Showing papers on "Rogue wave published in 2016"
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: The breather, rogue wave, and semirational solutions of the HGNLS equation can be converted into the nonpulsating soliton solutions and the novel interactions between the rogue waves and other nonlinear waves are displayed.
Abstract: We study the nonlinear waves on constant backgrounds of the higher-order generalized nonlinear Schrodinger (HGNLS) equation describing the propagation of ultrashort optical pulse in optical fibers. We derive the breather, rogue wave, and semirational solutions of the HGNLS equation. Our results show that these three types of solutions can be converted into the nonpulsating soliton solutions. In particular, we present the explicit conditions for the transitions between breathers and solitons with different structures. Further, we investigate the characteristics of the collisions between the soliton and breathers. Especially, based on the semirational solutions of the HGNLS equation, we display the novel interactions between the rogue waves and other nonlinear waves. In addition, we reveal the explicit relation between the transition and the distribution characteristics of the modulation instability growth rate.
193 citations
TL;DR: Recent numerical and experimental data suggest that the probability of the appearance of rogue waves in a chaotic wave state in such systems increases when the initial state is a random function of sufficiently high amplitude.
Abstract: Turbulence in dynamical systems is one of the most intriguing phenomena of modern science. Integrable systems offer the possibility to understand, to some extent, turbulence. Recent numerical and experimental data suggest that the probability of the appearance of rogue waves in a chaotic wave state in such systems increases when the initial state is a random function of sufficiently high amplitude. We provide explanations for this effect.
188 citations
TL;DR: This work analyzes several sets of field data in various European locations with various tools, and finds that the main generation mechanism for rogue waves is the constructive interference of elementary waves enhanced by second-order bound nonlinearities and not the modulational instability.
Abstract: Since the 1990s, the modulational instability has commonly been used to explain the occurrence of rogue waves that appear from nowhere in the open ocean. However, the importance of this instability in the context of ocean waves is not well established. This mechanism has been successfully studied in laboratory experiments and in mathematical studies, but there is no consensus on what actually takes place in the ocean. In this work, we question the oceanic relevance of this paradigm. In particular, we analyze several sets of field data in various European locations with various tools, and find that the main generation mechanism for rogue waves is the constructive interference of elementary waves enhanced by second-order bound nonlinearities and not the modulational instability. This implies that rogue waves are likely to be rare occurrences of weakly nonlinear random seas.
185 citations
TL;DR: Real-time measurements are performed in an optical fibre system of the unstable breakup of a continuous wave field, simultaneously characterizing emergent modulation instability breather pulses and their associated statistics, which allow quantitative comparison between experiment, modelling and theory.
Abstract: Modulation instability is a fundamental process of nonlinear science, leading to the unstable breakup of a constant amplitude solution of a physical system. There has been particular interest in studying modulation instability in the cubic nonlinear Schrodinger equation, a generic model for a host of nonlinear systems including superfluids, fibre optics, plasmas and Bose-Einstein condensates. Modulation instability is also a significant area of study in the context of understanding the emergence of high amplitude events that satisfy rogue wave statistical criteria. Here, exploiting advances in ultrafast optical metrology, we perform real-time measurements in an optical fibre system of the unstable breakup of a continuous wave field, simultaneously characterizing emergent modulation instability breather pulses and their associated statistics. Our results allow quantitative comparison between experiment, modelling and theory, and are expected to open new perspectives on studies of instability dynamics in physics.
178 citations
TL;DR: The modulational stability regions with zero growth rate coincide with the transition condition using rogue wave eigenvalues, which could be useful for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in diverse physical systems modeled by vc-NLS equation with higher-order effects.
Abstract: We study a variable-coefficient nonlinear Schrodinger (vc-NLS) equation with higher-order effects. We show that the breather solution can be converted into four types of nonlinear waves on constant backgrounds including the multipeak solitons, antidark soliton, periodic wave, and W-shaped soliton. In particular, the transition condition requiring the group velocity dispersion (GVD) and third-order dispersion (TOD) to scale linearly is obtained analytically. We display several kinds of elastic interactions between the transformed nonlinear waves. We discuss the dispersion management of the multipeak soliton, which indicates that the GVD coefficient controls the number of peaks of the wave while the TOD coefficient has compression effect. The gain or loss has influence on the amplitudes of the multipeak soliton. We further derive the breather multiple births and Peregrine combs by using multiple compression points of Akhmediev breathers and Peregrine rogue waves in optical fiber systems with periodic GVD modulation. In particular, we demonstrate that the Peregrine comb can be converted into a Peregrine wall by the proper choice of the amplitude of the periodic GVD modulation. The Peregrine wall can be seen as an intermediate state between rogue waves and W-shaped solitons. We finally find that the modulational stability regions with zero growth rate coincide with the transition condition using rogue wave eigenvalues. Our results could be useful for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in diverse physical systems modeled by vc-NLS equation with higher-order effects.
141 citations
TL;DR: In this paper, a generalized Darboux transformation was used to construct the N-bright soliton solution in a compact determinant form, the N -breather solution including the Akhmediev breather and a general higher order rogue wave solution.
114 citations
TL;DR: By injecting two colliding and modulated pumps with orthogonal states of polarization in a randomly birefringent telecommunication optical fiber, this work provides the first experimental demonstration of an optical dark rogue wave and introduces the concept of multi-component analog gravity.
Abstract: Photonics enables to develop simple lab experiments that mimic water rogue wave generation phenomena, as well as relativistic gravitational effects such as event horizons, gravitational lensing and Hawking radiation. The basis for analog gravity experiments is light propagation through an effective moving medium obtained via the nonlinear response of the material. So far, analogue gravity kinematics was reproduced in scalar optical wave propagation test models. Multimode and spatiotemporal nonlinear interactions exhibit a rich spectrum of excitations, which may substantially expand the range of rogue wave phenomena, and lead to novel space-time analogies, for example with multi-particle interactions. By injecting two colliding and modulated pumps with orthogonal states of polarization in a randomly birefringent telecommunication optical fiber, we provide the first experimental demonstration of an optical dark rogue wave. We also introduce the concept of multi-component analog gravity, whereby localized spatiotemporal horizons are associated with the dark rogue wave solution of the two-component nonlinear Schrodinger system.
109 citations
TL;DR: Breathers and rogue waves as exact solutions of a nonlocal partial differential equation of the third order are derived by a bilinear transformation and demonstrated analytically to exist for special classes of nonlocal equations relevant to optical waveguides.
89 citations
TL;DR: Based on a suitable ansatz approach and Hirota's bilinear form, kink solitary wave, rogue wave and mixed exponential-algebraic solitary wave solutions of (2+1)-dimensional Burgers equation are derived as discussed by the authors.
Abstract: Based on a suitable ansatz approach and Hirota’s bilinear form, kink solitary wave, rogue wave and mixed exponential–algebraic solitary wave solutions of (2+1)-dimensional Burgers equation are derived. The completely non-elastic interaction between kink solitary wave and rogue wave for the (2+1)-dimensional Burgers equation are presented. These results enrich the variety of the dynamics of higher dimensional nonlinear wave field.
85 citations
TL;DR: Based on the dynamical behavior of solution, the classification of the localized wave solutions on the nonzero background is given explicitly and the parameter conditions for breather, dark soliton and rogue wave solution of mCNLSE are given in detail.
TL;DR: In this article, a conceptually new approach to the formation of rogue waves (RWs) is proposed, in which appropriate, locally coherent structures are specifically isolated from a globally incoherent wave train to be subsequently analyzed by implementing a numerical inverse scattering transform (IST) procedure relying on a spatial periodization of the object under consideration.
Abstract: The nonlinear Schrodinger equation (NLSE) stands out as the dispersive nonlinear partial differential equation that plays a prominent role in the modeling and understanding of the wave phenomena relevant to many fields of nonlinear physics. The question of random input problems in the one-dimensional and integrable NLSE enters within the framework of integrable turbulence, and the specific question of the formation of rogue waves (RWs) has been recently extensively studied in this context. The determination of exact analytic solutions of the focusing 1D-NLSE prototyping RW events of statistical relevance is now considered as the problem of central importance. Here we address this question from the perspective of the inverse scattering transform (IST) method that relies on the integrable nature of the wave equation. We develop a conceptually new approach to the RW classification in which appropriate, locally coherent structures are specifically isolated from a globally incoherent wave train to be subsequently analyzed by implementing a numerical IST procedure relying on a spatial periodization of the object under consideration. Using this approach we extend the existing classifications of the prototypes of RWs from standard breathers and their collisions to more general nonlinear modes characterized by their nonlinear spectra.
TL;DR: In this article, the formation of 3D acoustic rogue waves in dusty plasmas was investigated. But the authors focused on one-dimensional (1D) wave behaviour through modulation instabilities, and to a lesser extent on higher-dimensional behaviour.
Abstract: Rogue waves have been observed in fluids and other wave contexts. Experiments now show the formation of 3D acoustic rogue waves in dusty plasmas; they result from wave–particle interactions driving the dust particles into high-amplitude dynamics. Rogue waves—rare uncertainly emerging localized events with large amplitudes—have been experimentally observed in many nonlinear wave phenomena, such as water waves1,2,3,4,5,6, optical waves7,8, second sound in superfluid He II (ref. 9) and ion acoustic waves in plasmas10. Past studies have mainly focused on one-dimensional (1D) wave behaviour through modulation instabilities1,3,4,5,7,11, and to a lesser extent on higher-dimensional behaviour5,6,8,11,12. The question whether rogue waves also exist in nonlinear 3D acoustic-type plasma waves, the kinetic origin of their formation and their correlation with surrounding 3D waveforms are unexplored fundamental issues. Here we report the direct experimental observation of dust acoustic rogue waves in dusty plasmas and construct a picture of 3D particle focusing by the surrounding tilted and ruptured wave crests, associated with the higher probability of low-amplitude holes for rogue-wave generation.
TL;DR: These findings provide the first experimental demonstration that the DRWs could be observed in the soliton explosion regime and further enhance the understanding of the physical mechanism of optical RW generation.
Abstract: We reported on the observation of dissipative rogue waves (DRWs) induced by soliton explosions in an ultrafast fiber laser. It was found that the soliton explosions could be obtained in the fiber laser at a critical pump power level. During the process of the soliton explosion, the high-amplitude waves that fulfill the rogue wave criteria could be detected. The appearance of the DRWs was identified by characterizing the intensity statistics of the time-stretched soliton profile based on the dispersive Fourier-transform method. Our findings provide the first experimental demonstration that the DRWs could be observed in the soliton explosion regime and further enhance the understanding of the physical mechanism of optical RW generation.
TL;DR: The role of multiple soliton and breather interactions in the formation of very high waves is disclosed within the framework of the integrable modified Korteweg-de Vries (MKdV) equation, when solitons possess "frozen" phases (certain polarities).
Abstract: The role of multiple soliton and breather interactions in the formation of very high waves is disclosed within the framework of the integrable modified Korteweg-de Vries (MKdV) equation. Optimal conditions for the focusing of many solitons are formulated explicitly. Namely, trains of ordered solitons with alternate polarities evolve to huge strongly localized transient waves. The focused wave amplitude is exactly the sum of the focusing soliton heights; the maximum wave inherits the polarity of the fastest soliton in the train. The focusing of several solitary waves or/and breathers may naturally occur in a soliton gas and will lead to rogue-wave-type dynamics; hence, it represents a new nonlinear mechanism of rogue wave generation. The discovered scenario depends crucially on the soliton polarities (phases), and is not taken into account by existing kinetic theories. The performance of the soliton mechanism of rogue wave generation is shown for the example of the focusing MKdV equation, when solitons possess "frozen" phases (certain polarities), though the approach is efficient in some other integrable systems which admit soliton and breather solutions.
TL;DR: In this paper, the authors studied the relations between modulational instability and several well-known nonlinear excitations in a nonlinear fiber such as bright soliton, nonlinear continuous wave, Akhmediev breather, Peregrine rogue wave, and Kuznetsov-Ma breather.
Abstract: We study the relations between modulational instability and several well-known nonlinear excitations in a nonlinear fiber such as bright soliton, nonlinear continuous wave, Akhmediev breather, Peregrine rogue wave, and Kuznetsov–Ma breather We present a quantitative correspondence between them based on the dominant frequency and propagation constant of each perturbation on a continuous-wave background We especially demonstrate that rogue wave comes from modulational instability with the resonance perturbation on continuous-wave background The numerical simulations are performed to test these theoretical results These results will deepen our understanding of rogue wave excitation and be helpful for controllable nonlinear wave excitations in nonlinear fiber and other nonlinear systems
TL;DR: In this paper, the authors proposed a semi-classical model of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schrodinger (NLS) equation with the initial condition in the form of a rectangular barrier (a "box").
Abstract: We propose a novel, analytically tractable, scenario of the rogue wave formation in the framework of the small-dispersion focusing nonlinear Schrodinger (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 large-amplitude quasi-periodic breather lattices whose amplitude profiles are closely approximated by the Akhmediev and Peregrine breathers within certain space-time 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.
TL;DR: An experimental study confirming that extreme localizations in an irregular oceanic Joint North Sea Wave Project wave field can be tracked back to originate from exact NLSE breather solutions, such as the Peregrine breather.
Abstract: Breather solutions of the nonlinear Schrodinger equation (NLSE) are known to be considered as backbone models for extreme events in the ocean as well as in Kerr media. These exact deterministic rogue wave (RW) prototypes on a regular background describe a wide range of modulation instability configurations. Alternatively, oceanic or electromagnetic wave fields can be of chaotic nature and it is known that RWs may develop in such conditions as well. We report an experimental study confirming that extreme localizations in an irregular oceanic Joint North Sea Wave Project wave field can be tracked back to originate from exact NLSE breather solutions, such as the Peregrine breather. Numerical NLSE as well as modified NLSE simulations are both in good agreement with laboratory experiments and highlight the significance of universal weakly nonlinear evolution equations in the emergence as well as prediction of extreme events in nonlinear dispersive media.
TL;DR: It is reported that there are possibilities to ‘catch’ rogue waves through manipulating nonlinear function and gain/loss coefficient.
TL;DR: It is demonstrated that each multipeak soliton exhibits the coexistence of shape change and conservation of the localized energy of a light pulse against the continuous wave background.
Abstract: We study symmetric and asymmetric optical multipeak solitons on a continuous wave background in the femtosecond regime of a single-mode fiber. Key characteristics of such multipeak solitons, such as the formation mechanism, propagation stability, and shape-changing collisions, are revealed in detail. Our results show that this multipeak (symmetric or asymmetric) mode could be regarded as a single pulse formed by a nonlinear superposition of a periodic wave and a single-peak (W-shaped or antidark) soliton. In particular, a phase diagram for different types of nonlinear excitations on a continuous wave background, including the unusual multipeak soliton, the W-shaped soliton, the antidark soliton, the periodic wave, and the known breather rogue wave, is established based on the explicit link between exact solution and modulation instability analysis. Numerical simulations are performed to confirm the propagation stability of the multipeak solitons with symmetric and asymmetric structures. Further, we unveil a remarkable shape-changing feature of asymmetric multipeak solitons. It is interesting that these shape-changing interactions occur not only in the intraspecific collision (soliton mutual collision) but also in the interspecific interaction (soliton-breather interaction). Our results demonstrate that each multipeak soliton exhibits the coexistence of shape change and conservation of the localized energy of a light pulse against the continuous wave background.
TL;DR: In this paper, a generalized (2+1)-dimensional Boussinesq equation is used to describe the propagation of small-amplitude, long wave in shallow water.
Abstract: Under investigation in this work is a generalized (2+1)-dimensional Boussinesq equation, which can be used to describe the propagation of small-amplitude, long wave in shallow water. By virtue of Bell's polynomials, an effective way is presented to succinctly construct its bilinear form. Furthermore, based on the bilinear formalism and the extended homoclinic test method, the breather wave solution, rogue-wave solution and solitary-wave solution of the equation are well constructed. Our results can be used to enrich the dynamical behavior of the generalized (2+1)-dimensional nonlinear wave fields.
TL;DR: The properties of the chaotic wave fields generated in the frame of the Kundu-Eckhaus equation (KEE) are studied and it is showed that the probability of rogue wave occurrence significantly depends on the quintic and the Raman-effect nonlinear terms of the KEE.
Abstract: In this paper we study the properties of the chaotic wave fields generated in the frame of the Kundu-Eckhaus equation (KEE). Modulation instability results in a chaotic wave field which exhibits small-scale filaments with a free propagation constant, $k$. The average velocity of the filaments is approximately given by the average group velocity calculated from the dispersion relation for the plane-wave solution; however, direction of propagation is controlled by the $\ensuremath{\beta}$ parameter, the constant in front of the Raman-effect term. We have also calculated the probabilities of the rogue wave occurrence for various values of propagation constant $k$ and showed that the probability of rogue wave occurrence depends on $k$. Additionally, we have showed that the probability of rogue wave occurrence significantly depends on the quintic and the Raman-effect nonlinear terms of the KEE. Statistical comparisons between the KEE and the cubic nonlinear Schr\"odinger equation have also been presented.
TL;DR: In this paper, the exact solution of the Kadomtsev-Petviashvili equation is obtained via the bilinear transformation method via the presence of breathing and rogue waves.
Abstract: Breathers and rogue waves as exact solutions of the three-dimensional Kadomtsev—Petviashvili equation are obtained via the bilinear transformation method. The breathers in three dimensions possess different dynamics in different planes, such as growing and decaying periodic line waves in the (x, y), (x, z) and (y, t) planes. Rogue waves are localized in time, and are obtained theoretically as a long wave limit of breathers with indefinitely larger periods. It is shown that the rogue waves possess growing and decaying line profiles in the (x, y) or (x, z) plane, which arise from a constant background and then retreat back to the same background again.
TL;DR: In this paper, the authors considered the dynamics of random multi-soliton fields within the framework of the modified Korteweg-de Vries equation and showed that the results sufficiently depend on the soliton gas properties, i.e., whether it is unipolar or bipolar.
TL;DR: Based on the generalized Darboux transformation and formal series method, Wang et al. as discussed by the authors obtained the high-order rogue wave solution without the special limitation on the wave vectors, and showed that the distribution patterns for vector rogue waves are much more abundant than the ones for scalar rogue waves, and also different from the ones obtained with the constrain conditions on background fields.
Abstract: We study on dynamics of high-order rogue wave in two-component coupled nonlinear Schrodinger equations. Based on the generalized Darboux transformation and formal series method, we obtain the high-order rogue wave solution without the special limitation on the wave vectors. As an application, we exhibit the first, second-order rogue wave solutions and the superposition of them by computer plotting. We find the distribution patterns for vector rogue waves are much more abundant than the ones for scalar rogue waves, and also different from the ones obtained with the constrain conditions on background fields. The results further enrich and deepen our realization on rogue wave excitation dynamics in such diverse fields as Bose-Einstein condensates, nonlinear fibers, and superfluids.
TL;DR: In this paper, a hierarchy of N t h -order semi-rational solutions with 3 N free parameters of SSE can be calculated theoretically through a modified dressing transformation associated with a novel expansion technique.
Abstract: Sasa–Satsuma equation (SSE) is one of the nontrivial integrable extensions of nonlinear Schrodinger equation including third order dispersion, self-frequency shift and self-steepening. The hierarchy of N t h -order semi-rational solutions with 3 N free parameters of SSE can be calculated theoretically through a modified dressing transformation associated with a novel expansion technique. The Bloch eigenfunction is introduced and Schur multinomial is invoked. When the free parameters satisfy some constraints, these solutions can be reduced to pure rational solutions so as to study the dynamics of rogue waves. In addition, triangular and elliptical multi-rogue wave patterns in either single-peak or double-peak case are examined smoothly. On the other hand, the semi-rational solutions allow us to investigate various interesting superimposed scenes between rogue waves and breathers. These results may contribute to demonstrating the rogue wave phenomenon emerging in water waves, optical fibers and generally in dispersive nonlinear media.
TL;DR: Turbulence in integrable systems exhibits a noticeable scientific advantage: it can be expressed in terms of the nonlinear modes of these systems, particularly when the wave field contains both solitons and breathers, thus revealing intermediate characteristics.
Abstract: Turbulence in integrable systems exhibits a noticeable scientific advantage: it can be expressed in terms of the nonlinear modes of these systems. Whether the majority of the excitations in the system are breathers or solitons defines the properties of the turbulent state. In the two extreme cases we can call such states ``breather turbulence'' or ``soliton turbulence.'' The number of rogue waves, the probability density functions of the chaotic wave fields, and their physical spectra are all specific for each of these two situations. Understanding these extreme cases also helps in studies of mixed turbulent states when the wave field contains both solitons and breathers, thus revealing intermediate characteristics.
TL;DR: In this article, the authors studied the nonlinear stage of the modulational instability (MI) of cnoidal waves in the framework of the focusing one-dimensional nonlinear Schrodinger (NLS) equation.
Abstract: We study numerically the nonlinear stage of the modulational instability (MI) of cnoidal waves in the framework of the focusing one-dimensional nonlinear Schrodinger (NLS) equation. Cnoidal waves are exact periodic solutions of the NLS equation which can be represented as the lattices of overlapping solitons. The MI of these lattices leads to the development of 'integrable turbulence' (Zakharov 2009 Stud. Appl. Math. 122 219–34). We study the major characteristics of turbulence for the dn-branch of cnoidal waves and demonstrate how these characteristics depend on the degree of 'overlapping' between the solitons within the cnoidal wave. Integrable turbulence, which develops from the MI of the dn-branch of cnoidal waves, asymptotically approaches its stationary state in an oscillatory way. During this process, kinetic and potential energies oscillate around their asymptotic values. The amplitudes of these oscillations decay with time as , , the phases contain nonlinear phase shift decaying as t −1/2, and the frequency of the oscillations is equal to the double maximal growth rate of the MI, . In the asymptotic stationary state, the ratio of potential to kinetic energy is equal to −2. The asymptotic PDF of the wave intensity is close to the exponential distribution for cnoidal waves with strong overlapping, and is significantly non-exponential for cnoidal waves with weak overlapping of the solitons. In the latter case, the dynamics of the system reduces to two-soliton collisions, which occur at an exponentially small rate and provide an up to two-fold increase in amplitude compared with the original cnoidal wave. For all cnoidal waves of the dn-branch, the rogue waves at the time of their maximal elevation have a quasi-rational profile similar to that of the Peregrine solution.
TL;DR: In this paper, the wavelet transform over the Fourier transform was used for early detection of rogue waves in a chaotic wave field, and the triangular wavelet spectra of the rogue waves can be detected at early stages of the development of the wave.
TL;DR: In this paper, a generalized coupled nonlinear Schrodinger system with higher-order terms was constructed to describe the propagation properties of solitons in two ultrashort optical fields, and the N-fold Darboux transformation (DT) has been constructed.
Abstract: Under investigation in this paper is a generalized coupled nonlinear Schrodinger system with higher-order terms, which describes the propagation properties of ultrashort solitons in two ultrashort optical fields. Based on the 3
$$\times $$
3 lax pair, the N-fold Darboux transformation (DT) has been constructed. Several kinds of solitons, breathers, and rogue wave solutions are generated on the vanishing and nonvanishing backgrounds by virtue of the DT. Figures are plotted to reveal the dynamic features of those solutions: (1) elastic interactions between two solitons; (2) mutual attractions and repulsions of bound solitons; (3) propagation properties of Ma-breathers, Akhmediev breathers, two-breathers, and rogue waves. The results show that the rogue waves can result from two different ways: the limit process of Ma-breathers and Akhmediev breathers.