Soliton Structures Research Articles

The Multi-Soliton Solutions for the (2+1)-Dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada Equation

The (2+1)-dimensional integrable Caudrey–Dodd–Gibbon–Kotera–Sawada equation is a higher-order generalization of the Kadomtsev–Petviashvili equation, which can be applied in some physical branches such as the nonlinear dispersive phenomenon. In this paper, we first present the bilinear form for this equation after constructing one Bäcklund transformation. As a result, the one-soliton solution, two-soliton solution, and three-soliton solution are shown successively and the corresponding soliton structures are constructed. These solitons and their interactions illustrate that the obtained solutions have powerful applications.

Dust acoustic soliton and shock structures with consequence of head-on collision in multi-component unmagnetized plasmas

Dust acoustic soliton and shock structures with consequence of head-on collision in multi-component unmagnetized plasmas

Propagation of optical spatial solitons in nematic liquid crystals with quadruple power law of nonlinearity appears in fluid mechanics

Abstract The present research explores nematicons in liquid crystals (LCs) with quadruple power law nonlinearity utilizing the modified extended Fan sub-equation technique as an analytical tool to investigate the optical spatial soliton solutions. For the inaugural time, a novel version of nonlinearity is investigated in relation to LCs. There are distinct applications for the several wave solutions that have been created in optical handling data. The aforementioned modified extended Fan subequation approach offers novel, comprehensive solutions that are relatively easy to deploy in comparison to earlier, regular methodologies. This approach translates a coupled non-linear partial differential equation into a coupled ordinary differential equation through implementing a traveling wave conversion. This approach indicates that a large variety of traveling and solitary solutions that rely upon five parameters can be incorporated by the nematicons in LCs. In addition, the investigation yields solutions of the single and mixed non-degenerate Jacobi elliptic function form. Novel solutions, such as the periodic pattern, kink and anti-kink patterns, N-pattern, W-pattern, anti-Z-pattern, M-pattern, V-pattern, complexion pattern and anti-bell pattern, or dark soliton solutions of nematic LCs, have been constructed by means of modified extended Fan subequation technique through granting suitable values for the parameters. The computer software Mathematica 14 is used to illustrate several modulus, real and imaginary solutions visually in the form of contour, 2D, and 3D visualizations that help understand the concrete importance of the nematicons in LCs. This research additionally offers a physical comprehension of the obtained solutions and applications of model. The imposed approach is ultimately thought to be more potent and effective than alternative approaches, and the solutions found in this work could be beneficial in our understanding of soliton structures in LCs.

Exploration of soliton structures and modulation instability analysis for the highly dispersive perturbed NLSE with sextic-power law refractive index

Exploration of soliton structures and modulation instability analysis for the highly dispersive perturbed NLSE with sextic-power law refractive index

Analysis of soliton wave structure for coupled Higgs equation via Lie symmetry, Paul Painlevé approach and the Unified method

Analysis of soliton wave structure for coupled Higgs equation via Lie symmetry, Paul Painlevé approach and the Unified method

An innovative method for solving the nonlinear fractional diffusion reaction equation with quadratic nonlinearity analysis

Abstract This study generates and investigates spreading solitons in the fractional DR quadratic equation (FDE) with fractional derivatives using the Extended Direct Algebraic Method (EDAM). In population growth, mathematical biology, and reaction-diffusion mechanisms, the FDE is crucial. Applying the series form solution to the NODE from the FDE conversion into a recommended EDAM yields many traveling soliton solutions. To characterize and explore soliton structure propagation, we draw shock and kink soliton solutions. Through reaction-diffusion mechanics and mathematical biology, we may explain complex processes in many academic subjects.

Study of complex dynamics and novel soliton solutions of the Kraenkel-Manna-Merle model describing saturated ferromagnetic materials

This paper introduces a framework for nonlinear short wave propagation in an external field for saturated ferromagnetic materials with zero conductivity: the Kraenkel-Manna-Merle system. New solutions are produced via the G′/(bG′+G+a)-expansion technique. The suggested method is easier to understand, more precise, and simpler to compute. Specifically, a set of singular-periodic and kink-shaped exact soliton solutions is produced. Contour plots, 3D, and 2D visualizations with suitable parametric values are employed to present the computed solutions, which are derived through constraint conditions. For the development of certain novel soliton structures, the arbitrary functions in the solutions are selected. Moreover, bifurcation and chaos theory are applied to the primary dynamical system in order to provide a qualitative analysis of the system under study. Phase portraits of bifurcation are presented at fixed locations to illustrate different situations about parameter values in the dynamic system. However, adopting techniques for identifying chaos in a dynamical system and applying an external force verifies the occurrence of chaotic behavior in system. These results offer new perspectives that can enhance understanding of the dynamics of the KMM system.

Dynamical structures of optical solitons for highly dispersive perturbed NLSE with β-fractional derivatives and a sextic power-law refractive index using a novel approach

In this paper, we investigate the highly dispersive perturbed nonlinear Schrödinger equation (NLSE) with β-fractional derivatives, generalized nonlocal laws and sextic-power law refractive index. This equation is crucial for modeling complex phenomena in nonlinear optics, such as soliton formation, light pulse propagation in optical fibers, and light wave control, with potential applications in designing efficient optical communication devices. Furthermore, it provides a framework for understanding the intricate interactions between high dispersion, nonlocality, and complex nonlinearity, contributing to the development of new theories in wave physics. To accomplish this, we use the modified extended direct algebraic method. A variety of distinct traveling wave solutions are furnished. The obtained solutions comprise dark, bright, combo bright-dark and singular soliton solutions. Additionally, singular periodic solutions, rational and exponential solutions. Furthermore, graphical simulations are presented that highlight the distinctive characteristics of these solutions. Compared to Nofal et al. (Optik 228:166120, 2021), the proposed technique produced novel and diversified results. The results showcase the significant influence of fractional derivatives in shaping the characteristics of the soliton solutions, which is crucial for accurately modeling the dispersive and nonlocal effects in optical fibers. The extracted solutions confirmed the efficacy and strength of the current approach. The parameter constraints ensure the existence of the obtained soliton solutions. It is worth noting that the proposed method, being effective, consistent, and influential, can be applied to solve various other physical models and related disciplines.

Geometric properties of almost pure metric plastic pseudo-Riemannian manifolds

This paper investigates the geometric and structural properties of almost plastic pseudo-Riemannian manifolds, with a specific focus on three-dimensional cases. We explore the interplay between an almost plastic structure and a pseudo-Riemannian metric, providing a comprehensive analysis of the conditions that define pure metric plastic P-Kählerian manifolds. In this context, the fundamental tensor field is symmetric and also represents another pure metric. A key finding is the necessary and sufficient condition for the integrability of the plastic structure by means of partial differential equations, which is characterized by a specific function depending solely on one variable. Furthermore, the necessary and sufficient condition for an almost pure metric plastic pseudo-Riemannian manifold to be pure metric plastic P-Kählerian is obtained. The study also reveals that the Riemannian curvature tensor vanishes under specific conditions, while the scalar curvature vanishes if a particular polynomial form is satisfied. The analysis extends to the properties of vector fields, identifying conditions under which they become Killing vector fields or form Ricci soliton structures. Additionally, we examine three-dimensional Walker manifolds, detailing the conditions for vanishing scalar curvature, Killing vector fields, and Ricci soliton structures. Our findings provide a detailed framework for understanding almost plastic manifolds, contributing to the broader field of differential geometry by clarifying the relationship between plastic structures and pseudo-Riemannian metrics.

Exploring Nonlinear Dynamics in Intertidal Water Waves: Insights from Fourth-Order Boussinesq Equations

The fourth-order nonlinear Boussinesq water wave equation, which describes the propagation of long waves in the intertidal zone, is investigated in this study. The exact wave patterns of the equation were computed using the tanh method. As stability decreased, soliton wave structures were derived using similarity transformations. Numerical simulations supported these findings. The tanh method introduced a Galilean modification, leading to the discovery of several new exact solutions. Subsequently, the fourth-order nonlinear Boussinesq wave equation was transformed into a planar dynamical system using the travelling wave transformation. The quasi-periodic, cyclical, and nonlinear behaviors of the analyzed equation were particularly examined. Numerical simulations revealed that varying the physical parameters impacts the system’s nonlinear behavior. Graphs represent all possible examples of phase portraits in terms of these parameters. Furthermore, the study was proven to be highly beneficial for addressing issues such as shock waves and highly active travelling wave processes. Sensitivity analysis theory and the Lyapunov exponent were employed, offering a wide variety of linear periodic and first-frequency periodic characteristics. Sensitivity analysis and multistability analysis of the Boussinesq water wave equation were thoroughly investigated.

Periodic, quasi-periodic, chaotic waves and solitonic structures of coupled Benjamin-Bona-Mahony-KdV system

Abstract In this paper, we mainly focus on studying the dynamical behaviour and soliton solution of the coupled Benjamin-Bona-Mahony-Korteweg–de Vries (BBM-KdV) system, which characterizes the propagation of long waves in weakly nonlinear dispersive media. The paper utilizes different tools to detect chaos, such as time series analysis, bifurcation diagrams, power spectra, phase portraits, Poincare maps, and Lyapunov exponents. This analysis helps in more accurate predictive modeling of the systems. This understanding can aid in the design of control strategies, resulting in enhancements in prediction, control, optimization, and design. Additionally, we construct the system’s solitary wave structures using the Jacobi elliptic function (JEF) method. We identify periodic wave solutions expressed in terms of rational, hyperbolic, and trigonometric functions. Certain parameter values can lead to periodic wave solutions, solitary waves (bell-shaped solitons), shock wave solutions (kink-shaped soliton solutions), and double periodic wave solutions.

The generalized soliton wave structures and propagation visualization for Akbota equation

Abstract This paper explores in detail the integrable Akbota equation, a Heisenberg ferromagnet-type problem that is essential to the study of surface and curve geometry. A variety of soliton families are represented by the generalized solitonic wave profiles that are produced using the improved modified Sardar sub-equation technique, which is renowned for its accuracy and dependability. There has never been a study that used this technique before the current one. As a result, the solitonic wave structures have kink, dark, brilliant, king-singular, dark-singular, dark-bright, exponential, trigonometric, and rational solitonic structures, among other characteristics. In order to check the energy conservation, the Hamiltonian function is created and energy level demonstrated. The sensitivity analysis is also presented at various initial conditions. The graphical representation is also depicted along with the appropriate parametric values.

Exploring the nonlinear behavior of solitary wave structure to the integrable Kairat-X equation

This study presented various types of soliton solutions for the nonlinear integrable Kairat-X equation by utilizing the improved F-expansion technique with symbolic computational software Mathematica. Explored results for the nonlinear integrable Kairat-X equation are interesting, novel, and more general with different physical structures of solitary waves and solitons, such as kink wave, mixed dark–bright, peakon, anti-kink wave, bright, anti-kink dark, periodic, and dark solitons. With numerical simulations, the secured soliton solutions visualized in two-dimensional, three-dimensional, and contour graphs represent the physical phenomena of the demonstrated results. The explored soliton solutions will be helpful to comprehend interesting physical structures in fiber optics, nonlinear optics, ferromagnetic dynamics, and many other scientific fields. The extracted soliton structure sheds light that the enhanced technique is effective, powerful, concise, and reliable. We can also investigate the soliton results of other nonlinear integrable partial and fractional equations.

Dynamic instabilities and turbulence of merged rotating Bose–Einstein condensates

We present the simulation results of merging harmonically confined rotating Bose–Einstein condensates in two dimensions. Merging of the condensate is triggered by positioning the rotation axis at the trap minima and moving both condensates toward each other while slowly ramping their rotation frequency. We analyze the dynamics of the merged condensate by letting them evolve under a single harmonic trap. We systematically investigate the formation of solitonic and vortex structures in the final, unified condensate, considering both nonrotating and rotating initial states. In both cases, merging leads to the formation of solitons that decay into vortex pairs through snake instability, and subsequently, these pairs annihilate. Soliton formation and decay-induced phase excitations generate sound waves, more pronounced when the merging time is short. We witness no sound wave generation at sufficiently longer merging times that finally leads to the condensate reaching its ground state. With rotation, we notice off-axis merging (where the rotation axes are not aligned), leading to the distortion and weakening of soliton formation. The incompressible kinetic energy spectrum exhibits a Kolmogorov-like cascade [E(k)∼k−5/3] in the initial stage for merging condensates rotating above a critical frequency and a Vinen-like cascade [E(k)∼k−1] at a later time for all cases. Our findings hold potential significance for atomic interferometry, continuous atomic lasers, and quantum sensing applications.

Classification of Vector Fields and Soliton Structures on a Tangent Bundle with a Ricci Quarter-Symmetric Metric Connection

Consider $TM$ as the tangent bundle of a (pseudo-)Riemannian manifold $M$, equipped with a Ricci quarter-symmetric metric connection $\overline{\nabla }$. This research article aims to accomplish two primary objectives. Firstly, the paper undertakes the classification of specific types of vector fields, including incompressible vector fields, harmonic vector fields, concurrent vector fields, conformal vector fields, projective vector fields, and $% \widetilde{\varphi }(Ric)$ vector fields, within the framework of $\overline{% \nabla }$ on $T\dot{M}$. Secondly, the paper establishes the necessary and sufficient conditions for the tangent bundle $TM$ to become as a Riemannian soliton and a generalized Ricci-Yamabe soliton with regard to the connection $\overline{\nabla }$.

Doubly warped product manifolds with the structure of hyperbolic Ricci solitons

In this paper we study hyperbolic Ricci solitons on doubly warped product manifolds. The necessary conditions are obtained for a hyperbolic Ricci soliton with the structure of a doubly warped product to be an Einstein manifold when we consider the potential field as a Killing or a conformal vector field. Also, we examined the behavior of hyperbolic Ricci solitons on doubly warped product space-times.

Solitonic wave Structures and Stablility Analysis for the M-fractional Generalized Coupled Nonlinear SchröDinger-KdV Equations

Solitonic wave Structures and Stablility Analysis for the M-fractional Generalized Coupled Nonlinear SchröDinger-KdV Equations

Supplementary optical solitonic expressions for Gerdjikov–Ivanov equations with three Kudryashov-based methods

The present manuscript uses three Kudryashov-based methods to analytically inspect the class of Gerdjikov–Ivanov equations, which comprises the standard Gerdjikov–Ivanov equation and the perturbed Gerdjikov–Ivanov equation. Various optical solitonic solutions have been constructed. Certainly, as the reported solitonic structures happened to be exponential functions, diverse true solitonic solutions can easily be resorted to upon suitably fixing the involving parameters, including mainly the bright and singular solitons. Lastly, the study graphically examined some of the constructed structures, which were then found to portray some interesting known shapes in the theory of solitary waves and nonlinear Schrödinger equations. Additionally, the Kudryashov-index d has been noted to play a significant role in the propagation of complex waves in the nonlinear media described by Gerdjikov–Ivanov equations.

Solitonic Analysis of the Newly Introduced Three-Dimensional Nonlinear Dynamical Equations in Fluid Mediums

The emergence of higher-dimensional evolution equations in dissimilar scientific arenas has been on the rise recently with a vast concentration in optical fiber communications, shallow water waves, plasma physics, and fluid dynamics. Therefore, the present study deploys certain improved analytical methods to perform a solitonic analysis of the newly introduced three-dimensional nonlinear dynamical equations (all within the current year, 2024), which comprise the new (3 + 1) Kairat-II nonlinear equation, the latest (3 + 1) Kairat-X nonlinear equation, the new (3 + 1) Boussinesq type nonlinear equation, and the new (3 + 1) generalized nonlinear Korteweg–de Vries equation. Certainly, a solitonic analysis, or rather, the admittance of diverse solitonic solutions by these new models of interest, will greatly augment the findings at hand, which mainly deliberate on the satisfaction of the Painleve integrability property and the existence of solitonic structures using the classical Hirota method. Lastly, this study is relevant to contemporary research in many nonlinear scientific fields, like hyper-elasticity, material science, optical fibers, optics, and propagation of waves in nonlinear media, thereby unearthing several concealed features.

Nonlinear Structures of Dispersive Electrostatic Solitary Waves in a Multi‐Ion Partially Ionized Plasma

ABSTRACTThe nonlinear structure and dynamics of dispersive solitons and breather waves described by Korteweg‐de Vries and nonlinear Schrödinger equations are studied. The theoretical and numerical study of the generalized hydrodynamic equations, accounting for wave dissipation and particle production‐loss mechanism, are considered. The reductive expansion method has been used in the context of the instability problem of multi‐fluid dynamics, applied to the study of electrostatic solitons and ion‐acoustic waves. A nonlocal model of interacting solitary‐breather waves has been presented. Applications of the theory, concerning the ion streaming instability in the framework of plasma physics, are presented.