Peregrine Soliton Research Articles

Dark beam excitations in the defocusing (2+1)-dimensional Zakharov system

The defocusing (2+1)-dimensional Zakharov system describes beams that involve three independent variables: the propagation distance and two orthogonal coordinates. Consequently, it can be inferred from the system characteristics that such beams exhibit considerably richer nonlinear behavior than the (1+1)-dimensional systems. In this paper, using the Hirota bilinear method, we found analytical dark beam solutions of the defocusing (2+1)-dimensional Zakharov system, describing beams in normal regions of nonlinear media. Based on the parameters describing dark beam solutions, some characteristics are discussed, and the impact of these parameters on the propagation of such beams is analyzed. Our results indicate that the shape of dark beams can be completely controlled by adjusting these physical parameters. In addition to finding order-1 dark beams, we also obtain the analytical order-2 dark beam solutions and the order-n dark beam solutions of the system mentioned above, and discuss those in special cases. Interestingly, it turned out that such solutions can be simplified to breathers and rogue waves. An interesting special case is that of the dark Peregrine soliton, which can be considered as the basic dark rogue wave. The basic idea of nonlinear wave control that was used in this paper can be extended to other higher-dimensional nonlinear systems.

Modulation of electromagnetic waves in a relativistic degenerate plasma at finite temperature

We study the modulational instability (MI) of a linearly polarized electromagnetic (EM) wave envelope in an intermediate regime of relativistic degenerate plasmas at a finite temperature (T≠0) where the thermal energy (KBT) and the rest-mass energy (mec2) of electrons do not differ significantly, i.e., βe≡KBT/mec2≲ (or ≳) 1, but the Fermi energy (KBTF) and the chemical potential energy (μe) of electrons are still a bit higher than the thermal energy, i.e., TF>T and ξe=μe/KBT≳1. Starting from a set of relativistic fluid equations for degenerate electrons at finite temperature, coupled to the EM wave equation and using the multiple scale perturbation expansion scheme, a one-dimensional nonlinear Schödinger (NLS) equation is derived, which describes the evolution of slowly varying amplitudes of EM wave envelopes. Then, we study the MI of the latter in two different regimes, namely, βe<1 and βe>1. Like unmagnetized classical cold plasmas, the modulated EM envelope is always unstable in the region βe>4. However, for βe≲1 and 1<βe<4, the wave can be stable or unstable depending on the values of the EM wave frequency, ω, and the parameter ξe. We also obtain the instability growth rate for the modulated wave and find a significant reduction by increasing the values of either βe or ξe. Finally, we present the profiles of the traveling EM waves in the form of bright (envelope pulses) and dark (voids) solitons, as well as the profiles (other than traveling waves) of the Kuznetsov–Ma breather, the Akhmediev breather, and the Peregrine solitons as EM rogue (freak) waves, and discuss their characteristics in the regimes of βe≲1 and βe>1.

Experimental study of extreme waves based on nonlinear Schrödinger equation under background of a random sea

The Peregrine breather (PB) solution of the nonlinear Schrödinger equation is used to model ocean extreme waves in a water wave flume. Triangular spectral features of wave elevations are observed over the nonlinear evolution of the extreme waves, which can be applied for early detection of the formation of extreme waves. To model a more realistic sea state, a background random wave is superposed to the PB in this study. We examine the spectral features of the nonlinear wave evolution with random background waves by the spectral analysis. It is found that the wave elevations show similar triangular spectral features for extreme waves with a relatively mild background wave. Moreover, we find that the second harmonic elevations of the extreme waves also show triangular spectral features, suggesting the potential use of the second harmonic elevations (in addition to the first) for detection of the formation of extreme waves.

Modified nonlinear Schrödinger equation for gravity waves with the influence of wind, currents, and dissipation

A new modified nonlinear Schrödinger (MNLS) equation is derived for gravity waves with the presence of wind, dissipation, and shear currents in finite water depth. Horizontal surface currents are assumed stationary and slowly varying spatially. Using the MNLS equation, the modulational instability (MI) of deep-water gravity wave trains affected by wind and dissipation is considered. It was demonstrated that the modulational perturbation of waves is unstable or becomes unstable after several wave periods, whereas the dissipation will suppress the MI. Then, a new theoretical formula for predicting the maximum amplitude is derived to take into account the effect of vorticity, dissipation, and wind. The effect of dissipation becomes significant in strong currents, while wind can increase the height amplification. Furthermore, an explicit analytical Peregrine breather (PB) solution that considers the effect of vorticity, dissipation, and wind is presented. Opposing currents and winds will increase the height of PB. However, following currents and dissipation have opposite effects. The effects of the shear current, dissipation, and wind on nondimensional maximum amplitudes during the evolution of the Akhmediev breather are similar to PB solution.

Two-dimensional toroidal breather solutions of the self-focusing nonlinear Schrödinger equation

We introduce the ring-like toroidal breather solutions of the (2+1)-dimensional nonlinear Schrödinger (NLS) equation with Kerr nonlinearity. When the width of the toroidal pulse is much smaller than its radius, the (2+1)-dimensional NLS equation in cylindrical coordinates can be approximated by the standard (1+1)-dimensional NLS equation. Based on this NLS equation, we obtain the first-order toroidal breather solutions, and study their dynamic characteristics. In particular, we discover and present for the first time the toroidal Peregrine soliton. This research can be extended to other high-dimensional nonlinear equations and provides a practical method for the study of other high-dimensional nonlinear localized wave structures.

Experimental generation and investigation of Peregrine breather waves in deep water

ABSTRACT The modulation instability is a prominent explanation for the occurrence of nonlinear freak waves. The modulation instability could be described by the famous nonlinear Schrödinger (NLS) equation. The Peregrine breather solution of the NLS equation is usually adopted as a prototype in freak wave simulation. The freak waves based on the Peregrine breather solution (Peregrine breather waves) are characterised by huge wave crests, large wave heights and concentrated energies. In this paper, the Peregrine breather waves are experimentally generated in deep water. The specific experimental wave-making procedure is first proposed. The measured wave elevations show great agreement with the analytical solutions. Moreover, the modulation instability scenario is observed by wavelet analysis. The effects of wave parameters on wave energy distributions are first analysed. The results indicate carrier wave amplitude and wave height determine the width of focusing energy frequency and carrier wave frequency determines the duration of focusing energy.

Evolution of Water Wave Groups in the Forced Benney–Roskes System

For weakly nonlinear waves in one space dimension, the nonlinear Schrödinger Equation is widely accepted as a canonical model for the evolution of wave groups described by modulation instability and its soliton and breather solutions. When there is forcing such as that due to wind blowing over the water surface, this can be supplemented with a linear growth term representing linear instability leading to the forced nonlinear Schrödinger Equation. For water waves in two horizontal space dimensions, this is replaced by a forced Benney–Roskes system. This is a two-dimensional nonlinear Schrödinger Equation with a nonlocal nonlinear term. In deep water, this becomes a local nonlinear term, and it reduces to a two-dimensional nonlinear Schrödinger Equation. In this paper, we numerically explore the evolution of wave groups in the forced Benney–Roskes system using four cases of initial conditions. In the one-dimensional unforced nonlinear Schrödinger equa tion, the first case would lead to a Peregrine breather and the second case to a line soliton; the third case is a long-wave perturbation, and the fourth case is designed to stimulate modulation instability. In deep water and for finite depth, when there is modulation instability in the one-dimensional nonlinear Schdrödinger Equation, the two-dimensional simulations show a similar pattern. However, in shallow water where there is no one-dimensional modulation instability, the extra horizontal dimension is significant in producing wave growth through modulation instability.

Fourth-order stability analysis for capillary-gravity waves on finite-depth currents with constant vorticity

We derive a fourth-order nonlinear evolution equation (NLEE) for narrow-banded Stokes wave in finite depth in the presence of surface tension and a mean flow with constant vorticity. The two-dimensional capillary-gravity wave motion on the surface of finite depth is considered here. The analysis is limited to one horizontal dimension, parallel to the direction of wave propagation, in order to take advantage of a formulation using potential flow theory. This evolution equation is then employed to examine the effect of vorticity on the Benjamin–Feir instability (BFI) of the Stokes capillary-gravity wave trains. It is found that the vorticity modifies significantly the modulational instability and in the case of finite depth, the combined effect of vorticity and capillarity is to enhance the instability growth rate influenced by capillarity when the vorticity is negative. The key point is that the present fourth-order analysis exhibits considerable deviations in the stability properties from the third-order analysis and gives better results consistent with the exact numerical results. Furthermore, the influence of linear shear current on Peregrine breather (PB) is studied.

Shallow ocean soliton and localized waves in extended (2 + 1)-dimensional nonlinear evolution equations

In recent years, the derivation and solution of integrable nonlinear evolution equations (NLEEs) in one, two, or more dimensions have been the apex in the field of applied mathematics. As is well-known, NLEEs are capable of describing a wide variety of nonlinear physical phenomena in the fields of applied science and engineering. In this article, the integrability, shallow ocean wave soliton, Peregrine soliton, multi-soliton, lumps, breathers (such as Akhmediev, general, and Kuznetsov-Ma breather) solutions, and wave interactions for four extended (2+1)-dimensional NLEEs are investigated. The Painlevé test is used to analyze the integrability of these models, and then their bilinear forms are constructed using the simplified bilinear approach. Furthermore, various forms of the aforementioned solutions are constructed using the Hirota bilinear method and several ansatz functions. The shallow ocean waves for the X-, Y-, and H-types, as well as other complex interactions are replicated by the two, three, and four soliton solutions. These structures, along with a few other intriguing structures are represented by graphs in three-dimensional and contour plots. Some two dimensional plots are also presented. We hope these findings might well be useful in explaining the complex dynamic behaviors of shallow water waves.

Modulation instability—rogue wave correspondence hidden in integrable systems

The bulk-boundary correspondence is a key feature of topological physics and is universally applicable to Hermitian and non-Hermitian systems. Here, we report a similar universal correspondence intended for the rogue waves in integrable systems, by establishing the relationship between the fundamental rogue wave solutions of integrable models and the baseband modulation instability of continuous-wave backgrounds. We employ an N-component generalized nonlinear Schrödinger equation framework to exemplify this modulation instability-rogue wave correspondence, where we numerically confirm the excitation of three coexisting Peregrine solitons from a turbulent wave field, as predicted by the modulation instability analysis. The universality of such modulation instability-rogue wave correspondence has been corroborated using various integrable models, thereby offering an alternative way of obtaining exact rogue wave solutions from the modulation instability analysis.

Transparent boundary condition for simulating rogue wave solutions in the nonlinear Schrödinger equation.

This paper addresses the construction of numerical boundary conditions for simulating rogue wave solutions in the nonlinear Schrödinger equation. While three kinds of commonly used boundary conditions require a big enough computational domain to reproduce solutions faithfully in the central domain, we propose transparent boundary conditions for the Peregrine soliton and Kuznetsov-Ma breather solutions, respectively. For both solutions, these boundary conditions require a smaller computational domain than other boundary conditions to attain the best accuracy of the Crank-Nicolson scheme and selected mesh size, which will be referred to as the "acceptable accuracy" below. In particular, the computational domain with these boundary conditions is only 1/16 as small as others in the simulations of the Peregrine soliton solution. As a result, they reduce both the memory requirement and the computing time for the Peregrine soliton solution.

Experimental Evidence of Nonlinear Focusing in Standing Water Waves.

Nonlinear wave focusing originating from the universal modulation instability (MI) is responsible for the formation of strong wave localizations on the water surface and in nonlinear wave guides, such as optical Kerr media and plasma. Such extreme wave dynamics can be described by breather solutions of the nonlinear Schrödinger equation (NLSE) like by way of example the famed doubly-localized Peregrine breathers (PB), which typify particular cases of MI. On the other hand, it has been suggested that the MI relevance weakens when the wave field becomes broadband or directional. Here, we provide experimental evidence of nonlinear and distinct PB-type focusing in standing water waves describing the scenario of two counterpropagating wave trains. The collected collinear wave measurements are in excellent agreement with the hydrodynamic coupled NLSE (CNLSE) and suggest that MI can undisturbedly prevail during the interplay of several wave systems and emphasize the potential role of exact NLSE solutions in extreme wave formation beyond the formal narrow band and unidirectional limits. Our work may inspire further experimental investigations in various nonlinear wave guides governed by CNLSE frameworks as well as theoretical progress to predict strong wave coherence in directional fields.

The distortion of the Peregrine soliton under the perturbation in initial condition

The Peregrine soliton of the nonlinear Schrödinger (NLS) equation, i.e., the lowest order rogue wave, can be observed and has great significance in nonlinear fiber optics. In this paper, we use the physics-informed neural networks (PINN) to investigate the distortion of the Peregrine soliton under a Gaussian-type perturbation in initial condition. We propose two quantities, MH – the displacement of maximum amplitude and ML – the displacement of minimum amplitude, to measure the degree of distortion for the Peregrine soliton. MH and ML are dominated by the amplitude (σ) of the Gaussian-type perturbation, the slope of MH is increasing along σ, but the slope of ML is decreasing. This result may be of practical significance to prevent or to exploit rogue waves in fiber optical system.

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.

Nonlinear wave transmission in a two-dimensional nonlinear electric transmission network with dissipative elements

A two-dimensional nonlinear discrete electric transmission network made of several well-known anharmonic modified Noguchi lines coupled transversely to one another with a linear inductor is considered and the dynamics of modulated waves are investigated. The linear analysis shows that increasing either the dispersive elements of the network, the coupling elements of the network, or the transverse wavenumber increases the propagating frequencies. Employing the reductive perturbation method, we establish that the dynamics of modulated waves in the network system are governed by a two-dimensional dissipative nonlinear Schrödinger (NLS) equation having, depending on both the longitudinal and transverse wavenumbers as well as on the network parameters, either the hyperbolic or the elliptic character. Through the derived dissipative NLS equation, the phenomenon of the baseband modulational instability (MI) of the electric system under consideration is investigated and the analytical expression for the baseband MI growth rate is presented. We show that the dispersive elements of the network system softens the baseband MI and enhances the baseband MI domain. Under the condition of the baseband MI, we find approximate modulated wave solutions of the governing equations which are then used for analytical investigation of the transmission of super rogue wave voltage, Peregrine soliton voltage, and bright solitary wave signals through the network system under consideration. The effects of the network and solution parameters such as the dispersive, coupling parameters, the carrier wavenumber, and the wavenumbers in both the longitudinal and transverse directions on the dissipative wave signals are presented.

Modulation effect of linear shear flow on interfacial waves in a two-layer fluid with finite layer depths

The modulation effect of linear shear flow (LSF) comprising uniform and shear flows with constant vorticity on interfacial waves in a two-layer fluid with finite layer depths is studied. Herein, lower-layer LSF is focused on. A nonlinear Schrödinger equation (NLSE) modified by lower-layer LSF is derived. By comparing and analyzing the dispersion relation of upper- and lower-layer LSFs, sufficient conditions of Kelvin–Helmholtz stability are afforded to ensure the applicability of NLSE. Based on the relationship between the modulational instability (MI) of NLSE and interfacial freak waves (IFWs) represented as Peregrine breather, existence conditions of IFWs affected by LSF are presented. The convergence effect of the flow field against the wave propagation direction, e.g., uniform up-flow and positive (negative) vorticity of the lower (upper) layer, increases the MI growth rate. This subsequently increases the wave height and promotes IFW generation. However, the divergent effect of the flow field along the wave propagation direction, including uniform down-flow and negative (positive) vorticity of lower (upper) layer, inhibits IFW generation. Moreover, eight kinds of LSFs are presented under uniform flow and vorticity, among which two promote generation, two inhibit generation, and the remaining four depend on the counteracting effect between uniform flow and vorticity.

Role of Homoclinic Breathers in the Interpretation of Experimental Measurements, With Emphasis on the Peregrine Breather

A class of generalized homoclinic solutions of the nonlinear Schrödinger (NLS) equation in 1+1 dimensions is studied. These are homoclinic breathers that are shown to be derivable from the ratio of Riemann theta functions for the genus-2 solutions of the nonlinear Schrödinger equation. We discuss how these solutions behave in the homoclinic limit for which a fundamental parameter ε goes to zero, ε→0 (such that two points of simple spectrum converge to double points at some particular lambda-plane eigenvalue). The homoclinic solutions cover the entire lambda plane (the Riemann surface of the NLS equation) and are given in terms of simple trigonometric functions. When the spectral eigenvalues converge to the carrier amplitude in the lambda plane we have the Peregrine breather. While the Peregrine solution is often called a soliton, it is in reality a breather, albeit occurring at the “singular point” corresponding to the carrier eigenvalue in the lambda plane and consequently “breathes” only once in its lifetime. The Peregrine breather separates small-amplitude modulations below the carrier from large amplitude modulation above the carrier. This fact means that the Peregrine breather has a “central” role in the lambda plane characterization of the NLS nonlinear spectrum. The Akhmediev breather occurs somewhat below the carrier (and is therefore a small-amplitude modulation) and the Kuznetsov-Ma breather occurs above the carrier (and is therefore a large-amplitude modulation). The general homoclinic solutions can be constructed everywhere in the lambda plane and are shown to be a useful tool to interpret the nonlinear Fourier spectrum of space and time series recorded in the laboratory and ocean environment. Nonlinear filtering is suggested as a way to extract breather trains from experimental time series. The generalized homoclinic breathers can be thought of as “extreme wave packets” or “rogue wave” solutions of water waves for scientific and engineering applications in various fields of physics including physical oceanography and nonlinear optics.

Amplification of Wave Groups in the Forced Nonlinear Schrödinger Equation

In many physical contexts, notably including deep-water waves, modulation instability in one space dimension is often studied by using the nonlinear Schrödinger equation. The principal solutions of interest are solitons and breathers which are adopted as models of wave packets. The Peregrine breather in particular is often invoked as a model of a rogue wave. In this paper, we add a linear growth term to the nonlinear Schrödinger equation to model the amplification of propagating wave groups. This is motivated by an application to wind-generated water waves, but this forced nonlinear Schrödinger equation potentially has much wider applicability. We describe a series of numerical simulations which in the absence of the forcing term would generate solitons and/or breathers. We find that overall the effect of the forcing term is to favour the generation of solitons with amplitudes growing at twice the linear growth rate over the generation of breathers.

Generation of high-power breathing pulses based on Peregrine soliton by using saturable absorber

We obtain high-power breathing pulses based on Peregrine soliton by using a saturable absorber. Due to the presence of background wave, Peregrine soliton cannot transmit stably in optical fiber. To obtain high-power breathing pulses, we use a saturable absorber to eliminate the background wave. The results show that the high-power breathing pulse with periodic oscillation can be generated from Peregrine soliton by the saturable absorber with appropriate modulation depth. And the high-power breathing pulses can transmit long-distance in optical fiber. The evolution behavior of breathing pulses can be controlled by the initial input pulse and the length of erbium-doped fiber. And the breathing pulse has strong anti-noise ability.

Electron Acoustic Peregrine Breathers in a Quantum Plasma With 1-D Temperature Anisotropy

In the work, we study the formation of electron acoustic Korteweg–de Vries (KdV) solitons and their gradual nonlinear evolution into a peregrine soliton (PS) in a dense plasma at finite temperature with quantum effects. The plasma under consideration has temperature anisotropy along the direction of wave propagation. Such an anisotropy is due to the initial disturbance incorporated in the system or due to the lack of homogeneity of field quantities. We have studied both the linear and nonlinear characteristics of their dispersion relation. By employing reductive perturbation technique, we have obtained the KdV equation and studied the characteristics of the solitary structures. The effect of external forcing term which leads to self-interaction among different parts of the system and the gradual conversion of the waveform into Peregrine breather mode is analytically studied. Finally, we study the dynamical properties of the system and its dependence on plasma parameters. This theoretical study will help in predicting possible characteristics in laser plasma interaction and beam plasma experiments in which there are instances of temperature anisotropy.