Search

In this paper, we primarily focus on the nonlinear paraxial wave equation in Kerr media, a generalization of the nonlinear Schrödinger equation utilized to characterize the dynamics of optical beam propagation. By using three potent analytical methods, namely, the Sine-Gordon expansion method, the functional variable method, and the Bernoulli (G′/G)-expansion method, numerous novel soliton solutions are derived. These solutions, comprising hyperbolic, trigonometric and exponential functions, represent significant additions to the field of optics. Furthermore, we elucidate the physical characteristics of these novel optical soliton solutions by presenting a series of three-dimensional (3D) and two-dimensional (2D) graphs with appropriate parameter values. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger–Poisson equation with gravitational potential field is examined, along with its many soliton interactions and travelling wave solutions. By applying Hirota Bilinear scheme, we will give a brief study on transformation of solitons, also by Sine–Gordon expansion scheme solitary travelling wave solutions will be obtained in terms of trigonometric and hyperbolic functions. The innovative Hirota bilinear method is a powerful and standard technique, play a crucial role in producing soliton solutions as well as lump solutions. With the assistance of this method one can transformed integrable equation into Hirota bilinear form under dependent variable transformation. Soliton solutions obtained by Hirota bilinear method are significant in mathematical physics, may be superimposed in fibers. These solitons are applicable in optical communications, which enable to produce faster, richer, more secure and more flexible communication systems. Also we will investigate the gravitational potential's behaviour on soliton solutions, such as parabolic, bright, dark, anti-dark, combination and S-shaped solitons. Contour plots and three-dimensional soliton solutions are provided. These solitons are important in nonlinear processes, magnetic devices, and lasers beams because of variations in gravitational field. [ABSTRACT FROM AUTHOR]

This research focuses on the nonlinear coupled Davey–Stewartson Fokas system, which models pulse propagation in monomode optical fibers. In order to find the novel periodic and solitary wave solutions of the nonlinear coupled Davey–Stewartson Fokas system, we have employed two effective mathematical techniques named as the Sine‐Gordon expansion method and the simple equation method. We found after researching into previous literature that these novel solutions are unique and have never been reported. Some 3D and 2D graphs are also used to discuss the dynamical behavior of these new solutions. [ABSTRACT FROM AUTHOR]

The generalized q-deformed sinh Gordon equation (GDSGE) serves as a significant nonlinear partial differential equation with profound applications in physics. This study investigates the GDSGE's mathematical and physical properties, examining its solutions and clarifying the essence of the q-deformation parameter. The Sardar sub-equation method (SSEM) and sine-Gordon expansion method (SGEM) are employed to solve this GDSGE. The synergistic application of these techniques improves our knowledge of the GDSGE and provides a thorough foundation for investigating different evolution models arising in various branches of mathematics and physics. A positive aspect of the proposed methods is that they offer a wide variety of solitons, including bright, singular, dark, combination dark-singular, combined dark-bright, and periodic singular solitons. Obtained solutions demonstrate the method's high degree of reliability, simplicity, and functionalization for various nonlinear equations. To better describe the physical characterization of solutions, a few 2D and 3D visualizations are generated by taking precise values for parameters using mathematical software, Mathematica.

The main objective of this research is to investigates the intricate realm of high stochastic solitons. The underlying model is an eighth-order nonlinear Schrödinger equation, considering the effects of spatio-temporal dispersion with elevated polynomial nonlinearity and multiplicative white noise. The influence of these factors on soliton's behavior is investigated using Itô calculus. To explore the impact of multiplicative white noise on the governing model more clearly, both bright and dark soliton profiles are generated that are associated with the given equation. Two valid techniques, the Sine-Gordon expansion method and modified sub-equation method that are widely employed in nonlinear dynamics and soliton theory, are utilized to create these solitonic shapes in the presence of white noise. It has been noted that the phase component of the solitons contains the white noise, examining the full spectrum of soliton solutions. These techniques offer a versatile approach to collecting various soliton solutions in this field. Using both 3D and 2D visualization tools, solutions in clear graphical representations are provided. This one-of-a-kind combination of problem and methodology has led to the discovery of a plethora of novel soliton solutions and their accompanying behaviors. [ABSTRACT FROM AUTHOR]

Fractional differential equations are being used to define numerous physical phenomena instead of conventional ordinary or partial differential equations. The secret is to get more generalization of the analysis, hence its solutions. The applications vary between electrical circuit modeling, oscillating circuits, shallow water behavior, viscous fluids, solution transportation, control theory and ultrafast optics. Sine–Gordon Expansion (SGE) Method is being employed hereafter to fully investigate the space–time conformable fractional Vakhnenko–Parkes equation and its modified version. Based on SGE method, the fractional models are transformed into an equivalent ordinary differential equation. The solutions are very important in studying the behavior of ultrafast optics in fibers or waveguides and the wave propagation throughout fiber optics. The solutions are formatted as solitons and are illustrated graphically. The illustrations show that the traveling waves inside fiber optics behave like solitons with traveling peak values according to the wave velocity. One other mode of propagation is to travel in kink shapes. Additionally, the ultrafast optical waves may propagate in shockwaves mode if the waves propagate at ultra-high speeds. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of exact traveling-wave solutions for nonlinear spatial and temporal fractional partial differential equations with conformable order derivatives arising in nonlinear propagation waves of small amplitude including nonlinear fractional modified Benjamin–Bona–Mahony equation, fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation and fractional (2+1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation as well. By utilizing the Sine-Gordon expansion method (SGEM), new real- and complex-valued exact traveling-wave solutions are reported by preferring suitable values of physical free parameters. The nonlinear governing equations are reduced into auxiliary nonlinear ordinary differential equations with aid of fractional traveling-wave transformation, in which the fractional derivative is evaluated in a conformable sense. The productivity process of the proposed method for predicting the desirable solutions is also provided. Some of the obtained solutions are simulated graphically in 3D and contour plots. Meanwhile, the effects of the fractional parameter α in the space and the time direction are illustrated in 2D plots to ensure the novelty, applicability and credibility of the SGEM. These results reveal that the suggested method is general and adequate for dealing with nonlinear models featuring fractional derivatives and can be employed to analyze wide classes of complex phenomena of partial differential equations occurring in engineering and nonlinear dynamics. [ABSTRACT FROM AUTHOR]

In this study, the closed-form wave solutions has been examined to the space–time fractional simplified Camassa-Holm equation through two potential techniques, namely the sine-Gordon expansion approach and the extended tanh function scheme. The equation explains the dispersion effects of several phenomena, including liquid drop patterning in plasma, fluid flow, fission and fusion processes, acoustics, control theory, etc. The fractional-order equation is transformed into a nonlinear equation through a complex transformation. Diverse solutions are determined, and for specified parametric values, the solutions are graphically assessed by depicting 3D and contour plots that provide kinks, singular kinks, plane-shaped, bell-shaped, and other types of solitons. All derived solutions are substituted into the original equation to ensure their accuracy, and the obtained solutions are compared with existing solutions in the literature to demonstrate their novelty. It is noteworthy that the suggested techniques are reliable, competent, and potential mathematical tools to establish closed-form wave solutions.

Nonlinear evolution equations play a crucial role in modelling various physical phenomena, such as fluid flow dynamics and plasma waves. Among them, the generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt (WBKBK) system, along with its special cases, has garnered significant attention in the study of soliton theory due to its intricate mathematical structure and broad range of applications. This study explores the properties and behaviour of soliton solutions in the WBKBK system. The study presents six new classes of wavelike solutions of the WBKBK system and investigate the unique characteristics exhibited by dark and anti-dark solitons, as well as the phenomenon of forward and backward propagating solitons in the system. These findings contribute to a deeper understanding of the fundamental dynamics of the WBKBK system and offer valuable insights into soliton solutions of the system.

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.

This paper explores some novel solutions to the generalized Schrödinger-Boussinesq (gSBq) equations, which describe the interaction between complex short wave and real long wave envelope. In order to derive some novel complex hyperbolic and complex trigonometric function solutions, the sine-Gordon equation method (sGEM) is applied to the gSBq equations. Novel complex hyperbolic and trigonometric function solutions are expressed by the dark, bright, combo dark-bright, W-shaped, M-shaped, singular, combo singular, and periodic wave solutions. The accuracy of the explored solitons is examined under the back substitution to the corresponding equations via the symbolic computation software Maple. It is found from the back substitution outcomes that all soliton solutions satisfy the original equations. The proper significance of the explored outcomes is demonstrated by the three-dimensional (3D) and two-dimensional (2D) graphs, which are presented under the selection of particular values of the free parameters. All the combo-soliton (W-shaped, M-shaped, and periodic wave) solutions are found to be new for the interaction between complex short wave and real long wave envelope in laser physics that show the novelty of the study. Moreover, the applied method provides an efficient tool for exploring novel soliton solutions, and it overcomes the complexities of the solitary wave ansatz method.

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. [ABSTRACT FROM AUTHOR]

In this article, the Fokas-system represents the nonlinear pulse propagation in monomode optical fibers. Soliton solutions of the Fokas-system has been determined using three algorithmic schemes including singular manifold method, the G ′ G 2 -expansion method and Sine-Gordon expansion. The proposed methods have been applied to extract various types of solutions such as bright, dark, singular, periodic and complexitons. Moreover, the constraint conditions for the existence of the solutions will be discussed. To grasp the debated approaches, the obtained solutions are discussed with 3D surface plots and 2D line plots to visualize and support the theoretical results. The extracted results indicate that the proposed methods are excellently explore nonlinear evolution equations. The presented paper provides full spectrum of solutions for the Fokas-System. These novel approaches are used through the symbolic computations impart a dynamical and potent mathematical implement for solving various benevolent nonlinear wave problems. The resulting solutions are novel, intriguing, and potentially helpful for understanding energy transit and diffusion processes in mathematical models of several disciplines of interest, including nonlinear optics. [ABSTRACT FROM AUTHOR]

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.

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 α values and these are given as tables. [ABSTRACT FROM AUTHOR]

The Kawahara equation (KE) and the modified Kawahara equation (MKE) are important modeling equations to interpret shallow water waves with surface tension, magneto-acoustic wave in plasma and gravity waves. In this study, the stable broad-ranging soliton solutions including the well-known bell-shape soliton, anti-bell shape soliton, periodic soliton, compacton, kink, etc. are established through the sine-Gordon expansion (SGE) method. The effect of the physical parameters, namely the dispersion and perturbed dispersive coefficients in the surface elevation are shown through the figure. The surface elevations remain ascertainable for diverse values of the physical and the associated free parameters. The 3D and contour plot of the ascertained solutions clarifies the surface wave properties. The achieved results for each modeling equation provide a significant contribution in analyzing the ion-acoustic waves in plasma, gravity waves, the surface waves in shallow water. This study asserts that the SGE method is reliable, competent, easy and powerful for extracting closed form soliton solutions. [ABSTRACT FROM AUTHOR]

In this article, the propagation of pulses in optical fiber has been studied by considering the nonlinear partial differential equation (NPDE). The proposed model is investigated using two analytical techniques namely the Sine-Gordon expansion (SGE) procedure and the modified auxiliary equation (MAE) method. The trigonometric function, hyperbolic function, and rational function solutions have been extracted from the proposed methods. The employed procedures are compatible in obtaining traveling wave solutions. Moreover, the obtained results are assisted with 3D graphs to demonstrate the physical significance and dynamical behaviors by using different parameter values. [ABSTRACT FROM AUTHOR]

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.

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.

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. [ABSTRACT FROM AUTHOR]

This article presents soliton solutions to a generalized nonlinear Fokas-Lenells equation via the Sine-Gordon expansion method. To uncover the clear picture of the gained solutions, the two and three-dimensional figures for the solutions are given. It is shown that the proposed methodology provides powerful mathematical tools for obtaining the exact traveling wave solutions of different nonlinear evolution equations.

The present study is devoted to using two analytic approaches to study the Triki-Biswas equation (TBE). The TBE model plays a vital role in propagation of short pulses of width around regions of sub-10 fs in optical. The analytic approaches used are the sine-Gordon expansion (SGEM) and the Riccatti Bernoulli sub-ODE (RBSO) methods. Chirped kink-type, bright envelope and singular solitons are formally derived.

Abstract In this paper, the sine-Gordon expansion method is used to obtain analytical solutions of the conformable space-time generalized reaction Duffing model and conformable space-time Eckhaus equation with the aid of symbolic computation. These equations can be reduced into ordinary differential equations (ODEs) using a suitable wave transformation with a predicted polynomial-type solution.

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. [ABSTRACT FROM AUTHOR]

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.

Discrete networks, nonlinear networks, high speed computer data buses, computer network connections, and cable signal distribution are analyzed using the modified Zakharov-Kuznetsov (mZK) equation as a model governing the propagation in the electrical transmission lines. In this study, we establish viable extensive traveling wave solutions with the familiar singular kink type, kink type, bell-type solution, singular bell-type solution, t-type solution, and some advanced soliton solutions through the modified simple equation (MSE) and the sine-Gordon expansion methods. The established solutions are highly stable, resilient and capable to travel long distance. The solutions obtained can determine the voltage and current flow that can be used to design the transmission lines. The density plot and three-dimensional diagrams of the solutions obtained illustrate the voltage and current distributions in the transmission line. The traveling wave solutions are found in closed-form which simplify their utilization. The implemented methods are compatible for the extraction of traveling wave solutions.

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.

In this work, (1 + 2)-dimensional chiral nonlinear Schrödinger's equation is considered. The optical soliton solutions for the aforementioned model are extracted in a systematic way. Two robust and efficient integration tools such as the first integral method and the sine-Gordon expansion method are used to extract abundant solution such as dark, bright soliton as well as complexitons. The parametric conditions for the validity of these solutions are also listed along with the solutions. The physical meaning of the geometrical structures for some solutions is discussed for different choices of the free parameters. [ABSTRACT FROM AUTHOR]

In this paper, we present the sine-Gordon expansion method to prepare the mixed dark bright wave patterns to the nonlinear partial differential equations arising in mathematical physics. Then, we apply the proposed method for a credible recourse of two nonlinear physical models: the modified Vakhnenko-Parkes and modified α -equation. These exact solutions comprise the hyperbolic, trigonometric, rational and exponential function with few licentious parameter. The analytical solutions have different physical structures and they are graphically analyzed in order to show their dynamical behavior by means of 2D, 3D and contour plots. [ABSTRACT FROM AUTHOR]

In this paper, we apply an analytical method, namely, the sine-Gordon expansion method and extract some complex optical soliton solutions to the (2+1)-dimensional extended shallow water wave model, which describes the evolution of shallow water wave propagation. We obtain some complex mixed-dark and bright soliton solutions to this nonlinear model. Considering some suitable values of parameters, we plot the various dimensional simulations of every results found in this manuscript. We observe that our result may be useful in detecting some complex wave behaviors. [ABSTRACT FROM AUTHOR]

This article presents soliton solutions to a generalized nonlinear Fokas-Lenells equation via the Sine-Gordon expansion method. To uncover the clear picture of the gained solutions, the two and three-dimensional figures for the solutions are given. It is shown that the proposed methodology provides powerful mathematical tools for obtaining the exact traveling wave solutions of different nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

The purpose of this study is to find some novel soliton solutions of Fokas–Lenells (FL) equation where the perturbation terms are taken into account with nonlinearity. The sine-Gordon expansion method (SGEM) and the (m + (G ′ / G)) -expansion method are used in this context. The dark, bright, dark–bright and singular optical soliton solutions are successfully obtained. Moreover, the constraint conditions for guaranteeing the existence of solutions are also given. [ABSTRACT FROM AUTHOR]

The Sine-Gordon expansion method is implemented to construct exact solutions some conformable time fractional equations in Regularized Long Wave (RLW)-class. Compatible wave transform reduces the governing equation to classical ordinary differential equation. The homogeneous balance procedure gives the order of the predicted polynomial-type solution that is inspired from well-known Sine-Gordon equation. The substitution of this solution follows the previous step. Equating the coefficients of the powers of predicted solution leads a system of algebraic equations. The solution of resultant system for coefficients gives the necessary relations among the parameters and the coefficients to be able construct the solutions. Some solutions are simulated for some particular choices of parameters. [ABSTRACT FROM AUTHOR]

In this study, the closed-form wave solutions has been examined to the space–time fractional simplified Camassa-Holm equation through two potential techniques, namely the sine-Gordon expansion approach and the extended tanh function scheme. The equation explains the dispersion effects of several phenomena, including liquid drop patterning in plasma, fluid flow, fission and fusion processes, acoustics, control theory, etc. The fractional-order equation is transformed into a nonlinear equation through a complex transformation. Diverse solutions are determined, and for specified parametric values, the solutions are graphically assessed by depicting 3D and contour plots that provide kinks, singular kinks, plane-shaped, bell-shaped, and other types of solitons. All derived solutions are substituted into the original equation to ensure their accuracy, and the obtained solutions are compared with existing solutions in the literature to demonstrate their novelty. It is noteworthy that the suggested techniques are reliable, competent, and potential mathematical tools to establish closed-form wave solutions. [ABSTRACT FROM AUTHOR]

This paper obtains the dark, bright, dark-bright, dark-singular optical and singular soliton solutions to the nonlinear Schrödinger equation with quadratic-cubic nonlinearity (QC-NLSE), which describes the propagation of solitons through optical fibers. The adopted integration scheme is the sine-Gordon expansion method (SGEM). Further more, the modulation instability analysis (MI) of the equation is studied based on the standard linear-stability analysis, and the MI gain spectrum is got. Physical interpretations of the acquired results are demonstrated. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the PNSE.

Abstract In this work, we investigate the optical solitons to the nonlinear Schrödinger equation (NLSE) with spatio-temporal dispersion (STD) arising in ferromagnetic materials and optical fiber. The integration technique adopted is the Sine–Gordon expansion method (SGEM). In addition, the stability analysis of the model is performed. [ABSTRACT FROM AUTHOR]

In this paper, we employ the powerful sine-Gordon expansion method in investigating the solitary wave solutions of the fifth-order nonlinear equation and the Date-Jimbo-Kashiwara-Miwa equation with symbolic computation. We obtain the hyperbolic, trigonometric and complex solutions and the corresponding plots of the solitary wave solutions are given out analytically and graphically. [ABSTRACT FROM AUTHOR]

Nonlinear evolution equations play a crucial role in modelling various physical phenomena, such as fluid flow dynamics and plasma waves. Among them, the generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt (WBKBK) system, along with its special cases, has garnered significant attention in the study of soliton theory due to its intricate mathematical structure and broad range of applications. This study explores the properties and behaviour of soliton solutions in the WBKBK system. The study presents six new classes of wavelike solutions of the WBKBK system and investigate the unique characteristics exhibited by dark and anti-dark solitons, as well as the phenomenon of forward and backward propagating solitons in the system. These findings contribute to a deeper understanding of the fundamental dynamics of the WBKBK system and offer valuable insights into soliton solutions of the system. • The generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt system is studied. • The sine-Gordon expansion method is used to obtain solitary waveform solutions to the system. • The system is characterized by forward and backward propagating kink, antikink and bell-shaped solitary waves. [ABSTRACT FROM AUTHOR]

In this paper, we present new exact solution sets of nonlinear conformable time-fractional coupled Drinfeld-Sokolov-Wilson equation which arise in shallow water flow models, when special assumptions are used to simplify the shallow water equations by means of Sine-Gordon expansion method. We also present an analytical-approximate method namely perturbation-iteration algorithm (PIA) for the system. Basic definitions of fractional derivatives are described in the conformable sense. An example is given and the results are compared to exact solutions. The results show that the presented methods are powerful, reliable, simple to use and ready to apply to wide range of fractional partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, the classification of conservation laws (Cls) of the long short-wave interaction system (LSWS) which appears in fluid mechanics as well as plasma physics is implemented using two Cls theorems, namely, the multipliers approach and the new conservation theorem. The LSWS describes the interaction between one long longitudinal wave and one short transverse wave propagating in a generalized elastic medium. The zeroth-order multipliers and the nonlinear self-adjoint substitutions of the model are derived. Considering the fact that the new conservation theorem needs Lie point symmetries in constructing Cls, we derive the point symmetries of a system of nonlinear partial differential equations (NPDEs) acquired by transforming the model into real and imaginary components. Moreover, we derive some kink-type, bell-shaped, singular and combined soliton solutions to the model using the powerful sine-Gordon expansion method (SGEM). Some figures are presented to show the physical interpretations of the acquired results. [ABSTRACT FROM AUTHOR]

In this paper, the sine–Gordon expansion method (SGEM) and the generalized tanh (GTM) methods are employed to find the optical solitary wave solutions to the perturbed nonlinear Schrödinger equation (PNLSE) with Kerr law nonlinearity describing the propagation of optical solitary waves in nonlinear optical fibers. The dark, bright, dark–bright and dark-singular optical solitary waves are retrieved. Additionally, singular solitary waves are also derived as by-products of the integration scheme. Physical interpretations of the obtained results are demonstrated with interesting figures. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the PNLSE. [ABSTRACT FROM AUTHOR]

In this article, we employed the powerful sine-Gordon expansion method in obtaining analytical solutions of the Benjamin-Bona-Mahony equation. We obtain some new solutions with the hyperbolic function structures. Benjamin-Bona-Mahony equation has a wide range of applications in modelling long surface gravity waves of small amplitude. We also plot the 2- and 3-dimensional graphics of all analytical solutions obtained in this paper. On the other hand, we analyze the finite difference method and operators, we obtain discretize equation using the finite difference operators. We consider one of the analytical solutions to the Benjamin-Bona-Mahony equation with the new initial condition. We observe that finite difference method is stable when Fourier-Von Neumann technique is used. We also analyze the accuracy of the finite difference method with terms of the errors $$L_{2}$$ and $$L_{\infty }$$ . We use the finite difference method in obtaining the numerical solutions of the Benjamin-Bona-Mahony equation. We compare the numerical results and the exact solution that are obtained in this paper, we support this comparison with the graphic plot. We perform all the computations and graphics plot in this study with the help of Wolfram Mathematica 9. [ABSTRACT FROM AUTHOR]

In this paper, using the Painleve property, the traveling wave transformation, and the sine-Gordon expansion method (SGEM), the Tzitzéica type evolution equations in nonlinear optics, including the Tzitzéica equation, the Dodd–Bullough–Mikhailov (DBM) equation, the Tzitzéica–Dodd–Bullough (TDB) equation, and the Liouville equation are studied. As a result, a series of traveling wave solutions for the Tzitzéica type equations are obtained. The efficiency of the solution methodology in exactly solving nonlinear evolution equations is confirmed. [ABSTRACT FROM AUTHOR]

In this paper, with the help of Wolfram Mathematica 9 we employ the powerful sine-Gordon expansion method in investigating the solution structures of the two well known nonlinear evolution equations, namely; Calogero-Bogoyavlenskii-Schiff and Kadomtsev-Petviashvili hierarchy equations. We obtain new solutions with complex, hyperbolic and trigonometric function structures. All the obtained solutions in this paper verified their corresponding equations. We also plot the three- and two-dimensional graphics of all the obtained solutions in this paper by using the same program in Wolfram Mathematica 9. We finally submit a comprehensive conclusion. [ABSTRACT FROM AUTHOR]

In this article, new solitary wave solutions of (2 + 1)−dimensional soliton equation are constructed. This equation is similar to integrable Zakharov equation in plasma physics. The sine-Gordon expansion method and ( G ′/ G )-expansion method are utilized for a reliable treatment of (2 + 1)-dimensional soliton equation. Many novel solutions such as rational, hyperbolic and complex function solutions are characterized with some free parameters to the problem studied. [ABSTRACT FROM AUTHOR]

In this paper, we seek for the new soliton solutions to the coupled nonlinear Maccari's system by using sine-Gordon expansion method. Sine-Gordon expansion method is based on sine-Gordon Eq. (2.1) and a travelling wave transformation. We obtain new complex hyperbolic functions solutions to this system with help of Wolfram Mathematica 9, we also plot 3-dimensional and 2-dimensional graphics to some obtained solution using the same program in Wolfram Mathematica 9. Finally, we submit a comprehensive conclusions. [ABSTRACT FROM AUTHOR]