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The standard nonlinear Schrödinger Equation (NLSE) is one of the elegant equations to find detailed information about the modulational instability criteria of dust-ion-acoustic (DIA) waves and associated DIA rogue waves (DIARWs) in a three-component dusty plasma medium with inertialess super-thermal kappa distributed electrons, and inertial warm positive ions and negative dust grains. It can be seen that the plasma system supports both fast and slow DIA modes under consideration of inertial warm ions along with inertial negatively charged dust grains. It is also found that the modulationally stable parametric regime decreases with κ. The numerical analysis has also shown that the amplitude of the first and second-order DIARWs decreases with ion temperature. These results are to be considered the cornerstone for explaining the real puzzles in space and laboratory dusty plasmas.

In this paper, we investigate breather positons and higher-order rogue waves for the nonlocal Fokas-Lenells equation. In this nonlocal optical system, rogue waves can be generated when periods of breather positons go to infinity. In addition, we find two very interesting phenomena: one is that rogue waves sitting on a periodic line wave background are derived; the other is that a hybrid of rogue waves and a periodic kink wave is also constructed. We believe that these interesting findings exist in the optical system corresponding to the nonlocal Fokas-Lenells equation.

This article considers additional phenomena that complement the earlier topics addressed by Ibrahim [(Liquid Sloshing Dynamics: Theory and Applications. Cambridge University Press, Cambridge, 2005), (ASME J Fluids Eng 137(9):090801, 2015)]. The first phenomenon is the localized Faraday waves known as oscillons, which were observed in granular materials and liquid layers subjected to parametric excitation. Extreme waves, known as rogue, generated in the Faraday surface ripples, are related to the increase in the horizontal mobility of oscillating solitons (oscillons), and their horizontal motion is random over a limited range of excitation acceleration amplitude. Parametric excitation of water in a Hele–Shaw cell and the associated localized standing surface waves of large amplitude will be discussed. The surface wave pattern exhibited a certain similarity with the three-dimensional axisymmetric oscillon. Faraday waves in superfluid Fermi–Bose mixtures and their wave function will be addressed in terms of position and time as described by the Schrodinger equation with time-dependent parabolic potential. The phenomenon of walking fluid droplets on Faraday waves constitutes the majority portion of this article. Different regimes of droplet motion in terms of droplet physical properties, the fluid bath excitation acceleration amplitude and frequency will be discussed. The droplet trajectory diffraction, when passes through a slit, shares the same random features of electron diffraction. The duality of the droplet-wave field together with the path-memory-driven nonlocality and other related topics will be assessed. This article is complemented with the fascinating phenomenon of the stone and bombs skipping/ricochet over water surface.

We define a quantized nonlinear pulse ‘volume’. This then allows us to determine the characteristic of a possible rogue wave by looking at its volume, found from a surface integral. Where higher powers are needed in the integrand, due to slow decay of a pulse, we describe the excitation as a ‘semi-rogue’ wave.

In this investigation, the characteristics of ion acoustic (IA) solitary, breathers, and freak waves have been examined by using spin evolution quantum hydrodynamic model in a degenerate quantum pl...

In present paper, multiple rogue wave solutions of (2+1)-dimensional Yu–Toda–Sasa–Fukuyama equation were studied by applying the traveling wave transformation and the Hirota bilinear method. We obt...

A theoretical investigation is made to study the properties of dust–acoustic (DA) waves, and corresponding dust–acoustic rogue waves (DARWs) in a magnetized electron-depleted dusty plasma that contains opposite polarity warm dust grains, external magnetic field, and nonextensive distributed ions. By using a multiscale reductive perturbation technique (RPT), the nonlinear Schrodinger (NLS) equation is derived in this model. The effects of magnetic field parameter and nonextensive ions are examined on the profiles of the modulational instability. It is found that with the increase of the magnetic field parameter, the instability growth rate and bandwidth increase (decrease) in the case of positive (negative) nonextensive distributed ions. Moreover, the characteristics (spatial wavelength, width, and amplitude) of nonlinear coherent structures involving the bright and dark solitons, the fundamental and second-order rogue waves are also presented in detail. It is observed that these characteristics are significantly modified by effects of magnetic field parameter and nonextensive ions. The results of the present investigation may be applicable to understanding the characteristics and basic nonlinear structures of magnetized plasma environments both in space and laboratory situations.

General rogue waves in (1+1)-dimensional three-wave resonant interaction systems are derived by the bilinear method. These solutions are divided into three families, which correspond to a simple root, two simple roots and a double root of a certain quartic equation arising from the dimension reduction respectively. It is shown that while the first family of solutions associated with a simple root exist for all signs of the nonlinear coefficients in the three-wave interaction equations, the other two families of solutions associated with two simple roots and a double root can only exist in the so-called soliton-exchange case, where the nonlinear coefficients have certain signs. Many of these rogue wave solutions, such as those associated with two simple roots, and higher-order solutions associated with a simple root, are new solutions which have not been reported before. Technically, our bilinear derivation of rogue waves for the double-root case is achieved by a generalization to the previous dimension reduction procedure in the bilinear method, and this generalized procedure allows us to treat roots of arbitrary multiplicities. Dynamics of the derived rogue waves is also examined, and new rogue-wave patterns are presented. Connection between these bilinear rogue waves and those derived earlier by Darboux transformation is also explained., A revised version, where Theorem 3 is further generalized, so that additional new rogue waves are obtained. In addition, boundary conditions for rational solutions in Theorem 2 are also proved

We investigate the vector breather waves and higher-order rouge waves of the coupled higher-order nonlinear Schrodinger equations via the Darboux-dressing transformation. Firstly, based on the Darb...

The parameter limit method on the basis of Hirota’s bilinear method is proposed to construct the rogue wave solutions for nonlinear partial differential equations (NLPDEs). Some real and complex differential equations are used as concrete examples to illustrate the effectiveness and correctness of the described method. The rogue waves and homoclinic solutions of different structures are obtained and simulated by three-dimensional graphics, respectively. More importantly, we find that rogue wave solutions and homoclinic solutions appear in pairs. That is to say, for some NLPDEs, if there is a homoclinic solution, then there must be a rogue wave solution. The twin phenomenon of rogue wave solutions and homoclinic solutions of a class of NLPDEs is discussed.

Based on the bilinear method, rational lump and mixed lump-solitary wave solutions to an extended (2+1)-dimensional KdV equation are constructed through the different assumptions of the auxiliary function in the trilinear form. It is found that the rational lump decays algebraically in all directions in the space plane and its amplitude possesses one maximum and two minima. One kind of the mixed solution describes the interaction between one lump and one line solitary wave, which exhibits fission and fusion phenomena under the different parameters. The other kind of the mixed solution shows one lump interacting with two paralleled line solitary waves, in which the evolution of the lump gives rise to a two-dimensional rogue wave. This shows that these three interesting phenomena exist in the corresponding physical model.

Based on the Hirota bilinear form for the (3 + 1)-dimensional Jimbo–Miwa equation, we constructed the first-order, second-order, third-order and fourth-order rogue waves for this equation using the symbolic computation approach. Also some properties of the higher-order rogue waves and their interaction are explained by some figures via some special choices of the parameters.

Quality control measures for ocean waves observations are necessary to give confidence of their accuracy. It is common practice to detect anomalies or outliers in surface displacement observations by applying a standard deviation threshold. Besides being a purely statistical method, this quality control procedure is likely to flag extreme wave events erroneously, thereby impacting higher-order descriptions of the wave field. In this paper we extend the use of the statistical phase-space threshold, an established outlier detection method in the field of turbulence, to detect anomalies in a wave record. We show that a wave record in phase space (here defined as a diagram of displacement against acceleration) can be enclosed by a predictable ellipse where the major and minor axes are defined by the spectral properties of the wave field. By using the parameterized ellipse in phase space as a threshold to identify wave anomalies, this is a semiphysical filtering method. Wave buoy data obtained from a mooring deployed near King George Island, Antarctica [as part of the Antarctic Modeling Observation System (ATMOS)], and laser altimeter data obtained at the Northwest Shelf of Australia were used to demonstrate the functioning of the filtering methodology in identifying wave anomalies. Synthetic data obtained using a high-order spectral model are used to identify how extreme waves are positioned in phase space.

Freak waves are a type of extreme waves with great wave heights and concentrated energies. When they encounter ships, large amounts of water flow onto deck and lead to strong impact pressures, thre...

In this paper, the semi-rational solutions that causes vector rogue waves and breathers can be obtained by using the Darboux dressing transformation. We studied vector rogue waves and the interaction between rogue waves and light-dark solitons, and observed that during the interaction, due to the interference between the light-dark components of the solitons, a respiration-like structures appears. Besides, it can be observed that the rogue waves and soliton merge together. Moreover, the main characteristics of the interactions between the breathers and bright-dark solitons are displayed with some graphics.

In this paper, the inverse scattering approach is applied to the Kundu-Eckhaus equation with two cases of zero boundary condition (ZBC) and nonzero boundary conditions (NZBCs) at infinity. Firstly, we obtain the exact formulae of soliton solutions of three cases of N simple poles, one higher-order pole and multiple higher-order poles via the associated Riemann-Hilbert problem (RHP). Moreover, given the initial data that allow for the presence of discrete spectrum, the higher-order rogue waves of the equation are presented. For the case of NZBCs, we can construct the infinite order rogue waves through developing a suitable RHP. Finally, by choosing different parameters, we aim to show some prominent characteristics of this solution and express them graphically in detail. Our results should be helpful to further explore and enrich the related nonlinear wave phenomena.

Nonlinear evolution equation is a research hot spot in the field of science and engineering. Recently, searching for breather and rogue wave solutions to nonlinear evolution equations has become a popular topic in nonlinear mathematical physics. In this paper, with the help of symbolic calculation, breather and rogue wave solutions to the two generalized (3+1)-dimensional Jimbo-Miwa equation are investigated, respectively. Those analytic solutions are presented, and the influences of the corresponding parameters on breathers and rogue waves are analyzed. Their dynamic properties are discussed based on graphical presentation. Amplitude of breathers and rogue waves are adjusted. Results of this study are helpful to the generation of breather and rogue waves in applied mechanics and physics.

Super rogue ion-acoustic waves are proposed as a physical catalyst for the heavy hydrocarbon ions formation in the Titan ionosphere. We justified that analytically and numerically by probing a Titan referenced plasma system, consists of the most abundant positive ions and superthermal electrons. A solution of the nonlinear Schrodinger equation (NLSE) has provided us by the plasma (un) stable regions at altitude 900–1200 km from the Titan surface with superthermal parameter values, relative ion to electron densities, and temperature ratio variations. Our results are not only agreed with the Cassini data but also predict a chemistry independent approach for the heavy hydrocarbons’ formation conditions.

In this paper, the nonlinear ion acoustic rogue waves (IARWs) are investigated in an unmagnetized, collisionless plasma which consist of positive ions, superthermal electrons and positrons, respectively. A nonlinear Schrodinger equation (NLSE) of IAW packets has been obtained by using the reductive perturbation technique. Furthermore, the effects of superthermal electrons and positrons on the nonlinear dispersion relation, the group velocity and the modulational instability of nonlinear ion acoustic wave (IAW) packets have been studied in detail. At the same time, the characteristics of first- and second-order IARW also have been discussed. It seems that there is some significant influence of superthermal electrons and positrons on the nonlinear ion acoustic waves. Meanwhile, it is also found that the first- and second-order IARW consisting of Peregrine breathers can be excited by using the modulational instability in this system.

The nonlinear propagation of dust acoustic (DA) waves (DAWs) in an unmagnetized electron depleted dusty plasma (containing opposite polarity warm dust grains and non-extensive positive ions) has been theoretically investigated. The nonlinear Schrodinger equation (NLSE) is derived by employing the reductive perturbation method. It is observed that the dusty plasma system under consideration supports two branches of modes, namely, fast and slow DA modes, and that both of these two modes can be modulationally stable or unstable depending on the sign of ratio of the dispersive and nonlinear coefficients of the NLSE. The conditions for the formation of bright and dark envelope solitons as well as the first- and second-order rogue waves have been examined. The numerical analysis has shown that the basic features (viz., stability/instability, amplitude and width of the envelope solitons and rogue structures, etc.) of the DAWs associated with the fast and slow DA modes are significantly modified by dust masses, dust charges, non-extensivity of ions, and other various plasma parameters. The implications of the results to space plasma research (viz., magnetosphere of Jupiter, upper mesosphere, Saturn’s F-ring, cometary tail, etc.) are briefly discussed.

Based on a direct variable transformation, we obtain multiple rogue wave solutions of a generalized (3 + 1)-dimensional variable-coefficient nonlinear wave equation, including first-order, two-order and three-order rogue wave solutions. Their dynamic behaviors are shown by some 3D plots. Compared with Zha’s symbolic computation approach, we do not need to resort to Hirota bilinear form, and it can be used to deal with variable-coefficient integrable equations. Interaction solution between rogue wave and periodic wave is obtained by using the Hirota bilinear form. Abundant breather wave solutions are presented by a direct test function.

In this paper, on the background of general periodic travelling wave solutions, we construct rogue wave solutions of the fifth-order Ito equation. In particular, those solutions cover the known results in the literature. By means of the Darboux transformation, we derive one-, two- and three-fold rogue wave solutions on the periodic travelling wave solutions background. We provide several illustrations of such rogue waves and analyze their generation mechanisms.

In this paper, outcomes of the study on the Backlund transformation, Lax pair, and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics, fluid mechanics, and plasma physics are presented. Via the Hirota bilinear method, a bilinear Backlund transformation is obtained, based on which a Lax pair is constructed. Via the symbolic computation, mixed rogue–solitary and rogue–periodic wave solutions are derived. Interactions between the rogue waves and solitary waves, and interactions between the rogue waves and periodic waves, are studied. It is found that (1) the one rogue wave appears between the two solitary waves and then merges with the two solitary waves; (2) the interaction between the one rogue wave and one periodic wave is periodic; and (3) the periodic lump waves with the amplitudes invariant are depicted. Furthermore, effects of the noise perturbations on the obtained solutions will be investigated.

In linear theory the formation of extreme waves their existence is interpreted as a local superposition of surface monochromatic waves. Natural water areas are resonators that have their own set of natural oscillations – standing waves of stable spatial structure and fixed period. In the spectra of waves of many water bodies of World Ocean observed double high waves, this is explained by the tidal-seiche resonance. During a storm, the energy of natural oscillations increases ten times the background energy, during a tsunami it can increase up to three orders of magnitude. Examples of the effects of natural oscillations on the coast are given, and it is reported about the increased probability of the occurrence on the coast freak waves. Additionally, it is noted that natural oscillations in water mass are a normal state for any body of water at any time of its existence. The corresponding indices of the water fluctuations of the water basins are given. The events of extreme waves during the accidents at DniproHES (Zaporizhia) on August 18, 1941, and the Kurenivsky dam (Kyiv) on March 13, 1961, are presented. The excitement of the freak wave can be interpreted as enhancing the natural oscillations of the water basin, represented by standing waves of stable spatial structure, fixed period and high probability of waves in the water body. This does not contradict the linear theory of the resonant formation of abnormally high waves. The purpose of the article is to investigate possible sources of the excitement of freak waves, the results are proposed to be implemented in the development of countermeasures to the destructive process. However, the waves carry out both destructive and creative work. A task is presented, which involves the development of measures to stimulate extreme waves. This will increase electricity generation. Affiliation of dam-break waves to freak waves can be doubtful. However, they formally correspond to the classical condition of double exceeding the significant wave height. Most water basins are integral anthropogenic sites. The variability of both natural and anthropogenic environments forces the overriding of systematization and definition. It is proposed to attribute extreme waves of dam-break waves to freak waves.

In this work, we use the inverse scattering transform method to consider the focusing Kundu–Eckhaus (KE) equation with nonzero background (NZBG) at infinity. Based on the analytical, symmetric, asymptotic properties of eigenfunctions, the inverse problem is solved via a matrix Riemann–Hilbert problem (RHP). The general multi-pole solutions are given in terms of the solution of the associated RHP. And the formula of the N simple-pole soliton solutions are obtained, too. We show that the Peregrine’s rational solution can be viewed as some appropriate limit of the simple-pole soliton solutions at branch point. Furthermore, by taking some other proper limits, the two and three simple-pole soliton solutions can yield to the double- and triple-pole solutions for focusing KE equation with NZBG. The effect of the parameter $$\beta $$ , characterizing the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effect, and some typical collisions between solutions of focusing KE equation are graphically displayed.

It is well known that most classical test functions to solve nonlinear partial differential equations can be constructed via single hidden layer neural network model by using Bilinear Neural Network Method (BNNM). In this paper, the neural network model of test function for the (3+1)-dimensional Jimbo–Miwa equation is extended to the “4-2-3” model. By giving some specific activation functions, new test function is constructed to obtain analytical solutions of the (3+1)-dimensional Jimbo–Miwa equation. Rogue wave solutions and the bright and dark solitons are obtained by giving some specific parameters. Via curve plots, three-dimensional plots, contour plots and density plots, dynamical characteristics of these waves are exhibited.

In this paper, we analyse two types of rogue wave solutions generated from two improved ansatzs, to the (2 + 1)-dimensional generalized Korteweg–de Vries equation. With symbolic computation, the first-order rogue waves, second-order rogue waves, third-order rogue waves are generated directly from the first ansatz. Based on the Hirota bilinear formulation, another type of one-rogue waves and two-rogue waves can be obtained from the second ansatz. In addition, the dynamic behaviours of obtained rogue wave solutions are illustrated graphically.

Purpose The purpose of this paper is to study the homoclinic breather waves, rogue waves and multi-soliton waves of the (2 + 1)-dimensional Mel’nikov equation, which describes an interaction of long waves with short wave packets. Design/methodology/approach The author applies the Hirota’s bilinear method, extended homoclinic test approach and parameter limit method to construct the homoclinic breather waves and rogue waves of the (2 + 1)-dimensional Mel’nikov equation. Moreover, multi-soliton waves are constructed by using the three-wave method. Findings The results imply that the (2 + 1)-dimensional Mel’nikov equation has breather waves, rogue waves and multi-soliton waves. Moreover, the dynamic properties of such solutions are displayed vividly by figures. Research limitations/implications This paper presents efficient methods to find breather waves, rogue waves and multi-soliton waves for nonlinear evolution equations. Originality/value The outcome suggests that the extreme behavior of the homoclinic breather waves yields the rogue waves. Moreover, the multi-soliton waves are constructed, including the new breather two-solitary and two-soliton solutions. Meanwhile, the dynamics of these solutions will greatly enrich the diversity of the dynamics of the (2 + 1)-dimensional Mel’nikov equation.

In this paper, a (3+1)-dimensional generalized Kadomtsev-Petviashvili Benjamin-Bona-Mahony equation are investigated. The first-order, second-order and third-order rogue wave solutions of this equation are derived based on a symbolic computation approach. Their dynamics features are shown in some 3D and contour plots. Compared with the previous literatures, our work does not require the Hirota bilinear form of the equation.

Under investigation in this work is the general three-component nonlinear Schrodinger equation, which is an important integrable system. The new localized wave solutions of the equation are derived using a Darboux-dressing transformation with an asymptotic expansion. These localized waves display rogue waves on a multisoliton background. Furthermore, the main characteristics of the new localized wave solutions are analyzed with some graphics. Our results indicate that more abundant and novel localized waves may exist in the multi-component coupled equations than in the uncoupled ones.

Wave field data are affected not only by the accuracy of instruments recording them but also by sampling variability, an uncertainty due to the limited number of observations. For stationary meteorological conditions, due to the randomness of the sea surface elevation, wave parameters derived from a temporal or spatial wave record will depend on which part of a wave record is used in an analysis as well as on the length of a wave record and/or size of the investigated ocean area. This study demonstrates, using numerical simulations, challenges that sampling variability brings in the interpretation of nonlinear wave characteristics of the surface elevation when single 20- or 30-min field wave records are used in an analysis. As examples, we use sea states in which rogue waves were observed in the North Sea and investigate them using linear, second-, and third-order numerical simulations. The third-order wave data are simulated by a numerical solver based on the higher order spectral method (HOSM) which includes the leading order nonlinear dynamical effects, accounting for the effect of modulational instability. Wave steepness, the maximum wave crest, skewness, and kurtosis are investigated in unidirectional and directional wave fields. The study shows that having a single 20- or 30-min wave record may make it difficult to determine on the degree of wave field nonlinearity and the accuracy of derived wave parameters, as well as to evaluate the validity of wave models. Both single-point temporal and stereo-video camera data are discussed. We demonstrate that numerical simulations represent important supporting tools for wave field measurements.

The modulational instability of ion-acoustic rogue waves in a three-dimensional magnetized, dissipative plasma consisting of inertial warm positive ions and nonextensive electrons has been theoreti...

Modulational instability of dust-acoustic waves, which propagate in an opposite polarity dusty plasma system containing inertial warm negatively and positively charged massive dust grains as well as nonextensive q-distributed electrons and ions has been theoretically investigated. The nonlinear Schrodinger equation is derived by employing the reductive perturbation method. The nonlinear Schrodinger equation predicts the conditions of the modulational instability of dust-acoustic waves and the formation of dust-acoustic rogue waves in a nonlinear and dispersive plasma medium. It is observed that the basic features of the dust-acoustic rogue waves (viz., amplitude and width) are significantly modified by the various plasma parameters such as nonextensivity of electrons and ions, electron number density, electron temperature, ion number density, ion temperature, the mass and number density of the dust grains, etc. The application of the results in space and laboratory opposite polarity dusty plasma is briefly discussed.

The amplitude modulation of electron acoustic waves (EAWs) is investigated in an unmagnetized four-component quantum plasma containing dynamic cold electron fluid, Thomas–Fermi distributed hot electrons and positrons and immobile ions in presence of exchange-correlation potentials. For this purpose, basic set of quantum hydrodynamic equations is reduced to a nonlinear Schrodinger equation (NLSE) adopting a reductive perturbation technique. The regions of the stable and unstable wave packets are obtained precisely in terms of the intrinsic plasma parameters. It is shown that the NLSE leads to the modulational instability (MI) of EAWs, and the formation of EA rogue waves, which are due to the effects of nonlinearity and dispersion in the propagation of EAWs. The effects of relevant plasma parameters on the MI criterion are numerically investigated. It is also found that the modulated EAW packets can propagate in the form of bright envelope solitons as well as dark envelope solitons. Within the modulational instability region, the dependence of first- and second-order EA rogue wave profiles on the system parameters is discussed.

In this paper, a modified symbolic computation approach is proposed. The multiple rogue wave solutions of a generalized (2+1)-dimensional Boussinesq equation are obtained by using the modified symbolic computation approach. Dynamics features of these obtained multiple rogue wave solutions are displayed in 3D and contour plots. Compared with the original symbolic computation approach, our method does not need to find Hirota bilinear form of nonlinear system., Comment: 11 pages, 18 figures

A new wave data set from the southern North Sea covering the period 2011–2016 and composed of wave buoy and radar measurements sampling the sea surface height at frequencies between 1.28 and 4 Hz was quality controlled and scanned for the presence of rogue waves. Here, rogue waves refer to waves whose height exceeds twice the significant wave height. Rogue wave frequencies were analyzed and compared to Rayleigh and Forristall distributions, and spatial, seasonal, and long-term variability was assessed. Rogue wave frequency appeared to be relatively constant over the course of the year and uncorrelated among the different measurement sites. While data from buoys basically correspond with expectations from the Forristall distribution, radar measurement showed some deviations in the upper tail pointing towards higher rogue wave frequencies. The amount of data available in the upper tail is, however, still too limited to allow a robust assessment. Some indications were found that the distribution of waves in samples with and without rogue waves was different in a statistical sense. However, differences were small and deemed not to be relevant as attempts to use them as a criterion for rogue wave detection were not successful in Monte Carlo experiments based on the available data.

In this paper, the interactions between extreme waves and a vertical cylinder are investigated through a 3-D two-phase flow model. The numerical model is verified and validated by experimental data. Then, two factors are considered, the global wave steepness and the frequency bandwidth of the wave groups, in the studies of the in-line wave forces and the wave run-up around a cylinder. It is found that both the in-line wave forces and the wave run-up are remarkably increased with the increase of the global wave steepness, whereas the effect of the frequency bandwidth on the in-line wave forces is relatively weak in comparison with its effect on the wave run-up. The minimum and maximum wave run-ups are located in the directions of 22.5° and 180° with respect to the direction of the incident waves, respectively. Additionally, a new empirical formula is proposed for predicting the in-line wave forces by using only the free surface elevations around the cylinder. The results of the formula agree well with the simulation results.

We consider a ( $$2+1$$ )-dimensional nonautonomous-coupled nonlinear Schrodinger equation, which includes the partially nonlocal nonlinearity under linear and harmonic potentials. Via a projecting expression between nonautonomous and autonomous equations, and utilizing the bilinear method and Darboux transformation method, we find diversified exact solutions. These solutions contain the nonlocal rogue wave and Akhmediev or Ma breather solutions, and the combined solution which describes a rogue wave superposed on an Akhmediev or Ma breather. By adjusting values of diffraction, width and phase chirp parameters of wave, the maximum value of the accumulated time can be modulated. When we compare the maximum value of the accumulated time with that of the excitation position parameters, we study the management of scalar and vector rogue waves, such as the excitations of full shape, early shape and climax shape for rogue waves.

We report the novel colliding dynamics of the rogue waves (RWs) for a three-component transient stimulated Raman scattering system arising from nonlinear optics. The N-order RW solutions are constructed by utilizing the generalized Darboux transformation scheme. Then, the dynamics of the RWs are analyzed. As $$N\ge 2$$ , the multiple RWs collide towards a central point. During the collisions and interactions, the peaks and troughs increase greatly. It is also interesting and magical that all the RWs possess exquisite symmetrical structures.

We construct matter rogue wave (RW) solutions for both attractive-attractive and repulsive-repulsive two-component Bose–Einstein condensates (BECs). With the help of similarity transformation, we t...

In this work, the $$(2+1)$$ -dimensional extended Kadomtsev–Petviashvili equation, which models the surface waves and internal waves in straits or channels, is investigated via the Hirota bilinear method. N-soliton and high-order breather solutions are obtained analytically. Furthermore, mixed solutions consisting of first-order breathers and solitons are also derived, and the corresponding dynamic behaviors are shown by three-dimensional plots. Additionally, based on the long-wave limit, we obtain line rogue waves, lumps and semi-rational solutions composed of lumps, line rogue waves and solitons. It is noteworthy that the semi-rational solutions derived in this paper exhibit elastic interactions.

In this work, we study the phenomena of Rogue waves (RW) on one-dimensional (1D) photonic lattices presenting diagonal and non-diagonal disorder. Our results show the appearance of extreme events coming from the superposition of different, extended and localized, linear waves for weak disorder. We perform experiments on femtosecond laser written waveguide arrays having disorder in coupling constants, which is originated from a random waveguide distribution. Both, numerics and experiments, are in good agreement and show that RW are generically present in 1D lattices for weak disorder only, after a mandatory data filtering process.

General rogue waves are derived for the generalized derivative nonlinear Schrodinger (GDNLS) equations by a bilinear Kadomtsev-Petviashvili (KP) reduction method. These GDNLS equations contain the Kaup-Newell equation, the Chen-Lee-Liu equation and the Gerdjikov-Ivanov equation as special cases. In this bilinear framework, it is shown that rogue waves to all members of these equations are expressed by the same bilinear solution. Compared to previous bilinear KP reduction methods for rogue waves in other integrable equations, an important improvement in our current KP reduction procedure is a new parameterization of internal parameters in rogue waves. Under this new parameterization, the rogue wave expressions through elementary Schur polynomials are much simpler. In addition, the rogue wave with the highest peak amplitude at each order can be obtained by setting all those internal parameters to zero, and this maximum peak amplitude at order $N$ turns out to be $2N+1$ times the background amplitude, independent of the individual GDNLS equation and the background wavenumber. It is also reported that these GDNLS equations can be decomposed into two different bilinear systems which require different KP reductions, but the resulting rogue waves remain the same. Dynamics of rogue waves in the GDNLS equations is also analyzed. It is shown that the wavenumber of the constant background strongly affects the orientation and duration of the rogue wave. In addition, some new rogue patterns are presented., 19 pages, 3 figures

By virtue of Hirota bilinear form of the coupled Higgs field equation, some higher-order rogue wave, the Ma breather, the Akhmediev breather and the general breather solutions are constructed through symbolic computation. With the help of the contour line method, we investigate the localization characters of the first-order rogue wave, which is different from the case of its in nonlinear Schrodinger equation. The Ma breather and the Akhmediev breather solutions of the coupled Higgs field equation possess time-periodic and space-periodic, respectively. The results of this paper show us the generation and evolutions of novel nontraveling wave with several interesting structures.

The coupled Gerdjikov–Ivanov (cGI) equation, associated with a $$3\times 3$$ matrix Lax pair, is an important integrable system that can be reduced to the third kind of derivative nonlinear Schrodinger equation, i.e., Gerdjikov–Ivanov equation. Based on the symmetric relations of the Lax pair, 2N-fold Darboux transformation for the cGI equation is constructed. As an application of the Darboux transformation, we obtain some exact solutions of the cGI equation, including bright–bright soliton, dark–bright soliton, Ma breather, breather fission, soliton fusion and dark–bright rogue wave.

Non-Kerr media possess certain applications in photonic lattices and optical fibers. Studied in this paper are the vector rational and semi-rational rogue waves in a non-Kerr medium, through the coupled cubic-quintic nonlinear Schrodinger system, which describes the effects of quintic nonlinearity on the ultrashort optical pulse propagation in the medium. Applying the gauge transformations, we derive the Nth-order Darboux transformation and Nth-order vector rational and semi-rational rogue wave solutions, where N is a positive integer. With such solutions, we present three types of the second-order rogue waves with the triangle structure: the one with each component containing three four-petaled rogue waves, the one with each component containing three eye-shaped rogue waves, and the other with one component containing three anti-eye-shaped rogue waves and the other component containing three eye-shaped rogue waves. We exhibit the third-order vector rogue waves with the merged, triangle and pentagon structures in each component. Moreover, we show the first- and second-order vector semi-rational rogue waves which display the coexistence of the rogue waves and the breathers.

In this manuscript, we consider a coupled AB system, which describes the baroclinic instability processes in the geophysical flows. Darboux-dressing transformation is used to derive the bright-dark soliton, breather and semirational rogue wave solutions for such a system. We observe that type of the solutions is relate to the spectral parameter $$\lambda $$ , amplitude $$a_{1}$$ and wave number q. Elastic collision between dark or bright solitons, propagations of the bright or dark breathers and rogue waves coexist with two dark or bright solitons are respectively illustrated in figures. The results about those localized wave phenomena are expected to have potential applications.

In this paper, the exact rogue wave solutions for the Hirota equation are investigated on the Jacobi elliptic cn-function background. Under the Bargmann constraint, the differential constraints on the potential are obtained by the nonlinearization of spectral problem. Further, the eigenvalues are determined based on the differential constraint equations. The non-periodic solution of spectral problem is constructed, which can produce rogue periodic wave solutions of the Hirota equation. Moreover, the exact rogue periodic wave solutions are presented via the one- and two-fold Darboux transformation formulas. Finally, the generation mechanism and characteristics of rogue periodic waves are analyzed from the viewpoint of two- and three-dimensional structures.

Wave focusing is one of the main mechanisms that can generate freak waves in deep water. The strongly nonlinear High-Level Irrotational Green–Naghdi (HLIGN) equations and the Harmonic Polynomial Cell (HPC) method are used here to simulate the focused waves. Numerical simulations are performed in intermediate water depth as well as in deep water. In intermediate water, the HLIGN free surface and wave kinematics are then compared with the well-accepted laboratory data and the results of the HPC method. The effect of water depth on focused waves is also carried out. It is demonstrated that both the HLIGN equations and the HPC method can accurately simulate the surface elevation and particle velocity of focused waves compared with the experiments. The velocity distribution is distinct for the focused waves with the same crest elevations but in different water depths. The HLIGN equations are considered to be an accurate and efficient wave model to investigate focused waves in the intermediate and deep water cases.

Tsunamis and storms cause considerable coastal flooding, numerous fatalities, destruction of structures, and erosion. The characterization of energy and frequency associated with each wave contribute to the risk assessment in coastal regions. Coastal boulder deposits represent a physical proof of extreme inundation and allow us to study the effects of marine floods further back in time than instrumental and historical records. Age estimation of these deposits is challenging due to lack of materials (such as sand, shells, corals, or organic matter) that retain information about the passage of time. Lichenometry, a simple age estimation method, which is cost-effective, quick to apply, and non-destructive, is here proposed as a solution. A lichen growth model for a calcium-tolerant lichen species was developed and used to estimate the age of a boulder deposit related to extreme marine inundation(s) in Portugal. Estimated ages indicate several very recent events (