Anti-kink Solitary Waves Research Articles

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.

Construction of new wave structures and stability analysis for the nonlinear Klein-Gordon equation

In this study, the nonlinear Klein–Gordon equation the relativistic equivalents of the nonlinear Schrödinger equations is presented which characterizes the connection between relativistic energy-momentum in a quantized manner. A variety of exact solutions are developed by employing the extended Fan sub-equation approach and the Sardar sub-equation method. The study includes three dimensional surface plots of some derived solutions like dark and bright solitary waves, kink solitary waves, anti-kink solitary waves, periodic solitary waves and hyperbolic functions, with several solutions being novel. Furthermore, the stability analysis of the observed solutions is also established to validate the scientific computations. The 3D, 2D and contour visualizations of the wave dynamics are demonstrated using Mathematica for a suitable choice of parameters. The applied strategies are recognized as a significant mathematical tool for obtaining analytical solutions to partial differential equations in mathematical physics.

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.

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.

Optical solitons and traveling wave solutions for the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms

This work mainly concerned with the optical solitons and other traveling wave solutions for the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms. With the assistance of the second-order and triple-order polynomial complete discriminant system method, we construct a series of optical soliton solutions which include bright soliton solutions, singular soliton solutions, dark soliton solutions, periodic solutions, kink solitary wave solutions, anti-kink solitary wave solutions and some other traveling wave solutions. What is more, numerical simulations are also given to visualize the mechanism of the higher-order nonlinear Schrödinger system by selecting some suitable parameters of optical soliton solutions. Finally, we give the comparison between the obtained works in this paper and other’s results.

Two component model of microtubules – subsonic and supersonic solitary waves

This work represents a contribution to modelling nonlinear dynamics of microtubules, the basic components of the eukaryotic cytoskeleton. Their dynamics can be explained in terms of kink and antikink solitary waves. Special attention was paid to the stability of solitonic solutions of differential equations describing the dynamics of microtubules. It is shown that subsonic solitons are stable, while supersonic ones are not.

Modeling bidirectional pedestrian flow with the perceived uncertainty of preceding pedestrian information

Pedestrian flow models contribute to the effective design and operation of pedestrian spaces. In real traffic, pedestrians usually adjust their walking state in the movement procession according to perceptual information. Subject to the human capability, certain errors would exist between the perceptual information and the real traffic information, which can greatly affect the pedestrian flow performance. This paper proposes a bidirectional pedestrian flow model considering the effect of perceived uncertainty of preceding pedestrian information. In the linear and nonlinear stability analysis, we derive the stability criterion and modified Korteweg–de Vries (mKdV) equation of the proposed model. The kink–antikink solitary wave solution can be acquired via solving the above mKdV equations, which provides insight into the essential property of the formation and propagation of traffic jams near the neutral stability curve. Numerical examples demonstrate that the uncertainty terms and the perception time of pedestrians directly affect the pedestrian congestion.

Bifurcation, Traveling Wave Solutions, and Stability Analysis of the Fractional Generalized Hirota–Satsuma Coupled KdV Equations

In this paper, the bifurcation, phase portraits, traveling wave solutions, and stability analysis of the fractional generalized Hirota–Satsuma coupled KdV equations are investigated by utilizing the bifurcation theory. Firstly, the fractional generalized Hirota–Satsuma coupled KdV equations are transformed into two-dimensional Hamiltonian system by traveling wave transformation and the bifurcation theory. Then, the traveling wave solutions of the fractional generalized Hirota–Satsuma coupled KdV equations corresponding to phase orbits are easily obtained by applying the method of planar dynamical systems; these solutions include not only the bell solitary wave solutions, kink solitary wave solutions, anti-kink solitary wave solutions, and periodic wave solutions but also Jacobian elliptic function solutions. Finally, the stability criteria of the generalized Hirota–Satsuma coupled KdV equations are given.

Two component model of microtubules and continuum approximation

In the present work, we study the nonlinear dynamics of microtubules, the basic components of the eukaryotic cytoskeleton. We introduce a two-component model describing tangential oscillations of dimers. A crucial nonlinear differential equation is solved using continuum approximation. We show that the dynamics of microtubules can be explained in terms of kink and antikink solitary waves. We used two mathematical procedures, that is the tangent hyperbolic function method and, more general, the simplest equation method. It is shown that both procedures bring about equal solutions.

A modified two-dimensional triangular lattice model under honk environment

The honk effect is not uncommon in the real traffic and may exert great influence on the stability of traffic flow. As opposed to the linear description of the traditional one-dimensional lattice hydrodynamic model, the high-dimensional lattice hydrodynamic model is a gridded analysis of the real traffic environment, which is a generalized form of the one-dimensional lattice model. Meanwhile, the high-dimensional traffic flow exposed to the open-ended environment is more likely to be affected by the honk effect. In this paper, we propose an extension of two-dimensional triangular lattice hydrodynamic model under honk environment. The stability condition is obtained via the linear stability analysis, which shows that the stability region in the phase diagram can be effectively enlarged under the honk effect. Modified Korteweg–de Vries equations are derived through the nonlinear stability analysis method. The kink–antikink solitary wave solution is obtained by solving the equation, which can be used to describe the propagation characteristics of density waves near the critical point. Finally, the simulation example verifies the correctness of the above theoretical analysis.

New Numerical Results for the Time-Fractional Phi-Four Equation Using a Novel Analytical Approach

This manuscript investigates the fractional Phi-four equation by using q -homotopy analysis transform method ( q -HATM) numerically. The Phi-four equation is obtained from one of the special cases of the Klein-Gordon model. Moreover, it is used to model the kink and anti-kink solitary wave interactions arising in nuclear particle physics and biological structures for the last several decades. The proposed technique is composed of Laplace transform and q -homotopy analysis techniques, and fractional derivative defined in the sense of Caputo. For the governing fractional-order model, the Banach’s fixed point hypothesis is studied to establish the existence and uniqueness of the achieved solution. To illustrate and validate the effectiveness of the projected algorithm, we analyze the considered model in terms of arbitrary order with two distinct cases and also introduce corresponding numerical simulation. Moreover, the physical behaviors of the obtained solutions with respect to fractional-order are presented via various simulations.

Study of bright–dark solitons of strain wave equation in micro-structured solids and its applications

In this study, several novel solutions of strain wave equation for micro-structured solids are obtained by using the powerful modified extended direct algebraic method. In micro-structured solids, the strain wave dynamical equation is utilized to model the wave propagation in micro-structured materials. We have profitably obtained exact traveling wave solutions, including solitons, solitary waves, kink and anti-kink solitary waves, periodic and rational solutions have been found. This approach can not only produce the same solution, but also find new solutions that we think other researchers have missed. These attained solutions help researchers for knowing the physical phenomena of this dynamical equation. The three-dimensional maps of some solutions obtained in this study are plotted through giving appropriate values of the parameters engaged in the solutions for knowing the physical interpretation. The computational work and obtained results demonstrate that the new extended technique, gives an efficient and direct mathematical tools for resolving the nonlinear models in engineering and mathematical physics.

Dispersive optical solitary wave solutions of strain wave equation in micro-structured solids and its applications

The new technique of modified extended mapping technique is an important, prevailing and direct Mathematical tool for constructing exact solutions in the form of traveling waves of non-linear evolution equations (NLEEs) in applied science and engineering. The dispersive effect of the combination of the microstructure and nonlinearity of the material can produce solitary waves. In micro-structured solids, the wave propagation in microstructured materials is modeled by strain wave mechanics equation. In this paper, several new solutions of strain wave dynamical model for micro-structured solids are obtained by using the strongly modified extended mapping technique. We obtain exact waves solutions, including solitons, solitary waves, kinks and anti-kink solitary waves, periodic and rational solutions which have key applications and useful for researchers to understand the phenomena. The three-dimensional graphs and contour plots of some solutions attained in this study are drawn by giving the suitable values of the parameters used to understand the solutions of physical interpretation. The model stability is studied via modulational instability analysis and dispersion relation is derived which verify that these solutions are stable. The computational work and the results demonstrate that the new extended technique is effective and influential.

Dispersive Solitary Wave Solutions of Strain Wave Dynamical Model and Its Stability

In the materials of micro-structured, the propagation of wave modeling should take into account the scale of various microstructures. The different kinds solitary wave solutions of strain wave dynamical model are derived via utilizing exp(−ϕ(ξ))-expansion and extended simple equation methods. This dynamical equation plays a key role in engineering and mathematical physics. Solutions obtained in this work include periodic solitary waves, Kink and anti-Kink solitary waves, bell-shaped solutions, solitons, and rational solutions. These exact solutions help researchers for knowing the physical phenomena of this wave equation. The stability of this dynamical model is examined via standard linear stability analysis, which authenticate that the model is stable and their solutions are exact. Graphs are depicted for knowing the movements of some solutions. The results show that the current methods, by the assist of symbolic calculation, give an effectual and direct mathematical tools for resolving the nonlinear problems in applied sciences.

Stability Analysis of Solitary Wave Solutions for Coupled and (2+1)-Dimensional Cubic Klein-Gordon Equations and Their Applications

The searching exact solutions in the solitary wave form of non-linear partial differential equations (PDEs) play a significant role to understand the internal mechanism of complex physical phenomena. In this paper, we employ the proposed modified extended mapping method for constructing the exact solitary wave and soliton solutions of coupled Klein-Gordon equations and the (2+1)-dimensional cubic Klein-Gordon (K-G) equation. The Klein-Gordon equations are relativistic version of Schrödinger equations, which describe the relation of relativistic energy-momentum in the form of quantized version. We productively achieve exact solutions involving parameters such as dark and bright solitary waves, Kink solitary wave, anti-Kink solitary wave, periodic solitary waves, and hyperbolic functions in which several solutions are novel. We plot the three-dimensional surface of some obtained solutions in this study. It is recognized that the modified mapping technique presents a more prestigious mathematical tool for acquiring analytical solutions of PDEs arise in mathematical physics.

Solitary and Jacobi elliptic wave solutions of the generalized Benjamin-Bona-Mahony equation

Exact bright, dark, antikink solitary waves and Jacobi elliptic function solutions of the generalized Benjamin-Bona-Mahony equation with arbitrary power-law nonlinearity will be constructed in this work. The method used to carry out the integration is the F-expansion method. Solutions obtained have fractional and integer negative or positive power-law nonlinearities. These solutions have many free parameters such that they may be used to simulate many experimental situations, and to precisely control the dynamics of the system.

Exact traveling discrete kink-soliton solutions for the discrete nonlinear electrical transmission lines

We investigate exact soliton solutions for the discrete nonlinear electrical transmission line by performing the simplest equation method from a trivial seed solution. Starting from the nonlinear propagation of signals in electrical transmission lines, we derive exact traveling kink and antikink solitary wave solutions. It is shown that under a safe range of parameter, the shape of kink soliton can be controlled well by adjusting the parameter of the line. The analytical solutions for the kink and antikink solitary waves are tested in direct simulations.

OPTICAL SOLITARY WAVE SOLUTIONS FOR THE FOURTH-ORDER DISPERSIVE CUBIC-QUINTIC NONLINEAR SCHRÖDINGER EQUATION

We present several kinds of optical solitary wave solutions for the fourth-order dispersive cubic-quintic nonlinear Schrödinger equation describing the propagation of optical pulses in a medium that exhibits a parabolic nonlinearity law. Among them, apart from some regular fundamental bright solitary wave solutions and dark solitary wave solutions given, there are also two kinds of combined solitary wave solutions, i.e., bright and dark solitary wave and W-shaped solitary wave solutions which describe bright and dark solitary wave properties. It is found that their amplitude may approach nonzero when the time variable approaches infinity. Moreover, when the parameters of the bright and dark solitary wave are selected appropriately, the kink and anti-kink solitary wave solutions, which are first reported in the fourth-order dispersive cubic-quintic nonlinear Schrödinger equation, are discovered.

A direct truncation method for finding abundant exact solutions and application to the one-dimensional higher-order Schrödinger equation

We suggest a direct truncation technique for finding exact solutions of nonlinear differential equation, this method is based on the WTC test. As an application, abundant new exact stationary solutions of the one-dimensional higher-order nonlinear Schrodinger equation are obtained. These solutions include bright, dark, kink or anti-kink solitary wave solutions, which are dependent of the model and free parameters in the solutions. Algebraic solitary-like solution and new periodic solutions are also obtained. An interesting fact is that some solitary solutions can convert into the periodic solutions and vice versa when the free parameters are changed. (C) 2004 Elsevier Ltd. All rights reserved.