Peregrine Soliton Research Articles

Investigation of Nonlinear Wave–Ice Interaction Using Parameter Study and Numerical Simulation

Abstract This paper investigates the question of the existence of nonlinear wave–ice interaction with the focus on nonlinear wave propagation and dispersion of waves. The scope of this investigation is to provide a better understanding of ice and wave conditions required to observe nonlinear wave effects under level ice. Direct numerical simulations of nonlinear waves in solid ice are performed within the weakly nonlinear Schrödinger equation (NLSE) framework, using the theoretical findings from Liu and Mollo-Christensen’s 1988 paper. Systematic variations of wave and ice parameters address the impact of the mechanical ice properties and ice thickness on traveling waves of certain wave lengths. The impacts of parameter characteristics on nonlinear focusing and wave dynamics, as well as possible constraints regarding physical consistency, are discussed. It is presented that nonlinear focusing in level ice occurs theoretically. Hereby, distinctive areas of validity with respect to nonlinear wave focusing are identified within the parameter study, which strongly depends on the material properties of the level ice. The results obtained in the parameter study are subsequently used to investigate wave focusing under level ice. Therefore, an exact solution of the NLSE, the Peregrine breather, is utilized. The analytical solution for level ice is compared to the open water solution and accompanied by direct numerical simulations. These investigations show that nonlinear wave focusing can be predicted under level ice for certain parameters. In addition, the agreement of the direct simulations and the analytic solution verifies the numerical approach for nonlinear waves in solid ice.

Mulitiscale spatiotemporal structures in mode-locked fiber lasers

Using an Er-doped fiber laser as a test bed, here we for the first time experimentally demonstrate the simultaneous effect of the fast scale (round-trip time) and slow scale (thousands round-trip time) instabilities on the emergence of breathers similar to the Akhmediev breathers, Peregrine solitons, and partially mode-locked chaotic solitons. The anomalous statistics of the laser output power justifies the connection of the observed spatiotemporal structures with bright and dark rogue waves. Apart from the interest in laser physics for revealing mechanisms of the multiscale dynamics, the obtained results can be of fundamental interest for studying spatiotemporal patterns induced by the interplay of the mechanisms mentioned above in various distributed systems.

Effect of local Peregrine soliton emergence on statistics of random waves in the one-dimensional focusing nonlinear Schrödinger equation.

The Peregrine soliton is often considered as a prototype of rogue waves. After recent advances in the semiclassical limit of the one-dimensional focusing nonlinear Schrödinger equation [M. Bertola and A. Tovbis, Commun. Pure Appl. Math. 66, 678 (2013)0010-364010.1002/cpa.21445] this conjecture can be seen from another perspective. In the present paper, connecting deterministic and statistical approaches, we numerically demonstrate the effect of the universal local emergence of Peregrine solitons on the evolution of statistical properties of random waves. Evidence of this effect is found in recent experimental studies in the contexts of fiber optics and hydrodynamics. The present approach can serve as a powerful tool for the description of the transient dynamics of random waves and provide new insights into the problem of the rogue waves formation.

Resonant radiation from Peregrine solitons

We investigate the phenomenon of resonant radiation emitted by Peregrine solitons. We show that, unlike bright or dark solitons of the nonlinear Schrödinger equation, the radiation process is affected by the intrinsic local longitudinal variation of the soliton wavenumber. We give a phase-matching condition that allows the prediction of the multiple spectral peaks of the resonant radiation.

Peregrine Soliton and Akhmediev Breathers in a Chameleon Electrical Transmission Line

We analyze the particular behavior exhibited by a chaotic waves field containing Peregrine soliton and Akhmediev breathers. This behavior can be assimilated to a tree with “roots of propagation” which propagate randomly. Besides, this strange phenomenon can be called “tree structures”. So, we present the collapse of dark and bright solitons in order to build up the above mentioned chaotic waves field. The investigation is done in a particular nonlinear transmission line called chameleon nonlinear transmission line. Thus, we show that this line acts as a bandpass filter at low frequencies and the impact of distance, frequency and dimensionless capacitor are also presented. In addition, the chameleon’s behavior is due to the fact that without modifying the appearance structure, it can present alternatively purely right- or left-handed transmission line. This line is different to the composite one.

Triangular Rogue Waves and Multi- Wave Trains Generation in a Chameleon Electrical Transmission Line

In this work, we present a particular nonlinear transmission line called chameleon nonlinear transmission line. In fact, the chameleon’s behavior is related to the fact that without changing its appearance, the line can exhibit alternatively purely right - or left - handed behavior. This transmission line is different to the composite one. So, the goal of this work is to demonstrate that this special line can support the propagation of soliton light pulse subjected to quintic – phase modulation, improve the freak wave’s mechanism of generation and verify the chameleon’s behavior of the line. Consequently, we employ collective coordinate’s theory in order to give a great characterization of the light pulse. Then, we introduce an upgraded function called “type II Ansatz function” with eight collective coordinates compared to the conventional Gaussian Ansatz with six collective coordinates. We show that the two additional coordinates will allow us to improve the technique of measurement of internal excitation leading to the generation of rogue events. Moreover, these coordinates will give supplemental details on frequency shift and chirp distortions during the generation of specific rogue events such as “wall of waves”, tree structures, multi - wave trains, Kuznetsov - Ma breathers, Akhmediev breathers, Peregrine solitons and triangular rogue waves. The stability of the soliton light pulse will be also investigated at specific frequency ranges.

Self-similarity transformation and two-dimensional rogue wave construction of non-autonomous Kadomtsev-Petviashvili equation

<sec>Rogue wave is a kind of natural phenomenon that is fascinating, rare, and extreme. It has become a frontier of academic research. The rogue wave is considered as a spatiotemporal local rational function solution of nonlinear wave model. There are still very few (2 + 1)-dimensional nonlinear wave models which have rogue wave solutions, in comparison with soliton and Lump waves that are found in almost all (2 + 1)-dimensional nonlinear wave models and can be solved by different methods, such as inverse scattering method, Hirota bilinear method, Darboux transform method, Riemann-Hilbert method, and homoclinic test method. The structure and evolution characteristics of the obtained (2 + 1)-dimensional rogue waves are quite different from the prototypes of the (1 + 1)-dimensional nonlinear Schrödinger equation. Therefore, it is of great value to study two-dimensional rogue waves.</sec><sec>In this paper, the non-autonomous Kadomtsev-Petviashvili equation is first converted into the Kadomtsev-Petviashvili equation with the aid of a similar transformation, then two-dimensional rogue wave solutions represented by the rational functions of the non-autonomous Kadomtsev-Petviashvili equation are constructed based on the Lump solution of the first kind of Kadomtsev-Petviashvili equation, and their evolutionary characteristics are illustrated by images through appropriately selecting the variable parameters and the dynamic stability of two-dimensional single rogue waves is numerically simulated by the fast Fourier transform algorithm. The obtained two-dimensional rogue waves, which are localized in both space and time, can be viewed as a two-dimensional analogue to the Peregrine soliton and thus are a natural candidate for describing the rogue wave phenomena. The method presented here provides enlightenment for searching for rogue wave excitation of (2 + 1)-dimensional nonlinear wave models.</sec><sec>We show that two-dimensional rogue waves are localized in both space and time which arise from the zero background and then disappear into the zero background again. These rogue-wave solutions to the non-autonomous Kadomtsev-Petviashvili equation generalize the rogue waves of the nonlinear Schrödinger equation into two spatial dimensions, and they could play a role in physically understanding the rogue water waves in the ocean.</sec>

Recent developments of extreme wave events in integrable resonant systems

From a microscopic perspective, the single extreme rogue wave event can be thought of as the spatiotemporally localized rational solutions of the underlying integrable model. A typical example is the fundamental Peregrine rogue wave, who in general entails a three-fold peak amplitude, while making its peak position arbitrary on a finite continuous-wave background. This kind of bizarre wave structure agrees well with the fleeting nature of realistic rogue waves and has been confirmed experimentally, first in nonlinear fibers, then in water wave tanks and plasmas, and recently in an irregular oceanic sea state. In this review, with a brief overview of the current state of the art of the concepts, methods, and research trends related to rogue wave events, we mainly discuss the fundamental Peregrine rogue wave solutions as well as their recent progress, intended for three typical integrable models, namely, the long-wave short-wave resonant equation, the three-wave resonant interaction equation, and the nonlinear Schrödinger and Maxwell–Bloch equation. Basically, while the first two models can describe the resonant interaction among optical waves, the latter governs the interaction between the optical waves and the resonant medium. For each integrable model, we present explicitly its Lax pair, Darboux transformation formulas, and fundamental Peregrine rogue wave solutions, in a self-consistent way. We confirm by convincing examples that these fundamental rogue wave solutions exhibit universality and can be applied to the multi-component or the higher-order versions of the current integrable models. By means of numerical simulations, we demonstrate as well several novel rogue wave dynamics such as coexisting rogue waves, complementary rogue waves, and Peregrine solitons of self-induced transparency.

Differential Relations for the Solutions to the NLS Equation and Their Different Representations

Solutions to the focusing nonlinear Schr\"odinger equation (NLS) of order $N$ depending on $2N-2$ real parameters in terms of wronskians and Fredholm determinants are given. These solutions give families of quasi-rational solutions to the NLS equation denoted by $v_{N}$ and have been explicitly constructed until order $N = 13$. These solutions appear as deformations of the Peregrine breather $P_{N}$ as they can be obtained when all parameters are equal to $0$. These quasi rational solutions can be expressed as a quotient of two polynomials of degree $N(N+1)$ in the variables $x$ and $t$ and the maximum of the modulus of the Peregrine breather of order $N$ is equal to $2N+1$. \\ Here we give some relations between solutions to this equation. In particular, we present a connection between the modulus of these solutions and the denominator part of their rational expressions. Some relations between numerator and denominator of the Peregrine breather are presented.

Study on the behavior of weakly nonlinear water waves in the presence of random wind forcing

Specific solutions of the nonlinear Schrodinger equation, such as the Peregrine breather, are considered to be prototypes of extreme or freak waves in the oceans. An important question is whether these solutions also exist in the presence of gusty wind. Using the method of multiple scales, a nonlinear Schrodinger equation is obtained for the case of wind-forced weakly nonlinear deep water waves. Thereby, the wind forcing is modeled as a stochastic process. This leads to a stochastic nonlinear Schrodinger equation, which is calculated for different wind regimes. For the case of wind forcing which is either random in time or random in space, it is shown that breather-type solutions such as the Peregrine breather occur even in strong gusty wind conditions.

A nonlinear Schrödinger equation for gravity waves slowly modulated by linear shear flow**Project supported by the National Key Research and Development Program of China (Grant Nos. 2016YFC1401404 and 2017YFA0604102) and the National Natural Science Foundation of China (Grant No. 41830533).

Assume that a fluid is inviscid, incompressible, and irrotational. A nonlinear Schrödinger equation (NLSE) describing the evolution of gravity waves in finite water depth is derived using the multiple-scale analysis method. The gravity waves are influenced by a linear shear flow, which is composed of a uniform flow and a shear flow with constant vorticity. The modulational instability (MI) of the NLSE is analyzed, and the region of the MI for gravity waves (the necessary condition for existence of freak waves) is identified. In this work, the uniform background flows along or against wave propagation are referred to as down-flow and up-flow, respectively. Uniform up-flow enhances the MI, whereas uniform down-flow reduces it. Positive vorticity enhances the MI, while negative vorticity reduces it. Hence, the influence of positive (negative) vorticity on MI can be balanced out by that of uniform down (up) flow. Furthermore, the Peregrine breather solution of the NLSE is applied to freak waves. Uniform up-flow increases the steepness of the free surface elevation, while uniform down-flow decreases it. Positive vorticity increases the steepness of the free surface elevation, whereas negative vorticity decreases it.

The linearly damped nonlinear Schrödinger equation with localized driving: spatiotemporal decay estimates and the emergence of extreme wave events

We prove spatiotemporal algebraically decaying estimates for the density of the solutions of the linearly damped nonlinear Schr\"odinger equation with localized driving, when supplemented with vanishing boundary conditions. Their derivation is made via a scheme, which incorporates suitable weighted Sobolev spaces and a time-weighted energy method. Numerical simulations examining the dynamics (in the presence of physically relevant examples of driver types and driving amplitude/linear loss regimes), showcase that the suggested decaying rates, are proved relevant in describing the transient dynamics of the solutions, prior their decay: they support the emergence of waveforms possessing an algebraic space-time localization (reminiscent of the Peregrine soliton) as first events of the dynamics, but also effectively capture the space-time asymptotics of the numerical solutions.

Space–time evolution of optical breathers and modulation instability patterns in metamaterial waveguides

We present numerical and theoretical investigations of the spontaneous emergence of noise-driven modulation instability patterns in a metamaterial waveguide, which involves the generation of optical breather waves such as the Peregrine soliton, Akhmediev breathers and Kuznetsov–Ma breathers. We show that the intrinsic properties of the metamaterial waveguide, e.g. self-steepening and the magnetooptic effects, offer the potential to control the formation and subsequent spectral and temporal dynamics of these localized nonlinear waves. Such internal or external perturbations break the symmetry of the spectrum of nonlinear waves, thus leading to the existence of a controllable characteristic group velocity in their space–time evolution.

Two-dimensional modulation instability of wind waves

It is widely known that deep-water waves are modulationally unstable and that this can be modelled by a nonlinear Schrödinger equation. In this paper, we extend the previous studies of the effect of wind forcing on this instability to water waves in finite depth and in two horizontal space dimensions. The principal finding is that the instability is enhanced and becomes super-exponential and that the domain of instability in the modulation wavenumber space is enlarged. Since the outcome of modulation instability is expected to be the generation of rogue waves, represented within the framework of the nonlinear Schrödinger equation as a Peregrine breather, we also examine the effect of wind forcing on a Peregrine breather. We find that the breather amplitude will grow at twice the rate of a linear instability.

Higher-order rogue waves and modulation instability of the two-component derivative nonlinear Schrödinger equation

The determinant representation of the n-th order different kinds of solutions to the two-component derivative nonlinear schrödinger (DNLS) equation is constructed by performing the n-fold Darboux transformation (DT). Based on this representation, the Peregrine-type, double, triple, quadruple, and sextuple vector rogue waves are obtained. Interestingly, the Peregrine-type rogue wave is not usual Peregrine soliton, but anomalous Peregrine soliton whose peak amplitude can go beyond threefold limit. Besides, coexisting rogue waves display more abundant dynamical behaviors which have not been found in scalar DNLS equation. Finally, we analyze modulation instability (MI) and find that there exist two types of MI: one MI has an usual M profile, but the other has a deformed double M-type profile. The parameter space for the two types of baseband MI regimes is agreement with the one which results in the generation of rogue waves. It hence further confirms that the baseband MI is a possible mechanism to generate the rogue wave in a nonlinear system from a plane wave background.

Nonlinear Ion Acoustic Wave and Group Dynamics Near Critical Density in a Plasma with Negative Ion

A wide class of nonlinear structures of ion acoustic wave and its group dynamics are investigated in a negative ion plasma through the Gardner equation. In presence of self-interaction (self-focusing effect), the wave group dynamics is governed by a nonlinear Schrödinger equation, which is shown to become modulationally unstable due to the focussing (de-focussing) effect for the positive value of the third order nonlinear interaction term. Interestingly, at the boundary of this parameter space, the wave group dynamics is shown to governed by a modified nonlinear Schrödinger equation. The analytical and computational results reveal that nonlinear wave exhibits soliton, double layer, table-top soliton, bipolar structure, envelope, Peregrine soliton and breather structures. The results are in qualitative agreement with experimental observations in plasmas with negative ions.

Numerical study of the interaction between peregrine breather based freak waves and twin-plate breakwater

Freak wave is a kind of extreme and unexpected surface wave with great wave height and energy, which might happen in different ocean areas such as the deep-sea zone and offshore zone. Coastal, offshore and marine structures such as harbors, ships, platforms and very large floating structures built in various ocean areas might be prone to be threatened by ocean waves, especially by extreme waves like freak waves. However, although researchers have already proposed and studied many types of breakwaters to protect structures from damage by ocean waves, few research has been done on the interaction between freak waves and breakwaters. This paper aims to study the interaction between freak waves based on Peregrine breather solution and twin-plate breakwaters, including the characteristics of interaction phenomena, transmitted wave surface, wave loads and structural responses. Firstly, a 2-D numerical wave tank is built in which the simulations are conducted using computational fluid dynamics (CFD) approach, while a strong-coupled fluid–structure interaction (FSI) algorithm is applied to consider the hydroelastic effects. Secondly, validations of the interaction between regular waves and single/twin-plate breakwaters are performed to show the effectiveness of the numerical method. Eventually, simulations of the interaction between freak waves and twin-plate breakwaters are carried out, in which the phenomena are shown and results are analyzed. Effects of carrier wave height and carrier wave length of freak waves are further considered respectively. Hydroelasticity excitation caused by severe slamming is observed and analyzed during the FSI process.

Super chirped rogue waves in optical fibers.

The super rogue wave dynamics in optical fibers are investigated within the framework of a generalized nonlinear Schrödinger equation containing group-velocity dispersion, Kerr and quintic nonlinearity, and self-steepening effect. In terms of the explicit rogue wave solutions up to the third order, we show that, for a rogue wave solution of order n, it can be shaped up as a single super rogue wave state with its peak amplitude 2n+1 times the background level, which results from the superposition of n(n+1)/2 Peregrine solitons. Particularly, we demonstrate that these super rogue waves involve a frequency chirp that is also localized in both time and space. The robustness of the super chirped rogue waves against white-noise perturbations as well as the possibility of generating them in a turbulent field is numerically confirmed, which anticipates their accessibility to experimental observation.

General rogue wave solutions of the coupled Fokas-Lenells equations and non-recursive Darboux transformation.

We formulate a non-recursive Darboux transformation technique to obtain the general nth-order rational rogue wave solutions to the coupled Fokas-Lenells system, which is an integrable extension of the noted Manakov system, by considering both the double-root and triple-root situations of the spectral characteristic equation. Based on the explicit fundamental and second-order rogue wave solutions, we demonstrate several interesting rogue wave dynamics, among which are coexisting rogue waves and anomalous Peregrine solitons. Our solutions are generalized to include the complete background-field parameters and therefore helpful for future experimental study.

Possibility of the existence of the rogue wave and the super rogue wave in granular matter.

By using the traditional perturbation technique, a focusing nonlinear Schrödinger equation (NLSE) for the one-dimensional bead chain with the initial prestress is first obtained. The Peregrine soliton, called the rogue wave in the present paper, and the super rogue wave are investigated both numerically and analytically. It is noted that both the rogue wave and the super rogue wave do exist in the one-dimensional bead chain. The solutions from the NLSE can correctly describe the real rogue wave as well as the real super rogue wave in the limiting case of small amplitude. Both the rogue wave and the super rogue wave propagate in the granular bead chain as if they are solitary waves.