Rogue waves emerging from the resonant interaction of three waves

Rogue wave excitations and hybrid wave structures of the Heisenberg ferromagnet equation with time-dependent inhomogeneous bilinear interaction and spin-transfer torque

In this paper, we focus on the localized rational waves of the variable-coefficient Heisenberg spin chain equation, which models the local magnetization in ferromagnet with time-dependent inhomogeneous bilinear interaction and spin-transfer torque. First, we establish the iterative generalized (m,N-m)-fold Darboux transformation of the Heisenberg spin chain equation. Then, the novel localized rational solutions (LRSs), rogue waves (RWs), periodic waves, and hybrid wave structures on the periodic, zero, and nonzero constant backgrounds with the time-dependent coefficients α(t) and β(t) are obtained explicitly. Additionally, we provide the trajectory curves of magnetization and the variation of the magnetization direction for the obtained nonlinear waves at different times. These phenomena imply that the LRSs and RWs play the crucial roles in changing the circular motion of the magnetization. Finally, we also numerically simulate the wave propagations of some localized semi-rational solutions and RWs.

Triad resonance for internal waves in a uniformly stratified fluid: Rogue waves and breathers

Three-wave (triad) resonance in a uniformly stratified fluid is investigated as a case study of energy transfer among oscillatory modes. The existence of a degenerate triad is demonstrated explicitly, where two components have identical group velocity. An illuminating example is a resonance involving waves from modes 1, 3, 5 families, but many other combinations are possible. The physical applications and nonlinear dynamics of rogue waves derived analytically in the literature are examined. Exact solutions with four free parameters (two related to the amplitudes of the background plane waves, two related to the frequencies of slowly varying envelopes) describe motions localized in both space and time. The differences between rogue waves of the degenerate versus the nondegenerate cases are highlighted. The phase and profile of the degenerate case rogue waves are correlated. The volume or energy of the rogue wave (defined as the total extent or energy contents of the fluid set in motion for the duration of the rogue wave) may change drastically, if the wave envelope parameters vary. Pulsating modes (breathers) have been studied previously by layered-fluid and modified Korteweg-de Vries models. Here we extend the consideration to stratified fluids but for the simpler case of nondegenerate triads. Instabilities of fission and fusion of breathers are confirmed computationally with Floquet analysis. This knowledge should prove useful for energy transfer processes in the oceans.

Coexistence of Gaussian and non-Gaussian statistics in vector integrable turbulence

Integrable turbulence studies the complex dynamics of random waves for the nonlinear integrable systems, and it has become an important element in exploring the sophisticated turbulent phenomena. In the present work, based on the coupled nonlinear Schrödinger models, we have shown the coexistence of Gaussian and non-Gaussian single-point statistics in multiple wave components, which might be viewed as an exclusive feature for the vector integrable turbulence. This coexistent statistic can relate to different distributions of the vector solitonic excitations depending on the time-invariant nonlinear spectra. Our results are expected to shed light on a deeper understanding of the turbulent behaviors of vector waves and may motivate relevant experiments in the coupled optical or atomic systems.

Formation of Rogue Waves and Modulational Instability with Zero-Wavenumber Gain in Multicomponent Systems with Coherent Coupling

It is known that rogue waves (RWs) are generated by the modulational instability (MI) of the baseband type. Starting with the Bers-Kaup-Reiman system for three-wave resonant interactions, we identify a specific RW-building mechanism based on MI which includes zero wavenumber in the gain band. An essential finding is that this mechanism works solely under a linear relation between the MI gain and a vanishingly small wavenumber of the modulational perturbation. The same mechanism leads to the creation of RWs by MI in other multicomponent systems-in particular, in the massive Thirring model.

Fermi-Pasta-Ulam-Tsingou recurrence and cascading mechanism for resonant three-wave interactions

Evolution of resonant three-wave interaction is governed by quadratic nonlinearities. While propagating localized modes and inverse scattering mechanisms have been studied, transient states such as rogue waves and breathers are not fully understood. Modulation instability modes can trigger growth of disturbances and the eventual development of breathers. Here we study computationally the dynamics beyond the first formation of breathers, and demonstrate repeating patterns of breathers as a manifestation of the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT). While nonlinearity governs the actual dynamics, the range of wave numbers for modulation instability remains a useful indicator. Depending on the stability characteristics of the fundamental mode and the higher-order harmonics ("sidebands"), "regular" and "staggered" FPUT patterns can arise. A "cascading mechanism" provides analytical verification, as the fundamental and sideband modes attain the same magnitude at one particular instant, signifying the first occurrence of a breather. A triangular spectrum is also computed, similar to experimental observations of optical pulses. Such spectra can elucidate the spreading of energy among the sidebands and components of the triad resonance. The concept of "effective energy" is examined and the eigenvalues of the inverse scattering mechanism are computed. Both approaches are utilized to correlate with the occurrence of regular or staggered FPUT. These numerical and analytical studies can enhance our understanding of wave interactions in fluid mechanics and optics.

Exact analytical soliton solutions of N-component coupled nonlinear Schrödinger equations with arbitrary nonlinear parameters

Exact analytical soliton solutions play an important role in soliton fields. Soliton solutions were obtained with some special constraints on the nonlinear parameters in nonlinear coupled systems, but they usually do not hold in real physical systems. We successfully release all usual constrain conditions on nonlinear parameters for exact analytical vector soliton solutions in N-component coupled nonlinear Schrödinger equations. The exact soliton solutions and their existence condition are given explicitly. Applications of these results are discussed in several present experimental parameter regimes. The results would motivate experiments to observe more novel vector solitons in nonlinear optical fibers, Bose-Einstein condensates, and other nonlinear coupled systems.

Super rogue waves: Collision of rogue waves in Bose-Einstein condensate

An important and incompletely answered question is whether a high-order rogue wave (RW) can be excited by the collision of first-order RWs, especially for its generation and propagation mechanisms. In this paper, the evolution properties of collisions between RWs are studied numerically for two-component coupled Bose-Einstein condensates. We find that whether a second-order RW can be successfully excited strongly depends on the location and time of the first-order RWs encountered. Only when the highest peaks of two RWs meet at the same position at the same time, can the standard second-order RW be triggered and its structure is in good agreement with the exact solution. Further, we demonstrate that such results are also applicable to the collision of subordinate RWs. This paper not only reveals the collision properties of RWs, but also provides a feasible scheme to generate higher-order RWs for experimental realization.

Measuring the rogue wave pattern triggered from Gaussian perturbations by deep learning

Weak Gaussian perturbations on a plane wave background could trigger lots of rogue waves (RWs), due to modulational instability. Numerical simulations showed that these RWs seemed to have similar unit structure. However, to the best of our knowledge, there are no relative results to prove that these RWs have the similar patterns for different perturbations, partly due to that it is hard to measure the RW pattern automatically. In this work, we address these problems from the perspective of computer vision via using deep neural networks. We propose a rogue wave detection network (RWD-Net) model to automatically and accurately detect RWs in the images, which directly indicates they have the similar computer vision patterns. For this purpose, we herein meanwhile have designed and release the corresponding dataset, termed as rogue wave dataset-10K (RWD-10K), which has 10191 RW images with bounding box annotations for each RW unit. In our detection experiments, we get 99.29% average precision on the test splits of the proposed dataset. Finally, we derive our metric, termed as the density of RW units, to characterize the evolution of Gaussian perturbations and obtain the statistical results on them.

Strong and weak interactions of rational vector rogue waves and solitons to any n -component nonlinear Schrödinger system with higher-order effects

The higher-order effects play an important role in the wave propagations of ultrashort (e.g. subpicosecond or femtosecond) light pulses in optical fibres. In this paper, we investigate any n -component fourth-order nonlinear Schrödinger ( n -FONLS) system with non-zero backgrounds containing the n -Hirota equation and the n -Lakshmanan–Porsezian–Daniel equation. Based on the loop group theory, we find the multi-parameter family of novel rational vector rogue waves (RVRWs) of the n -FONLS equation starting from the plane-wave solutions. Moreover, we exhibit the weak and strong interactions of some representative RVRW structures. In particular, we also find that the W-shaped rational vector dark and bright solitons of the n -FONLS equation as the second- and fourth-order dispersion coefficients satisfy some relation. Furthermore, we find the higher-order RVRWs of the n -FONLS equation. These obtained rational solutions will be useful in the study of RVRW phenomena of multi-component nonlinear wave models in nonlinear optics, deep ocean and Bose–Einstein condensates.

Parity-time-symmetric rational vector rogue waves of the n-component nonlinear Schrödinger equation

Extreme events are investigated in the integrable n-component nonlinear Schrödinger (NLS) equation with focusing nonlinearity. We report novel multi-parametric families of rational vector rogue wave (RW) solutions featuring the parity-time ( PT) symmetry, which are characterized by non-identical boundary conditions for the components that are consistent with the degeneracy of n branches of Benjamin-Feir instability. Explicit examples of PT-symmetric rational vector RWs are presented. Subject to the specific choice of the parameters, high-amplitude RWs are generated. The effect of a small non-integrable deformation of the 3-NLS equation on the excitation of vector RWs is discussed. The reported results can be useful for the design of experiments for observation of high-amplitude RWs in multi-component nonlinear physical systems.

Triads and Rogue Events for Internal Waves in Stratified Fluids with a Constant Buoyancy Frequency

Internal waves in a stratified fluid with a constant buoyancy frequency were studied, with special attention given to rogue modes, extreme waves, dynamical evolution, and Fermi–Pasta–Ulam–Tsingou type recurrence phenomena. Rogue waves for triads in a general physical setting have recently been derived analytically, but the implications in a fluid mechanics context have not yet been fully assessed. Numerical simulations were conducted for cases of coupled triads where the common member is a daughter wave mode. In sharp contrast with previous studies, rogue modes instead of plane waves were used as the initial condition. Furthermore, spatial dependence was incorporated. Rogue or extreme waves in one set of triads provided a possible mechanism for significant energy transfer among modes of the internal wave spectrum, in addition to the other known theories, e.g., weak turbulence. Remarkably, Fermi–Pasta–Ulam–Tsingou recurrence types of growth and decay cycles arose, similar to those observed for surface gravity wave groups governed by the cubic nonlinear Schrödinger equation. These mechanisms will enhance our understanding of transport processes in oceans.

Rogue waves of ultra-high peak amplitude: a mechanism for reaching up to a thousand times the background level

We unveil a mechanism enabling a fundamental rogue wave, expressed by a rational function of fourth degree, to reach a peak amplitude as high as a thousand times the background level in a system of coupled nonlinear Schrödinger equations involving both incoherent and coherent coupling terms with suitable coefficients. We obtain the exact explicit vector rational solutions using a Darboux-dressing transformation. We show that both components of such coupled equations can reach extremely high amplitudes. The mechanism is confirmed in direct numerical simulations and its robustness is confirmed upon noisy perturbations. Additionally, we showcase the fact that extremely high peak-amplitude vector fundamental rogue waves (of about 80 times the background level) can be excited even within a chaotic background field .

Optical polarization rogue waves and their identifications

Optical rogue waves are a class of pulses with extremely large amplitudes, whose probability of occurrence unexpectedly deviates from Gaussian-law statistics. To date, the mechanisms of rogue wave generation are still debated: investigations are under way, exploring the statistics of various pulse dimensions across different physical domains. Although polarization is one of the fundamental parameters of optical rogue waves, its statistics have received little attention until recently. Here, we review recent process of the polarization-dependent properties of optical rogue waves in ultrafast optics. Based on a two-dimensional statistical model, we introduce the concept of optical polarization rogue waves. Specifically, we consider the frequency of generation of waves with freak or rogue state of polarization, with a probability of occurrence deviating from a normal distribution. We demonstrate three nonlinear optical laser systems: a partially mode-locked laser, a dissipative soliton laser, and supercontinuum generation within a highly nonlinear fiber. Further, we identify optical polarization rogue waves in nonlinear laser systems, and discuss their generation mechanisms. Related results reveal that optical polarization rogue waves are embedded in optical systems with a deteriorated degree of coherence, which originates from vector four-wave-mixing processes. Polarization-dependent investigations will provide additional insight for our understanding of optical rogue waves.

Fundamental Peregrine Solitons of Ultrastrong Amplitude Enhancement through Self-Steepening in Vector Nonlinear Systems

We report the universal emergence of anomalous fundamental Peregrine solitons, which can exhibit an unprecedentedly ultrahigh peak amplitude comparable to any higher-order rogue wave events, in the vector derivative nonlinear Schrödinger system involving the self-steepening effect. We present the exact explicit rational solutions on either a continuous-wave or a periodical-wave background, for a broad range of parameters. We numerically confirm the buildup of anomalous Peregrine solitons from strong initial harmonic perturbations, despite the onset of competing modulation instability. Our results may stimulate the experimental study of such Peregrine soliton anomaly in birefringent crystals or other similar vector systems.

High-order rogue waves of a long-wave-short-wave model of Newell type

The long-wave-short-wave (LWSW) model of Newell type is an integrable model describing the interaction between the gravity wave (long wave) and the capillary wave (short wave) for the surface wave of deep water under certain resonance conditions. In the present paper, we are concerned with rogue-wave solutions to the LWSW model of Newell type. By combining the Hirota's bilinear method and the KP hierarchy reduction, we construct its general rational solution expressed by the determinant. It is found that the fundamental rogue wave for the short wave can be classified into three different patterns: bright, intermediate, and dark states, whereas the one for the long wave is always a bright state. The higher-order rogue wave corresponds to the superposition of fundamental ones. The modulation instability analysis shows that the condition of the baseband modulation instability where an unstable continuous-wave background corresponds to perturbations with infinitesimally small frequencies, coincides with the condition for the existence of rogue-wave solutions.

Bright and dark rogue internal waves: The Gardner equation approach

We have found "bright and dark" solutions of the Gardner equation which can model internal rogue waves in three-layer fluids. We provide the first four "bright" and "dark" exact rational solutions to the Gardner equation. These are the lowest-order solutions of the corresponding hierarchies of rogue-wave solutions of this equation. They have been obtained from the rogue-wave solutions of a modified Korteweg-de Vries equation by using a Lorentz-type transformation. The maximal (and minimal) amplitudes and the background levels of these solutions for arbitrary order are deduced, based on the lowest-order examples. These solutions can be useful for explanations of extremely large amplitude internal waves in the ocean, as well as for abnormally large-amplitude waves in other areas of nonlinear physics, such as optics and dusty plasmas.

Doubly periodic solutions of the class-I infinitely extended nonlinear Schrödinger equation

We present doubly periodic solutions of the infinitely extended nonlinear Schrödinger equation with an arbitrary number of higher-order terms and corresponding free real parameters. Solutions have one additional free variable parameter that allows one to vary periods along the two axes. The presence of infinitely many free parameters provides many possibilities in applying the solutions to nonlinear wave evolution. Being general, this solution admits several particular cases which are also given in this article.

The general coupled Hirota equations: modulational instability and higher-order vector rogue wave and multi-dark soliton structures

The general coupled Hirota equations are investigated, which describe the wave propagations of two ultrashort optical fields in a fibre. Firstly, we study the modulational instability for the focusing, defocusing and mixed cases. Secondly, we present a unified formula of high-order rational rogue waves (RWs) for the focusing, defocusing and mixed cases, and find that the distribution patterns for novel vector rational RWs of focusing case are more abundant than ones in the scalar model. Thirdly, the Nth-order vector semirational RWs can demonstrate the coexistence of Nth-order vector rational RWs and N breathers. Fourthly, we derive the multi-dark-dark solitons for the defocsuing and mixed cases. Finally, we derive a formula for the coexistence of dark solitons and RWs. These results further enrich and deepen the understanding of localized wave excitations and applications in vector nonlinear wave systems.

Peregrine Solitons Beyond the Threefold Limit and Their Two-Soliton Interactions

Within the coupled Fokas-Lenells equations framework, we show explicitly that, in contrast to the expected threefold-amplitude magnification, Peregrine solitons can reach a peak amplitude as high as 5 times the background level. Besides, the interaction of two such anomalous Peregrine solitons can generate a spikelike rogue wave of extremely high peak amplitude, depending on the parameters used. We numerically confirm that the Peregrine soliton beyond the threefold limit can be reproduced from either a deterministic initial profile or a chaotic background field, hence anticipating the feasibility of its experimental observation.

Growth rate of modulation instability driven by superregular breathers

We report an exact link between Zakharov-Gelash super-regular (SR) breathers (formed by a pair of quasi-Akhmediev breathers) with interesting different nonlinear propagation characteristics and modulation instability (MI). This shows that the absolute difference of group velocities of SR breathers coincides exactly with the linear MI growth rate. This link holds for a series of nonlinear Schrödinger equations with infinite-order terms. For the particular case of SR breathers with opposite group velocities, the growth rate of SR breathers is consistent with that of each quasi-Akhmediev breather along the propagation direction. Numerical simulations reveal the robustness of different SR breathers generated from various non-ideal single and multiple initial excitations. Our results provide insight into the MI nature described by SR breathers and could be helpful for controllable SR breather excitations in related nonlinear systems.

Statistics of vector Manakov rogue waves

We present a statistical analysis based on the height and return-time probabilities of high-amplitude wave events in both focusing and defocusing Manakov systems. We find that analytical rational or semirational solutions, associated with extreme, rogue wave (RW) structures, are the leading high-amplitude events in this system. We define the thresholds for classifying an extreme wave event as a RW. Our results indicate that there is a strong relationship between the type of RW and the mechanism which is responsible for its creation. Initially, high-amplitude events originate from modulation instability. Upon subsequent evolution, the interaction among these events prevails as the mechanism for RW creation. We suggest a strategy for confirming the basic properties of different extreme events. This involves the definition of proper statistical measures at each stage of the RW dynamics. Our results point to the need for redefining criteria for identifying RW events.

Combined effects of nonparaxiality, optical activity, and walk-off on rogue wave propagation in optical fibers filled with chiral materials

The generalized nonparaxial nonlinear Schrödinger (NLS) equation in optical fibers filled with chiral materials is reduced to the higher-order integrable Hirota equation. Based on the modified Darboux transformation method, the nonparaxial chiral optical rogue waves are constructed from the scalar model with modulated coefficients. We show that the parameters of nonparaxiality, third-order dispersion, and differential gain or loss term are the main keys to control the amplitude, linear, and nonlinear effects in the model. Moreover, the influence of nonparaxiality, optical activity, and walk-off effect are also evidenced under the defocusing and focusing regimes of the vector nonparaxial NLS equations with constant and modulated coefficients. Through an algorithm scheme of wider applicability on nonparaxial beam propagation methods, the most influential effect and the simultaneous controllability of combined effects are underlined, showing their properties and their potential applications in optical fibers and in a variety of complex dynamical systems.

Sasa-Satsuma hierarchy of integrable evolution equations

We present the infinite hierarchy of Sasa-Satsuma evolution equations. The corresponding Lax pairs are given, thus proving its integrability. The lowest order member of this hierarchy is the nonlinear Schrödinger equation, while the next one is the Sasa-Satsuma equation that includes third-order terms. Up to sixth-order terms of the hierarchy are given in explicit form, while the provided recurrence relation allows one to explicitly write all higher-order terms. The whole hierarchy can be combined into a single general equation. Each term in this equation contains a real independent coefficient that provides the possibility of adapting the equation to practical needs. A few examples of exact solutions of this general equation with an infinite number of terms are also given explicitly.

Solitons and rogue waves in spinor Bose-Einstein condensates

We present a general classification of one-soliton solutions as well as families of rogue-wave solutions for F=1 spinor Bose-Einstein condensates (BECs). These solutions are obtained from the inverse scattering transform for a focusing matrix nonlinear Schrödinger equation which models condensates in the case of attractive mean-field interactions and ferromagnetic spin-exchange interactions. In particular, we show that when no background is present, all one-soliton solutions are reducible via unitary transformations to a combination of oppositely polarized solitonic solutions of single-component BECs. On the other hand, we show that when a nonzero background is present, not all matrix one-soliton solutions are reducible to a simple combination of scalar solutions. Finally, by taking suitable limits of all the solutions on a nonzero background we also obtain three families of rogue-wave (i.e., rational) solutions.

Mechanism of Kuznetsov-Ma breathers

We discuss how to understand the dynamical process of Kuznetsov-Ma breather, based on some basic physical mechanisms. It is shown that the dynamical process of Kuznetsov-Ma breather involves at least two distinctive mechanisms: modulational instability and the interference effects between a bright soliton and a plane-wave background. Our analysis indicates that modulational instability plays dominant roles in the mechanism of Kuznetsov-Ma breather admitting weak perturbations, and the interference effect plays a dominant role for the Kuznetsov-Ma breather admitting strong perturbations. For intermediate cases, the two mechanisms are both greatly involved. These characters provide a possible way to understand the evolution of strong perturbations on a plane-wave background.

Matter rogue waves for the three-component Gross-Pitaevskii equations in the spinor Bose-Einstein condensates

To show the existence and properties of matter rogue waves in an F=1 spinor Bose-Einstein condensate (BEC), we work on the three-component Gross-Pitaevskii (GP) equations. Via the Darboux-dressing transformation, we obtain a family of rational solutions describing the extreme events, i.e. rogue waves. This family of solutions includes bright-dark-bright and bright-bright-bright rogue waves. The algebraic construction depends on Lax matrices and their Jordan form. The conditions for the existence of rogue wave solutions in an F=1 spinor BEC are discussed. For the three-component GP equations, if there is modulation instability, it is of baseband type only, confirming our analytic conditions. The energy transfers between the waves are discussed.

Optical Peregrine rogue waves of self-induced transparency in a resonant erbium-doped fiber

The resonant interaction of an optical field with two-level doping ions in a cryogenic optical fiber is investigated within the framework of nonlinear Schrödinger and Maxwell-Bloch equations. We present explicit fundamental rational rogue wave solutions in the context of self-induced transparency for the coupled optical and matter waves. It is exhibited that the optical wave component always features a typical Peregrine-like structure, while the matter waves involve more complicated yet spatiotemporally balanced amplitude distribution. The existence and stability of these rogue waves is then confirmed by numerical simulations, and they are shown to be excited amid the onset of modulation instability. These solutions can also be extended, using the same analytical framework, to include higher-order dispersive and nonlinear effects, highlighting their universality.

Weak-light vector rogue waves, breathers, and their Stern-Gerlach deflection via electromagnetically induced transparency

We propose a scheme for generating and manipulating vector (or two-component) optical rogue waves using Akhmediev and Kuznetsov-Ma breathers in a coherent atomic system with an M-type five-level configuration via electromagnetically induced transparency (EIT). We show that the propagation velocity of these nonlinear excitations can be reduced to 10^-4c and their generation power can be lowered to microwatts. We also show that the motion trajectories of the two polarization components in these excitations can be deflected significantly by using a transversal gradient magnetic field, similar to the Stern-Gerlach effect of an atomic beam. We find that the deflection angle can reach to 10^-4 radian within the propagation distance of only several centimeters; at variance with the atomic Stern-Gerlach effect, the deflection angle can be made different for different polarization components and may be actively adjusted in a controllable way. The results obtained may have promising applications, including the precise measurement of gradient magnetic fields.

Semi-rational solutions of the third-type Davey-Stewartson equation

General dark solitons and mixed solutions consisting of dark solitons and breathers for the third-type Davey-Stewartson (DS-III) equation are derived by employing the bilinear method. By introducing the two differential operators, semi-rational solutions consisting of rogue waves, breathers, and solitons are generated. These semi-rational solutions are given in terms of determinants whose matrix elements have simple algebraic expressions. Under suitable parametric conditions, we derive general rogue wave solutions expressed in terms of rational functions. It is shown that the fundamental (simplest) rogue waves are line rogue waves. It is also shown that the multi-rogue waves describe interactions of several fundamental rogue waves, which would generate interesting curvy wave patterns. The higher order rogue waves originate from a localized lump and retreat back to it. Several types of hybrid solutions composed of rogue waves, breathers, and solitons have also been illustrated. Specifically, these semi-rational solutions have a new phenomenon: lumps form on dark solitons and gradual separation from the dark solitons is observed.

Generation mechanisms of fundamental rogue wave spatial-temporal structure

We discuss the generation mechanism of fundamental rogue wave structures in N-component coupled systems, based on analytical solutions of the nonlinear Schrödinger equation and modulational instability analysis. Our analysis discloses that the pattern of a fundamental rogue wave is determined by the evolution energy and growth rate of the resonant perturbation that is responsible for forming the rogue wave. This finding allows one to predict the rogue wave pattern without the need to solve the N-component coupled nonlinear Schrödinger equation. Furthermore, our results show that N-component coupled nonlinear Schrödinger systems may possess N different fundamental rogue wave patterns at most. These results can be extended to evaluate the type and number of fundamental rogue wave structure in other coupled nonlinear systems.

Vector semirational rogue waves and modulation instability for the coupled higher-order nonlinear Schrödinger equations in the birefringent optical fibers

We report the existence and properties of vector breather and semirational rogue-wave solutions for the coupled higher-order nonlinear Schrödinger equations, which describe the propagation of ultrashort optical pulses in birefringent optical fibers. Analytic vector breather and semirational rogue-wave solutions are obtained with Darboux dressing transformation. We observe that the superposition of the dark and bright contributions in each of the two wave components can give rise to complicated breather and semirational rogue-wave dynamics. We show that the bright-dark type vector solitons (or breather-like vector solitons) with nonconstant speed interplay with Akhmediev breathers, Kuznetsov-Ma solitons, and rogue waves. By adjusting parameters, we note that the rogue wave and bright-dark soliton merge, generating the boomeron-type bright-dark solitons. We prove that the rogue wave can be excited in the baseband modulation instability regime. These results may provide evidence of the collision between the mixed ultrashort soliton and rogue wave.

Rogue-wave bullets in a composite (2+1)D nonlinear medium

We show that nonlinear wave packets localized in two dimensions with characteristic rogue wave profiles can propagate in a third dimension with significant stability. This unique behavior makes these waves analogous to light bullets, with the additional feature that they propagate on a finite background. Bulletlike rogue-wave singlet and triplet are derived analytically from a composite (2+1)D nonlinear wave equation. The latter can be interpreted as the combination of two integrable (1+1)D models expressed in different dimensions, namely, the Hirota equation and the complex modified Korteweg-de Vries equation. Numerical simulations confirm that the generation of rogue-wave bullets can be observed in the presence of spontaneous modulation instability activated by quantum noise.

Chirped Peregrine solitons in a class of cubic-quintic nonlinear Schrödinger equations

We shed light on the fundamental form of the Peregrine soliton as well as on its frequency chirping property by virtue of a pertinent cubic-quintic nonlinear Schrödinger equation. An exact generic Peregrine soliton solution is obtained via a simple gauge transformation, which unifies the recently-most-studied fundamental rogue-wave species. We discover that this type of Peregrine soliton, viable for both the focusing and defocusing Kerr nonlinearities, could exhibit an extra doubly localized chirp while keeping the characteristic intensity features of the original Peregrine soliton, hence the term chirped Peregrine soliton. The existence of chirped Peregrine solitons in a self-defocusing nonlinear medium may be attributed to the presence of self-steepening effect when the latter is not balanced out by the third-order dispersion. We numerically confirm the robustness of such chirped Peregrine solitons in spite of the onset of modulation instability.

Complementary optical rogue waves in parametric three-wave mixing

We investigate the resonant interaction of two optical pulses of the same group velocity with a pump pulse of different velocity in a weakly dispersive quadratic medium and report on the complementary rogue wave dynamics which are unique to such a parametric three-wave mixing. Analytic rogue wave solutions up to the second order are explicitly presented and their robustness is confirmed by numerical simulations, in spite of the onset of modulation instability activated by quantum noise.

Integrable Turbulence and Rogue Waves: Breathers or Solitons?

Turbulence in dynamical systems is one of the most intriguing phenomena of modern science. Integrable systems offer the possibility to understand, to some extent, turbulence. Recent numerical and experimental data suggest that the probability of the appearance of rogue waves in a chaotic wave state in such systems increases when the initial state is a random function of sufficiently high amplitude. We provide explanations for this effect.

Rogue waves, rational solitons, and modulational instability in an integrable fifth-order nonlinear Schrödinger equation

We analytically study rogue-wave (RW) solutions and rational solitons of an integrable fifth-order nonlinear Schrödinger (FONLS) equation with three free parameters. It includes, as particular cases, the usual NLS, Hirota, and Lakshmanan-Porsezian-Daniel equations. We present continuous-wave (CW) solutions and conditions for their modulation instability in the framework of this model. Applying the Darboux transformation to the CW input, novel first- and second-order RW solutions of the FONLS equation are analytically found. In particular, trajectories of motion of peaks and depressions of profiles of the first- and second-order RWs are produced by means of analytical and numerical methods. The solutions also include newly found rational and W-shaped one- and two-soliton modes. The results predict the corresponding dynamical phenomena in extended models of nonlinear fiber optics and other physically relevant integrable systems.

The Darboux transformation of the Kundu–Eckhaus equation

We construct an analytical and explicit representation of the Darboux transformation (DT) for the Kundu–Eckhaus (KE) equation. Such solution and n -fold DT T n are given in terms of determinants whose entries are expressed by the initial eigenfunctions and ‘seed’ solutions. Furthermore, the formulae for the higher order rogue wave (RW) solutions of the KE equation are also obtained by using the Taylor expansion with the use of degenerate eigenvalues λ 2 k − 1 → λ 1 = − 1 2 a + β c 2 + i c , k =1,2,3,…, all these parameters will be defined latter. These solutions have a parameter β , which denotes the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effects. We apply the contour line method to obtain analytical formulae of the length and width for the first-order RW solution of the KE equation, and then use it to study the impact of the β on the RW solution. We observe two interesting results on localization characters of β , such that if β is increasing from a /2: (i) the length of the RW solution is increasing as well, but the width is decreasing; (ii) there exist a significant rotation of the RW along the clockwise direction. We also observe the oppositely varying trend if β is increasing to a /2. We define an area of the RW solution and find that this area associated with c =1 is invariant when a and β are changing.

Modulational instability, higher-order localized wave structures, and nonlinear wave interactions for a nonautonomous Lenells-Fokas equation in inhomogeneous fibers

In this paper, the nonautonomous Lenells-Fokas (LF) model is investigated. The modulational instability analysis of the solutions with variable coefficients in the presence of a small perturbation is studied. Higher-order soliton, breather, earthwormon, and rogue wave solutions of the nonautonomous LF model are derived via the n-fold variable-coefficient Darboux transformation. The solitons and earthwormons display the elastic collisions. It is found that the nonautonomous LF model admits the higher-order periodic rogue waves, composite rogue waves (rogue wave pair), and oscillating rogue waves, whose dynamics can be controlled by the inhomogeneous nonlinear parameters. Based on the second-order rogue wave, a diamond structure consisting of four first-order rogue waves is observed. In addition, the semirational solutions (the mixed rational-exponential solutions) of the nonautonomous LF model are obtained, which can be used to describe the interactions between the rogue waves and breathers. Our results could be helpful for the design of experiments in the optical fiber communications.

Optical rogue waves associated with the negative coherent coupling in an isotropic medium

Optical rogue waves of the coupled nonlinear Schrödinger equations with negative coherent coupling, which describe the propagation of orthogonally polarized optical waves in an isotropic medium, are reported. We construct and discuss a family of the vector rogue-wave solutions, including the bright rogue waves, four-petaled rogue waves, and dark rogue waves. A bright rogue wave without a valley can split up, giving birth to two bright rogue waves, and an eye-shaped rogue wave can split up, giving birth to two dark rogue waves.

Watch-hand-like optical rogue waves in three-wave interactions

We investigate the resonant interaction of three optical pulses of different group velocity in quadratic media and report on the novel watch-hand-like super rogue wave patterns. In addition to having a giant wall-like hump, each rogue-wave hand involves a peak amplitude more than five times its background height. We attribute such peculiar structures to the nonlinear superposition of six Peregrine-type solitons. The robustness has been confirmed by numerical simulations. This stability along with the non-overlapping distribution property may facilitate the experimental diagnostics and observation of these super rogue waves.

Dark three-sister rogue waves in normally dispersive optical fibers with random birefringence

We investigate dark rogue wave dynamics in normally dispersive birefringent optical fibers, based on the exact rational solutions of the coupled nonlinear Schrödinger equations. Analytical solutions are derived up to the second order via a nonrecursive Darboux transformation method. Vector dark "three-sister" rogue waves as well as their existence conditions are demonstrated. The robustness against small perturbations is numerically confirmed in spite of the onset of modulational instability, offering the possibility to observe such extreme events in normal optical fibers with random birefringence, or in other Manakov-type vector nonlinear media.

Coexisting rogue waves within the (2+1)-component long-wave-short-wave resonance

The coexistence of two different types of fundamental rogue waves is unveiled, based on the coupled equations describing the (2+1)-component long-wave-short-wave resonance. For a wide range of asymptotic background fields, each family of three rogue wave components can be triggered by using a slight deterministic alteration to the otherwise identical background field. The ability to trigger markedly different rogue wave profiles from similar initial conditions is confirmed by numerical simulations. This remarkable feature, which is absent in the scalar nonlinear Schrödinger equation, is attributed to the specific three-wave interaction process and may be universal for a variety of multicomponent wave dynamics spanning from oceanography to nonlinear optics.

Rogue-wave pattern transition induced by relative frequency

We revisit a rogue wave in a two-mode nonlinear fiber whose dynamics is described by two-component coupled nonlinear Schrödinger equations. The relative frequency between two modes can induce different rogue wave patterns transition. In particular, we find a four-petaled flower structure rogue wave can exist in the two-mode coupled system, which possesses an asymmetric spectrum distribution. Furthermore, spectrum analysis is performed on these different type rogue waves, and the spectrum relations between them are discussed. We demonstrate qualitatively that different modulation instability gain distribution can induce different rogue wave excitation patterns. These results would deepen our understanding of rogue wave dynamics in complex systems.

Vector rogue waves and baseband modulation instability in the defocusing regime

We report and discuss analytical solutions of the vector nonlinear Schrödinger equation that describe rogue waves in the defocusing regime. This family of solutions includes bright-dark and dark-dark rogue waves. The link between modulational instability (MI) and rogue waves is displayed by showing that only a peculiar kind of MI, namely baseband MI, can sustain rogue-wave formation. The existence of vector rogue waves in the defocusing regime is expected to be a crucial progress in explaining extreme waves in a variety of physical scenarios described by multicomponent systems, from oceanography to optics and plasma physics.

High-order rogue waves in vector nonlinear Schrödinger equations

We study the dynamics of high-order rogue waves (RWs) in two-component coupled nonlinear Schrödinger equations. We find that four fundamental rogue waves can emerge from second-order vector RWs in the coupled system, in contrast to the high-order ones in single-component systems. The distribution shape can be quadrilateral, triangle, and line structures by varying the proper initial excitations given by the exact analytical solutions. The distribution pattern for vector RWs is more abundant than that for scalar rogue waves. Possibilities to observe these new patterns for rogue waves are discussed for a nonlinear fiber.

Optical rogue waves generated on Gaussian background beam

We study optical rogue waves (RWs) in a nonlinear graded-index waveguide with variable coefficients. An exact RW solution on Gaussian background beam is presented, in contrast to the previous studies about RWs, on plane wave background. It is shown that the characteristics of RWs are maintained on Gaussian background beam and that the beam's width is even a bit smaller than the RWs scale. These results may raise the possibility of related experiments and potential applications in nonlinear optics.