Kuznetsov-Ma Breather 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.

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.

Lump, Breather, Ma-Breather, Kuznetsov–Ma-Breather, Periodic Cross-Kink and Multi-Waves Soliton Solutions for Benney–Luke Equation

The goal of this research is to utilize some ansatz forms of solutions to obtain novel forms of soliton solutions for the Benney–Luke equation. It is a mathematically valid approximation that describes the propagation of two-way water waves in the presence of surface tension. By using ansatz forms of solutions, with an appropriate set of parameters, the lump soliton, periodic cross-kink waves, multi-waves, breather waves, Ma-breather, Kuznetsov–Ma-breather, periodic waves and rogue waves solutions can be obtained. Breather waves are confined, periodic, nonlinear wave solutions that preserve their amplitude and shape despite alternating between compression and expansion. For some integrable nonlinear partial differential equations, a lump soliton is a confined, stable solitary wave solution. Rogue waves are unusually powerful and sharp ocean surface waves that deviate significantly from the surrounding wave pattern. They pose a threat to maritime safety. They typically show up in solitary, seemingly random circumstances. Periodic cross-kink waves are a particular type of wave pattern that has frequent bends or oscillations that cross at right angles. These waves provide insights into complicated wave dynamics and arise spontaneously in a variety of settings. In order to predict the wave dynamics, certain 2D, 3D and contour profiles are also analyzed. Since these recently discovered solutions contain certain arbitrary constants, they can be used to describe the variation in the qualitative characteristics of wave phenomena.

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.

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.

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.

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.

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.

Data-driven localized waves and parameter discovery in the massive Thirring model via extended physics-informed neural networks with interface zones

In this paper, we study data-driven localized wave solutions and parameter discovery in the massive Thirring (MT) model via the deep learning in the framework of physics-informed neural networks (PINNs) algorithm. Abundant data-driven solutions including soliton of bright/dark type, breather and rogue wave are simulated accurately and analyzed contrastively with relative and absolute errors. For higher-order localized wave solutions, we employ the extended PINNs (XPINNs) with domain decomposition to capture the complete pictures of dynamic behaviors such as soliton collisions, breather oscillations and rogue-wave superposition. In particular, we modify the interface line in domain decomposition of XPINNs into a small interface zone and introduce the pseudo initial, residual and gradient conditions as interface conditions linked adjacently with individual neural networks. Then this modified approach is applied successfully to various solutions ranging from bright-bright soliton, dark-dark soliton, dark-antidark soliton, general breather, Kuznetsov-Ma breather and second-order rogue wave. Experimental results show that this improved version of XPINNs reduce the complexity of computation with faster convergence rate and keep the quality of learned solutions with smoother stitching performance as well. For the inverse problems, the unknown coefficient parameters of linear and nonlinear terms in the MT model are identified accurately with and without noise by using the classical PINNs algorithm.

Excitation and quasi-transition of rogue waves in a one-dimensional quantum droplet

We study the dynamics of rogue wave excitations and quasi-transitions in one-dimensional quantum droplets, whose homogeneous-density flat top provides an applicable background wave for nonlinear excitation. Using a quenching technique and a localized space modulation technique for the nonlinear effect coefficient, the rogue wave is excited on droplets. The excitation modes of rogue waves can be transitioned to dispersive shock waves or Kuznetsov–Ma breathers by regulating their modulation amplitude and width, and their maximal density and width can be regulated by modulation their amplitude. Meanwhile, two Gaussian modulations for droplets also bring the second-order rogue wave’s dynamics, whose maximal amplitude can approach 3.6 times that of the amplitude of droplets. Our results provide guidance for rogue wave excitation and transition on self-bound background waves and deepen our understanding of quantum droplets.

Multi-soliton interactions, comb behavior, strange mixed breathers and other exotic rogue waves generated in a nonlinear left-handed transmission line

We establish a cubic–quintic-nonlinear Schrödinger equation (CQNLSE) including cubic- and quintic-nonlinearities, second-, third- and fourth-order dispersions in a left-handed nonlinear transmission line. We use the collective coordinates’ theory and the Gaussian Ansatz function with six coordinates to improve the comprehension of the system. Besides, the following innovations have been found when linear and nonlinear effects interact. (i) Some phenomena have emerged such as comb behavior, fragmentation of waves and wall of waves. (ii) Some solutions have also emerged such as the dark soliton, multi-peak waves, Kuznetsov–Ma breathers, periodical and non periodical mixed wave trains, Sasa–Satsuma waves, Peregrine waves, and crossed solitons. Otherwise, the internal perturbations leading to all aforementioned exotic rogue events are well studied in detail.

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

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

Multi-breather and high-order rogue waves for the quintic nonlinear Schrödinger equation on the elliptic function background

In this paper, we derive the multi-breather and higher-order rogue wave solutions of the quintic nonlinear Schrödinger (NLS) equation on the elliptic function background by using the Darboux transformation (DT) and the modified squared wave (MSW) function approach. Firstly, we construct the elliptic function solution and solve the corresponding associated Lax pair with the given elliptic function potential by the traveling wave transformation. Secondly, we use the theta functions to represent the elliptic function solution and the fundamental solution of the Lax pair. Thirdly, we present several types of breathers on the dn- and cn-periodic backgrounds including the general breather (GB), the Kuznetsov–Ma breather (KMB) and the Akhmediev breather (AB). In addition, we illustrate the nonlinear dynamics of the breathers on the effect of elliptic function background. Finally, at branch points for breather solutions we obtain the higher-order rogue wave solutions by using the generalized DT. By several examples, we provide the first-, second- and second second-order rogue waves on the elliptic function background. It is expected that our results are helpful to understand the dynamics of breathers and rogue waves on the periodic background in various physical systems with higher-order effects.

Generation of double Sasa-Satsuma, double Kuznetsov-Ma and other exotic solutions of cubic-quintic Ginzburg-Landau equation in a left-handed material

We present new exotic numerical solutions and a strange periodical mixed Sasa-Satsuma and Kuznetsov-Ma waves’ train. Much more, we study a nonlinear Schrödinger (NLS) equation for absorption regime in a left-handed material. The complex coefficients coming from absorption regime transform this NLS equation into a Cubic-Quintic Ginzburg–Landau (CQGL) equation. Moreover, we employ variational theory with Type I Ansatz function containing seven coordinates compared to six coordinates often used. Some interesting results are found such as one, two, three, four, five, six and seven peak solutions. Among those solutions specific ones are outlined as follows. (i) Double Kuznetsov-Ma breathers and tree structure. (ii) Conventional, random, double, triple and new forms of Sasa-Satsuma solutions. (iii) Stair-, Pyramid-, flower-shaped three and five peak solutions. (iv) First-and second-forms of seven peak solutions. Further, we also present the physical conditions and the internal excitation leading to the generation of those new exotic solutions.

Nonlinear waves coupled by magnetosonic wave and ion acoustic wave in overdense plasma

The coupling of the magnetosonic wave and ion acoustic wave in the presence of the ponderomotive force of the magnetosonic wave is studied by using a nonlinear Schrödinger equation. The several nonlinear waves are proposed to explain the nonlinear coupling of magnetosonic wave and ion acoustic wave in overdense plasma. It is shown that there are two wave lengths in the x direction for Kuznetsov-Ma breather solution. One is due to the background wave whose wave length is much shorter, while the other is much longer. Similar characteristics are also given for Akhmediev breather solution, bright soliton solution and the Peregrine rogue wave solution. The present analytical results have potential applications in explaining the recently reported experimental observations.

A special two-dimensional lattice by Blaszak and Szum: Solitons, breathers, lump solutions, and their interactions and dynamics

In this paper, we study a special two-dimensional lattice equation proposed by Blaszak and Szum. Upon utilizing the bilinear method, we derive solitons, breathers and rational solutions to the lattice equation both on the constant and periodic background. These solutions are given in terms of determinants whose matrix elements have simple algebraic expressions. In particular, we find three types of breather solutions, including Kuznetsov-Ma breather, Akhmediev breather and general one. By introducing two differential operators applied to the soliton solutions, we obtain rational solutions in terms of Schur polynomials. We demonstrate that rational solutions can exhibit algebraic solitons and lump solitons. We explore the asymptotic analysis to the algebraic solitons and find asymptotic algebraic solitons localized in the straight. By taking higher-order differential operators, we present multiple and higher-order rational solutions. The dynamical behaviors of these obtained solutions are illustrated and analyzed.

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

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

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

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