Anti-kink Wave Research Articles

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.

EXACT SOLUTIONS AND BIFURCATION OF A MODIFIED GENERALIZED MULTIDIMENSIONAL FRACTIONAL KADOMTSEV–PETVIASHVILI EQUATION

In this paper, we investigate the exact solutions of a modified generalized multidimensional fractional Kadomtsev–Petviashvili (KP) equation by the bifurcation method. First, the equation is converted into a planar dynamical system through fractional complex wave transformation. The phase portraits of the equation and qualitative analysis are presented under different bifurcation conditions. Then, the bounded and unbounded traveling wave solutions, including periodic, kink, anti-kink, dark-solitary, bright-solitary and breaking wave solutions, are acquired by integrating along different orbits. Finally, numerical simulations of the dynamic behaviors of the solutions obtained are graphically illustrated by choosing appropriate parameters.

Analysis and simulation of vehicle following behavior with consideration of multiple time delays

With the application of intelligent transportation system, drivers can receive more traffic status information in the process of driving, but they may have multiple time delays when they are presented with different information. As a result, to further research the influence of multiple time delays on vehicle following behavior, this study comes up with a modified car following model, which takes into account multiple time delays in sensing both distance and velocity information, based on the backward looking and velocity difference model. The linear stability theory is used to explore the influence of multiple time delays on the stability of the extended model. Different nonlinear equations are derived using nonlinear theory, which include Burgers, Korteweg–de Vries, and Modified Korteweg–de Vries equations and are used to depict triangular wave, soliton wave and kink–antikink wave corresponding to stable, metastable and unstable regions respectively. In the end, numerical simulation is used to further study the improved model to depict the dynamic evolution of traffic stream. The results display that the improvement of time delay in sensing distance can induce traffic congestion, conversely, the time delay in sensing velocity offsets the negative effect of time delay in sensing distance. The theoretical results and numerical simulation results are consistent with each other.

Soliton solutions of a (2+1)-dimensional nonlinear time-fractional Bogoyavlenskii equation model

In this analysis, we propose a mathematical approach named the extended (ℵ, ℜ) expansion scheme to integrate nonlinear fractional and classical evolution models. We utilize the technique to the time fractional Bogoyavlenskii equation, which signifies the (2 + 1)-dimensional interaction of propagating Riemann waves along a definite axis and a wave normal to it. Additionally, we integrate the model using the new extended (G′/G) expansion scheme. Consequently, we derive exact wave solutions, including singular and multiple periodic soliton solutions, kink waves, anti-kink waves, bell-shaped waves, lump waves, rogue waves, periodic lump waves, and interactions between lump and kink wave profiles. The properties of the fractional parameter on the achieved outcomes are also analyzed. We have created 3D plots, 3D plots with contour lines, and 2D plots of our attained solutions using the computational software, Maple. These systems can also represent various solutions for other fractional models in the domains of nonlinear science and engineering.

Further physical research about soliton structures and phase portraits in nonlinear fractional electrical transmission line model

This paper presents several novel contributions in the field of nonlinear fractional low-pass electrical transmission line model (NFLETLM). Firstly, using the modified (G′G2)-expansion method and the extended modified Jacobi elliptic expansion method, we discovered new and accurate solutions for NFLETLM, which have not been reported in the existing literature (Tala-Tcbuc et al. 2014; Nuruzzaman et al. 2021). These solutions, denoted as U3,U4,U7,U8,U9,U10,U13,U14 and UJ1,UJ2,UJ3,UJ4 represent novel contributions to the fields. Secondly, by utilizing computer simulations, we observed various intriguing phenomena in the wave solution graphs, such as anti-kink waves, periodic waves, intense singular periodic waves, bright singular wave solutions, multi-periodic waves, intense double periodic waves, and alternating patterns of light and shade waves. These findings shed light on previously unexplored aspects of the problem. Thirdly, through an extensive study on the newly discovered solutions, we provided a comprehensive understanding of the solitons inherent in the NFLETLM, including comparison with other derivative solutions. Our conformable fractional derivative solutions showed similarities to beta derivative solutions, distinguishing them from Riemann–Liouville derivative solutions. Lastly, we explored the phase portrait, bifurcation analysis, sensitivity, and potential chaotic behaviors of the NFLETLM, which have not been addressed in previous works (Tala-Tcbuc et al. 2014; Nuruzzaman et al. 2021). This innovative contribution expands our understanding of the NFLETLM model and uncovers new dynamics and phenomena.

Similarity reductions and new exact solutions for (3+1)-dimensional B–B equation

The main objective of this investigation is to obtain new closed form solutions for the (3+1)-dimensional Bateman–Burgers equation (B–B equation) by the symmetry method and developed Exp-function method (DEFM). The symmetry method is utilized to infer the infinitesimal generators and build the commutation relations. The similarity reductions of the governing partial differential equations are obtained. The governing partial differential equations are reduced to different ordinary differential equations. The ordinary differential equations are solved by using DEFM. The solutions are analyzed numerically to show the behaviors of the solutions. Solution behaviors include the solitary wave solution, the singular wave solution, the kink wave solution, and the anti-kink wave solution.

Abundant time-wavering solutions of a modified regularized long wave model using the EMSE technique

This research article presents the application of the enhanced modified simple equation integral technique to obtain abundant time-wavering solutions of the modified regularized long wave (RLW) model. The model is known for its significance in exploring wave phenomena in various fields, including shallow water dynamics, pressure waves in liquids and gas bubbles, nonlinear transverse waves in magneto-hydrodynamics, and ion-acoustic waves in plasma. By implementing the enhanced modified simple equation integral technique, we construct arbitrary time-varying wave solutions for the model, resulting in a diverse range of solitonic waveforms. These solutions include kink waves, anti-kink waves, bright and dark bell waves, double periodic waves, and combinations of solitons and periodic waves. The obtained solutions are visually presented through 2D and 3D density and contour plots. Our findings demonstrate the effectiveness and potential of the enhanced modified simple equation integral technique in identifying innovative time-varying soliton solutions within the modified RLW model.

The Painlevé integrability and abundant analytical solutions for the potential Kadomtsev–Petviashvili (pKP) type coupled system with variable coefficients arising in nonlinear physics

Acknowledging the inhomogeneities of media and non-uniformity of boundaries, the nonlinear evolution equations with variable coefficients may show more realistic situations dealing with time-varying environment, inhomogeneous medium, etc. In this order, this article deals with the variable-coefficients (2+1)-dimensional potential Kadomtsev–Petviashvili (pKP) type coupled equations. The Painlevé analysis approach is being used to assess the integrability of the governing model. The Laurent series, which appears when using the Painlevé analysis technique, has been truncated in order to build an auto-Bäcklund transformation (ABT). Three different analytical solutions in the form of hyperbolic, exponential, periodic and rational have been evaluated using the ABT approach. The multi-soliton solutions are also derived by using Hirota bilinear method for the aforesaid equation. All the solutions are displayed as 3D plots in which the variable coefficients and parameters can be varied to see the effects. In these plots, one can see the many different behaviors of the examined coupled system, such as periodic waves, kink-soliton, anti-kink soliton, kink anti-kink wave, complex periodic waves, complex kink anti-kink wave, and multi-soliton surfaces.

Heterogeneous lattice hydrodynamic model and jamming transition mixed with connected vehicles and human-driven vehicles

As communication and perception technologies develop, connected vehicles (CVs) advance rapidly. In the process of popularization of CVs, it is bound to lead to the coexistence of CVs and human-driven vehicles (HDVs) on the road network for some time to come, and the previous study assumes that road vehicle is homogeneous, which could not quantify the difference between mixed vehicular flow, based on this, we consider the differences between CAVs and CVs in the way of information acquisition, a heterogeneous lattice hydrodynamics model mixing the HDVs and CVs are presented. Subsequently, by using the small perturbation method, we analyze the linear stability of the proposed model and derive the corresponding stability criteria; when the above stability norm does not hold, in order to investigate the nonlinear phenomenon, nonlinear stability analysis is performed and a modified Korteweg–de Vries equation (mKdV) corresponding to the proposed model is derived. By solving the mKdV equation, we obtain the kink - antikink solitary wave solution which can be used to explain how traffic jams form and propagate. At last, several numerical simulations were conducted to test the impact of the penetration rate of CVs and the bi-directional visual field for both forward-looking and backward-looking on the heterogeneous traffic flow stability, i.e., the first two are positive, while the last is negligible, this conclusion is consistent with the conclusion of theoretical derivation.

Novel dynamics of the Zoomeron model via different analytical methods

We apply the novel Kudryashov scheme, the 1G′ approach, and the G′G′+G+A technique to handle the Zoomeron model for the first time, which yields several kinds of soliton solutions with some novel dynamic properties. The proposed model handles specific incidents of soliton structures with distinctive characteristics that arise in laser sciences, fluid mechanics, and nonlinear optics. We observe dark soliton, bright soliton, periodic waves, kink waves, anti-kink waves, and breather waves for the mentioned equation by symbolic calculation. The suggested model also produces lump-type breather waves. Density, 3D, and 2D images are used to exhibit the dynamics of the adopted outcomes. The results will be crucial for future research on higher-order nonlinear models.

Diversity of soliton solutions to the (3 + 1)-Dimensional Wazwaz-Benjamin-Bona-Mahony equations arising in mathematical physics

The current study scrutinizes exact singular solitary, kink, anti-kink, bell-shaped, and periodic traveling wave solutions for the newly implemented 3-dimensional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations family using a modified Exp-function approach. In spite of the presence of several trigonometric, hyperbolic, and rational functions, some new exact solitary, hyperbolic, and periodic traveling wave solutions to the considered equations are obtained by putting the modified Exp-function method into practice using the computer program Maple. The arrangement of the functions tanh, sech2, tan, and sec2 expresses the specific solutions produced by the defined procedure. The information was obtained to evaluate how well the suggested method could compute the exact solutions of the WBBM equations that could be applied to nonlinear water model applications in the ocean and coastal engineering. To show the physical composition and properties of the discovered solitons, contour, three-, and two-dimensional graphs are plotted using the computer program Mathematica. The acquired results imply that the proposed method can be applied to find various, enhanced, useful, and compatible solutions for other important nonlinear partial differential equations. All of the solutions given have been qualified by swapping the respective equations with those generated via the computer program Maple. Furthermore, we compared our obtained results with the solutions already put forward in the literature.

Bifurcations and Exact Bounded Solutions of Some Traveling Wave Systems Determined by Integrable Nonlinear Oscillators withq-Degree Damping

For a class of nonlinear diffusion–convection–reaction equations, the corresponding traveling wave systems are well-known nonlinear oscillation type of systems. Under some parameter conditions, the first integrals of these nonlinear oscillators can be obtained. In this paper, the bifurcations, exact solutions and dynamical behavior of these nonlinear oscillators are studied by using methods of dynamical systems. Under some parametric conditions, exact explicit parametric representations of the monotonic and nonmonotonic kink and anti-kink wave solutions, as well as limit cycles, are obtained. Most important and interestingly, a new global bifurcation phenomenon of limit bifurcation is found: as a key parameter is varied, so that singular points (except the origin) disappear, a planar dynamical system can create a stable limit cycle.

Numerical Investigation of Time-Fractional Phi-Four Equation via Novel Transform

This paper examines two methods for solving the nonlinear fractional Phi-four problem with variable coefficients. One of the distinct states of the Klein–Gordon model yields the Phi-four equation. It is also used to simulate the kink and anti-kink solitary wave connections that have recently emerged in biological systems and nuclear particle physics. The approaches that are being suggested consist of the Yang transform, the homotopy perturbation approach, the decomposition approach, and the fractional derivative as stated by Caputo. The advantages of the proposed techniques are their capability of combining two dominant approaches for attaining precise and approximate solutions of nonlinear equations. It is important to keep in mind that the suggested methods can perform better in general as they need less computational effort than the alternative methods, while keeping a high level of numerical precision. The actual and estimated outcomes are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed approaches. Additionally, several simulations are used to show the physical behaviors of the found solutions with regard to fractional order. The article’s results possess complimentary properties that relate to the symmetry of partial differential equations.

Bifurcation of Some Novel Wave Solutions for Modified Nonlinear Schrödinger Equation with Time M-Fractional Derivative

In this paper, we investigate the time M-fractional modified nonlinear Schrödinger equation that describes the propagation of rogue waves in deep water. Periodic, solitary, and kink (or anti-kink) wave solutions are discussed using the bifurcation theory for planar integrable systems. Some new wave solutions are constructed using the first integral for the traveling wave system. The degeneracy of the obtained solutions is investigated by using the transition between orbits. We visually explore some of the solutions using graphical representations for different values of the fractional order.

Bifurcations and the Exact Solutions of the Time-Space Fractional Complex Ginzburg-Landau Equation with Parabolic Law Nonlinearity

This paper studies the bifurcations of the exact solutions for the time–space fractional complex Ginzburg–Landau equation with parabolic law nonlinearity. Interestingly, for different parameters, there are different kinds of first integrals for the corresponding traveling wave systems. Using the method of dynamical systems, which is different from the previous works, we obtain the phase portraits of the the corresponding traveling wave systems. In addition, we derive the exact parametric representations of solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions, peakon solutions, periodic peakon solutions and compacton solutions under different parameter conditions.

Persistence of kink and anti-kink wave solutions for the perturbed double sine-Gordon equation

Based on the geometric singular perturbation theory and the Melnikov method, we study the persistence of kink and anti-kink wave solutions for the perturbed double sine-Gordon equation. The explicit expression of the Melnikov function is given. Moreover, the monotonicity of the period function for unperturbed double sine-Gordon equation is investigated.

The applications of symbolic computation to exact wave solutions of two HSI-like equations in (2+1)-dimensional

It is renowned that Hirota–Satsuma–Ito (HSI) equation is widely used to study wave dynamics of shallow water. In this work, two new HSI-like equations are investigated which could be extended to diversify problems in natural phenomena and give admirable contributions by applying the generalized exponential rational function method (GERFM). With the aid of symbolic calculations, various constraints on the free parameters are given, while classes of wave solutions are explicitly constructed from the coefficients of the combined non-linear and dissipative terms. After specifying values for free parameters, singular, periodic singular and anti-kink waves are demonstrated in 3D figures to exhibit different kinds of wave propagations. The fact that parameters directly influence the wave amplitude and speed of traveling waves is illustrated. The derived results are innovative and have important applications in the current field of mathematical physics research. Eventually, we show that generalized exponential rational function method is effective and straightforward to solve higher-order and high-dimensional non-linear evolution equations.

Lump and soliton on certain spatially-varying backgrounds for an integrable (3+1) dimensional fifth-order nonlinear oceanic wave model

The dynamics of nonlinear waves with controllable dispersion, nonlinearities, and background continues to be an exciting line of active research in recent years. In this work, we focus to investigate an integrable (3+1)-dimensional nonlinear model describing the evolution of water waves with higher-order temporal dispersion by characterizing the dynamics of lump and soliton waves on different spatially-varying backgrounds. Particularly, we construct some explicit lump wave and kink-soliton solutions for the considered model through its trilinear and bilinear equations with appropriate forms of the polynomial and exponential type initial seed solutions, respectively. Additionally, we obtain hyperbolic and periodic waves through bilinear Bäcklund transformation. We explore their propagation and transformation/modulation dynamics due to different spatial backgrounds through categorical analysis and clear graphical demonstrations for a complete understanding of the resulting solutions. Our analysis shows that the lump solution results into a simultaneous existence of well-localized spike and declining (coupled bright-dark wave) structures on a constant background, while the soliton solution exhibits kink and anti-kink wave patterns along different spatio-temporal domains with controllable properties. When the arbitrary spatial background is incorporated appropriately, we can witness the manifestation of soliton into manipulated periodic and interacting dynamical structures. On the other hand, the lump wave results into the coexistence, interacting wave and breather formation due to periodic and localized type arbitrary spatial backgrounds. The present results will be an important addition to the context of engineering nonlinear waves with non-vanishing controllable backgrounds in higher-dimensional models. Further, the present study can be extended to investigate several other nonlinear systems to understand the physical insights of the variable background in their dynamics.

Bifurcations and exact traveling wave solutions of the (n+1)-dimensional q-deformed double sinh-Gordon equations

<p style='text-indent:20px;'>We consider an (n+1)-dimensional q-deformed double sinh-Gordon equation. The dynamical system approach is employed to study the model problem, from which we are able to obtain all possible bounded solutions, such as solitary wave solutions, kink and anti-kink wave solutions, and periodic wave solutions under different parameter conditions. Eleven exact parametric representations are provided explicitly.</p>