sine-Gordon Expansion Method Research Articles

Novel Solutions of Perturbed Boussinesq Equation

In this article, we have worked on the perturbed Boussinesq equation. We have applied the generalized Kudryashov method (GKM) and sine-Gordon expansion method (SGEM) to the perturbed Boussinesq equation. So, we have obtained some new soliton solutions of the perturbed Boussinesq equation. Furthermore, we have drawn some 2D and 3D graphics of these results by using Wolfram Mathematica 12.

Sine-Gordon expansion method to construct the solitary wave solutions of a family of 3D fractional WBBM equations

The Sine-Gordon expansion (SGE) method is used to develop exact traveling wave solutions of a family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations. The governing equations are reduced to standard ordinary differential equations via companionable wave transformation. The order of the projected polynomial-type solution is figured out using the homogeneous stability approach based on the renowned Sine-Gordon equation. This solution's replacement corresponds to the earlier stage. A system of algebraic equations (SAE) is formed by comparing the coefficients of the powers of the projected solution. A robust coefficient scheme supplies the necessary relationships between parameters and coefficients to develop solutions. Some solutions are modeled for various parameter combinations. Several hyperbolic function solutions are created using the mentioned method. The MATLAB program displays the answers' graphical exemplifications in three-dimensional (3D) surface plots, contour plots, and 2D line plots. The resulting solutions to the equation, with the appropriate parameters, are used to depict the absolute wave configurations in all displays. Additionally, it can conclude that the discovered solutions and their physical characteristics could aid in understanding how shallow water waves propagate in nonlinear dynamics.

Invariance properties, exact and explicit solutions of time-fractional Gear–Grimshaw model

In this study, we consider a time-fractional shallow water wave model which was introduced by Gear and Grimshaw. This model represents the dynamics of two-layered shallow water waves in oceans and rivers. Also it covers many shallow water wave models with special values taken to parameters. By performing the group invariant analysis approach for the governing system with Riemann–Liouville derivative, symmetry generators were found. Similarity reduction was constructed by Erdelyi–Kober sense. In addition, by adapting the Sine-Gordon expansion method, we find some exact traveling wave solutions. Furthermore, explicit solutions in form of power series are examined via the residual power series method. Exact and approximate solutions are compared for some [Formula: see text] values and these are given as tables.

On the new hyperbolic wave solutions to Wu-Zhang system models

In this study, some solitary wave solutions of the Wu-Zhang system are analyzed using the modified expansion function method and the sine–Gordon expansion method. Solitary wave solutions of this non-linear mathematical model consisting of hyperbolic and trigonometric function structures are get. Two and three dimensional, density graphics of the solitary solutions of the mathematical models are drawn by choosing the appropriate parameters. It was seen that all solution functions found as a result provide the mathematical model. In this study, Wolfram Mathematica software program was used for all mathematical calculations.

The Unstable Analysis of Soliton Compression in Silicon Photonic Crystal

Soliton solutions to the perturbed nonlinear Schrodinger equation, for describing the wave in the photonic crystal, can be found by the sine-Gordon expansion method. This method provides both bright and dark soliton solutions. The instability of the perturbed bright soliton in the perpendicular direction to the wave motion cab be expressed in terms of the growth rate. The time evolution of the unstable soliton can be visualized from the transformation to the higher dimensional solitons by the spectral method.

Newly developed analytical method and its applications of some mathematical models

This paper presents a newly developed method, namely, the rational sine-Gordon expansion method to find novel exact solutions to nonlinear differential equations. This method is based on the sine-Gordon expansion method. To generalize the approach, we utilize the ansatz, which is a rational function as different to a polynomial function. In this way, we have more general wave solutions for nonlinear dynamic systems. We apply this method to the ([Formula: see text])-dimensional conformable Zakharov–Kuznetsov modified equal width (ZK-MEW) equation and the modified regularized long wave (MRLW) equation. Some new solutions are reported. Consequently, we submit the new soliton solutions to the literature.

Soliton theory and modulation instability analysis: The Ivancevic option pricing model in economy

In this projected paper, we study on the Ivancevic option pricing model. We apply two important methods, namely, rational sine-Gordon expansion method which is recently developed, and secondly, modified exponential method. Via these methods, we obtain some important properties of Ivancevic option pricing model. We extract many solutions such as complex, periodic, dark bright, mixed dark-bright, singular, travelling and hyperbolic functions. We investigate the option price wave functions of dependent variable, and also, observe the modulation instability analysis in detail. Furthermore, we report the strain conditions for the valid solutions under the family conditions, as well. We simulate the 2D, 3D and counter plots by choosing the suitable values of the parameters involved. Finally, we present the top and low points of pricing in the mentioned intervals via contour simulations.

The exact solutions of conformable time-fractional modified nonlinear Schrödinger equation by Direct algebraic method and Sine-Gordon expansion method

<abstract><p>In this article, we used direct algebraic method (DAM) and sine-Gordon expansion method (SGEM), to find the analytical solutions of conformable time-fractional modified nonlinear Schrödinger equation (CTFMNLSE) and finally, we present numerical results in tables and charts.</p></abstract>

Sine-Gordon Expansion Method to Construct the Solitary Wave Solutions of a Family of 3d Fractional Wbbm Equations

Sine-Gordon Expansion Method to Construct the Solitary Wave Solutions of a Family of 3d Fractional Wbbm Equations

Newly Developed Analytical Scheme and Its Applications to the Some Nonlinear Partial Differential Equations with the Conformable Derivative

This paper presents a novel and general analytical approach: the rational sine-Gordon expansion method and its applications to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. Some trigonometric, periodic, hyperbolic and rational function solutions are extracted. Physical meanings of these solutions are also presented. After choosing suitable values of the parameters in the results, some simulations are plotted. Strain conditions for valid solutions are also reported in detail.

Application of The Sine-Gordon Expansion Method on Nonlinear Various Physical Models

In this paper, by utilizing the Sine-Gordan expansion method, soliton solutions of the higher-order improved Boussinesq equation, Kuramoto-Sivashinsky equation, and seventh-order Sawada-Kotera equation are obtained. Given partial differential equations are reduced to ordinary differential equations, by choosing the compatible wave transformation associated with the structure of the equation. Based on the solution of the Sine-Gordan equation, a polynomial system of equations is obtained according to the principle of homogeneous balancing. The solution of the outgoing system gives the parameters which are included by the solution. Plot3d and Plot2d graphics are given in detail. As a result, many different graphic models are obtained from soliton solutions of equations that play a very important role in mathematical physics and engineering.

Construction of Soliton Solutions for Chaffee-Infante Equation

In this article, has been studied on the Chaffee-Infante equation and soliton solutions of these equation are examined. In accordance with this purpose, The sine-Gordon expansion method, which is one of the solution methods of nonlinear partial differential equations, was used. Also graphical representation of the obtained results of the specified equation is made using Wolfram Mathematica 12 for certain values and thus the conformity of the founded results has been demonstrated.

ON THE COMPLEX MIXED DARK-BRIGHT WAVE DISTRIBUTIONS TO SOME CONFORMABLE NONLINEAR INTEGRABLE MODELS

In this research paper, we implement the sine-Gordon expansion method to two governing models which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equation and the Caudrey–Dodd–Gibbon–Sawada–Kotera equation. We use conformable derivative to transform these nonlinear partial differential models to ordinary differential equations. We find some wave solutions having trigonometric function, hyperbolic function. Under the strain conditions of these solutions obtained in this paper, various simulations are plotted.

Conformable space-time fractional nonlinear (1+1)-dimensional Schrödinger-type models and their traveling wave solutions

• A family of fractional nonlinear ( 1 + 1 ) dimensions Schrödinger-type models is introduced. • The space-time fractional equation is in the sense of the conformable derivative. • A variety of optical soliton solutions for these equations are obtained. Space-time conformable fractional nonlinear ( 1 + 1 )-dimensional Schrödinger-type models are investigated in this paper. Traveling wave solutions using the sine-Gordon expansion approach for these models are presented. The sine-Gordon expansion method is used to obtain exact solutions for three types of space-time conformable fractional nonlinear Schrödinger-type equations which some of them are new.

Wave profile analysis of a couple of (3+1)-dimensional nonlinear evolution equations by sine-Gordon expansion approach

The (3+1)-dimensional Kadomtsev-Petviashvili and the modified KdV-Zakharov-Kuznetsov equations have a significant impact in modern science for their widespread applications in the theory of long-wave propagation, dynamics of shallow water wave, plasma fluid model, chemical kinematics, chemical engineering, geochemistry, and many other topics. In this article, we have assessed the effects of wave speed and physical parameters on the wave contours and confirmed that waveform changes with the variety of the free factors in it. As a result, wave solutions are extensively analyzed by using the balancing condition on the linear and nonlinear terms of the highest order and extracted different standard wave configurations, containing kink, breather soliton, bell-shaped soliton, and periodic waves. To extract the soliton solutions of the high-dimensional nonlinear evolution equations, a recently developed approach of the sine-Gordon expansion method is used to derive the wave solutions directly. The sine-Gordon expansion approach is a potent and strategic mathematical tool for instituting ample of new traveling wave solutions of nonlinear equations. This study established the efficiency of the described method in solving evolution equations which are nonlinear and with higher dimension (HNEEs). Closed-form solutions are carefully illustrated and discussed through diagrams.

Extension of the sine-Gordon expansion scheme and parametric effect analysis for higher-dimensional nonlinear evolution equations

Different wave solutions assist to interpret phenomena in different aspects of optics, physics, plasma physics, engineering, and other related subjects. The higher dimensional generalized Boussinesq equation (gBE) and the Klein-Gordon (KG) equation have remarkable applications in the field of quantum mechanics, recession flow analysis, fluid mechanics etc. In this article, the soliton solutions of the higher-dimensional nonlinear evolution equations (NLEEs) have been extracted through extending the sine-Gordon expansion method and we analyze the effect of the associated parameters and the phenomena establishing the lump, kink, rogue, bright-dark, spiked, periodic wave, anti-bell wave, singular soliton etc. Formerly, the sine-Gordon expansion (sGE) method was used only to search for lower-dimensional NLEEs. In order to illustrate the latency, we have portrayed diagrams for different values of parameters and it is noteworthy that the properties of the features change as the parameters change.

Optical soliton solutions for the coupled conformable Fokas–Lenells equation with spatio-temporal dispersion

New optical soliton solutions for the coupled conformable Fokas–Lenells equation with spatio-temporal dispersion are obtained via Atangana’s derivative operator using an integration method. The used method in this article is sine–Gordon expansion method. The obtained solutions include dark, bright, singular as well as periodic waves solitons which are graphically discussed through 2D- and 3D-plots.

On the optical solutions to nonlinear Schrödinger equation with second-order spatiotemporal dispersion

Abstract In this article, the sine-Gordon expansion method is employed to find some new traveling wave solutions to the nonlinear Schrödinger equation with the coefficients of both group velocity dispersion and second-order spatiotemporal dispersion. The nonlinear model is reduced to an ordinary differential equation by introducing an intelligible wave transformation. A set of new exact solutions are observed corresponding to various parameters. These novel soliton solutions are depicted in figures, revealing the new physical behavior of the acquired solutions. The method proves its ability to provide good new approximate solutions with some applications in science. Moreover, the associated solution of the presented method can be extended to solve more complex models.

Soliton solutions of the generalized Davey-Stewartson equation with full nonlinearities via three integrating schemes

This paper considers the generalized Davey-Stewartson equation that is used to investigate the dynamics of wave propagation in water of finite depth under the effects of gravity force and surface tension. The model is considered in the presence of full nonlinearity. The main objective of this paper is to extract soliton solutions of the generalized Davey-Stewartson equation. Three state-of-the-art integration schemes, namely exp(-Φ(ξ))-expansion method, the first integral method and the Sine-Gordon expansion method have been employed for obtaining the desired soliton solutions. The proposed methods successfully attain different structures of explicit solutions such as bright, dark, singular, rational and periodic solitary wave solutions. All the newly found solutions are discussed with their existence criteria. The 2D and 3D portraits are also shown for some of the reported solutions.

The sine-Gordon expansion method for higher-dimensional NLEEs and parametric analysis

The Estevez-Mansfield-Clarkson (EMC) equation and the (2+1)-dimensional Riemann wave (RW) equation are important mathematical models in nonlinear science, engineering and mathematical physics which have remarkable applications in the field of plasma physics, fluid dynamics, optics, image processing etc. Generally, through the sine-Gordon expansion (SGE) method only the lower-dimensional nonlinear evolution equations (NLEEs) are examined. However, the method has not yet been extended of finding solutions to the higher-dimensional NLEEs. In this article, the SGE method has been developed to rummage the higher-dimensional NLEEs and established steady soliton solutions to the earlier stated NLEEs by putting in use the extended higher-dimensional sine-Gordon expansion method. Scores of soliton solutions are figure out which confirms the compatibility of the extended SGE method. The solutions are analyzed for both lower and higher-dimensional nonlinear evolution equations through sketching graphs for alternative values of the associated parameters. From the figures it is notable to perceive that the characteristic of the solutions depend upon the choice of the parameters. This study might play an impactful role in analyzing higher-dimensional NLEEs through the extended SGE approach.