Akhmediev Breathers Research Articles

Study of Fifth-Order Nonlinear Caudrey–Dodd–Gibbon Model for Multiwave, M-Shaped Solitons, Lumps, Generalized Breathers, Manifold Periodic, Rogue Wave and Kink Wave Interactions

Numerous novel wave structures, such as lump, lump with one-kink, lump with two-kink soliton, generalized breather, Ma breather, Kuznetsov-Ma breather, Akhmediev breather, manifold periodic, and rogue wave solutions to [Formula: see text]-dimensional fifth-order nonlinear Caudrey–Dodd–Gibbon (FNCDG) model via symbolic computation are studied based on the ansatz function scheme. Studying the lump-type solution involves selecting the function as the generic quadratic polynomial function. The lump with one- and two-kink solutions is created by mixing a quadratic function with one or two exponential functions. By adding hyperbolic function with a quadratic function, the rogue wave solutions are manipulated. In addition, the multiwave, [Formula: see text]-shaped rational solitons and their interactions with kink waves are analytically evaluated. Furthermore, different 3D, 2D, and contour profiles are created in different dimensions by changing the parameter values.

Wave patterns of the coupled nonlinear Schrödinger equations in photonic crystal fibers with four-wave mixing

Abstract In this paper, we examine the behavior of modulation instability within photonic crystals. The model employed is the coherent coupled nonlinear Schrödinger equation, incorporating weak birefringence and four-wave mixing, which arises at the edge of the optical mode. The linear analysis is used to derive the modulation instability spectrum. Throughout the modulation instability spectrum, we identify both stable and unstable modes, thereby confirming the breakdown of the plane wave. For certain four-wave mixing parameters, the amplitude of the modulation instability spectrum and its bandwidths expand, creating an opening for localized structures to emerge. Another aspect of this study has been demonstrated in normal and anomalous dispersion regimes where an increasing initial amplitude of the plane wave is fulfilled. Specifically, numerical simulations highlight the occurrence of Benjamin-Feir instability, where wave patterns emerge under the influence of four-wave mixing. Additionally, solitonic waves are generated, demonstrating the presence of Akhmediev breathers and other modulated structures, confirming that photonic crystals with four-wave mixing are conducive to these formations. The findings from this study could inform future research in the development of nonlinear photonic waveguides.

Families of exact similaritons with flexible modulations in N-coupled homogeneous and inhomogeneous systems

In this paper, a family of exact solutions to N-coupled nonlinear Schrödinger (N-CNLS) equations including bright and dark solitons, Akhmediev breathers, Kuznetsov-Ma breathers, and rogue waves are derived with the aid of a generic nonlinear wave assumption. As an application of the solutions, we present a scheme to realize amplitude coding in an identical waveguide system. Furthermore, by a general self-similar transformation method, exact controllable similaritons of N-coupled inhomogeneous NLS (N-CINLS) equations are constructed under relaxed compatibility conditions. Flexible modulations of similaritons are investigated in a typical inhomogeneous system. Based on the general self-similar transformation, we further study various and novel modulations of composite similariton of 2-CINLS regimes. The various similaritons presented here may be expected to find application in the control and transmission of nonlinear waves in realizable complex systems including those based on optical fibers and Bose-Einstein condensates.

Akhmediev Breathers and Kuznetsov–Ma Solitons in the Cubic-Quintic Nonlinear Schrödinger Equation

Akhmediev Breathers and Kuznetsov–Ma Solitons in the Cubic-Quintic Nonlinear Schrödinger Equation

Generation and control of optical rogue waves by super-Gaussian pulses with varying steepness

We investigate the dynamics of optical rogue waves induced by super-Gaussian (SG) pulses in fiber optics. SG pulses with controlled steep leading and trailing edges can produce high-intensity self-focusing when propagating in nonlinear fibers with weak dispersion and strong nonlinearity. We observe the emergence of various rogue waves, including Peregrine breathers, Akhmediev breathers (ABs), and second-order ABs. By adjusting the steepness of the SG pulses, the generation of these rogue waves can be controlled. As the steepness of the SG pulse increases, we find the fundamental-order AB can stably coexist. Simultaneously, we reveal that the period of these ABs can be effectively controlled by adjusting the steepness of the SG pulse. At a certain steepness, the second-order AB with high power can also be observed. We also investigated the propagation of SG pulses with an initial chirp in fiber optics and found that they can also control AB and second-order Peregrine breathers. To our knowledge, our findings present a new approach to generating controllable rogue waves.

Rogue and solitary waves of a system of coupled nonlinear Schrödinger equations in a left-handed transmission line with second-neighbors coupling

The effects of the second-order neighbor interaction have been effective on the coherent localized waves and modulation instability spectrum. A coupled nonlinear Schrödinger equation is derived in the nonlinear left-handed electrical lattice, and the linear stability is used to formulate the modulation instability spectrum expression. Unstable modes are displayed to show symmetric lobes and increasing bandwidths under the influence of the neighbor coupling. The Benjamin-Feir instability has prospered in the network, and for weak perturbed wave numbers, the unstable modes increase. The numerical simulation is used to develop localized waves, including rogue waves with one crest and two humps, Akhmediev breathers, and other modulated structures. We notice that, for strong values of the perturbed wave number, an exponential growth of the continuous wave arises to confirm the fact that the nonlinear left-handed electrical lattice with coupling strength is opened to coherent localized waves. The long-lived nature of the obtained structures has also been demonstrated for specific times of propagation.

Fundamental breathers and their physical spectra in vector fields with self-steepening

We study the dynamics of fundamental breathers in vector fields with self-steepening. Exact multi-parametric vector solutions of Akhmediev breathers, Kuznetsov-Ma solitons, and beating solitons are obtained. Their existence conditions are analyzed by using the square spectrum parameters. The analytic physical spectra of breathers are presented by using a residue theorem associated with second-order singularity. In contrast to the Manakov limit without the self-steepening term, these spectra exhibit asymmetry in both the vector wave component and the total wave field. We characterize explicitly such asymmetry. The correctness of our exact solutions is confirmed by numerical simulations.

Resonant Akhmediev breathers

Modulation instability is a phenomenon in which a minor disturbance within a carrier wave gradually amplifies over time, leading to the formation of a series of compressed waves with higher amplitudes. In terms of frequency analysis, this process results in the generation of new frequencies on both sides of the original carrier wave frequency. We study the impact of fourth-order dispersion on this modulation instability in the context of nonlinear optics that lead to the formation of a series of pulses in the form of Akhmediev breather. The Akhmediev breather, a solution to the nonlinear Schrödinger equation, precisely elucidates how modulation instability produces a sequence of periodic pulses. We observe that when weak fourth-order dispersion is present, significant resonant radiation occurs, characterized by two modulation frequencies originating from different spectral bands. As an Akhmediev breather evolves, these modulation frequencies interact, resulting in a resonant amplification of spectral sidebands on either side of the breather. When fourth-order dispersion is of intermediate strength, the spectral bandwidth of the Akhmediev breather diminishes due to less pronounced resonant interactions, while stronger dispersion causes the merging of the two modulation frequency bands into a single band. Throughout these interactions, we witness a complex energy exchange process among the phase-matched frequency components. Moreover, we provide a precise explanation for the disappearance of the Akhmediev breather under weak fourth-order dispersion and its resurgence with stronger values. Our study demonstrates that Akhmediev breathers, under the influence of fourth-order dispersion, possess the capability to generate infinitely many intricate yet coherent patterns in the temporal domain.

Breathers, rogue waves, and interaction solutions for the variable coefficient Kundu-nonlinear Schrödinger equation

With the inhomogeneity of optical fiber media taken into account, under investigation in this paper is the variable coefficient Kundu-nonlinear Schrödinger equation, which describes the pulses propagation in optical fibers. Based on Lax pair, the Nth-order Darboux transformation is constructed. Depending on plane wave solution, the first- and second-order breather solutions are derived and the interactions between breathers are graphically analyzed. The Kuznetsov–Ma breather, Akhmediev breather, and spatial-temporal breather have been obtained. Moreover, the first-, second-, and third-order rogue wave solutions have been constructed. The usual rogue waves and first- and second-order line rogue waves are observed. The weak and strong interactions between the first-, second-order rogue waves, and spatial-temporal period breather are studied. Furthermore, variable coefficient δ(t) causes rogue waves to produce some interesting evolutionary phenomena, which have been systematically analyzed. In addition, the influences of parameters for the properties of solutions are discussed.

Breathers for the sixth-order nonlinear Schrödinger equation on the plane wave and periodic wave background

In this paper, we study the breathers in the framework of the sixth-order nonlinear Schrödinger equation by using the Darboux transformation. The primary objective of this research is twofold. First, we consider the nonlinear superposition of breathers on the plane wave background. Based on the concept that rogue waves are formed from colliding Akhmediev breathers, we obtain rogue wave sequences and a first-order Akhmediev breather with a central second-order rogue wave peak. Second, we consider the formation of breathers on the periodic wave background. The difficulty of solving the Lax pair is overcome, and we successfully construct the breathers on the cn- and dn-periodic wave background.

Numerical investigation of turbulence generation using Zakharov-like model equation

This study investigates the turbulence generation behavior with a Zakharov-like (ZL) equation in a fluid system. The model equation is derived using conservation equations (mass and momentum conservation), and the source of nonlinearity is the high amplitude of the acoustic wave. The Zakharov-like equation has been derived and then solved numerically, then turned into a modified nonlinear Schrödinger equation. Furthermore, modulation instability, or Benjamin–Feir instability, of the model equations, which leads to the emergence of Akhmediev breathers, is discussed. The numerical simulation uses a finite difference method for temporal evolution and a pseudo-spectral approach to determine spatial regimes. The outcomes indicate that the situation involving the nonlinear Schrödinger equation case displays a periodic pattern in space and time. The findings also demonstrate that the localization of structure and the Fermi, Pasta, and Ulam (FPU) recurrences are disrupted for the modified nonlinear Schrödinger equation and Zakharov-like equation cases. The energy spectrum exhibits a power law behavior that approximately follows k−1.65 in the ZL model equation case, and it is steeper than Kolmogorov's spectrum within the inertial sub-range.

Multi-breathers and higher-order rogue waves on the periodic background in a fourth-order integrable nonlinear Schrödinger equation

In this paper, we present a systematic formulation of multi-breathers and higher-order rogue wave solutions of a fourth-order nonlinear Schrödinger equation on the periodic background. First of all, we compute a complete family of elliptic solution of this higher-order equation, which can degenerate into two particular cases, i.e., the dnoidal and cnoidal solutions. By using the modified squared wavefunction approach, we solve the spectral problem on the elliptic function background. Then, we derive multi-breather solutions in terms of the theta functions, particular examples of which are the Kuznetsov-Ma breather and the Akhmediev breather. Furthermore, taking the limit of the breather solutions at branch points, we construct higher-order rogue wave solutions by employing a generalized Darboux transformation technique. On the periodic background, we present the first-order, second-order and second-second-order rogue waves. With aid of the theta functions, we explicitly characterize the resulting breathers and rogue waves, and demonstrate their dynamic behaviors by illustrative examples. Finally, we discuss how the parameter of the higher-order effects affects the breathers and rogue waves.

Solitonic rogue and modulated wave patterns in the monoatomic chain with anharmonic potential

Modulation instability and rogue wave structures have been investigated in this work. This study is an extension of the work in Abbagari (2023), where a nonlinear Schrödinger equation with higher-order dispersion is derived to show only the development of the modulated waves bounded to bright soliton as a nonlinear exhibition of modulation instability. Here, the coupled nonlinear Schrödinger equation is derived by using the multi-scale scheme. An overview of the analytical calculations of the perturbed plane wave is carried out to show the effects of the nonlinear chain parameters on the modulation instability growth rate and bandwidths. The interest of this study lies equally in the nonlinear modes of excitation, where solitonic waves are generated under certain conditions in lower and upper frequency bands. On the other hand, relevant results have been developed to show the features of the type I and type II rogue waves of the Manakov system. Such investigations are obtained under the variation of the interaction potential parameters and the free parameter of the similarity method. Via a numerical simulation, rogue wave structures have been generated as a consequence of the long-time evolution of the perturbed plane wave. At a specific time of propagation, another localized object has been obtained to show the Akhmediev breathers and Kuznetsov-Ma solitons clusters under a strong perturbed wave number. These results have opened up new features, and many applications could follow in the future.

Breathers of Gerdjikov-Ivanov equation under background of elliptic function

As one specific type of local solutions of nonlinear evolution equation, the breathers have the characteristic of envelope oscillation structure. This kind of oscillation is periodic. According to the periodicity of the distribution and evolution directions, there are three kinds of breathers, namely, the Kuznetsov-Ma breather (KMB), the Akhmediev breather (AB), and the general breather (GB). In recent years, the propagation of envelope breathers under the periodic background has been observed in many nonlinear physical fields, including nonlinear optical fibers and hydrodynamics. It is believed that the breathers can arise due to the modulational instability of the periodic waves, and they demonstrate many rich physical properties and dynamic behaviors of interactions. Therefore, recently great attention has been paid to the breathers under the periodic background in nonlinear science. As an important integrable model, the Gerdjikov-Ivanov (GI) equation can be used to describe various nonlinear phenomena in many physical fields such as in the quantum field theory, weak nonlinear dispersive water wave, and nonlinear optics. It is very meaningful to solve various types of solutions of this model to describe the propagation of nonlinear waves. As far as we know, the breather solutions for the GI equation have not been given under the elliptic function background. In this study, firstly, elliptic function solutions of the GI equation are solved by the modified squared wave (MSW) function approach and the traveling wave transformation. Then, we obtain the basic solution of the Lax pair corresponding to the Jacobi elliptic function seed solution. Based on the elliptic function transformation formulas and the integral formulas, the potential function solution can be expressed in terms of the Weierstrass elliptic function. Secondly, by the once iterated Darboux transformation, three types of breather solutions under the elliptic function background are constructed including the GB, the KMB and the AB. In addition, we analyze the dynamic behaviors of these three kinds of breathers, and present their three-dimensional space-time structures. By the twice iterated Darboux transformation, under the dn-periodic background we exhibit three types of interactions between two breathers, i.e. a GB and a KMB, an AB and a KMB, and a GB and an AB. Finally, we also present three types of interactions between two breathers under the general periodic background.

Dust acoustic inertial Alfvénic nonlinear structures in an electron depleted dusty plasma

In this investigation, the propagation properties of dust inertial Alfvén wave (DIAW) solitons, rogons, and breather structures have been studied in a magnetised electron-depleted two fluids plasma consisting of negatively charged dust and positively charged ions. The Korteweg–de Vries (KdV) equation has been derived by using the reductive perturbation method (RPM) to study the properties of solitons. Further, the nonlinear Schrödinger equation (NLSE) has also been derived by employing a suitable transformation in the KdV equation. The evolution of various kinds of nonlinear structures such as rogons, Akhmediev breathers (AB), and Kuznetsov–Ma breathers (KM) has been elucidated from the different rational solutions of the nonlinear Schrödinger equation. The impact of various physical parameters, such as the dust plasma beta and the angle of propagation of different species, on the salient properties of DIAW solitons, rogons, and breather structures has been studied numerically.

Higher-order matrix nonlinear Schrödinger equation with the negative coherent coupling: binary Darboux transformation, vector solitons, breathers and rogue waves

Optical fiber communication plays a crucial role in modern communication. In this work, we focus on the higher-order matrix nonlinear Schrödinger equation with negative coherent coupling in a birefringent fiber. For the slowly varying envelopes of two interacting optical modes, we construct a binary Darboux transformation using the corresponding Lax pair. With vanishing seed solutions and the binary Darboux transformation, we investigate vector degenerate soliton and exponential soliton solutions. By utilizing these soliton solutions, we demonstrate three types of degenerate solitons and double-hump bright solitons. Furthermore, considering non-vanishing seed solutions and applying the binary Darboux transformation, we obtain vector breather solutions, and present the vector single-hump beak-type Akhmediev breather, Kuznetsov-Ma breathers, double-hump beak-type Akhmediev breather, Kuznetsov-Ma breathers, and vector degenerate beak-type breathers. Additionally, we take the limit in the breather solutions and derive vector rogue wave solutions. We illustrate the beak-type rogue waves and bright-dark rogue waves. Humps of these vector double-hump waves can separate into two individual humps. The results obtained in this work may potentially provide valuable insights for experimentally manipulating the separation of two-hump solitons, breathers, and rogue waves in optical fibers.

Solitonic rogue waves induced by the modulation instability in a split-ring-resonator-based left-handed coplanar waveguide

Modulation instability and nonlinear excitation modes in lower and upper frequency bands are investigated in the nonlinear left-handed coplanar waveguide, where nonlinear capacitance (voltage-dependent) is used. The multi-scale method is employed to derive a system of two coupled nonlinear Schödinger equations governing the evolution of the kink and bright solitons of these modes. A modulation instability is investigated to show the effects of the left-handed and right-handed influences on the modulation instability gain and the intensity of the plane wave. In the lower and upper frequency modes, the breathing structures have strong values for the left-handed component. At the end, the numerical simulation of the continuous wave is used to show the development of the modulated waves and localized objects. Another relevant localized structure has been obtained to display the second-order breathers and Akhmediev breathers of types A and B under the variation of the perturbed wave number. Both linear inductances and perturbed wave numbers are tools for controlling the amplitude of the localized structures, and modulation instability is sensitive to these parameters.

Nonlinear internal gravity waves in the atmosphere: Rogue waves, breathers and dark solitons

We study nonlinear internal gravity waves (IGWs) in the atmosphere. The reductive perturbation method is used to derive a system of two-dimensional nonlinear equations for the envelope of velocity stream function and the mean flow. In the one-dimensional case, we obtain a nonlinear Schrödinger (NLS) equation corresponding to both horizontal and vertical propagation of IGWs. Depending on the characteristic wavelengths, the NLS equation is focusing or defocusing. In the focusing case, non-stationary solutions in the form of the Peregrine soliton, the Akhmediev breather and the Kuznetsov–Ma breather are considered as potential candidates for the modeling of rogue waves in the atmosphere. In the defocusing case, stationary nonlinear IGWs are considered in the form of nonlinear periodic waves and dark solitons.

Vector ring-like combined Akhmediev breathers for partially nonlocal nonlinearity under external potentials

Via the link between φp of a (3＋1)-dimensional variable-coefficient coupled partially nonlocal nonlinear Schrödinger system (NLSS) and Φ of constant-coefficient NLSS, considering the ring approximation and by means of the Darboux approach, we derive vector ring-like combined Akhmediev breathers(AB), including ring-like combined AB-doublet and ring-like combined AB with giant wave. From this breather solution, we find that physical quantities such as amplitude, width, center and phase are strongly affected by parameters of diffraction, linear potential and phase chirp. Moreover, we study the influence of parameters for radius and thickness on the form of vector ring-like combined AB.

Superposition of modulated nonlinear waves in inhomogeneous systems with negative coherent coupling

The superimposed wave solutions of the variable coefficient nonlinear Schrödinger equations with negative coherent coupling are derived under a more relaxed constraint condition than those in previous literatures. For the benefit of the more relaxed constraint, the dispersion, nonlinearity, and gain/loss can be designed freely, and the obtained solutions can describe the nonlinear waves in general inhomogeneous optical fiber systems. The obtained solutions with two free phase parameters can be deemed to be the superposition of the typical simple modulated solutions, and the arbitrary of the optical parameters and the free phase parameters be expected to give the rise of abundant forms of modulation functions, that leads to the diverse characteristics of superimposed waves. Take the kink dispersion fiber systems with constant gain/loss and trigonometric gain/loss as examples, rich dynamics of the superimposed waves are displayed. By changing the gain/loss, the physical features of superimposed waves, such as the amplitudes of solitons and Kuznetsov-Ma breathers, the widths of solitons, the distances between Kuznetsov-Ma breathers, and the backgrounds of Akhmediev breathers and rogue waves can be controlled. The interaction of solitons or Kuznetsov-Ma breathers, and the number of the rogue waves or Akhmediev breathers can also be manipulated by selecting appropriate value of gain/loss. The results presented here may be useful to explore the diverse dynamics of superimposed waves and prove significance for the control of nonlinear waves in weakly birefringent fibers.