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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.

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, 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.

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.

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.

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]

This aim of this paper is threefold. A survey of trigonometric-hyperbolic function methods for constructing solitary wave solutions of nonlinear equations. Further analysis of the sine-Gordon expansion method. Studying the solution ol the generalized PHI-four equation and the generalized regularized long wave equation by using a special transformation derived from the Sine-Gordon equation. [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. • The coupled conformable Fokas–Lenells equation with spatio-temporal dispersion is investigated. • The Sine–Gordon expansion method is applied to that equation. • A graphical representation of the obtained soliton solutions is presented. [ABSTRACT FROM AUTHOR]

• Abundant solutions of the electrical transmission lines in mZK equation are found. • Performance was done using the strategy of two different techniques. • Physical explanations are discussed for the obtained solutions. 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. [ABSTRACT FROM AUTHOR]

• The broad-ranging general stable soliton solutions to some nonlinear equations have been established associated with parameters and integral constants in physics and gas dynamics. • The characteristics of gas flow ensuing shock fronts and nonlinear optics have been analysed. • The 3D and contour plots of the obtained solutions are depicted to illustrate profile of the phenomena. 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. [ABSTRACT FROM AUTHOR]

This paper presents many new complex combined dark-bright soliton solutions obtained with the help of the accurate sine-Gordon expansion method to the B-type Kadomtsev-Petviashvili-Boussinesq equation with binary power order nonlinearity. With the use of some computational programs, we plot many new surfaces of the results obtained in this paper. In addition, we present the interactions between complex travelling wave patterns and their solitons. [ABSTRACT FROM AUTHOR]