 citations
10 Aug 2017
TL;DR: In this paper, the Wigner transform and the Penrose condition are used to recover spatially periodic unstable wave modes, which are called unstable Penrose modes, and their parameters are obtained by resolving the penrose condition, a system of nonlinear equations involving P(k).
Abstract: In this paper, we model Rogue Waves as localized instabilities emerging from homogeneous and stationary background wavefields, under NLS dynamics. This is achieved in two steps: given any background Fourier spectrum P(k), we use the Wigner transform and Penrose’s method to recover spatially periodic unstable modes, which we call unstable Penrose modes. These can be seen as generalized Benjamin–Feir modes, and their parameters are obtained by resolving the Penrose condition, a system of nonlinear equations involving P(k). Moreover, we show how the superposition of unstable Penrose modes can result in the appearance of localized unstable modes. By interpreting the appearance of an unstable mode localized in an area not larger than a reference wavelength $$\lambda _0$$
as the emergence of a Rogue Wave, a criterion for the emergence of Rogue Waves is formulated. Our methodology is applied to $$\delta $$
spectra, where the standard Benjamin–Feir instability is recovered, and to more general spectra. In that context, we present a scheme for the numerical resolution of the Penrose condition and estimate the sharpest possible localization of unstable modes.
13 citations
References
More filters
Journal Article•
2,856 citations
TL;DR: In this article, a systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering, where the form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator.
Abstract: A systematic method is developed which allows one to identify certain important classes of evolution equations which can be solved by the method of inverse scattering The form of each evolution equation is characterized by the dispersion relation of its associated linearized version and an integro-differential operator A comprehensive presentation of the inverse scattering method is given and general features of the solution are discussed The relationship of the scattering theory and Backlund transformations is brought out In view of the role of the dispersion relation, the comparatively simple asymptotic states, and the similarity of the method itself to Fourier transforms, this theory can be considered a natural extension of Fourier analysis to nonlinear problems
2,746 citations
"Wave amplification in the framework..." refers methods in this paper
...hion as it was applied to the Korteweg – de Vries equation by Pelinovskii, 1971 and Johnson, 1973); this problem is solvable by means of the Inverse Scattering Transform (Zakharov & Shabat, 1972; Ablowitz et al, 1974). The solution to the scattering problem for sech-like pulse was found by Satsuma & Yajima (1974) and discussed in details in Kharif et al (2001). It follows that the amplified pulse will result i...
[...]
TL;DR: In this article, a number of ases in which these equations reduce to a one dimensional nonlinear Schrodinger (NLS) equation are enumerated, and several analytical solutions of NLS equations are presented, with discussion of their implications for describing the propagation of water waves.
Abstract: Equations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrodinger (NLS) equation are enumerated.Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.
1,318 citations
TL;DR: A review of physical mechanisms of the rogue wave phenomenon is given in this article, where the authors demonstrate that freak waves may appear in deep and shallow waters and demonstrate that these mechanisms remain valid but should be modified.
Abstract: 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 Schrodinger 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.
962 citations
TL;DR: This work presents the first experimental results with observations of the Peregrine soliton in a water wave tank, and proposes a new approach to modeling deep water waves using the nonlinear Schrödinger equation.
Abstract: The conventional definition of rogue waves in the ocean is that their heights, from crest to trough, are more than about twice the significant wave height, which is the average wave height of the largest one-third of nearby waves. When modeling deep water waves using the nonlinear Schr\"odinger equation, the most likely candidate satisfying this criterion is the so-called Peregrine solution. It is localized in both space and time, thus describing a unique wave event. Until now, experiments specifically designed for observation of breather states in the evolution of deep water waves have never been made in this double limit. In the present work, we present the first experimental results with observations of the Peregrine soliton in a water wave tank.
950 citations
"Wave amplification in the framework..." refers methods in this paper
...oup) in laboratory and numerical simulations by Slunyaev et al (2013); ii) the NLS rational multi-breather solutions were shown to be replicable in experimental conditions for rather steep waves (see Chabchoub et al (2011) and the sequel of works by this team); iii) the NLS model is capable of adequate description of statistics of steep nonlinear wave groups evolution in a flume (Shemer et al, 2010b). Thus, we believe ...
[...]