Kink Waves Research Articles

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.

Fast magnetohydrodynamic oscillations of a coronal loop embedded in a potential coronal arcade

Observations indicate variable widths exhibited by fan coronal loops and flare loops that tend to widen towards the apex. Short-period, quasi-periodic pulsations in solar flares are often interpreted in terms of the fast-sausage oscillations of flare loops and the collective vertical vibrations of arcade loops are attributed with the vertical kink mode. Both phenomena are used as a seismological tool to estimate the physical parameters in the corona. We performed an analytical study of fast sausage and kink oscillations in coronal loops, given the effects of loop curvature, expansion, and Alfv \'e n speed variation. We modelled a coronal loop as a dense expanding curved magnetic slab embedded within a potential coronal arcade, using a zero-beta plasma limit. We obtained the dispersion relation that governs fast waves in the model and studied it both numerically and analytically. The effects of loop expansion and variable Alfv \'e n speed reduce the cut-off frequency and increase the cut-off wavenumbers for fast sausage and kink waves. Moreover, the principal vertical kink mode has a cut-off and strongly attenuates in the leaky regime. The frequency increase is found to be minor for the global sausage mode both in the trapped and leaky regimes, with a frequency shift within a few percent. We found that in our model, where the Alfv \'e n speed increases from the footpoints to the loop top, the spatial profile of the longitudinal fundamental is broadened and the antinodes of the first overtone are shifted towards the footpoints. Using the classical expression for the cut-off wavenumber of the global sausage mode in a straight waveguide results in an underestimation of the density contrast constraint in flare loops. Instead, the suggested formula accounting for variations in loop widths provides more accurate results. The frequency of the global sausage mode can be correctly determined with the straight slab model.

Study of Fifth-Order Nonlinear Caudrey–Dodd–Gibbon Model for Multiwave, M-Shaped Solitons, Lumps, Generalized Breathers, Manifold Periodic, Rogue Wave and Kink Wave Interactions

Numerous novel wave structures, such as lump, lump with one-kink, lump with two-kink soliton, generalized breather, Ma breather, Kuznetsov-Ma breather, Akhmediev breather, manifold periodic, and rogue wave solutions to [Formula: see text]-dimensional fifth-order nonlinear Caudrey–Dodd–Gibbon (FNCDG) model via symbolic computation are studied based on the ansatz function scheme. Studying the lump-type solution involves selecting the function as the generic quadratic polynomial function. The lump with one- and two-kink solutions is created by mixing a quadratic function with one or two exponential functions. By adding hyperbolic function with a quadratic function, the rogue wave solutions are manipulated. In addition, the multiwave, [Formula: see text]-shaped rational solitons and their interactions with kink waves are analytically evaluated. Furthermore, different 3D, 2D, and contour profiles are created in different dimensions by changing the parameter values.

New periodic solutions and solitary wave solutions for the time-fractional differential equations

In this paper, we obtain many different types of exact solutions to the time-fractional Klein–Gordon equation and the time-fractional generalized Hirota-Satsuma coupled KdV system by using the modified rational function approach. Some new solutions such as the kink-periodic solution, the anti-kink-periodic solution and the concave-convex-periodic solution are constructed. Furthermore, the kink and the singular kink waves, the bell shaped soliton and the singular soliton solutions of the two equations also are found. Some numerical simulations are presented, these works can effectively reflect the propagation phenomena of time-fractional nonlinear systems, and also enable us to understand time-fractional nonlinear physical phenomena more clearly.

Kink Wave Phenomena in the Nonlinear Partial Differential Equation Representing the Transmission Line Model of Microtubules for Nanoionic Currents

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps.

Generation of Solitary Waves with Analytical Solution for The (3+1)-dimensional pKP-BKP Equation and Reductions

In this study, new solitary wave solutions are obtained for the combination of the B-type Kadomtsev-Petviashvili (BKP) equation and the potential Kadomtsev-Petviashvili (pKP) equation, called the integrable (3+1)-dimensional coupled pKP-BKP equation, and its two reduced forms. For this purpose, the Bernoulli auxiliary equation method, which is an ansatz-based method, is used. As a result, kink, lump, bright soliton and breather wave solutions are observed. It is concluded that this method and the results obtained for the considered pKP -BKP equations are an important step for further studies in this field.

Solitary wave solutions and their limits to the fractional Schrödinger system

This study is concerned with solitary wave solutions and the dynamic behavior of the (2+1)-dimensional nonlinear fractional Schrödinger system. By exploring the dynamic properties of the equilibrium levels to the corresponding Hamiltonian, the expressions of exact solutions of the above system are obtained, including solitary wave solutions, periodic wave solutions, singular periodic wave solutions, singular wave solutions, kink wave solutions, and anti-kink wave solutions. Moreover, the linear stability, geometric characteristics, and limiting behavior of these solutions to the nonlinear fractional Schrödinger system were investigated.

Role of Cairns–Gurevich ion distribution on nonlinear wave propagation in negatively charged dusty plasma

The study of dust-acoustic (DA) nonlinear electrostatic waves in dusty plasma is crucial for understanding plasma behavior in both astrophysical and laboratory environments. Dusty plasmas, characterized by the presence of negatively charged dust particles, exhibit complex dynamics influenced by various factors, including nonthermal electron and ion populations following Cairns–Gurevich distributions. This study addresses the problem of understanding wave propagation under these specific conditions by employing a set of fluid hydrodynamic equations for dust fluid, alongside appropriate electron and Cairns–Gurevich ion distributions, to model the dynamics and propagation of DA nonlinear electrostatic waves under specific plasma conditions with negatively charged dust particles. We derived a nonlinear Zakharov–Kuznetsov (ZK) equation to model the evolution of these waves and further transformed it into the nonlinear Schrödinger (NLS) equation to analyze rogue wave formation and identify unstable and stable zones within the plasma. We focused our analysis on the effects of key parameters, such as the nonthermal index, magnetic field strength, negative dust temperature ratio, polarization force, and density ratio, on the behavior of dust-acoustic waves (DAWs). The results demonstrated that soliton waves, explosive waves, and kink waves exhibit distinct behaviors depending on these parameters and the spatial context of Earth's magnetosphere. These findings provide new insights compared to previous studies into the conditions under which these waveforms emerge and evolve, especially in magnetized environments where earlier models were less effective at accurately characterizing or predicting wave behavior. By applying two different analytical methods, a direct integration approach and a generalized Kudryashov method, we expanded our understanding of nonlinear wave phenomena in dusty plasmas. Our results not only advance theoretical knowledge but also have practical implications for interpreting space plasma observations and guiding experimental design in plasma physics research. This study thus contributes to a more comprehensive understanding of plasma dynamics, with potential applications to astrophysical observation and laboratory plasma experimentation.

M-shaped rational, homoclinic breather, kink-cross rational, multi-wave and interactional soliton solutions to the fifth-order Sawada-Kotera equation

In this work, we develop multi-wave, homoclinic breathers, M-shaped rational, 1-kink interactions with M-shaped, periodic-cross rational and kink-cross rational solutions for the fifth-order Sawada-Kotera equation, which represents the motion of long waves under gravity in shallow water, using several ansatz transformations. A three-wave approach is used to identify multi-wave solitons. Additionally, novel forms of exact solutions are constructed using the homoclinic breathers technique. The kink-cross rational and periodic-cross rational solitons are investigated using appropriate transformations. We develop M-shaped solitons and demonstrate their behavior by selecting suitable parameter values. Furthermore, the interactions between kink waves and M-shaped solitons are also studied. The obtained solutions are determined in 3D, 2D, and contour profiles, where the free parameters involved in the solutions are assigned specific values. Their physical relevance is explored to highlight the inner context of tangible incidents in the natural domain. Since these recently discovered solutions contain a few arbitrary constants, they can be used to explain the variation in qualitative characteristics of wave phenomena.

Soliton-based modeling of nano-ionic currents in transmission line

Many nonlinear evolution equations, such as the nano-ionic currents (NIC) equation, are used extensively in many scientific and technological domains particularly in nanoelectronics and bioelectronics. The mathematical modeling of NIC phenomena is vital for understanding their behavior and optimizing device performance. Our research leverages an array of mathematical methods, including multi-wave analysis, periodic wave solutions, lump soliton dynamics, breather wave phenomena, homoclinic breathers, M-shaped waveforms, and rogue wave analysis. Additionally, our investigation encompasses the exploration of single kink and double kink configurations, interactions between periodic and kink waves, interaction between M shaped with kink and rogue, interaction between M shaped with one kink, interaction between M shaped with kink and periodic, interaction between M shaped with two kinks as well as periodic wave interactions with lump waves. To further emphasize the structure of solutions derived from particular parameter choices, we include three-dimensional, two-dimensional, streamplot, and contour graphs.

Propagating kink waves in an open coronal magnetic flux tube with gravitational stratification: Magnetohydrodynamic simulation and forward modelling

Context. In coronal open-field regions, such as coronal holes, there are many transverse waves propagating along magnetic flux tubes, which are generally interpreted as kink waves. Previous studies have highlighted their potential role in coronal heating, solar wind acceleration, and seismological diagnostics of various physical parameters. Aims. This study aims to investigate propagating kink waves, considering both vertical and horizontal density inhomogeneity, using 3D magnetohydrodynamic (MHD) simulations. Methods. We established a 3D MHD model of a gravitationally stratified open flux tube, incorporating a velocity driver at the lower boundary to excite propagating kink waves. Forward modelling was conducted to synthesise observational signatures of the Fe IX 17.1 nm line. Results. Resonant absorption and density stratification both affect the wave amplitude. When diagnosing the relative density profile with velocity amplitude, resonant damping needs to be properly considered to avoid a possible underestimation. In addition, unlike standing modes, propagating waves are believed to be Kelvin-Helmholtz stable. In the presence of vertical stratification, however, the phase mixing of transverse motions around the tube boundary can still induce small-scale structures, partially dissipating wave energy and leading to a temperature increase, especially at higher altitudes. Moreover, we conducted forward modeling to synthesise observational signatures, which revealed the promising potential of future coronal imaging spectrometers such as MUSE in resolving these wave-induced signatures. Also, the synthesised intensity signals exhibit apparent periodic variations, offering a potential method for indirectly observing propagating kink waves with current extreme ultraviolet imagers.

Investigation of Analytical Soliton Solutions to the Non-Linear Klein–Gordon Model Using Efficient Techniques

Nonlinear distinct models have wide applications in various fields of science and engineering. The present research uses the mapping and generalized Riccati equation mapping methods to address the exact solutions for the nonlinear Klein–Gordon equation. First, the travelling wave transform is used to create an ordinary differential equation form for the nonlinear partial differential equation. This work presents the construction of novel trigonometric, hyperbolic and Jacobi elliptic functions to the nonlinear Klein–Gordon equation using the mapping and generalized Riccati equation mapping methods. In the fields of fluid motion, plasma science, and classical physics the nonlinear Klein–Gordon equation is frequently used to identify of a wide range of interesting physical occurrences. It is considered that the obtained results have not been established in prior study via these methods. To fully evaluate the wave character of the solutions, a number of typical wave profiles are presented, including bell-shaped wave, anti-bell shaped wave, W-shaped wave, continuous periodic wave, while kink wave, smooth kink wave, anti-peakon wave, V-shaped wave and flat wave solitons. Several 2D, 3D and contour plots are produced by taking precise values of parameters in order to improve the physical description of solutions. It is noteworthy that the suggested techniques for solving nonlinear partial differential equations are capable, reliable, and captivating analytical instruments.

Bifurcation analysis and solitary wave solution of fractional longitudinal wave equation in magneto-electro-elastic (MEE) circular rod

In this study, we investigate the longitudinal wave equation (LWE) with the M−fractional derivative, which describes the propagation of longitudinal waves along a rod while incorporating interactions between mechanical, electrical, and magnetic fields within the material. Initially, we apply bifurcation analysis to examine the critical points or phase portraits where the system transitions to new behaviors, such as stability shifts or the emergence of chaos, and observe the mechanism of static soliton formation through a saddle-node bifurcation. Subsequently, we utilize a modified simple equation (MSE) technique to find solitary wave solutions. Depending on the relationships between free parameters, the solutions are expressed as hyperbolic, trigonometric, and exponential functions. The numerical form of the obtained solutions reveals complex phenomena, including dark and bright bell waves, kink periodic lump waves, kink periodic waves, periodic lump waves, interaction waves between kink and periodic waves, linked lump waves, and interactions of periodic and lump waves. Additionally, we compare our results with previously published work, demonstrating that the discussed methods are valuable tools for providing distinct, accurate soliton solutions relevant to nonlinear science and technology applications.

Multi-lump, resonant Y-shape soliton, complex multi kink solitons and the solitary wave solutions to the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation for incompressible fluid

The major contribution in this paper is to inquire into some new exact solutions to the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation (BLMPE) which plays a major role in area of the incompressible fluid. Taking advantage of the Cole-Hopf transform, we extract its bilinear form. Then two different kinds of the multi-lump solutions are probed by applying the new homoclinic approach. Secondly, the Y-shape soliton solutions are explored via assigning the resonance conditions to the N-soliton solutions. Additionally, the complex multi kink soliton solutions (CMKSSs) are investigated through the Hirota bilinear method. Lastly, some other wave solutions including the kink and anti-kink solitary wave solutions are developed with the aid of two efficacious approaches, namely the variational method and Kudryashov method. In the meantime, the profiles of the accomplished solutions are displayed graphically via Maple.

Resonant Y-type soliton, interaction wave and other wave solutions to the (3+1)-dimensional shallow water wave equation

The current study aims to develop some novel wave solutions of the (3+1)-dimensional shallow water wave equation (SWWE). First, the Y-type soliton (YTS) solutions are developed via imposing the certain resonant condition to the N-solitons solutions (NSSs) that extracted by the Hirota bilinear approach. Then, the symbolic computation and ansatz function scheme are employed to explore the interaction wave solutions. Eventually, the sub-equation approach is utilized to find the diverse wave solutions which include the kink wave, singular wave and the singular periodic wave solutions. The dynamics of the obtained solutions are presented graphically for revealing the nonlinear physical characteristics. The investigative solutions in this article have not been probed elsewhere and can help us comprehend the nonlinear behaviors of the (3+1)-dimensional SWWE better.

Abundant Soliton Solutions to the Generalized Reaction Duffing Model and Their Applications

The main aim of this study is to obtain soliton solutions of the generalized reaction Duffing model, which is a generalization for a collection of prominent models describing various key phenomena in science and engineering. The equation models the motion of a damped oscillator with a more complex potential than in basic harmonic motion. Two effective techniques, the mapping method and Bernoulli sub-ODE technique, are used for the first time to obtain the soliton solutions of the proposed model. Initially, the traveling wave transform, which comes from Lie symmetry infinitesimals, is applied, and a nonlinear ordinary differential equation form is derived. These approaches effectively retrieve a hyperbolic, Jacobi function as well as trigonometric solutions while the appropriate conditions are applied to the parameters. Numerous innovative solutions, including the kink wave, anti-kink wave, bell shape, anti-bell shape, W-shape, bright, dark and singular shape soliton solutions, were produced via the mapping and Bernoulli sub-ODE approaches. The research includes comprehensive 2D and 3D graphical representations of the solutions, facilitating a better understanding of their physical attributes and proving the effectiveness of the proposed methods in solving complex nonlinear equations. It is important to note that the proposed methods are competent, credible and interesting analytical tools for solving nonlinear partial differential equations.

Dynamical Behaviors and Abundant New Soliton Solutions of Two Nonlinear PDEs via an Efficient Expansion Method in Industrial Engineering

In this study, we discuss the dynamical behaviors and extract new interesting wave soliton solutions of the two significant well-known nonlinear partial differential equations (NPDEs), namely, the Korteweg–de Vries equation (KdVE) and the Jaulent–Miodek hierarchy equation (JMHE). This investigation has applications in pattern recognition, fluid dynamics, neural networks, mechanical systems, ecological systems, control theory, economic systems, bifurcation analysis, and chaotic phenomena. In addition, bifurcation analysis and the chaotic behavior of the KdVE and JMHE are the main issues of the present research. As a result, in this study, we obtain very effective advanced exact traveling wave solutions with the aid of the proposed mathematical method, and the solutions involve rational functions, hyperbolic functions, and trigonometric functions that play a vital role in illustrating and developing the models involving the KdVE and the JMHE. These new exact wave solutions lead to utilizing real problems and give an advanced explanation of our mentioned mathematical models that we did not yet have. Some of the attained solutions of the two equations are graphically displayed with 3D, 2D, and contour panels of different shapes, like periodic, singular periodic, kink, anti-kink, bell, anti-bell, soliton, and singular soliton wave solutions. The solutions obtained in this study of our considered equations can lead to the acceptance of our proposed method, effectively utilized to investigate the solutions for the mathematical models of various important complex problems in natural science and engineering.

Exact solutions of the (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-dimensional Zoomeron model arising in nonlinear optics via mapping method

The Zoomeron model covers particular kinds of solitons with distinctive properties that appear in several physical scenarios, such as, fluid dynamics, nonlinear optics and laser physics. First time utilising the mapping method, we determine the analytical solution to the described model, including several novel dynamical behaviours. Through symbolic computation, we are able to derive the breather waves, kink waves, dark soliton, singular soliton, periodic soliton and bright soliton of this model. Additionally, we encounter single kink waves and single breather waves. We find novel hyperbolic trigonometric, rational and elliptic functions. Modelling our observations with MATLAB tools and producing many 3D graphs. The results obtained will be crucial for further research on complicated nonlinear models.

Solitonic solutions and stability analysis of Benjamin Bona Mahony Burger equation using two versatile techniques

This study aims to explore the precise resolution of the nonlinear Benjamin Bona Mahony Burgers (BBMB) equation, which finds application in a variety of nonlinear scientific disciplines including fluid dynamics, shock generation, wave transmission, and soliton theory. Within this paper, we employ two versatile methodologies, specifically the extended exp(-Ψ(χ))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (-\\Psi (\\chi ))$$\\end{document} expansion technique and the novel Kudryashov method, to identify the exact soliton solutions of the nonlinear BBMB equation. The solutions we discovered involve trigonometric functions, hyperbolic functions, and rational functions. The uniqueness of this research lies in uncovering the bright soliton, kink wave solution, and periodic wave solution, and conducting stability analysis. Furthermore, the solutions’ graphical characteristics were explored through the utilization of the mathematical software Maple 2022 (https://maplesoft.com/downloads/selectplatform.aspx?hash=61ab59890f2313b2241fde3423fd975e). The system’s physical interpretation is defined through various types of graphs, including contour graphs, 3D-surface graphs, and line graphs, which use appropriate parameter values. These recommended techniques hold significant importance and are applicable in diverse nonlinear evolutionary equations found in the field of nonlinear sciences for illustrating nonlinear physical models.

Kinks and soliton solutions to the coupled Burgers equation by Lie symmetry approach

The current research employs a novel class of invariant solutions to Painlevé integrable coupled Burgers equations. Many mathematical physics domains such as fluid dynamics, traffic flow, nonlinear acoustics, turbulence phenomena, and the interaction of convection and diffusion processes, use this fundamental model. The presented investigations utilize the Lie point symmetry to yield a class of exact solutions unknown in previous findings. Lie point symmetry reduces the number of independent variables in coupled Burgers equations. For the physical visualizations of the solutions, their profiles are analysed. Since arbitrary functions and constants are available in the solutions, the derived solutions have the potential to reveal rich physical structures. We next go over kink waves, multisoliton, line multisoliton and annihilation profiles in detail. We compute conserved vectors to demonstrate the integrability of CBEs. The results demonstrate their novelty, as they diverge completely from previous findings.