Generation Of Rogue Waves Research Articles

Bifurcation of Some Novel Wave Solutions for Modified Nonlinear Schrödinger Equation with Time M-Fractional Derivative

In this paper, we investigate the time M-fractional modified nonlinear Schrödinger equation that describes the propagation of rogue waves in deep water. Periodic, solitary, and kink (or anti-kink) wave solutions are discussed using the bifurcation theory for planar integrable systems. Some new wave solutions are constructed using the first integral for the traveling wave system. The degeneracy of the obtained solutions is investigated by using the transition between orbits. We visually explore some of the solutions using graphical representations for different values of the fractional order.

On the possibility of rogue wave generation based on the dynamics of modified Burridge-Knopoff model of earthquake fault

In this study we propose a modified Burridge‐Knopoff model of earthquake fault, in which two tectonic plates are strongly coupled by nonlinear springs. By minimizing the effects of the veloci‐ ty‐weakening stick‐slip friction force between the masses and the moving surface, and in the limit of low amplitude oscillations; the system exhibits both stick‐slip and damped oscillatory motions as the values of some parameters are varied. Such motions usually characterize the dynamics of an earthquake fault, even though it is not always felt because of the low amplitude of vibrations. However when enough stress builds up in the subduction zones to overcome the frictional forces between tectonic plates, the oceanic rocks suddenly slip and there is violent release of energy at the epicentre. This outburst of energy simply signifies the generation of a very large amplitude and localized nonlinear wave. Such wave profile exactly fits the Peregrine solution of the damped/ forced nonlinear Schrodinger amplitude equation, derived from the modified one‐dimensional Burridge‐Knopoff equation of motion. In the regime of minimal or no frictional forces, these mon‐ ster waves suddenly appear and disappear without traces as shown by the numerical investigations. Our results strongly suggest that rogue waves emanates from the dynamics of nonlinearly coupled tectonic plates in subduction zones. This is further complemented by the fact that these giant waves were initially observed in Pacific and Atlantic oceans, which play hosts to the world’s largest oceanic subduction zones.

Dynamical analysis of higher-order localized waves for a three-component coupled nonlinear Schrödinger equation

This paper examines the three-component coupled nonlinear Schrödinger equation, which has various applications in deep ocean, nonlinear optics, Bose–Einstein (BE) condensates, and more. On the basis of seed solutions and a Lax pair, the Nth-order iterative expressions for the solutions are derived by using the generalized Darboux transformation. The evolution plots of dark-bright-rogue wave or breather-rogue wave are then obtained via numerical simulation. Particularly, a novel rogue wave propagation trajectory is found in the second and third order localized wave solutions. Moreover, by increasing the value of the free parameter α and β, the nonlinear waves merge with each other distinctly. The results further reveal the abundant dynamical patterns of localized waves in the three-component coupled system.

Rogue waves in quantum lattices with correlated disorder

We investigate the outbreak of anomalous quantum wave-function amplitudes in a one-dimensional tight-binding lattice featuring correlated diagonal disorder. Such rogue-wave-like behavior is fostered by a competition between localization and mobility. The effective correlation length of the disorder is ultimately responsible for bringing the local disorder strength to a minimum, fueling the occurrence of extreme events of much higher amplitudes, especially when compared to the case of uncorrelated disorder. Our findings are valid for a class of discrete one-dimensional systems and reveal profound aspects of the role of randomness in rogue-wave generation.

New perceptions for the bright and dark soliton solutions to the modified nonlinear Schrödinger equation

In this study, we will implement new bright and dark perceptions for the solitary wave solutions to the modified nonlinear Schrödinger equation. The achieved solutions will describe new vision of the following forms. The first form is the rogue wave modes for a derivative nonlinear Schrodinger model with positive linear dispersion which describe the propagation of rogue waves in ocean engineering as well as all similar waves such as dynamics waveguides that have unexpected large displacements. The second form is the waves which occur only in the regime of positive cubic nonlinearity. The third form is the waves that also occur in the regime that coincides exactly with the existence of instabilities of plane waves. The fourth form is the long-wave limit of a breather (a pulsing mode). Two famous different schemas are involved for this purpose. The first schema is the solitary wave ansatz method, while the second schema is the extended simple equation method. The two schemas are implemented in the same vein and parallel to construct new perceptions to the soliton solutions of this model. A comparison between the obtained new perceptions with the old perceptions that were realized before they have been documented.

Machine learning control of complex nonlinear dynamics in fibre lasers - INVITED

We review our recent work on the use of genetic algorithms to control non-stationary nonlinear wave dynamics in ultrafast fibre lasers, including the generation of breathing-soliton dynamics with controlled characteristics, the disclosure of the fractal dynamics of breathers, and the generation of rogue waves with controlled intensity.

Rogue Waves in the Southern North Sea—The Role of Modulational Instability

Abstract The role of the modulational instability for rogue wave generation in the ocean is still under debate. We investigated a continuous dataset, consisting of buoy and radar wave elevation data of different frequency resolutions, from eight measurement stations in the southern North Sea. For periods with rogue waves, we evaluated the presence of conditions for the modulational instability to work, that is, a narrow-banded wave spectrum in both frequency and angular direction. We found rogue waves that exceed twice the significant wave height indeed occur at slightly lower frequency bandwidths than usual. For rogue waves that are defined only by high crests, this was, however, not the case. The results were dependent on the measurement frequency. The directional spreading of the buoy spectra yielded no information on the presence of a rogue wave. In general, all spectra estimated from the dataset were found to be broad in frequency and angular direction, while the Benjamin–Feir index yielded no indication on a high nonlinearity of the sea states. These are unfavorable conditions for the evolution of a rogue wave through modulational instability. We conclude that the modulational instability did not play a substantial role in the formation of the rogue waves identified in our dataset from the southern North Sea. Significance Statement This work investigates whether rogue waves measured in the southern North Sea may have been generated by a modulational instability. The latter is a nonlinear mechanism of wave energy focusing that has been proven mathematically and confirmed in laboratory experiments. However, it is still unclear whether this mechanism is responsible for rogue wave generation under realistic ocean conditions. The modulational instability primarily arises when waves have similar frequencies and directions. In our data, these conditions were not satisfied. This finding leads to the insight that the modulational instability is not the most probable mechanism to generate rogue waves in our dataset.

Controllable optical rogue waves in inhomogeneous media

This paper investigates new rogue wave solutions of the nonlinear Schrödinger equation with variable coefficients, utilizing the self-similar transformation method. A new rogue wave family is introduced, which displays different wave structures. When one chooses appropriate variable coefficients, a series of first-order, second-order, and third-order rogue waves are obtained. We explore the generation and interaction of rational polynomial rogue waves in detail, and discuss some characteristics of their structures. The paper provides theoretical basis for studying the dynamic behavior of optical rogue wave in inhomogeneous media, and presents potential application value for control of rogue waves in fibers, plasmas and other fields of physics.

Spatial rogue waves in the actively Q-switched Nd:YAG laser under low Kerr nonlinearity.

We report the generation of spatial rogue waves in an actively Q-switched Nd:YAG laser. We observed that spatial rogue waves can emerge when the laser operates in a low-power regime below the self-focusing limit, while the lasing takes place at a large number of high-order transverse modes. These results suggest that the main mechanism leading to rogue waves generation is the modal overlap while large Kerr nonlinearity in the cavity is not a necessary factor in the process of spatial rogue waves formation. We have also investigated the spatio-temporal and coherence properties of the observed rogue waves.

Numerical study of rogue wave forces on flat decks based on the Peregrine breather solution under finite water depth

Rogue waves are extreme gravity waves that occur sporadically in the sea and vanish subsequently without a trace, posing great threats to marine, offshore and coastal structures. This paper presents a numerical study of rogue wave forces on submerged and elevated flat decks based on the Peregrine breather solution under finite water depth through a total of 294 simulation cases. A two-dimensional numerical wave tank is built and an in-house solver is applied to solve the wave–deck interactions. Peregrine breather solution of nonlinear Schrödinger equation (NLSE) is used for the rogue wave generation. Validations on nonlinear wave forces and rogue wave generations are conducted. Comparisons between rogue wave and solitary wave cases are made to reveal the characteristics of the interaction phenomena, free surface elevations and nonlinear wave forces induced by rogue waves. Parametric study on the influences of water depth, deck submergence (elevation), carrier wave number and wave crest height on rogue wave forces are performed, from which simple empirical formulae are proposed for the preliminary prediction of the maximum rogue wave forces. The consideration of Peregrine breather type rogue waves might be innovative, while the simulation results and proposed formulae might have reference values in engineering practice.

Dynamics study of integrable turbulence with fourth-order nonlinear Schrödinger equation.

In this paper, we focus on the fourth-order nonlinear Schrödinger equation, which can describe the optical system and the Heisenberg spin system. We consider a continuous wave perturbed by the one-dimensional random rough surface as the initial condition. First, we numerically resolve the eigenvalues under different control parameters utilizing the Fourier collocation method. Then, we simulate the evolution of this equation under the above initial conditions via the symmetrical split-step Fourier method. Moreover, we investigate the "steady" chaotic state by evolving a large number of initial conditions for the same control parameters. We find that the control parameters of the initial condition affect the number and intensity of rogue waves (RWs) in integrable turbulence. In particular, we locate the inflection point where the control parameter affects the velocities of solitons and the inconsistency within the parameter of the contribution to the generation of RWs. We further verify that the collision between breathers, solitons, and breathers and solitons can generate RWs. These results will enable us to understand the turbulent state and the formation mechanism of RWs.

Numerical generation and investigation of rogue waves for discrete nonlinear Schrödinger equations

We have numerically generated the rogue waves supported by discrete nonlinear Schrödinger equation (DNLSE). The investigation has been made on pure cubic DNLSE as well as on cubic-quintic discrete nonlinear Schrödinger equation (CQDNLSE). Runge–Kutta fourth (RK4) order method has been employed to obtain these results. For the cubic DNLSE breathers are found to be localized in one of the dimensions and periodic in the other dimensions, whereas rogue waves are found to be appearing from nowhere and disappearing without a trace. For CQDNLSE it is reported that the sign of quintic term possesses a significant effect on the intensity of the rogue waves. Further these waves are appearing from nowhere but instead of disappearing these show periodic reappearance. For larger value of quintic coefficient this phenomenon is visible at shorter length scale. Breathers are reported for the first time for the cubic DNLSE. The obtained results for CQDNLSE are unobserved previously in any constant coefficient discrete nonlinear equation.

Investigation of new solitons and other solutions to the modified nonlinear Schrödinger equation in ocean engineering

This article deals with the new solitons and other solutions to the modified nonlinear Schrödinger equation (MNLSE) that represents the propagation of rogue waves in deep water. The solutions are achieved in the shape of trigonometric and hyperbolic functions as well as solitary wave solutions by the mean of extended sinh-Gordon equation expansion and (G′G2)-expansion function methods. Furthermore, the singular periodic wave solutions are obtained and the constraint conditions for the existence of soliton solutions are also listed. We also plot 3D graphs for a better explanation of the achieved solutions by choosing the appropriate values of the parameters involved in solutions. We can assert from the obtained results that the applied techniques are simple, vibrant, and quite well, and will be helpful tool for addressing more highly nonlinear issues in various of fields, especially in ocean and coastal engineering. Furthermore, our findings are first step toward understanding the structure and physical behavior of complicated structures. We anticipate that our results will be highly valuable in better understanding the waves that occur in the ocean. We feel that this work is timely and will be of interest to a wide spectrum of experts working on ocean engineering models.

Rogue waves in nonlinear optics

Understanding the phenomenon of rogue wave formation, often called extreme waves, in diverse branches of nonlinear science has become one of the most attractive domains. Given the great richness of the new results and the increasing number of disciplines involved, we are focusing here on two pioneering fields: hydrodynamics and nonlinear optics. This tutorial aims to provide basic background and the recent developments on the formation of rogue waves in various systems in nonlinear optics, including laser physics and fiber optics. For this purpose we first discuss their formation in conservative systems, because most of the theoretical and analytical results have been realized in this context. By using a multiple space–time scale analysis, we review the derivation of the nonlinear Schrödinger equation from Maxwell’s equations supplemented by constitutive equations for Kerr materials. This fundamental equation describes the evolution of a slowly varying envelope of dispersive waves. This approximation has been widely used in the majority of systems, including plasma physics, fluid mechanics, and nonlinear fiber optics. The basic property of this generic model that governs the dynamics of many conservative systems is its integrability. In particular, we concentrate on a nonlinear regime where classical prototypes of rogue wave solutions, such as Akhmediev breathers, Peregrine, and Ma solitons are discussed as well as their experimental evidence in optics and hydrodynamics. The second part focuses on the generation of rogue waves in one- and two-dimensional dissipative optical systems. Specifically, we consider Kerr-based resonators for which we present a detailed derivation of the Lugiato–Lefever equation, assuming that the resonator length is shorter than the space scales of diffraction (or the time scale of the dispersion) and the nonlinearity. In addition, the system possesses a large Fresnel number, i.e., a large aspect ratio so that the resonator boundary conditions do not alter the central part of the beam. Dissipative structures such as solitons and modulational instability and their relation to frequency comb generation are discussed. The formation of rogue waves and the control employing time-delayed feedback are presented for both Kerr and semiconductor-based devices. The last part presents future perspectives on rogue waves to three-dimensional dispersive and diffractive nonlinear resonators.

The pump and seed wavelength difference control of rogue wave generation under the group velocity matching condition

We numerically simulate the effects of the wavelength difference between the pump light and the seed light on the mid infrared supercontinuum and rogue wave under the condition of fiber group velocity matching. The results show that with the increase of the wavelength difference between the two pulses, the filling effect of the spectrum and the control of rogue wave first increase and then decrease. The appropriate wavelength difference can realize the effective control of rogue wave. In addition, the modulation depth and pulse width of seed light play an important role in the control of rogue wave. The control effect of rogue wave first increases and then decreases with the increase of modulation depth and pulse width.

Generation mechanism and prediction of an observed extreme rogue wave

Rogue waves are individual ocean surface waves with crest height eta or trough-to-crest height H that are large compared to the significant wave height H_s of the underlying sea state: H/H_s>2.2 or eta /H_s>1.25. The physics of rogue wave generation and the potential of predicting the rogue wave risk are open questions. Only a few rogue waves in high sea states have been observed directly, but they can pose a danger to marine operations, onshore and offshore structures, and beachgoers. Here we report on a 17.6m high rogue wave in coastal waters with eta /H_s=1.98 and H/H_s=2.9 which are likely the largest normalized heights ever recorded. Simulations of random superposition of Stokes waves in intermediate water depth show good agreement with the observation. Non-linear wave modulational instability, a well known cause for rogue waves in laboratory settings, did not contribute significantly to the rogue wave generation. A parameter obtained from a routine spectral wave forecast provides a practical risk prediction for rogue waves. These results confirm that probabilistic prediction of oceanic rogue waves based on random superposition of steep waves are possible and should replace predictions based on modulational instability.

Modulational instability and statistical properties of irregular waves in finite water depth

Modulational instability is one of the possible basic mechanisms for the generation of rogue waves in deep water. Under the condition of finite depth, this instability is reduced and disappears as k1h<1.363 (k1 is the wavenumber of the carrier wave and h is the water depth). In this study, we addressed this phenomenon through three approaches, theoretical analysis of the Zakharov’s equation and the nonlinear Schrödinger equation and direct numerical simulations of the Euler equation. Their results on the modulational instability over arbitrary depths, particularly the growth rates of sidebands, are analysed and compared. In addition, the consequences of the suppression of modulational instability for realistic ocean waves are illustrated by direct simulations of long-crested, irregular waves at finite water depth. The results show that many phenomena related to the third-order nonlinearity (i.e. the downshift of the spectral peak frequency, the enhancement of kurtosis of surface elevations and rouge wave occurrence probability) are suppressed with decreasing depth k1h and eventually cease to exist as k1h<1.363. Considering the dependence of wave evolution on water depth, two Benjamin–Feir indices for arbitrary depths derived from Zakharov’s equation and nonlinear Schrödinger equation are used as indicators to estimate the modulational instability effect on nonlinear wave fields. Our results confirm that both parameters provide reasonable predictions of the modulational instability and probability of rogue wave occurrence.

The effect of coherent coupling nonlinearity on modulation instability and rogue wave excitation

We study modulation instability (MI) in both anomalous and normal dispersion regimes of a coherently coupled system. It is found that there exist three types of MI spectra with distinct characteristics termed baseband, passband, and zero-baseband based on the instability analysis of the in-phase and out-of-phase CW solutions. The coherent coupling nonlinearity is the source of the passband and zero-baseband variants. Guided by analytical predictions, we investigate numerically the excitation of rogue waves on weakly perturbed in-phase and out-of-phase continuous wave solutions in the parameter space where different kinds of MI exist. Simulations provide supporting evidence that rogue waves can only emerge in regimes where baseband or zero-baseband MI occurs. Moreover, the peak intensity of rogue waves in the case of baseband MI is greater than in the zero-baseband case. Finally, a combination of analysis and numerics uncovers the parameter conditions necessary for the generation of rogue waves.

Numerical Study of Statistical Properties for Semiconductor Laser Chaos by Exploring the Injection Parameter Space

The statistical properties of semiconductor laser chaos are of great importance for practical applications. By exploring the injection parameter space of an optically injected semiconductor laser, the associated probability density functions (PDFs) are numerically investigated for generation of chaotic waveforms with rogue waves (RWs) or Gaussian distributions. First, generation of RWs are demonstrated by considering chaotic waveforms with long-tailed PDFs. The dynamical behaviors for RW generations are systematically investigated by exploring the entire injection parameter space, where a pattern is identified for parameter regions of nonzero RWs. The occurrence of RWs is also investigated for different injection parameters by evaluating the effects of laser bias current, spontaneous emission noise, and detection bandwidth. Alternatively, Gaussian PDFs are generated by bandwidth-limited detection of chaotic waveforms. The bandwidth-dependent statistical properties are investigated through tuning the injection parameters for different detection bandwidths. Generation of Gaussian PDFs are demonstrated for a large region of injection parameters and for a detection bandwidth as comparable to the laser relaxation resonance frequency. This numerical study reveals the importance of injection parameters for tuning the statistics associated with chaotic lasers, and confirms the feasibility of generating RWs and Gaussian PDFs on demand for potential applications in various fields.

Rogue Waves generation by Using higher order rational solutions of Discrete Nonlinear Schrödinger Equation

The higher-order discrete nonlinear Schrödinger equation has been demonstrated to be related to cubic-quintic nonlinearity. This executes the specified number of periodic solutions. The localized solution of the homogeneous cubic-quintic discrete nonlinear Schrödinger equation has been researched. By employing scaling transformations, it was possible to convert cubic-quintic discrete nonlinear Schrödinger equations to constant-coefficient higher-order rational solutions with discrete nonlinear Schrödinger equations (DNLSE). We investigated our solution in the periodic form of the gain/loss term and discovered some intriguing results. Even the solution is interconnected, and its features are extensively exploited in a variety of physical phenomena, including optical waveguides, B-E condensates in optical media, non-linearity in photonic crystals, Kerr-type nonlinearity and photo refracting medium.