sine-Gordon Expansion Research Articles

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.

A variety of novel closed‐form soliton solutions to the family of Boussinesq‐like equations with different types

This paper deals with the closed-form solutions to the family of Boussinesq-like equations with the effect of spatio-temporal dispersion. The sine-Gordon expansion and the hyperbolic function approaches are efficiently applied to the family of Boussinesq-like equations to explore novel solitary, kink, anti-kink, combo, and singular-periodic wave solutions. The attained solutions are expressed by the trigonometric and hyperbolic functions including tan, sec, cot, csc, tanh, sech, coth, csch, and of their combination. In addition, the mentioned two approaches are applied to the aforesaid models in the sense of Atangana conformable derivative or Beta derivative to attain new wave solutions. Three-dimensional and two-dimensional graphs of some of the obtained novel solutions satisfying the corresponding equations of interest are provided to understand the underlying mechanisms of the stated family. For the bright wave solutions in terms of Atangana’s conformable derivative, the amplitudes of the bright wave gradually decrease, but the smoothness increases when the fractional parameters α and β increase. On the other hand, the periodicities of periodic waves increase. The attained new wave solutions can motivate applied scientists for engineering their ideas to an optimal level and they can be used for the validation of numerical simulation results in the propagation of waves in shallow water and other nonlinear cases. The performed approaches are found to be simple and efficient enough to estimate the solutions attained in the study and can be used to solve various classes of nonlinear partial differential equations arising in mathematical physics and engineering.

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.

Optical soliton with Kudryashov’s equation via sine-Gordon expansion and Kudryashov methods

Optical soliton with Kudryashov’s equation via sine-Gordon expansion and Kudryashov methods

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.

Solitary wave solutions in two-Core optical fibers with coupling-coefficient dispersion and Kerr nonlinearity

In this paper, we studies chirped solitary waves in two-Core optical fibers with coupling-coefficient dispersion and intermodal dispersion. To construct chirp soliton, the couple of nonlinear Schrödinger equation which describing the pulses propagation along the two-core fiber have been reduced to one equivalent equation. By adopting the traveling-waves hypothesis, the exact analytical solutions of the GNSE were obtained by using three relevant mathematical methods namely the auxiliary equation method, the modified auxiliary equation method and the Sine-Gordon expansion approach. Lastly, the behavior of the chirped like-soliton solutions were discussed and some contours of the plot evolution of the bright and dark solitons are obtained.

Soliton solutions to the Boussinesq equation through sine-Gordon method and Kudryashov method

The Boussinesq equation simulates weakly nonlinear and long wave approximation that can be used in water waves, coastal engineering, and numerical models for water wave simulation in harbors and shallow seas. In this article, the sine-Gordon expansion (SGE) approach and the generalized Kudryashov (GK) scheme are used to establishbroad-spectral solutions including unknown parameters and typical analytical solutions are recovered as a special case. The well-known bell-shape soliton, kink, singular kink, compacton, contracted bell-shape soliton, periodic soliton, anti-bell shape soliton, and other shape solitons are retrieved for the definite value of these constraints. The 3D and contour plots of some of the results obtained are sketched by assigning individual values of the parameter and analyzed the dynamical behavior of the waves. Furthermore, the compatibility of the two approaches has been compared and examined the efficiency to ascertain soliton solutions.

The M-fractional improved perturbed nonlinear Schrödinger equation: Optical solitons and modulation instability analysis

In order to obtain some novel analytical solutions for fractional improved nonlinear Schrodinger equation with perturbation, the polynomial expansion, simplest equation and extended sine-Gordon expansion schemes are utilized. A collection of different types of solitons are reported with a physical and important perspective, containing optical dark solitons. The methods reveal the solutions in a structure of rapidly converging series. This research highlights additional important features of the methods being considered. In addition, the modulation instability analysis is carried out to discuss the stability analysis of the solutions reached, and the movement role of the waves is examined, which affirms that all the established solutions are exact and stable.

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.

Linear and nonlinear effects analysis on wave profiles in optics and quantum physics

In the field of quantum mechanics and fluid physics, especially in the study of nonlinear geometric optics and superconductivity, the Landau-Ginzburg-Higgs (LGH) and the (2 + 1)-dimensional Novikov-Veselov (NV) equations are two significant models. In this article, we have affirmed that wave profile changes with the change of the free parameters associated with them and are mainly dominated by linear effects. The effects of nonlinearity and wave speed on the wave contours have also been analyzed. On account of this, wave solutions are computed concerning rational, hyperbolic, and trigonometric structures balancing the exponents of linear and nonlinear terms of the highest order, from which scores of typical wave profiles including kink, bell-shape soliton, lump, and periodic waves have been extracted. The wave solutions are designated through extending the typical concept of the sine-Gordon expansion (SGE) method from the lower dimensional to the higher dimensional nonlinear evolution equations. This study establishes the capability of the stated method in solving both lower and higher dimensional nonlinear evolution equations. The solutions are analyzed by sketching figures for different values of related variables, and it is observed that the attributes of these solutions are pivotal in the selection of parameters.

Practical analytical approaches for finding novel optical solitons in the single-mode fibers

In this work, under consideration is a (2 + 1)-dimensional generic model, known as the Hirota-Maccari system, which was studied in plasma physics, fluid dynamics and fiber optics communication systems. The article aims to extract novel soliton solutions of this model with the aid of symbolic computation. Three robust and versatile integration tools of the exp(−Ω(χ))-expansion, the first integral, and the sine-Gordon expansion methods are used to extract dark, bright, and singular solitons solutions. Constraints conditions are explicitly presented for the resulting solutions.

Study of the parametric effect of self-control waves of the Nizhnik-Novikov-Veselov equation by the analytical solutions

A significant model of the preserved scalar nucleons associated with neutral scalar masons is the (2 + 1)-dimensional Nizhnik-Novikov-Veselov equation. Wave solutions are computed concerning exponential, hyperbolic and trigonometric structures balancing the powers in linear and nonlinear terms of the highest order from which scores of typical wave profiles including kink, bell-shape soliton, singular soliton, bright soliton, periodic soliton, dark soliton and dark-bright waves have been extracted. In this article, we have demonstrated that the nature of wave profiles fluctuates regarding the change of the free parameters associated with it and are principally controlled by the linear and nonlinear effect. The effects of other free parameters and wave velocity in the wave profile have also been discussed. The wave solutions are arranged through the improve Bernoulli sub-equation function (IBSEF) method and extending higher dimensional sine-Gordon expansion (SGE) method. Comparing the results obtained from both methods and analyzed the solutions by outlining figures for different values of corresponding variables, and it is detected that the attributes of those solutions are pivotal on the selection of parameters.

Chirped W-shape bright, dark and other solitons solutions of a conformable fractional nonlinear Schrödinger’s equation in nonlinear optics

In this work, two types of integration architecture, namely the sine-Gordon expansion and modified $$\exp (-\psi (\xi )$$ -expansion function methods are used for constructing chirped W-shape bright, dark and other solitons solutions to perturbed nonlinear Schrodinger’s equation with conformable fractional order. The paper addresses optical solitons which are serviced in fields of nonlinear optics. As results, various types of new chirped soliton solutions and other solitons solutions not yet reported in the literature are obtained. These solutions associated with their chirp are included dark, bright optical solitons, hyperbolic, periodic and other types of solitons solutions. Moreover, some graphical representations were done successfully to stress out the virtue of the fractional order derivative.

Soliton solutions for non-linear Kudryashov's equation via three integrating schemes

This paper considers the non-linear Kudryashov's equation, that is an extension of the well-known dual-power law of refractive index and is analog to the generalized version of anti-cubic non-linearity. The model is considered in the presence of full non-linearity. The main objective of this paper is to extract soliton solutions of the proposed model. Three state-of-the-art integration schemes, namely modified auxiliary equation method, the sine-Gordon expansion method and the tanhcoth expansion method have been employed for obtaining the desired soliton solutions.

Onset of the broad-ranging general stable soliton solutions of nonlinear equations in physics and gas dynamics

Stable soliton solutions for the nonlinear Klein–Gordon equation in condensed matter physics, particle physics, nonlinear optics, solid state physics and the gas dynamics equation ensuing in shock fronts have been established by putting use of the sine-Gordon expansion procedure. We establish the broad-ranging familiar stable wave solutions related with some free parameters, for particular values of which some of the subsisting solutions in the literature are re-established and several fresh solutions are formulated. The contour and 3D shapes are depicted to describe the physical phenomena of the established solutions. Scores of solitary wave solutions are obtained such as, compacton, smooth soliton, anti-bell type solitary wave, bell-type solitary wave, kink and some other new types of solitary wave solutions. The obtained results might play important role in optics, quantum mechanics, mathematical physics and engineering. It is noticeable to inspect that the sine-Gordon expansion procedure is an efficient, competent and straightforward mathematical tool to ascertain exact solitary wave solutions.