Kink Waves Research Articles

Bifurcations and Exact Solutions of Optical Soliton Models in Fifth-Order Weakly Nonlocal Nonlinear Media

For the optical soliton model in fifth-order weakly nonlocal nonlinear media, to find its exact explicit solutions, the corresponding traveling wave system is formulated as a planar dynamical system with a singular straight line. Then, by using techniques from dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] to analyze the planar system and find the corresponding phase portraits, the dynamical behavior of the amplitude component can be assessed. Under different parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, kink, and anti-kink wave solutions, compacton solutions, as well as peakons and periodic peakons are found with precise formulations.

Existence of Kink and Antikink Wave Solutions of Singularly Perturbed Modified Gardner Equation

In this paper, the singularly perturbed modified Gardner equation is considered. Firstly, for the unperturbed equation, under certain parameter conditions, we obtain the exact expressions of kink wave solution and antikink wave solution by using the bifurcation method of dynamical systems. Then, the persistence of the kink and antikink wave solutions of the perturbed modified Gardner equation is studied by exploiting the geometric singular perturbation theory and the Melnikov function method. When the perturbation parameter is sufficiently small, we obtain the sufficient conditions to guarantee the existence of kink and antikink wave solutions.

Investigation of the dynamical structures of double-chain deoxyribonucleic acid model in biological sciences

The present research investigates the double-chain deoxyribonucleic acid model, which is important for the transfer and retention of genetic material in biological domains. This model is composed of two lengthy uniformly elastic filaments, that stand in for a pair of polynucleotide chains of the deoxyribonucleic acid molecule joined by hydrogen bonds among the bottom combination, demonstrating the hydrogen bonds formed within the chain’s base pairs. The modified extended Fan sub equation method effectively used to explain the exact travelling wave solutions for the double-chain deoxyribonucleic acid model. Compared to the earlier, now in use methods, the previously described modified extended Fan sub equation method provide more innovative, comprehensive solutions and are relatively straightforward to implement. This method transforms a non-linear partial differential equation into an ODE by using a travelling wave transformation. Additionally, the study yields both single and mixed non-degenerate Jacobi elliptic function type solutions. The complexiton, kink wave, dark or anti-bell, V, anti-Z and singular wave shapes soliton solutions are a few of the creative solutions that have been constructed utilizing modified extended Fan sub equation method that can offer details on the transversal and longitudinal moves inside the DNA helix by freely chosen parameters. Solitons propagate at a consistent rate and retain their original shape. They are widely used in nonlinear models and can be found everywhere in nature. To help in understanding the physical significance of the double-chain deoxyribonucleic acid model, several solutions are shown with graphics in the form of contour, 2D and 3D graphs using computer software Mathematica 13.2. All of the requisite constraint factors that are required for the completed solutions to exist appear to be met. Therefore, our method of strengthening symbolic computations offers a powerful and effective mathematical tool for resolving various moderate nonlinear wave problems. The findings demonstrate the system’s potentially very rich precise wave forms with biological significance. The fundamentals of double-chain deoxyribonucleic acid model diffusion and processing are demonstrated by this work, which marks a substantial development in our knowledge of double-chain deoxyribonucleic acid model movements.

Families of propagating soliton solutions for (3+1)-fractional Wazwaz-BenjaminBona-Mahony equation through a novel modification of modified extended direct algebraic method

The article presents a new modification to the modified Extended Direct Algebraic Method (mEDAM) namely r+mEDAM to effectively and precisely acquire propagating soliton and other travelling wave solutions to the Fractional Wazwaz-Benjamin-Bona-Mahony (FWBBM) equation. By using this updated approach, we are able to find more and new families of propagating soliton solutions for the FWBBM problem, such as soliton, kink, lump-like singular, trigonometric, hyperbolic, periodic, shock, singular & non-singular wave solutions. We also provide 3D and 2D graphs that visually illustrate the obtained solutions. By obtaining accurate propagating soliton solutions, our r+mEDAM proves to be practical while also revealing important details about the dynamics of the equation and suggesting possible applications in the fields of optics, materials research, and water waves.

Dynamical study and diverse soliton collisions for the Kraenkel–Manna–Merle system in ferrites

We investigate the rational solitons of the Kraenkel–Manna–Merle (KMM) system in ferrites via symbolic computation with the ansatz functions technique and logarithmic transformation. The KMM system describes the propagations of the ferromagnetic particles in ferrite materials. Multiwave solitons are reported by using the three-wave technique. Homocentric breathers (HBs) approach is also applied to build new forms of exact solutions. We construct M-shaped solitons and their dynamics are shown via the appropriate values of parameters. Two types of interactions for rational soliton with kink wave are investigated. Additionally, kink cross-rational solitons (KCRSs) and periodic cross-rational solitons (PCRSs) are examined via suitable transformation.

Nonlinear behavior of dust acoustic periodic soliton structures of nonlinear damped modified Korteweg–de Vries equation in dusty plasma

In this article, the nonlinear plasma model known the damped modified Korteweg–de Vries equation under consideration on the base of analytical technique to know the novel behavior of this nonlinear model. We formulated the nonlinear plasma model by utilizing the RP (reductive perturbation) technique with basic set of fluid system which consisted on cold negative charge dust fluid, immobile background neutral charges, positive charge ion fluid, fast (ionized) electrons and thermally electrons. We also constructed various kinds of novel soliton solutions including kink wave solitons, periodic solitary waves, anti-kink wave solitons for the damped modified KdV equation. The determined soliton solutions are novel, interested and did not determined in the previous research. The physical interpretation of demonstrated solutions represent in contour, two-dim and three-dim with computational software. The study prove that our proposed approach is powerful, efficient and can also be fruitful for study of other nonlinear models.

Non-reflective Propagation of Kink Waves in Magnetic-Flux Tubes in the Solar Atmosphere

We study the non-reflective propagation of kink waves in inhomogeneous magnetic-flux tubes. We use the thin-tube and zero-beta plasma approximations. The wave equation with the variable velocity is reduced to the Euler–Poisson–Darboux equation. This equation contains one dimensionless parameter. There are two infinite sequences of this parameter, one monotonically increasing and the other monotonically decreasing, when exact analytical solutions for the Euler–Poisson–Darboux equation can be obtained. For the monotonically increasing sequences the Euler–Poisson–Darboux equation becomes the equation describing spherically symmetric waves in multi-dimensional spaces. The general results are applied to kink-wave propagation in coronal magnetic loops. We consider a coronal magnetic loop of a half-circular shape. We find that for a fixed loop height there is a one-parametric family of dependences of the loop cross-sectional radius on the coordinate along the loop corresponding to the non-reflective kink-wave propagation.

Study of explicit travelling wave solutions of nonlinear (2 + 1)-dimensional Zoomeron model in mathematical physics

Abstract This article studeis the nonlinear (2 + 1)-dimensional Zoomeron equation by utilizing the various prominent analytical approaches namely the unified method and the extended hyperbolic function approach. The analysis in the current paper demonstrates the presence of travelling wave solutions. The applied methods are utilized as powerful tools to investigate and solve the model. The results obtained through these analytical methods reveal insightful patterns in the behavior of the Zoomeron equation. The significance of our work lies in the uniqueness of the methods employed. The two methods are applied to systematically analyze the equation, revealing hidden patterns and structures within its solution space. This leads to the discovery of a collection of solitary wave solutions such as kink waves, singular kink waves, periodic waves and dark soliton using contour plots, 3D and 2D graphics. In this article, we definitely prove that as the free parameters change, the wave amplitude changes as well. It is shown that the applied strategies are more effective and may be implemented to a variety of contemporary nonlinear evolution models emerging in mathematical physics.

Exploration of unexpected optical mixed, singular, periodic and other soliton structure to the complex nonlinear Kuralay-IIA equation

In this research work, we explored novel soliton solutions to the complex nonlinear Kuralay-II (K-IIA) equation on the bases of computational simulation. This complex nonlinear equation is used in many fields such as, optical fibers, ferromagnetic materials and nonlinear optics. We utilized the extended simple equation method on complex nonlinear Kuralay-IIA equation and extracted the unexpected soliton solutions. The extracted soliton solutions having various kinds of physical structure such as bright solitons, periodic wave solitons, kink wave solitons, peakon solitons, dark solitons, anti-kink wave solitons, mixed bright and dark solitons, periodic travelling waves, singular solitons and solitary waves. The graphically analysis of obtained solutions demonstrated with absolute, real and imaginary values of the function visualizing in three dimensional, two dimensional and contour graphics under the numerical simulation by utilizing the Mathematica software. The extracted soliton solutions in this research provide valuable insights into this complex nonlinear equation. The results offer a framework for further analysis, making the solutions effective, easy to apply, and reliable for future complex nonlinear problems development, making them valuable for future. The method used in this work effective, reliable, powerful, easy to apply for other nonlinear partial differential equations.

Bifurcation, chaotic pattern and traveling wave solutions for the fractional Bogoyavlenskii equation with multiplicative noise

This paper presents a new study that incorporates the Stratonovich integral and conformal fractional derivative into the fractional stochastic Bogoyavlenskii equation with multiplicative noise. The study exposes the behavior of the system, including sensitivity, chaos and traveling wave solutions, by using the planar dynamical systems approach. Time series, periodic perturbation, phase portraits, and the Poincaré section are used to comprehensively study the dynamic properties. Notably, the research uses the planar dynamic systems method to build multiple traveling wave solutions, including kink wave, dark soliton, and double periodic solutions. Furthermore, a comparative study approach is applied to investigate the effects of fractional derivative and multiplicative noise on the traveling wave solutions, which demonstrate a significant influence of both variables. This work demonstrates the creative application of the planar dynamic system method to the analysis of nonlinear wave equations, offering insightful information that may be generalized to more complex wave phenomena.

Novel kink and multi wave soliton solutions to the stochastic Phi-4 equation driven by the Weiner process

Novel kink and multi wave soliton solutions to the stochastic Phi-4 equation driven by the Weiner process

Electron-acoustic anti-kink, kink and periodic waves in a collisional superthermal plasma

Electron-acoustic anti-kink, kink and periodic waves in a collisional superthermal plasma

Novel dynamical behaviors of interaction solutions of the new (3+1)-dimensional integrable fourth-order nonlinear equation

The breather, rogue and interaction waves of the (3+1)-dimensional integrable fourth-order nonlinear equation are reviewed. The breather and rouge waves are attained with the aid of the extended homoclinic test. Thereafter, the interaction solutions between a lump wave and a 1-kink or 2-kink soliton are researched. Additionally, four kinds of interaction solutions between lump, kink, and periodic waves through a ” rational-cosh-cos” test function are established. Furthermore, the dynamic attributions of the attained solutions are presented utilizing the graphical analysis.

Novel closed-form analytical solutions and modulation instability spectrum induced by the Salerno equation describing nonlinear discrete electrical lattice via symbolic computation

A nonlinear electrical lattice is an effective experimental tool for dealing with nonlinear dispersive media and creating possibilities for the realistic modeling of electrical soliton. In this work, we discuss the dynamical behavior of the governing (1+1)-dimensional time–space Salerno equation, which describes the discrete electrical lattice with nonlinear dispersion. The different kinds of solutions in the form of periodic, Jacobi elliptic, and exponential functions are obtained by the two analytical approaches, namely the modified generalized exponential rational function method (MGERFM) and the extended sinh-Gordon equation expansion method (shGEEM). The MGERF approach is applied to construct exact solitary wave solutions containing trigonometry, hyperbolic and exponential functions, while the extended shGEE technique is employed to obtain periodic wave solutions containing hyperbolic and Jacobi elliptic functions via performing the symbolic computation in the software package MATHEMATICA. Moreover, by means of standard linear-stability analysis, the modulation instability (MI) and the MI gain spectrum is analytically computed for the considered equation. By selecting suitable parametric values, solution profiles are portrayed in order to make the solutions physically relevant. These results yield periodic waves, kink waves, multi-soliton, and their interactions. By comparing the obtained solutions to the previous findings, we can reveal the novelty of the solutions. In addition, all the obtained solutions were verified by substituting them back into the governing equation using the Mathematica software. The acquired results are significant for the study of wave propagation, signal transmission, and applications of super-transmission phenomena.

Dynamic wave solutions for (2+1)-dimensional DJKM equation in plasma physics

<abstract><p>In this paper, we attempt to obtain exact and novel solutions for Date-Jimbo-Kashiwara-Miwa equation (DJKM) via two different techniques: Lie symmetry analysis and generalized Kudryashov method (GKM). This equation has applications in plasma physics, fluid mechanics, and other fields. The Lie symmetry method is applied to reduce the governing equation to five different ordinary differential equations (ODEs). GKM is used to obtain general and various periodic solutions. These solutions have different behaviors such as kink wave, anti-kink wave, double soliton, and single wave solution. The physical behavior of the solutions was reviewed through 2-D and 3-D graphs.</p></abstract>

Bifurcation, chaos, and stability analysis to the second fractional WBBM model.

This manuscript investigates bifurcation, chaos, and stability analysis for a significant model in the research of shallow water waves, known as the second 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) model. The dynamical system for the above-mentioned nonlinear structure is obtained by employing the Galilean transformation to fulfill the research objectives. Subsequent analysis includes planar dynamic systems techniques to investigate bifurcations, chaos, and sensitivities within the model. Our findings reveal diverse features, including quasi-periodic, periodic, and chaotic motion within the governing nonlinear problem. Additionally, diverse soliton structures, like bright solitons, dark solitons, kink waves, and anti-kink waves, are thoroughly explored through visual illustrations. Interestingly, our results highlight the importance of chaos analysis in understanding complex system dynamics, prediction, and stability. Our techniques' efficiency, conciseness, and effectiveness advance our understanding of this model and suggest broader applications for exploring nonlinear systems. In addition to improving our understanding of shallow water nonlinear dynamics, including waveform features, bifurcation analysis, sensitivity, and stability, this study reveals insights into dynamic properties and wave patterns.

Analysis of bifurcation, chaotic structures, lump and $ M-W $-shape soliton solutions to $ (2+1) $ complex modified Korteweg-de-Vries system

<abstract><p>This research focuses on the fascinating exploration of the $ (2+1) $-dimensional complex modified Korteweg-de Vries (CmKdV) system, exhibiting its complex dynamics and solitary wave solutions. This system is a versatile mathematical model that finds applications in various branches of physics, including fluid dynamics, plasma physics, optics, and nonlinear dynamics. Two newly developed methodologies, namely the auxiliary equation (AE) method and the Hirota bilinear (HB) method, are implemented for the construction of novel solitons in various formats. Numerous novel soliton solutions are synthesised in distinct formats, such as dark, bright, singular, periodic, combo, $ W $-shape, mixed trigonometric, exponential, hyperbolic, and rational, based on the proposed methods. Furthermore, we also find some lump solutions, including the periodic cross rational wave, the homoclinic breather (HB) wave solution, the periodic wave solution, the $ M $-shaped rational wave solution, the $ M $-shaped interaction with one kink wave, and the multiwave solution, which are not documented in the literature. In addition, we employ the Galilean transformation to derive the dynamic framework for the presented equation. Our inquiry includes a wide range of topics, including bifurcations, chaotic flows, and other intriguing dynamic properties. Also, for the physical demonstration of the acquired solutions, 3D, 2D, and contour plots are provided. The resulting structure of the acquired results can enrich the nonlinear dynamical behaviors of the given system and may be useful in many domains, such as mathematical physics and fluid dynamics, as well as demonstrate that the approaches used are effective and worthy of validation.</p></abstract>

Modulation Instability and Dynamical Analysis of New Abundant Closed‐Form Solutions of the Modified Korteweg–de Vries–Zakharov–Kuznetsov Model With Truncated M‐Fractional Derivative

In this work, we study the modulation instability (MI) and closed‐form soliton solution of the modified Korteweg–de Vries–Zakharov–Kuznetsov (mKdV‐ZK) equation with a truncated M‐fractional derivative. The mKdV‐ZK equation can be used to describe the behavior of ion‐acoustic waves in plasma and the propagation of surface waves in deep water with nonlinear and dispersive effects in fluid dynamics. To execute a closed soliton solution, we implement two dominant techniques, namely, the improved F‐expansion scheme and unified solver techniques for the mKdV‐ZK equation. Under the condition of parameters, the obtained solutions exhibit hyperbolic, trigonometric, and rational functions with free parameters. Using the Maple software, we present three‐dimensional (3D) plots with density plots and two‐dimensional (2D) graphical representations for appropriate values of the free parameters. Under the conditions of the numerical values of the free parameters, the obtained closed‐form solutions provided some novel phenomena such as antikink shape wave, dark bell shape, collision of kink and periodic lump wave, periodic wave, collision of antikink and periodic lump wave, collision of linked lump wave with kink shape, periodic lump wave by using improved F‐expansion method and kink shape, diverse type of periodic wave, singular soliton, and bright bell and dark bell‐shape wave phenomena by using unified solver method. The comparative effects of the fractional derivative are illustrated in 2D plots. We also provided a comparison between the results obtained through the suggested scheme and those obtained by other approaches, showing some similar solutions and some that are different. Besides, to check of stability and instability of the solution, the MI analysis of the given system is investigated based on the standard linear stability analysis and the MI gain spectrum analysis. With the use of symbolic calculations, the applied approach is clear, simple, and elementary, as demonstrated by the more broad and novel results that are obtained. It may also be applied to more complex phenomena.

Analytical insights into solitary wave solutions of the fractional Estevez-Mansfield-Clarkson equation

<abstract><p>This study delved into the dynamics of wave solutions within the Estevez-Mansfield-Clarkson equation in fractional nonlinear space-time. Utilizing conformable fractional derivatives, the equation governing shallow water phenomena and fluid dynamics was transformed into a nonlinear ordinary differential equation. Applying the Riccati Bernoulli sub-ODE approach yielded a finite series representation. Notably, our findings revealed novel solitary wave solutions characterized by kink, anti-kink, periodic, and shock functions. Visualized through 3D and contour graphs, kink and periodic waves emerged as distinct observable manifestations. Intriguingly, the diversity of results surpassed previous results, contributing to a deeper understanding of the intricate dynamics inherent in the system.</p></abstract>

Wave dynamics for the new generalized (3+1)-D Painlevé-type nonlinear evolution equation using efficient techniques

<p>In this paper, diverse wave solutions for the newly introduced (3+1)-dimensional Painlevé-type evolution equation were derived using the improved generalized Riccati equation and generalized Kudryashov methods. This equation is now widely used in soliton theory, nonlinear wave theory, and plasma physics to study instabilities and the evolution of plasma waves. Using these methods, combined with wave transformation and homogeneous balancing techniques, we obtained concise and general wave solutions for the Painlevé-type equation. These solutions included rational exponential, trigonometric, and hyperbolic function solutions. Some of the obtained solutions for the Painlevé-type equation were plotted in terms of 3D, 2D, and contour graphs to depict the various exciting wave patterns that can occur. As the value of the amplitude increased in the investigated solutions, we observed the evolution of dark and bright solutions into rogue waves in the forms of Kuztnetsov-Ma breather and Peregrine-like solitons. Other exciting wave patterns observed in this work included the evolution of kink and multiple wave solitons at different time levels. We believe that the solutions obtained in this paper were concise and more general than existing ones and will be of great use in the study of solitons, nonlinear waves, and plasma physics.</p>