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This work explores the mathematical technique known as the Hirota bilinear transformation to investigate different wave behaviors of the nonlinear Rosenau equation, which is fundamental in the study of wave occurrences in a variety of physical systems such as fluid dynamics, plasma physics, and materials science, where nonlinear dynamics and dispersion offer significant functions. This equation was suggested to describe the dynamic behaviour of dense discrete systems. We use Mathematica to investigate these wave patterns and obtained variety of wave behaviors, such as M-shaped waves, mixed waves, multiple wave forms, periodic lumps, periodic cross kinks, bright and dark breathers, and kinks and anti-kinks. These patterns each depict distinct qualities and behaviors of waves, offering insights into the interactions and evolution of waves. The results found that free parameters have a substantial impact on travelling waves, including their form, structure, and stability. With the aid of this software, we potray the dynamics of these waves in 3Ds, contours and densities plots, which enables us to comprehend how waves move and take on various forms. The novel component is the application of Hirota's bilinear approach to generate new form of solutions as highlighted above, analyse their interactions, and give better visualisations, which goes beyond prior soliton-focused investigations of the Rosenau problem. All things considered, our work advances our understanding of waves and nonlinear systems and demonstrates the value of mathematical techniques for understanding intricate physical phenomena. These results may have implications for a wide range of fields, including environmental science, engineering, and physics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a damped variable-coefficient fifth-order modified Korteweg-de Vries equation in fluid mechanics. By virtue of the simplified Hirota method, the N-soliton solutions are derived under certain variable-coefficient constraints, where N is a positive integer. Based on the N-soliton solutions, the Hth-order breather and multi-pole solutions are determined through the complex conjugated transformations and limit approach, respectively, where H is a positive integer. Furthermore, we construct the hybrid solutions composed of the first-order breather and one soliton, first-order breather and two solitons, double-pole wave and one soliton, triple-pole wave and one soliton, and first-order breather and double-pole wave. Moreover, we discuss the influences of variable coefficients in the equation on those nonlinear waves graphically. [ABSTRACT FROM AUTHOR]

Korteweg-de Vries-type equations have occurred in the fields of planetary oceans, atmospheres, cosmic plasmas and so on, while nonlinear evolution equations with the variable coefficients have provided a realistic perspective on the inhomogeneities of media and non-uniformities of boundaries. In this paper, we investigate a (3+1)-dimensional Korteweg-de Vries equation with the time-dependent coefficients in a fluid. Based on the Hirota method, we obtain a bilinear form via the binary Bell polynomial approach. Based on the bilinear form, we derive the N-soliton, breather and periodic-wave solutions, where N is a positive integer. Besides, we investigate the asymptotic behaviors of the breather and periodic-wave solutions. Breather waves and periodic waves are graphically displayed. Finally, relation between the periodic-wave solutions and one-soliton solutions is discussed. This paper provides an intuitive understanding for the nonlinear phenomena of those obtained solutions, and those nonlinear phenomena have potential application value in fluid dynamics and other fields. [ABSTRACT FROM AUTHOR]

This paper presents theoretical investigation on the formation of elliptical q -Gaussian breather solitons in diffraction managed optical media. The optical nonlinearity of the medium has been modeled by cubic–quintic nonlinearity. To obtain the physical insight into the propagation dynamics of the laser beam, semi-analytical solution of the wave equation for the laser beam has been obtained by using variational theory approach in W.K.B approximation. Emphasis is put on investigating evolutions of transverse dimensions and axial phase of the optical beam. [ABSTRACT FROM AUTHOR]

Fluid mechanics concerns the mechanisms of liquids, gases and plasmas and the forces on them. We aim to investigate a (2 + 1) -dimensional generalized nonlinear evolution system in fluid mechanics and plasma physics in this paper. With the help of the Painlevé analysis, we find that the above system has Painlevé-integrable property. A set of the auto-Bäcklund transformations and some solutions are derived by the virtue of the truncated Painlevé method. We obtain certain bilinear forms via some seed solutions. According to the mentioned bilinear form, we derive the multiple-soliton solutions on some nonzero backgrounds. Based on the soliton solutions and conjugation transformations, the higher-order breather solutions on certain nonzero backgrounds have been obtained. Via some conjugation transformations, hybrid solutions formed from the breathers and solitons on certain nonzero backgrounds have been derived. We also graphically show the interactions between those solitons and breathers. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is a generalized Heisenberg ferromagnet (HF) equation which is named the Zhanbota-IIA equation. It is one of the integrable generalizations of the HF equation that plays an important role in nonlinear magnetization dynamics. Through the establishment of the N-fold Darboux transformation, a series of solutions will be obtained, including multi-solitons, one- and two-breathers, first- and higher-order rogue waves. Dynamic behaviors of those solutions will be analyzed, including several structures of rogue waves such as fundamental structure, triangular structure, ring structure and ring-fundamental structure, the coexistence of rogue waves and breathers, i.e. semi-rational solution and the interaction of two breathers. [ABSTRACT FROM AUTHOR]

Purpose of this work is to use analytical and numerical methods to consider the problem of the structure and dynamics of the kinks in the sine-Gordon model with “impurities” (or spatial inhomogeneity of the periodic potential). Methods. Using the method of collective variables for the case of three identical point impurities located at the same distance from each other, a system of differential equations is obtained. Resulting system of equations makes it possible to describe the dynamics of the kink taking into account the excitation of localized waves on impurities. To analyze the dynamics of the kink in the case of extended impurities, a numerical finite difference method with an explicit integration scheme was applied. Frequency analysis of kink oscillations and localized waves calculated numerically was performed using a discrete Fourier transform. Results. For the kink dynamics, taking into account the excitation of oscillations in modes, a system of equations for the coordinate of the kink center and the amplitudes of waves localized on impurities is obtained and investigated. Significant differences are observed in the dynamics of the kink when interacting with a repulsive and attractive impurity. The dynamics of the kink in a model with three identical extended impurities, taking into account possible resonant effects, was solved numerically. It is established that the found scenarios of kink dynamics for an extended rectangular impurity are qualitatively similar to the scenarios obtained for a point impurity described using a delta function. All possible scenarios of kink dynamics were determined and described taking into account resonant effects. Conclusion. The analysis of the influence of system parameters and initial conditions on possible scenarios of kink dynamics is carried out. Critical and resonant kink velocities are found as functions of the impurity parameters.

Under investigation in this paper is the variable coefficient nonlocal Fokas–Lenells equation. On the basis of the Lax pair, the infinitely-many conservation laws and Nth-fold Darboux transformation are constructed. Depending on zero seed solution, soliton solutions are derived via the Darboux transformation. Based on nonzero seed solution, breather solutions and rogue wave solutions are obtained. The behaviors of solutions are clearly analyzed graphically. The influences of variable coefficient for solutions are discussed. The different profiles of solitons, breathers and rogue waves are observed via selecting different variable coefficients. Furthermore, the interaction of solitons and the interaction of breathers for the variable coefficient nonlocal Fokas–Lenells equation are both elastic. [ABSTRACT FROM AUTHOR]

We have obtained the first-order solution of a three-coupled nonlinear Schrödinger equation based on the modified Darboux transformation. In addition, we have derived an expression for the distance between the rogue wave and breather. We have find that the amplitude of the rogue wave, the period of the breather, the distance between the rogue wave and breather, the transformation between the bright and dark rogue wave, and the transformation between the soliton and breather which are all affected by the values of the free parameters. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is the integrable Lakshmanan–Porsezian–Daniel (LPD) equation, which was proposed as a model for the nonlinear spin excitations in the one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin. Our main purpose was to construct soliton and breather solutions on the nonconstant background for the integrable local and nonlocal LPD equations. Firstly, the Bäcklund transformations are constructed based on the pseudopotential of equations. Secondly, starting from the nonconstant initial solution sech and applying the obtained transformation, various nonlinear wave solutions of the local LPD equation are provided, including the time-periodic breather, W-shaped soliton, M-type soliton and two-soliton solutions, the elastic interactions between the two-soliton solutions are shown and the relationship between parameters and wave structures is discussed. Thirdly, beginning with the nonconstant initial solutions sech and tanh , the time-periodic breather, bell-shaped one-soliton and anti-bell-shaped one-soliton solutions of the nonlocal LPD equation are generated and these solutions possess no singularity. What is more, the time-periodic breather solutions exhibit the x-periodic background and double-periodic background, which is different from the previous results. The corresponding dynamics of these solutions related to the integrable local and nonlocal LPD equations are illustrated graphically. The results in this paper might be helpful for us to understand the nonlinear characteristics of magnetic materials. [ABSTRACT FROM AUTHOR]

We consider the modified Korteweg–de Vries equation, and prove that given any sum P of solitons and breathers (with distinct velocities), there exists a solution p such that p(t) - P(t) → 0 when t → + ∞, which we call multi-breather. In order to do this, we work at the H² level (even if usually solitons are considered at the H¹level). We will show that this convergence takes place in any Hs space and that this convergence is exponentially fast in time. We also show that the constructed multi-breather is unique in two cases: in the class of solutions which converge to the profile P faster than the inverse of a polynomial of a large enough degree in time (we will call this a super polynomial convergence), or when all the velocities are positive (without any hypothesis on the convergence rate) [ABSTRACT FROM AUTHOR]

In this paper, the spatial discrete Hirota equation is investigated by Darboux–Bäcklund transformation. Firstly, the pseudopotential of the spatial discrete Hirota equation is proposed for the first time, from which a Darboux–Bäcklund transformation is constructed. Comparing it with the corresponding onefold Darboux transformation, we find that they are equivalent because there is no difference except for a constant times. We believe that this equivalence may hold universal if these two transformations are all derived from the same discrete spectral problem and using the similar technique in the references. Secondly, starting from vanishing and plane wave backgrounds, a variety of nonlinear wave solutions, including bell-shaped one-soliton, three types of breathers, W-shaped soliton, periodic solution and rogue wave are given, and the relevant dynamical properties and evolutions are illustrated by plotting figures. The relationship between parameters and solutions' structures is studied in detail, and the related method and technique can also be extended to other nonlinear integrable equations. Finally, we show that the continuum limit of breather and rogue wave solutions of the spatial discrete Hirota equation yields the counterparts of the Hirota equation. The results in this paper might be useful for understanding some physical phenomena in nonlinear optics. [ABSTRACT FROM AUTHOR]

Investigated in this paper is an extended (3+1)-dimensional Kadomtsev-Petviashvili equation. We determine the N-soliton solutions of that equation via an existing bilinear form, and then construct the Mth-order breather and Hth-order lump solutions from the N-soliton solutions using the complex conjugated transformations and long-wave limit method, where N, M, and H are the positive integers. In addition, we develop the hybrid solutions composed of the first-order breather and one soliton, the first-order lump and one soliton, as well as the first-order lump and first-order breather. Through those solutions, we demonstrate the (1) one breather or lump, (2) interaction between the two breathers or lumps, (3) interaction between the one breather and one soliton, (4) interaction between the one lump and one soliton, and (5) interaction between the one lump and one breather. We observe that the amplitude, shape, and velocity of the one breather or lump remain unchanged during the propagation. We also find that the amplitudes, shapes, and velocities of the solitons, breathers, and lumps remain unchanged after the interactions, suggesting that those interactions are elastic. [ABSTRACT FROM AUTHOR]

Purpose of this work is to use analytical and numerical methods to consider the problem of the structure and dynamics of coupled localized nonlinear waves in the sine-Gordon model with impurities (or spatial inhomogeneity of the periodic potential). Methods. Using the analytical method of collective coordinates for the case of the arbitrary number the same point impurities on the same distance each other, differential equation system was got for localized waves amplitudes as the functions on time. We used the finite difference method with explicit scheme for the numerical solution of the modified sine-Gordon equation. We used a discrete Fourier transform to perform a frequency analysis of the oscillations of localized waves calculate numerically. Results. We found of the differential equation system for three harmonic oscillators with the elastic connection for describe related oscillations of nonlinear waves localized on the three same impurity. The solutions obtained from this system of equations for the frequencies of related oscillation well approximate the results of direct numerical modeling of a nonlinear system. Conclusion. In the article shows that the related oscillation of nonlinear waves localized on three identical impurities located at the same distance from each other represent the sum of three harmonic oscillations: in-phase, in-phase-antiphase and antiphase type. The analysis of the influence of system parameters and initial conditions on the frequency and type of associated oscillations is carried out.

The first- and second-order localized waves for a variable-coefficient coupled Hirota equation describe the vector optical pulses in inhomogeneous optical fiber and are investigated via generalized Darboux transformation in this work. Based on the equation's Lax pair and seed solutions, the localized wave solutions are calculated, and the dynamics of the obtained localized waves are shown and analyzed through numerical simulation. A series of novel dynamical evolution plots illustrating the interaction between the rogue waves and dark-bright solitons or breathers are provided. It is found that functions have an influence on the propagation of shape, period, and velocity of the localized waves. The presented results contribute to enriching the dynamics of localized waves in inhomogeneous optical fiber. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear propagation dynamics of circular–linear edge dislocation beams (CLED beams) in nonlocal media are studied theoretically. The transport formulae of CLED beams in nonlocal nonlinear media is derived. The evolution of the beam width, curvature, and intensity distribution are discussed in detail. Two different incident cases of CLED beams, namely beam waist and non-beam waist incidences, are discussed and compared. It is found that the off-axial distance can effectively adjust the position of linear edge dislocation and play a key role on beam propagation. The initial beam power and non-beam waist distance also play a key role in propagation. In particular, when specific parameters are selected, the CLED beam can form a transmission state with beam width and transverse pattern both invariant, i.e., CLED soliton forms. For general parameters, the transverse multi-peak patterns of the CLED beam evolutes periodically, i.e., a generalized higher-order soliton or breather forms.

In this article, we prove that a sum of solitons and breathers of the modified Kortewegâ€"de Vries equation (mKdV) is orbitally stable. The orbital stability is shown in H 2. More precisely, we will show that if a solution of mKdV is close enough to a sum of solitons and breathers with distinct velocities at t = 0 in the H 2 sense, then it stays close to this sum of solitons and breathers for any time t â©ľ 0 in the H 2 sense, up to space translations for solitons or space and phase translations for breathers, provided the condition that the considered solitons and breathers are sufficiently decoupled from each other and that the velocities of the considered breathers are all positive, except possibly one. The constants that appear in this stability result do not depend on translation parameters. From this, we deduce the orbital stability of any multi-breather of mKdV, provided the condition that the velocities of the considered breathers are all positive, except possibly one (the condition about the decoupling of the considered solitons and breathers between each other is not required in this setting). The constants that appear in this stability result depend on translation parameters of the considered solitons and breathers. [ABSTRACT FROM AUTHOR]

Nonlinear differential-difference equations (nonlinear lattices) exhibit single soliton solutions in the form of 1-breathers. These solutions are time-periodic and exponentially localized in space. Necessary condition for their existence is the upper bounds on the linear spectrum of small perturbations around stationary point of the system. Unlike continuous models, integrability property do not help nonlinear lattices to have multi-breather solutions. The C-integrable Liouville lattice is discussed in view to get moving breathers, i.e. true time-periodic multi-solitons with free velocity and amplitude. We construct analogs of instanton solutions which are localized both in space and time. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the evolution characteristics of periodically revived elliptical cos-Gaussian solitons and breathers based on nonlocal nonlinear Schrödinger equation, which can be applied into describing the beam evolution in nonlocal nonlinear media. The elliptical cos-Gaussian solitons can present a variety of intensity distribution modes. With different incident energies, the statistical spot size can remain unchanged during the process of evolution, namely the soliton state; otherwise, the statistical spot size changes periodically, namely the breathing state. The transverse intensity mode always changes periodically which is similar to the higher-order temporal solitons. That is, they can be revived to the original mode at the end of each evolution period. Mathematical expressions are derived to describe the soliton propagation, the intensity pattern, the statistical spot size and the axial intensity etc. Various evolution characteristics are discussed in details and illustrated by numerical simulations.

Under investigation in this paper is a discrete reduced integrable nonlinear Schrödinger system on a triangular-lattice ribbon, which may have some prospective applications in modern nanoribbon. First, we construct the infinitely many conservation laws and discrete N-fold Darboux transformation for this system based on its known Lax pair. Then bright–bright multi-soliton and breather solutions in terms of determinants are obtained by means of the resulting Darboux transformation. Moreover, we investigate soliton interactions through asymptotic analysis and analyze some important physical quantities such as amplitudes, wave numbers, wave widths, velocities, energies and initial phases. Finally, the dynamical evolution behaviors are discussed via numerical simulations. It is found that soliton interactions in this system are elastic, and their evolutions are stable against a small noise in a short period of time. Results obtained in this paper may have some prospective applications for understanding some physical phenomena. [ABSTRACT FROM AUTHOR]

A quantum-chemical study of the atomic charges and bond orders in the cations of the linear conjugated systems was performed. It is shown that total charge in the collective system of the π-electrons generates the soliton-like wave of the alternated partial charges along the conjugated chain not only in ground state but also in the excited state. The excitation is accompanied by the change of the soliton phase and the wave dimension. Additionally, it is established that the electron density redistribution at the atoms and bonds also forms the soliton-like wave. In paper, the dependence of the solitonic wave shape on the dimension and section of the polymethine is studied; established regularities in the charge distribution in excited state could be used for the molecular design of organic semiconducting materials. [ABSTRACT FROM AUTHOR]

This article presents a study of bound periodically oscillating coherent structures arising on the free surface of deep water. Such structures resemble the well known bi-soliton solution of the nonlinear Schrödinger equation. The research was carried out in the super-compact Dyachenko-Zakharov equation model for unidirectional deep water waves and the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. The special numerical algorithm that includes a damping procedure of radiation and velocity adjusting was used for obtaining such bound structures. The results showed that in both nonlinear models for deep water waves after the damping is turned off, a periodically oscillating bound structure remains on the fluid surface and propagates stably over hundreds of thousands of characteristic wave periods without losing energy.

The paper presents the long-time dynamics with multiple collisions of breathers in the super compact Zakharov equation for unidirectional deep water waves. Solutions in the form of breathers were found numerically by the Petviashvili method. In the terms of envelope and the assumption of the narrow spectral width the super compact equation turns into the well known exact integrable model—nonlinear Schrödinger equation, and the breather solution in this case turns into envelope soliton. The results of numerical simulations show that two main scenarios of long-time dynamics occur during numerous collisions of breathers. In the first case, one of the breathers regularly takes a number of particles from the other one at each collision and in the second one a structure resembling the bi-soliton solution of nonlinear Schrödinger equation arises during the collision. Despite these scenarios, it is shown that after numerous collisions the only one breather having initially a larger number of particles remains.

In this paper, based on the nonlocal nonlinear Schrödinger equation, the evolution of complex-valued hyperbolic-sine-Gaussian beams (CVHSGBs) is investigated in nonlinear media with a spatial nonlocality. It is found that the evolution of CVHSGBs is variable depending on the parameters of complex-valued hyperbolic sine function. Choosing special parameters, the pattern of CVHSGBs can keep unchanged during propagation, and they propagate as solitons or breathers. Furthermore, for the general case, the CVHSGB evolutes periodically, and it recovers into its initial pattern at the end of each evolution period, namely it can be revivable periodically, which can be regarded as a generalized high-order breather. A series of analytical expressions are derived to describe the beam evolution, the intensity pattern, the beam spot size, the real beam curvature, etc. Some numerical simulations are also performed to demonstrate the typical evolution properties. [ABSTRACT FROM AUTHOR]

We use the KdV–Burgers equation to model a behaviour of a soliton which, while moving in non-dissipative medium encounters a barrier with dissipation. The modelling included the case of a finite width dissipative layer as well as a wave passing from a non-dissipative layer into a dissipative one. The dissipation results in reducing the soliton amplitude/velocity, and a reflection and refraction occur at the boundary(s) of a dissipative layer. In the case of a finite width barrier on the soliton path, after the wave leaves the dissipative barrier it retains a soliton form and a reflection wave arises as small and quasi-harmonic oscillations (a breather). The first order approximation in the expansion by the small dissipation parameter is studied. [ABSTRACT FROM AUTHOR]

The Darboux transformation of the three-component coupled derivative nonlinear Schrödinger equations is constructed. Based on the special vector solution generated from the corresponding Lax pair, various interactions of localized waves are derived. Here, we focus on the higher-order interactional solutions among higher-order rogue waves, multi-solitons, and multi-breathers. It is defined as the identical type of interactional solution that the same combination appears among these three components q1,q2, and q3, without considering different arrangements among them. According to our method and definition, these interactional solutions are completely classified as six types, among which there are four mixed interactions of localized waves in these three different components. In particular, the free parameters μ and ν play the important roles in dynamics structures of the interactional solutions. For example, different nonlinear localized waves merge with each other by increasing the absolute values of these two parameters. Additionally, these results demonstrate that more abundant and novel localized waves may exist in the multi-component coupled systems than in the uncoupled ones. [ABSTRACT FROM AUTHOR]