Periodic Breather Research Articles

Periodic solutions, breather, lump and interaction solutions of a generalized (2+1)-dimensional Hirota bilinear equation via bilinear neural network method

Abstract The main focus of this paper is to address a generalized (2+1)-dimensional Hirota bilinear equation utilizing the bilinear neural network method. The paper presents the periodic solutions through a single-layer model of [3-4-1], followed by breather, lump and their interaction solutions by using double-layer models of [3-3-2-1] and [3-3-3-1], respectively. A significant innovation introduced in this work is the computation of periodic cross-rational solutions through the design of a novel [3-(2+2)-4-1] model, where a specific hidden layer is partitioned into two segments for subsequent operations. Three-dimensional and density figures of the solutions are given alongside an analysis of the dynamics of these solutions.

Dynamics of fundamental and double-pole breathers and solitons for a nonlinear Schrödinger equation with sextic operator under non-zero boundary conditions

Abstract We study the dynamics of fundamental and double-pole breathers and solitons for the focusing and defocusing nonlinear Schrödinger equation with the sextic operator under non-zero boundary conditions. Our analysis mainly focuses on the dynamical properties of simple- and double-pole solutions. Firstly, through verification, we find that solutions under non-zero boundary conditions can be transformed into solutions under zero boundary conditions, whether in simple-pole or double-pole cases. For the focusing case, in the investigation of simple-pole solutions, temporal periodic breather and the spatial-temporal periodic breather are obtained by modulating parameters. Additionally, in the case of multi-pole solitons, we analyze parallel-state solitons, bound-state solitons, and intersecting solitons, providing a brief analysis of their interactions. In the double-pole case, we observe that the two solitons undergo two interactions, resulting in a distinctive “triangle” crest. Furthermore, for the defocusing case, we briefly consider two situations of simple-pole solutions, obtaining one and two dark solitons.

Mixed localized waves and their interaction structures for a spatial discrete Hirota equation

Mixed localized wave solutions and interactions are of great significance in nonlinear physical systems. This paper aims to investigating the generalized (m,N-m)-fold Darboux transformation and the mixed localized wave solutions of a spatial discrete Hirota equation. First, we construct the generalized (m,N-m)-fold Darboux transformation for the spatial discrete Hirota equation, which can produce the interactions between the breathers, degenerate breathers and rogue waves. For the Darboux transformation formula, we discuss the above order-1,2,3 localized wave solutions, as well as their dynamics by choosing the number of m = 1. We plot some specific examples such as the spatial (time)-periodic breather, second-order and third-order degenerate breathers and higher-order rogue waves with novel patterns. Furthermore, when m > 1, we give several kinds of mixed interaction solutions between the first-order rogue waves and first (second)-order (degenerate) breathers, between the first-order breather and second-order degenerate breathers, between second-order rogue waves and first-order breathers. At last, we also sum up the various mathematical features of the degenerate breathers and the mixed localized wave solutions.

Exploration of soliton structures in the Hirota–Maccari system with stability analysis

In this research, the modified extended tanh-function (METF) and the extended Jacobi elliptic function expansion (EJEFE) techniques are used to investigate the generation and detection of soliton structures in the Hirota–Maccari (HM) model. Consequently, we obtain soliton solutions with advanced structures, including singular bright soliton, dark soliton, periodic waves, breather waves, periodic breather waves, and multiple bright and dark breather waves. In addition, a lump-type breather wave is also included in the presented solutions. Stability analysis of the obtained solutions is addressed by employing the Hamiltonian technique. 3D surfaces and 2D visuals of the outcomes are represented with the help of a computer application. These findings contribute to understanding nonlinear wave phenomena with potential applications in optics, fluid dynamics, and plasma physics.

Novel dynamics of the Fokas-Lenells model in Birefringent fibers applying different integration algorithms

Abstract The Fokas-Lenells model has the broad applications in nonlinear physics to study various soliton phenomena. Employing the direct algebraic scheme, the modified rational sine-cosine technique, and the (1/G′) expansion scheme, the analytical solutions to this model are derived. Double periodic waves, bright soliton, dark soliton, single and multiple breather waves, and periodic breather waves are extracted from this model using symbolic computation. The dynamic behaviors of the acquired outcomes are vividly illustrated through density, two-dimensional (2D), and three-dimensional (3D) graphical representations. These discoveries are strategically positioned to significantly contribute to the advancement of exploring nonlinear models, standing as a fundamental pillar for forthcoming research endeavors.

Breather solutions for a radially symmetric curl-curl wave equation with double power nonlinearity

This paper is concerned with breather solutions of a radially symmetric curl-curl wave equation with double power nonlinearity ρ(x)utt+∇×(M(x)∇×u)+μ(x)u+vp(x)|u|p−1u+vq(x)|u|q−1u=0,where (x,t)∈R3×R, u:R3×R→R3 is the unknown function, M:R3→R3×3 and ρ,μ,vp,vq:R3→(0,+∞) are radially symmetric coefficient functions with 1<p<q. By considering the solutions with a special form u=y(|x|,t)x|x|, we obtain a family of ordinary differential equations (ODEs) parameterized by the radial variable r=|x|. Then we characterize periodic behaviors and analyze the joint effects of the double power nonlinear terms on the minimal period and the maximal amplitude. Under certain conditions, we construct a 2πρ(0)/μ(0)-periodic breather solution for the original curl-curl wave equation and find such a solution which can generate a continuum of phase-shifted breathers.

Breather waves, periodic cross-lump waves and complexiton type solutions for the (2 + 1)-dimensional Kadomtsev-Petviashvili equation in dispersive media

The manuscript under consideration explores the (2+1)-dimensional Kadomtsev-Petviashvili (KP) model using its bilinear representation. The focus is on studying the characteristics of periodic cross-lump waves and breather wave solutions. Breathers, which are localized periodic solutions of continuous media equations or discrete lattice equations, are of particular interest. The research also delves into complexiton solutions using the extended transformed rational function approach. The study graphically analyzes the dynamics of these interaction solutions using appropriate parameters. It's truly fascinating to learn about the application of breathers in continuous media equations or discrete lattice equations.

Dynamical analysis of higher-order rogue waves on the various backgrounds for the reverse space–time Fokas–Lenells equation

In this paper, we will focus on the reverse space–time Fokas–Lenells equation. According to the Lax pair, the semi-degenerate Darboux transformation will be constructed by Taylor expansions. Based on the determinant form, some rogue wave solutions will be derived on the three types of backgrounds, including M-shaped periodic waves, mixed M-shaped periodic waves and breather, mixed single-periodic and double-periodic waves. And the rogue waves in various backgrounds can be transformed into the corresponding backgrounds waves with parameters selection. The dynamic features of those solutions will be discussed via graphical illustration. It is worth mentioning that these results are novel.

Interaction solution to the (3+1)-D negative-order KdV first structure

We derive N-solitons and interaction solution for the (3+1)-D negative-order KdV first structure that arises in shallow-water waves. We use the bilinear scheme and the simplified Hirota technique for this solution. From the multiple solitons solution, we obtain a lump-shaped breather wave, a lump-shaped breather with a kink wave, and two lump-shaped breather waves from 2-solitons, 3-solitons, and 4-solitons, sequentially, by choosing complex conjugates of related parametric constants. Additionally, we show some novel interactions of the Jacobian elliptic transformation with periodic function, single soliton, and two solitons. Owing to these collisions, periodic waves with a kink wave, periodic breathers with a kink wave, and numerous bright and dark breather waves are created. We illustrate the properties of these solutions with 3D, density, and contour plots by selecting the appropriate values for the parameters. The obtained solutions will serve as milestones for studying the properties of nonlinear structures in the physical sciences and engineering fields.

Diversity of exact solutions to the (2+1)-dimensional Ito equation via bilinear neural network method

Recently, searching for exact solutions to nonlinear partial differential equations has gradually become a hot research topic. It is of great scientific research and application value to reveal the law of wave propagation, explain natural phenomena accurately and apply related technologies scientifically. In this paper, bilinear neural network method (BNNM) was employed to obtain some new exact analytical solutions to the (2 + 1)-dimensional Ito equation. Based on the Hirota form of Ito equation, we constructed activation functions f in various forms containing the exp(ξ), sin(ξ), cos(ξ), cosh(ξ) and squares of polynomial functions in multi-layer neurons such as [3-2-2] and [3-2-3] neural network models. The test function f in this work is a new expression. On the other hand, these solutions have not been studied yet. As a result, we obtained several new interaction solutions, such as periodic wave solution, breather solution and bright-dark soliton solution, etc. In addition, the corresponding 3D, density and contour plots of all the solution forms were drawn and their characters and dynamic behaviors were vividly demonstrated.

Localized waves and their novel interaction solutions for a dimensionally reduced (2 + 1)-dimensional Kudryashov Sinelshchikov equation

Propagation of the pressure waves in a liquid with gas bubbles is an important topic in the field of fluid dynamics and mathematical physics. The Kudryashov-Sinelshchikov equation is one of the models that describe the propagation of nonlinear waves in a bubbly liquid taking into consideration the viscosity of the liquid and the heat transfer. To explain such behaviors, we mainly focus in this study to explain the dynamics of localized waves and their variety of interaction solutions to a dimensionally reduced (2 + 1)-dimensional Kudryashov-Sinelshchikov equation with the aid of the Hirota bilinear method from N-soliton solutions. Four different forms of localized waves, including solitons, lumps, breathers, and rogues, are derived from the aforesaid equation based on the long wave limit approach. In particular, the localized waves can be used to find interaction solutions, which are the single breather or single lump formed by two solitons; interaction between one line soliton and one breather, as well as one line soliton and one lump soliton among the three solitons; interaction of the two-line soliton and one periodic breather, two periodic breathers, periodic breather and one lump soliton from the four solitons. The direction of propagation, phase shifts, shape, energy, and the variety of interaction solutions of localized waves are affected by these parameters. Moreover, analytical and graphical illustrations of these interaction solutions and their propagation properties are shown by the three-dimensional and density plots with the help of Maple 17. These newly discovered solutions in this study can be used to illustrate the interaction phenomenon of localized waves on ocean surfaces.

An envelope system with third order dispersion: ‘Unconventional’ modulation instability and Floquet analysis

Modulation instability of plane waves of the Hirota equation, an ‘integrable’ system with third order dispersion, arises from the interplay of dispersive and nonlinear effects. The conventional analysis cannot account for finite amplitude disturbances and suffers from a scenario of indefinite growth. In an attempt to remove these constraints, we consider, as an illustrative example, the exact doubly periodic solutions expressed via the Jacobi elliptic functions. Wavy profiles at the minima of the intensity are interpreted as finite amplitude disturbances on a plane wave background. The profiles are amplified and will saturate at the maxima of the intensity. Such periodic states and breathers can be generated from finite amplitude disturbances with wavenumbers falling outside the linear instability band. This growth phase thus qualifies as ‘unconventional’ or ‘extraordinary’ modulation instability. Floquet analysis is performed to investigate the stability of the periodic patterns.

Emergent soliton-like solutions in the parametrically driven 1-D nonlinear Schrödinger equation

We numerically investigate the long time dynamics of spatially periodic breather solutions of the 1-D nonlinear Schrödinger equation under parametric forcing of the form along with dissipation. In the absence of dissipation, robust soliton-like excitations are observed that travel with constant amplitude and velocity. With dissipation, these solitons lose energy (and amplitude) yet gain speed - a characteristic not observed in an ordinary soliton. Moreover, these novel solitons are found to be stable against random perturbations.

Some Exact Solutions to Generalized Kadomtsev-Petviashvili Equation

Most of the papers have explored the interactions between solitons with a zero background, while reports about exact solutions for nonzero background are rare. Hence, this paper is aimed at exploring the breather, lump, and interaction solutions with a small perturbation to ( 2 + 1 )-dimensional generalized Kadomtsev-Petviashvili (gKP) equation. General high-order periodic breather solutions are obtained using Hirota’s bilinear method with a small perturbation. At the same time, combining the use of long wave limit methods and module resonance constraints, general lump solutions and mixed solutions to gKP equation are generated. Finally, the space-time structures of the breather solutions, lump solutions, and interaction solutions are investigated and discussed.

Nth-order smooth positon and breather-positon solutions for the generalized integrable discrete nonlinear Schrödinger equation

In this paper, we investigate the smooth positon and breather-positon solutions of the generalized integrable discrete nonlinear Schrödinger (NLS) equation by the degenerate Darboux transformation (DT). Starting from the zero seed solution, the Nth-order smooth positon solutions are obtained by degenerate DT. The breather solutions including Akhmediev breather, Kuznetsov-Ma breather and space-time periodic breather are derived from the nonzero seed solution. Then the breather-positon solutions are constructed by gradual Taylor series expansion of the eigenfunctions in breather solutions. We study the effect of the coefficient of nonlinear term on these discrete smooth positon solutions and breather-positon solutions, which demonstrates that the interacting region of soliton-positon and breather-positon are highly compressed by higher-order nonlinear effects, but the distance between the two positons has an opposite effect in two waveforms.

Doubly periodic solutions and breathers of the Hirota equation: recurrence, cascading mechanism and spectral analysis

Doubly periodic solutions and breathers of the Hirota equation: recurrence, cascading mechanism and spectral analysis

Interactions of rogue and solitary wave solutions to the (2 + 1)-D generalized Camassa–Holm–KP equation

This research explores a (2 + 1)-D generalized Camassa–Holm–Kadomtsev–Petviashvili model. We use a probable transformation to build bilinear formulation to the model by Hirota bilinear technique. We derive a single lump waves, multi-soliton solutions to the model from this bilinear form. We present various dynamical properties of the model such as one, two, three and four solitons. The double periodic breather waves, periodic line rogue wave, interaction between bell soliton and double periodic rogue waves, rogue and bell soliton, rogue and two bell solitons, two rogues, rogue and periodic wave, double periodic waves, two pair of rogue waves as well as interaction of double periodic rogue waves in a line are established. Among the results, most of the properties are unexplored in the prior research. Furthermore, with the assistance of Maple software, we put out the trajectory of the obtained solutions for visualizing the achieved dynamical properties.

A study of nonlinear extended Zakharov–Kuznetsov dynamical equation in (3+1)-dimensions: Abundant closed-form solutions and various dynamical shapes of solitons

In this work, we execute the generalized exponential rational function (GERF) method to construct numerous and a large number of exact analytical solitary wave solutions of the nonlinear extended Zakharov–Kuznetsov (EZK) dynamical equation in (3+1)-dimensions. The implemented method is one of the best, most reliable, and efficient techniques in the present time for determining numerous closed-form wave analytic solutions to NPDEs. We have accomplished a variety of solitary wave solutions related to some arbitrary parameters under various family cases. These solutions take the following forms based on the free parameters chosen: exponential functions form, trigonometric functions form, and hyperbolic functions form. The obtained solutions are dissimilar and entirely new from the previous findings available in the literature. The dynamics of obtained solutions, namely, soltion, singular soliton wave, a periodic wave, bell-shape, anti-bell-shape wave, breather wave, and multisoliton wave solutions by the special-choice of parameters, are shown graphically in 3D, 2D, and corresponding density profiles. The results demonstrate that the employed computational strategy is efficient, direct, concise, and can be executed in various complex phenomena with symbolic computations. Furthermore, it is revealed that the generalized exponential rational function technique can be effectively utilized for several other NPDEs in engineering, sciences, and mathematical physics.

Nonlinear damped spatially periodic breathers and the emergence of soliton-like rogue waves

The spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger equation, prominent in modeling rogue waves, are unstable. In this paper we numerically investigate the routes to stability of the SPBs and related rogue wave activity in the framework of the nonlinear damped higher order nonlinear Schrödinger (NLD-HONLS) equation. The NLD-HONLS solutions are initialized with single-mode and two-mode SPB data at different stages of their development. The Floquet spectral theory of the NLS equation is applied to interpret and provide a characterization of the perturbed dynamics in terms of nearby solutions of the NLS equation. A broad categorization of the routes to stability of the SPBs is determined. Novel behavior related to the effects of nonlinear damping is obtained: tiny bands of complex spectrum develop in the Floquet decomposition of the NLD-HONLS data, indicating the breakup of the SPB into either a one or two “soliton-like” structure. For solutions initialized in the early to middle stage of the development of the modulational instability, we find that all the rogue waves in the NLD-HONLS flow occur when the spectrum is in a one or two soliton-like configuration. When the solutions are initialized as the modulational instability is saturating, rogue waves also occur when the spectrum is not in a soliton-like state. Another distinctive feature of the nonlinear damped dynamics is that the growth of instabilities can be delayed and expressed at higher order due to permanent frequency downshifting.

Various forms of lumps and interaction solutions to generalized Vakhnenko Parkes equation arising from high-frequency wave propagation in electromagnetic physics

This article retrieves multiple lump, periodic cross kink, rogue wave, multiwave, breathers, M-shaped and interactional solutions for generalized Vakhnenko Parkes (gVP) equation via appropriate transformations. This leads to a transformed equation for which it is straightforward to evaluate the exact solutions. The gVP equation gives a description of high-frequency waves in the relaxation medium possessing periodic and solitary traveling wave solutions some of which are loop-like in nature. These types of solutions have been observed in wave propagations, rogue waves and many other physical systems. We also interpret some of our solutions in distinct dimensions via 3D, 2D, contours and density profiles.