Peregrine Soliton Research Articles

Galilean-transformed solitons and supercontinuum generation in dispersive media

The Galilean transformation is a universal operation connecting the coordinates of a dynamical system, which move relative to each other with a constant speed. In the context of exact solutions of the universal nonlinear Schrödinger equation (NLSE), inducing a Galilean velocity (GV) to the pulse involves a frequency shift to satisfy the symmetry of the wave equation. As such, the Galilean transformation has been deemed to be not applicable to wave groups in nonlinear dispersive media. In this paper, we demonstrate that in a wave tank generated Galilean-transformed envelope and Peregrine solitons show clear variations from their respective pure dynamics on the water surface. The type of deviations depends on the sign of the GV and can be captured by the modified NLSE or the Euler equations. Moreover, we show that positive Galilean-translated envelope soliton pulses exhibit self-modulation. While designated GV and wave steepness values expedite multi-soliton dynamics, the strong focusing of such higher-order coherent waves inevitably lead to the generation of supercontinua as a result of soliton fission. We anticipate that kindred experimental and numerical studies might be implemented in other dispersive wave guides governed by nonlinearity.

Theoretical and numerical evidence for the potential realization of the Peregrine soliton in repulsive two-component Bose-Einstein condensates

The present work is motivated by the recent experimental realization of the Townes soliton in an effective two-component Bose-Einstein condensate by B. Bakkali-Hassan et al. [Phys. Rev. Lett. 127, 023603 (2021)]. Here, we use a similar multicomponent platform to exemplify theoretically and numerically, within the mean-field Gross-Pitaevskii framework, the potential toward the experimental realization of a different fundamental wave structure, namely the Peregrine soliton. Leveraging the effective attractive interaction produced within the mixture's minority species in the immiscible regime, we illustrate how initialization of the condensate with a suitable power-law decaying spatial density pattern yields the robust emergence of the Peregrine wave in the absence and in the presence of a parabolic trap. We then showcase the spontaneous emergence of the Peregrine soliton via a suitably crafted wide Gaussian initialization, again both in the homogeneous case and in the trap scenario. It is also found that narrower wave packets may result in periodic revivals of the Peregrine soliton, while broader ones give rise to a cascade of Peregrine solitons arranged in a so-called Christmas-tree structure. Strikingly, the persistence of these rogue-wave structures is demonstrated in certain temperature regimes as well as in the presence of transversal excitations through three-dimensional computations in a quasi-one-dimensional regime. This proof-of-principle illustration is expected to represent a practically feasible way to generate and observe this rogue wave in realistic current ultracold atom experimental settings.

Higher-order rogue wave solutions of the Sasa–Satsuma equation

Up to the third-order rogue wave solutions of the Sasa–Satsuma (SS) equation are derived based on the Hirota’s bilinear method and Kadomtsev–Petviashvili hierarchy reduction method. They are expressed explicitly by rational functions with both the numerator and denominator being the determinants of even order. Four types of intrinsic structures are recognized according to the number of zero-amplitude points. The first- and second-order rogue wave solutions agree with the solutions obtained so far by the Darboux transformation. In spite of the very complicated solution form compared with the ones of many other integrable equations, the third-order rogue waves exhibit two configurations: either a triangle or a distorted pentagon. Both the types and configurations of the third-order rogue waves are determined by different choices of free parameters. As the nonlinear Schrödinger equation is a limiting case of the SS equation, it is shown that the degeneration of the first-order rogue wave of the SS equation converges to the Peregrine soliton.

Prediction and manipulation of hydrodynamic rogue waves via nonlinear spectral engineering

In this work, we realize the mathematically predicted universal mechanism of the local emergence of Peregrine solitons in water tank experiments, with a particular aim to control the point of the soliton occurrence in space-time by employing the inverse scattering transform for the synthesis of the initial data. Using this approach, we are able to engineer a localized wave packet with a prescribed solitonic and radiative content, evolving in a rogue wave at a predicted position from the wave maker.

Quadratic Peregrine solitons resonantly radiating without higher-order dispersion.

We show that two-color Peregrine solitary waves in quadratic nonlinear media can resonantly radiate dispersive waves even in the absence of higher-order dispersion, owing to a phase-matching mechanism that involves the weaker second-harmonic component. We give very simple criteria for calculating the radiated frequencies in terms of material parameters, finding excellent agreement with numerical simulations.

The closeness of localized structures between the Ablowitz–Ladik lattice and discrete nonlinear Schrödinger equations: Generalized AL and DNLS systems

The Ablowitz–Ladik system, being one of the few integrable nonlinear lattices, admits a wide class of analytical solutions, ranging from exact spatially localized solitons to rational solutions in the form of the spatiotemporally localized discrete Peregrine soliton. Proving a closeness result between the solutions of the Ablowitz–Ladik system and a wide class of Discrete Nonlinear Schrödinger systems in a sense of a continuous dependence on their initial data, we establish that such small amplitude waveforms may be supported in nonintegrable lattices for significantly large times. Nonintegrable systems exhibiting such behavior include a generalization of the Ablowitz–Ladik system with power-law nonlinearity and the discrete nonlinear Schrödinger equation with power-law and saturable nonlinearities. The outcome of numerical simulations illustrates, in excellent agreement with the analytical results, the persistence of small amplitude Ablowitz–Ladik analytical solutions in all the nonintegrable systems considered in this work, with the most striking example being that of the Peregine soliton.

Neural networks enforcing physical symmetries in nonlinear dynamical lattices: The case example of the Ablowitz–Ladik model

In this work we introduce symmetry-preserving, physics-informed neural networks (S-PINNs) motivated by symmetries that are ubiquitous to solutions of nonlinear dynamical lattices. Although the use of PINNs have recently attracted much attention in data-driven discovery of solutions chiefly to partial differential equations, we demonstrate that they fail at enforcing important physical laws including symmetries of solutions and conservation laws. Through the correlation of parity symmetries in both space and time of solutions to differential equations with their group equivariant representation, we construct group-equivariant NNs which respect spatio-temporal parity symmetry. Moreover, we adapt the proposed architecture to enforce different types of periodicity (or localization) of solutions to nonlinear dynamical lattices. We do so by applying S-PINNs to the completely integrable Ablowitz–Ladik model, and performing numerical experiments with a special focus on waveforms that are related to rogue structures. These include the Kuznetsov–Ma soliton, and Akhmediev breather as well as the Peregrine soliton. Our numerical results demonstrate the superiority and robustness of the proposed architecture over standard PINNs.

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 three-dimensional green-water event study on a fixed simplified wall-sided ship under freak waves

As non-linear extreme waves, freak waves have huge wave crests, large wave heights and concentrated energies. The freak waves usually lead to severe impacts on ships and marine structures in green water events. Nevertheless, the general mechanisms, water motion scenarios and impact loads on bow deck and superstructure in these events remain unclear. In the study, the freak waves based on Peregrine breather solution of NLS equation are generated and the green water events of a fixed ship model are numerically investigated in three-dimensional wave flume by CFD method. First, the accuracy of green water simulation method is validated with experimental data. Then, the propagation and evolution of freak waves in front of ship bow are observed. After that, the water motion scenarios, velocity fields and impact pressures on ship bow in different stages are investigated. The necessity of three-dimensional simulation is verified. The effects of wave heights, wave periods and breaking waves are analyzed. The results indicate the water elevations and largest peak pressures increase nonlinearly with freak wave heights and the short-period freak waves cause more severe impacts than long-period ones.

Characteristic behavior and modulation instability of Peregrine solitons of the ion acoustic waves: a perspective of Titan’s ionosphere

The propagational properties of solitons along with peregrine solitonic modulational instability of ion acoustic (IA) waves are investigated taking into account the plasma properties of the Titan’s ionosphere. The Titan’s ionosphere typically consists of positive (, , and ) and negative (, , and ) ionic species along with electrons. In this study, it is assumed that the positive ions and electrons follow the Lorentzian-like kappa distribution. The fluid model equations are considered for negative ions. The Korteweg-de Vries (KdV) equation is derived using the reductive perturbation method to study the propagational properties and the characteristic behavior of solitons. In addition, the multiple-scale perturbation method is also used to derive the non-linear Schrödinger (NLS) equation for the analyses of modulational instability along with the production of rogue waves considering Peregrine solitonic solution. The dark solitons appear for the smaller wave numbers, whereas the bright solitons appear for the larger wave numbers. On the other hand, instability due to modulation is excited when the wave number is greater than its critical value, whereas it becomes stable when the wave number is smaller. The Peregrine soliton carries a single pick that decays to the plane wave asymptotic background either in space or time domain.

General rogue wave solutions under SU(2) transformation in the vector Chen–Lee–Liu nonlinear Schrödinger equation

We obtain analytically the exact explicit rogue wave solutions up to the second order of the vector Chen–Lee–Liu nonlinear Schrödinger equation, using the generalized non-recursive Darboux transformation method. In terms of these solutions, we demonstrate the fundamental Peregrine solitons as well as their doublet, triplet, quartet, and sextet counterparts on the general periodic backgrounds caused by SU(2) transformation. We numerically confirm that, although the Peregrine solitons sitting on SU(2) periodic backgrounds may suffer from larger disturbances than what they experience on SO(2) periodic backgrounds, such rational solitons developed on either kind of backgrounds can manifest clearly in spite of strong non-integrable perturbations. Other rogue wave topics such as rogue wave coexistence and the related parametric conditions are also discussed.

Multi-Raman soliton self-frequency shifts and dissipative rogue wave trains induced by cubic-quintic-Raman contributions in a double-negative material

We examine the cubic-quintic-Raman influences on the system when the second-order dispersion and the cubic-quintic-nonlinearities act in a double-negative material. This research is done for both absorption and negative index regimes. Using the collective coordinates’ technique, the quintic-nonlinearity induces the generation of Kuznetsov-Ma breathers and Peregrine soliton. Thereafter, the cubic-Raman effect comes into play and breeds the chaotic Peregrine waves’ field, the complex rogue event, and the modified complex rogue event. Therefore, the quintic-Raman effect induces first-, second-, and third-forms of the dissipative rogue waves and the twin parallel wave trains. Some fascinating phenomena such as deletion of waves, damping effect, tree structure, and disintegration of the Raman soliton self-frequency shift leading to the multi-Raman soliton self-frequency shifts, are also scrutinized. Further, internal strife leading to these above-mentioned effects, are likewise presented in detail.

Rational W-shape solitons on a nonlinear electrical transmission line with Josephson junction

This paper examines envelope solitons, rational W-shape solitons to a Nonlinear Electrical Transmission Line (NETL) with Josephson junction (JJ) that consist of the Nth cells of circuits. By employing the reductive perturbation approach in the semi-discrete approximation, we obtain the nonlinear Schrödinger equation (NLSE). From this equation, the frequency ranges of propagation of bright and dark solitons were obtained. As in most of the works, the NLSE obtained is well known as the seat of rogue waves and Peregrine solitons. The nonlinearity provided by the JJ, combined with the dispersion, made it possible to study the Modulation Instability (MI) gain spectrum. The 3D and 2D graphical representations illustrated the instability zones and the W-rational solutions have followed by using numerical simulation. The results obtained in this paper are of a very capital contribution for the study of rogue waves in nonlinear transmission lines or other physical problems.

Photonic rogue waves in a strongly dispersive coupled-cavity array involving self-attractive Kerr nonlinearity

Based on the combined Dicke, Bose-Hubbard, and Lipkin-Meshkov-Glick Hamiltonian, we investigate the formation of photonic rogue waves, including their Akhmediev and Kuznetsov-Ma cousins, in the strongly dispersive regime of a generalized atom-cavity array configuration. By means of the multiple-scale approach and mean-field approximation, we unveil that, due to the interplay between the photon hopping-induced tunneling effect and the self-attractive nonlinearity, the photons can redistribute in the whole array, which in turn leads to the redistribution (creation or annihilation) of excited atoms in each cavity and, consequently, the occurrence of Peregrine soliton states in both photonic and atomic fields. This interplay also yields an interesting intensity-clamping effect that the sum of the peak intensities of both rogue waves will be constant, although this quantity varies from site to site. We numerically confirm the analytical predictions, showing that the Peregrine rogue waves generated are stable against perturbations, despite the onset of modulation instability.

Two-dimensional self-similarity transformation theory and line rogue waves excitation

A two-dimensional self-similarity transformation theory is established, and the focusing (parabolic) (2 + 1)-dimensional NLS equation is taken as the model. The two-dimensional self-similarity transformation is proposed for converting the focusing (2 + 1)-dimensional NLS equation into the focusing (1 + 1) dimensional NLS equations, and the excitation of its novel line-rogue waves is further investigated. It is found that the spatial coherent structures induced by the Akhmediev breathers (AB) and Kuznetsov-Ma solitons (KMS) also have the short-lived characteristics which are possessed by the line-rogue waves induced by the Peregrine solitons, and the other higher-order rogue waves and the multi-rogue waves of the (1 + 1) dimensional NLS equations. This is completely different from the evolution characteristics of spatially coherent structures induced by bright solitons (including multi-solitons and lump solutions), with their shapes and amplitudes kept unchanged. The diagram shows the evolution characteristics of all kinds of resulting line rogue waves. The new excitation mechanism of line rogue waves revealed contributes to the new understanding of the coherent structure of high-dimensional nonlinear wave models.

Self-similar transformation and excitation of rogue waves for (2+1)-dimensional Zakharov equation

The search for the excitation of two-dimensional rogue wave in a (2+1)-dimensional nonlinear evolution model is a research hotspot. In this paper, the self-similar transformation of the (2+1)-dimensional Zakharov equation is established, and this equation is transformed into the (1+1)-dimensional nonlinear Schrödinger equation. Based on the similarity transformation and the rational formal solution of the (1+1)-dimensional nonlinear Schrödinger equation, the rogue wave excitation of the (2+1)-dimensional Zakharov equation is obtained by selecting appropriate parameters. We can see that the shape and amplitude of the rogue waves can be effectively controlled. Finally, the propagation characteristics of line rogue waves are diagrammed visually. We also find that the line-type characteristics of two-dimensional rogue wave are present in the <i>x-y</i> plane when the parameter <inline-formula><tex-math id="M5">\begin{document}$ \gamma = 1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20211181_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20211181_M5.png"/></alternatives></inline-formula>. The line rogue wave is converted into discrete localized rogue wave in the <i>x-y</i> plane when the parameter <inline-formula><tex-math id="M6">\begin{document}$ \gamma \ne 1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20211181_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20211181_M6.png"/></alternatives></inline-formula>. The spatial localized rogue waves with short-life can be obtained in the required <i>x-y</i> plane region. This is similar to the Peregrine soliton (PS) first discovered by Peregrine in the (1+1)-dimensional NLS equation, which is the limit case of the “Kuznetsov-Ma soliton” (KMS) or “Akhmediev breather” (AB). The proposed approach to constructing the line rogue waves of the (2+1) dimensional Zakharov equation can serve as a potential physical mechanism to excite two-dimensional rogue waves, and can be extended to other (2+1)-dimensional nonlinear systems.

Editorial: Peregrine Soliton and Breathers in Wave Physics: Achievements and Perspectives

Editorial: Peregrine Soliton and Breathers in Wave Physics: Achievements and Perspectives

Spatial self-similar transformation and novel line rogue waves in the Fokas system

We first propose a two-dimensional spatial self-similar transformation of the Fokas system, which is translated into the (1+1)-dimensional nonlinear Schrödinger equation, and is construct its abundant line rogue waves excitation. It found that the line rogue waves induced by the Akhmediev breathers and Kuznetsov-Ma solitons also have the short life characteristics possessed for the line rogue waves induced by the Peregrine solitons and other higher-order rogue waves and multi-rogue waves of the (1+1) dimensional standard NLS equations. This is completely different from the evolution characteristics of line soliton induced by bright solitons and the multi-solitons, which keeping their shape and amplitude unchanged. The diagram shows the evolution characteristics of the resulting all kinds of line rogue waves. The new excitation mechanism of line rogue waves for the Fokas system revealed contributes to the new understanding of the localized coherent structure of high-dimensional nonlinear wave models.

Electron-Acoustic (Un)Modulated Structures in a Plasma Having (r, q)-Distributed Electrons: Solitons, Super Rogue Waves, and Breathers

The propagation of electron-acoustic waves (EAWs) in an unmagnetized plasma, comprising (r,q)-distributed hot electrons, cold inertial electrons, and stationary positive ions, is investigated. Both the unmodulated and modulated EAWs, such as solitary waves, rogue waves (RWs), and breathers are discussed. The Sagdeev potential approach is employed to determine the existence domain of electron acoustic solitary structures and study the perfectly symmetric planar nonlinear unmodulated structures. Moreover, the nonlinear Schrödinger equation (NLSE) is derived and its modulated solutions, including first order RWs (Peregrine soliton), higher-order RWs (super RWs), and breathers (Akhmediev breathers and Kuznetsov–Ma soliton) are presented. The effects of plasma parameters and, in particular, the effects of spectral indices r and q, of distribution functions on the characteristics of both unmodulated and modulated EAWs, are examined in detail. In a limited cases, the (r,q) distribution is compared with Maxwellian and kappa distributions. The present investigation may be beneficial to comprehend and predict the modulated and unmodulated electron acoustic structures in laboratory and space plasmas.

Peregrine Soliton as a Limiting Behavior of the Kuznetsov-Ma and Akhmediev Breathers

This article discusses a limiting behavior of breather solutions of the focusing nonlinear Schrödinger equation. These breathers belong to the family of solitons on a non-vanishing and constant background, where the continuous-wave envelope serves as a pedestal. The rational Peregrine soliton acts as a limiting behavior of the other two breather solitons, i.e., the Kuznetsov-Ma breather and Akhmediev soliton. Albeit with a phase shift, the latter becomes a nonlinear extension of the homoclinic orbit waveform corresponding to an unstable mode in the modulational instability phenomenon. All breathers are prototypes for rogue waves in nonlinear and dispersive media. We present a rigorous proof using the ϵ-δ argument and show the corresponding visualization for this limiting behavior.