Kink Solitons Research Articles

New Bright and Kink Soliton Solutions for Fractional Complex Ginzburg–Landau Equation with Non-Local Nonlinearity Term

In this paper, we aim to discuss a fractional complex Ginzburg–Landau equation by using the parabolic law and the law of weak non-local nonlinearity. Then, we derive dynamic behaviors of the given model under certain parameter regions by employing the planar dynamical system theory. Further, we apply the ansatz method to derive soliton, bright and kinked solitons and verify their existence by imposing certain conditions. In addition, we integrate our solutions in appropriate dimensions to explain their behavior at various groups of parameters. Moreover, we compare the graphical representations of the established solutions at different fractional derivatives and illustrate the impact of the fractional derivative on the investigated soliton solutions as well.

Solitons Solution of Riemann Wave Equation via Modified Exp Function Method

In the areas of tidal and tsunami waves in oceans, rivers, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, homogeneous and stationary media, etc., the Riemann wave equations are attractive nonlinear equations. The modified exp(−Φ(η))-function method is used in this article to show how well it can be applied to extract travelling and solitary wave solutions from higher-order nonlinear evolution equations (NLEEs) using the equations mentioned above. Trigonometric, hyperbolic, and exponential functions solitary wave solutions can be extracted using the above-mentioned technique. By changing specific values of the embedded parameters, we can obtain bell-form soliton, consolidated bell-shape soliton, compacton, singular kink soliton, flat kink shape soliton, smooth singular soliton, and other sorts of soliton solutions. The solutions are graphically illustrated in 3D and 2D for the accuracy of the outcome by using the Wolfram Mathematica 10. The verification of numerical solvers on the stability analysis of the solution is substantially aided by the analytic solutions.

On the multiple explicit exact solutions to the double‐chain DNA dynamical system

The nonlinear dynamics of deoxyribonucleic acid (DNA) is the subject of this research. For an organism to flourish and reproduce, it needs the genetic information that is contained in its DNA. In the DNA molecule, the hydrogen bonds between the base pairs of the two polynucleotide chains are represented by an elastic membrane (or a series of linear springs) that connects the model's two long elastic homogeneous strands (or rods). Using two integration norms, namely, the new auxiliary equation method (NAEM) and the extended sinh‐Gordon equation expansion method (ShGEEM), we obtain a variety of nonlinear dynamical solitary wave structures in various shapes, including hyperbolic, trigonometric, and exponential function solutions, as well as some unique solitary wave solutions, including bell‐shaped, kink, singular, combined, and multiple solitons. Using the logarithmic transformation assistant, we examine a variety of wave structures and recover singular periodic wave solutions with unknown parameters, along with the validity constraints. According to the results, the governing model contains an unusually large number of theoretically exact solitary wave solutions. The physical characterization of some of the presented results can be visualized in 3D and 2D profiles by selecting the appropriate parameter values.

The soliton solutions and combined solutions of a high-dimensional wave soliton equation

In this paper, a high-dimensional wave soliton equation is considered and the simple Hirota method and bilinear backlund transformation are applied to construct the new soliton solutions, lump soliton solutions, breather solutions and their combined solutions. Then, through long-wave limit method and appropriate parameter constraints, one-soliton, one-breather, combined soliton-lump and lump-breather and soliton-breather solutions to nonlinear wave equation are obtained. Moreover, the physical interaction and frontal collision phenomena to the solutions of this equation are explored. In addition, a kind of hyperbolic and trigonometric ansatz is employed to derive other solutions such as kink soliton solution, periodic solutions for the high-dimensional wave equation.The obtained results verify the proposed solutions.

Construction of Exact Solutions for Gilson–Pickering Model Using Two Different Approaches

In this paper, the extended simple equation method (ESEM) and the generalized Riccati equation mapping (GREM) method are applied to the nonlinear third-order Gilson–Pickering (GP) model to obtain a variety of new exact wave solutions. With the suitable selection of parameters involved in the model, some familiar physical governing models such as the Camassa–Holm (CH) equation, the Fornberg–Whitham (FW) equation, and the Rosenau–Hyman (RH) equation are obtained. The graphical representation of solutions under different constraints shows the dark, bright, combined dark–bright, periodic, singular, and kink soliton. For the graphical representation, 3D plots, contour plots, and 2D plots of some acquired solutions are illustrated. The obtained wave solutions motivate researchers to enhance their theories to the best of their capacities and to utilize the outcomes in other nonlinear cases. The executed methods are shown to be practical and straightforward for approaching the considered equation and may be utilized to study abundant types of NLEEs arising in physics, engineering, and applied sciences.

Abundant invariant and classical solutions with the conservation laws of a new (3+1)-dimensional fifth-order nonlinear Wazwaz equation with the third-order dispersion terms in ocean physics

Ocean plays key roles in the earth’s system in shaping the planet’s climate as well as weather by absorbing, storing, and also transporting huge quantities of carbon-dioxide, water, heat and moisture. This paper presents the analytical examination of a new (3+1)-dimensional fifth-order nonlinear Wazwaz equation with third-order dispersion terms which is applicable in ocean physics and other nonlinear sciences. Ocean physics is the study of various conditions as well as processes that are physical within the ocean, especially the physical features and motions of ocean waters. Thus, the diverse outcomes of the investigations carried out on the model under study possesses significant applications in physical oceanography (Adeyemo, 2022). We first apply Lie group analysis to obtain various infinitesimal generators admitted by the equation. This is followed by invoking the generators to reduce the understudy equation for the purpose of achieving copious group-invariant solutions. We solve the resultant ordinary differential equations in order to obtain diverse classical solutions of the underlying equation. The outcome yields several solutions with arbitrary functions, quadratures, rational and hyperbolic functions alongside various topological kink solitons. Moreover, numerical simulations of the obtained results are performed so as to give some physical interpretations to the solutions and these yield various solitary waves as well as interactions between soliton waves of interest. Furthermore, we outline the applications of our results in ocean physics. Conclusively, conservation laws are constructed for the equation understudy with the use of Ibragimov’s theorem.

Impacts of nonlinearity and wave dispersion parameters on the soliton pulses of the (2+1)-dimensional Kundu–Mukherjee–Naskar equation

In this study, we explain the impacts of nonlinearity and wave dispersion parameters on the soliton pulses of the (2+1)-dimensional Kundu–Mukherjee–Naskar equation (KMNE). In this regard, some new optical solitons are received via the unified method to the aforesaid equation to explain such impacts of the soliton pulses. The presenting optical solitons are expressed by the dark, periodic, bell, bright, kink, and singular soliton solutions. Taking into account the two impacts help stabilize the soliton pulses during their propagation by generating new dynamics depending upon the nonlinearity and the wave dispersion parameters of the studied equation. All the characteristics of the soliton pulses are exhibited graphically. It is found from the graphical outputs that the soliton profiles are decreasing and increasing with the increase of nonlinearity and dispersion parameters, respectively. The outcomes reveal that the soliton pulses are balanced due to the influences of nonlinearity and wave dispersion parameters of the aforementioned equation. It is mentioned that the impact of wave dispersion and nonlinearity parameters on the soliton pulses has not been discussed in the past. Therefore, the applied method permits the explanation of the various wave dynamics by analyzing the attained soliton solutions in nonlinear optical fibers systems, which can be used for further studies.

Nonlocal conservation laws and dynamics of soliton solutions of (2 + 1)-dimensional Boiti–Leon–Pempinelli system

In this article, we obtain several new exact solutions of (2 + 1)-dimensional Boiti–Leon–Pempinelli system of nonlinear partial differential equations (PDEs) which describes the evolution of horizontal velocity component of water waves propagating in two directions. We perform the Lie symmetry analysis to the given system and construct a one-dimensional optimal subalgebra which involves some arbitrary functions of spatial variables. Symmetry group classifications of infinite-dimensional Lie algebra for higher-dimensional system of PDEs are very interesting and rare in the literature. Several new exact solutions are obtained by symmetry reduction using each of the optimal subalgebra and these solutions have not been reported earlier in the previous studies to the best of our knowledge. We then study the dynamical behavior of some exact solutions by numerical simulations and observed many interesting phenomena, such as traveling waves, kink and anti-kink type solitons, and singular kink type solitons. We construct several conservation laws of the system by using a multiplier method. As an application, we study the nonlocal conservation laws of the system by constructing potential systems and appending gauge constraints. In fact, determining nonlocal conservation laws for higher-dimensional nonlinear system of PDEs arising from divergence type conservation laws is very rare in the literature and have huge consequences in the study of nonlocal symmetries.

Analysis of interaction of lump solutions with kink-soliton solutions of the generalized perturbed KdV equation using Hirota-bilinear approach

Using the Hirota bilinear method (HBM) and Cole-Hopf transformation, we extract the interaction between the lump and kink-soliton solutions for the generalized perturbed Korteweg–de Vries (KdV) equation in this manuscript. In the literature [15], the authors have applied the Cole-Hopf transformation with a value of R to study the particular case of the generalized perturbed KdV equation, which is found to be an inappropriate value. Alqarni et al., computed an appropriate value of R in the literature [29]. We use the correct value of R and get more clearly the outcomes for a lump-single kink soliton, a lump-two kink soliton, and a lump-periodic soliton solution. We display these solutions in 2D and 3D space using Mathematica by selecting the proper values for the parameters.

Various localized wave structures and dynamical analysis for a (2+1)-dimensional version of fifth-order equation

Under investigation is the (2+1)-dimensional version of fifth-order equation, which is an extension of the (1+1)-dimensional fifth-order equation. First, the [Formula: see text]-soliton solutions of this (2+1)-dimensional equation are constructed by using its Hirota bilinear form. Second, through using long-wave limit technique and selecting appropriate parameters for the [Formula: see text]-soliton solutions, diverse localized wave structures including kink soliton, breather, lump solution and their mixed interaction behaviors are obtained, and relevant dynamics features are discussed and analyzed graphically. Finally, some mathematical properties and special physical phenomena are summarized. The results obtained in this paper might be helpful to deepen the understanding of the interaction evolution of nonlinear localized waves.

Exact Solutions of (2+1) Dimensional Cubic Klein-Gordon (cKG) Equation

In current study, (2+1)-dimensional cubic Klein-Gordon (cKG) equation illustrating dislocation propagation in crystals as well as the behaviour of elementary particles is investigated to establish a variety of new analytic exact solitary wave solutions. Modified exponential expansion method has been implemented to unfold certain wave solutions of considered model. As a result, three sorts of solutions emerge in a fairly systematic manner in the shape of hyperbolic, trigonometric, and rational functions. The kink and periodic wave solitons are acquired and presented geometrically, some 3D plots are simulated and displayed to respond the dynamic behavior of these obtained solutions. In this work we have used symbolic package maxima to obtained our solutions. Our acquired solutions might be most helpful to analyze physical issues that arise from nonlinear complicated dynamical systems.

Variational and non-variational approaches with Lie algebra of a generalized (3 + 1)-dimensional nonlinear potential Yu-Toda-Sasa-Fukuyama equation in Engineering and Physics

Nonlinear partial differential equations emerge in an extensive variants of physical problems inclusive of fluid dynamics, solid mechanics, plasma physics, quantum field theory as well as mathematics and engineering. It has also been noticed that systems of nonlinear partial differential equations arise in biological and chemical applications. This article presents the analytical investigation of a completely generalized (3 + 1)-dimensional nonlinear potential Yu-Toda-Sasa-Fukuyama equation which has applications in the fields of engineering and physics. The generalized version of the potential Yu-Toda-Sasa-Fukuyama equation is more comprehensively studied in this paper compared to other research work previously done on the equation, with various new solutions of interests achieved. The theory of Lie group is applied to the nonlinear partial differential equation to basically reduce the equation to an integrable form which consequently allows for direct integration of the result. The rigorous process involved in performing a comprehensive reduction of the model under consideration using its Lie algebra makes it possible to achieve various nontrivial solutions. Besides, more general solutions are found via a well-known standard technique. In consequence, we secured diverse solitons and solutions of great interest including topological kink solitons, singular solitons, algebraic functions, exponential function, rational function, Weierstrass function, Jacobi elliptic function as well as series solutions of the underlying equation. Moreover, the completeness of the result was ascertained by presenting the solutions graphically. In addition, discussions of the pictorial representations of the results are done. Conclusively, we constructed conserved quantities of the underlying equation via both the variational and non-variational approaches using the classical Noether’s theorem as well as the standard multiplier technique respectively. In addition, some pertinent observations made from the secured results via both techniques are analyzed.

Dynamical Analysis of Nonlocalized Wave Solutions of 2 + 1 -Dimensional CBS and RLW Equation with the Impact of Fractionality and Free Parameters

This study retrieves some new exact solutions to the 2 + 1 -dimensional Calogero-Bogoyavlenskii-Schilf (CSB) equation and regularized long wave (RLW) equation in the context of nonlinear traveling wave phenomena. In this regard, the advanced exp − φ ξ -expansion method is imposed to the 2 + 1 -dimensional CBS and RLW equation, and consequently, rogue, kink, singular kink, periodic, singular, and multiple soliton solutions are exhibited in terms of trigonometric, hyperbolic, and rational function solutions. To enucleate the underlying nonlocalized traveling wave features, accomplished exact solutions are established by making their dynamic behavior of the exact solutions exhibited in three-dimensional (3D) and two-dimensional (2D) combined chart with the help of computational software Maple 18. All of our accomplished solutions are claimed to be new in the sense of conformable derivative, chosen a unique fractional type wave transformation, dynamical behavior of fractionally and free variable, and the imposed method on our preferred equations.

RETRACTED: Lie symmetry analysis, optimal system and exact solutions for variable-coefficients (2 + 1)-dimensional dissipative long-wave system

In this article, the symmetry, optimal system, exact solutions and conservation laws of (2+1)-dimensional dissipative long-wave system with variable-coefficients (vcDLW) are investigated. These are important for mathematicians and physicists to solve problems about shallow water waves. The infinitesimal generators of independent variables are solved using Lie symmetry analysis method firstly. Based on the linear combination of vector fields, the representative elements of one-dimensional subalgebras optimal system are introduced by Olver's method. Some new exact solutions of the (2+1)-dimensional vcDLW are obtained by using Riccati equation method and tanh expansion method, such as kink solutions and soliton solutions. It is shown that this system has different solutions for different reductions. These solutions are important for describing a number of physical phenomena. The difference in this article is that the coefficients of some terms are functions. The unique feature of this article is that we can obtain many different types of equations when the coefficient functions change. This makes the (2+1)-dimensional vcDLW describe more general physical phenomena. In addition, the dynamical behaviour of the exact solutions of the (2+1)-dimensional vcDLW is analysed. And the evolution of several types of solutions is studied depending on the change in time. This process can be described better with the help of figures. Finally, The conservation laws of the (2+1)-dimensional vcDLW are obtained using the new conservation theorem with two modificatory rules.

Optical solitons to the Ginzburg–Landau equation including the parabolic nonlinearity

The major goal of the present paper is to construct optical solitons of the Ginzburg–Landau equation including the parabolic nonlinearity. Such an ultimate goal is formally achieved with the aid of symbolic computation, a complex transformation, and Kudryashov and exponential methods. Several numerical simulations are given to explore the influence of the coefficients of nonlinear terms on the dynamical features of the obtained optical solitons. To the best of the authors’ knowledge, the results reported in the current study, classified as bright and kink solitons, have a significant role in completing studies on the Ginzburg–Landau equation including the parabolic nonlinearity.

Lump and interaction dynamics of the (2+1)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili equation

This paper mainly considers the (2+1)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili equation describing propagation dynamics of nonlinear waves appearing in physics, biology and electrical networks. A simple method to derive the multiple lumps and lump molecules is proposed by means of the solutions of Grammian form and polynomial functions. The interaction dynamical behaviors of the lumps and lump molecules are analyzed with the aid of numerical simulation. The interaction solutions including lumps, lump molecules and kink solitons are obtained, where the lumps can be emitted and absorbed by solitons. The phenomenon that one lump wave exchanges between two bound solitons is presented. A special Grammian τ-function is employed to construct the lump chain. Based on an analytical method related to dominant domain, we systematically analyze the interactions between lump chain and solitons. The lump chain can be emitted by a soliton and then absorbed by other solitons when the parameter is in a certain range. A large number of new solutions obtained in this paper are helpful to study the wave phenomena existing in the theory of water wave, soliton, optical fiber communication and other relevant scientific fields.

Analysis of some new wave solutions of DNA-Peyrard–Bishop equation via mathematical method

The main goal of this paper is to extract the closed-form wave solutions by means of [Formula: see text]-expansion method to the DNA flow known as the Peyrard–Bishop model which describes the nonlinear interaction between neighboring displacements and hydrogen bonds. The aforementioned model is investigated for the time-fractional order and the fractional derivatives are used in the sense of conformable derivatives. The projected method provides additional generic and comprehensive wave solutions incorporated with physical parameters whose specific values reveal different shapes of the waveform such as periodic soliton, singular kink soliton, singular anti-kink soliton, and other different types of solitons. The obtained soliton solutions are novel, advanced, distinct, and relevant based on the available information in the literature which may be applied to more complex phenomena. The three-dimensional, two-dimensional, and contour plots of some of the attained results are sketched by assigning individual values of the parameters to scrutinize the dynamic behavior of the waves.

Bright, Dark, and Rogue Wave Soliton Solutions of the Quadratic Nonlinear Klein–Gordon Equation

This article reflects on the Klein–Gordon model, which frequently arises in the fields of solid-state physics and quantum field theories. We analytically delve into solitons and composite rogue-type wave propagation solutions of the model via the generalized Kudryashov and the extended Sinh Gordon expansion approaches. We obtain a class of analytically exact solutions in the forms of exponential and hyperbolic functions involving some arbitrary parameters with the help of Maple, which included comparing symmetric and non-symmetric solutions with other methods. After analyzing the dynamical behaviors, we caught distinct conditions on the accessible parameters of the solutions for the model. By applying conditions to the existing parameters, we obtained various types of rogue waves, bright and dark bells, combing bright–dark, combined dark–bright bells, kink and anti-kink solitons, and multi-soliton solutions. The nature of the solitons is geometrically explained for particular choices of the arbitrary parameters. It is indicated that the nonlinear rogue-type wave packets are restricted in two dimensions that characterized the rogue-type wave envelopes.

Assorted soliton solutions to the nonlinear dispersive wave models in inhomogeneous media

The Sharma-Tasso-Olver and Klein–Gordon equations are significant models to interpret plasma physics, relativistic physics, quantum mechanics, nonlinear optics, etc. In the present article, an advanced generalized approach, namely the generalized (G′/G)-expansion approach is scheduled to unfold certain wave solutions to the nonlinear evolution equations earlier described. The soliton solutions exposed are estimated in the form of hyperbolic, rational and trigonometric functions. The physical implications of the ascertained solutions are summed up by setting specific values of the integrated constraints and delineating the profiles to decode the physical phenomena. The numerical simulation of the solutions extracted is standard kink, rogue wave, bright-dark soliton, periodic soliton, breather type soliton, compaction, singular kink soliton, etc. This study affirms that the introduced scheme is robust, efficient in searching for nonlinear evolution equations, computer algebra compatible, and capable of finding further inclusive wave solutions.

Dynamics of mixed lump-soliton solutions to the (3+1)-dimensional Boiti–Leon–Manna–Pempinelli like equation

In this paper, a (3+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) like equation is proposed based on the generalized bilinear operators in a prime number p = 3 . By combining an exponential function with a quadratic function, an interaction between a lump wave and one kink soliton is generated. Furthermore, the lump and the interaction solutions are illustrated by three-dimensional plots, density plots, contour plot and propagation pattern along x -axis by selecting the appropriate parameters.