#### Abstract

This article deals with the application of well-known (*G*′/*G*)-expansion to investigate the travelling wave solutions of nonlinear evolution problems including the Boussinesq equation, Klein–Gordon equation, and sine-Gordon equation as these problems appear frequently in mathematical physics. The beauty of the suggested method is to transform the highly nonlinear evolution equation into a system of nonlinear algebraic equations by means of trial solution and auxiliary equation. It is found that the presented approach is simple, efficient, has a less computational cost, and produced rational trigonometric solutions. In order to investigate the novel results, various simulations have been executed. It is renowned that all solutions are in the form of soliton with a single hump and singular which is travelling as time increases gradually. It travels as time travel with the same shape. Moreover, as *A*_{2} decreases the amplitude of the solutions decreases. A comparative study illustrates that some of the obtained solution matches with the existing results against particular values of parameters and various new travelling wave solution attained the first time. The method seems more appropriate by means of a computational work. It can also be extended to demonstrate the behavior of other physical models of physical nature.

#### 1. Introduction

In recent times, the modeling and study of nonlinear complications governed by differential, integrodifferential or integral equations is an authentic area of research. The purpose of such models to seek the physical behavior of natural phenomena that arise in various disciplines of engineering and science including charge transporter, optical and semiconductor devices, game-theory, and dynamics of stochastic and quantum [1, 2]. In the recent century, prominent development is made to comprehend the phenomena of evolution type partial differential equations that are actually a space-time model and arise in electrodynamics, image processing, thermoelasticity, quantum, relativistic field theory, and some other scientific areas including biology, chemistry, physics, material engineering, and sciences [3, 4]. The family of evolution type differential equations are mathematically expressed in the equation (1) [5].where *U* and *H* (*U*) are functions, *U* depends on *x* (space) and *t* (time) while *H* (*U*) is a nonlinear term representing the various physical models [3, 5, 6] and given as follows. Similarly, the same model can be used for two and three dimensional problems where the function *U* will be, respectively, expressed as *U* (*x*, *y*, *t*) and *U* (*x*, *y*, *z*, *t*) while details are given in references [7,8].

The expression (1) along with the above function turned to various models for particular values of *H* (*U*); for example, if we take sin (*U*) then the model will be knowns as the sine-Gordon equation. The model will be converted to sinh-Gordon and cosh-Gordon for the values of *H* (*U*) as sinh (*U*) or cosh (*U*), respectively. The Klein–Gordon equation of quadratic and cubic nonlinearity will be constructed, respectively, by taking *H* (*U*) as *U* − *U*^{2}and *U* − *U*^{3}. The readers can see the cited references [5, 7, 9] to understand the detailed physical aspects of the models and historical overview. However, solutions of these PDEs are of great significance and importance in physical mechanisms as mentioned above.

The advancement in the mathematical techniques to deal with such nonlinear problems is of great importance and several analytical [10, 11], numerical [12–14], spectral [9, 15, 16], and soliton [17–20] based methods has been extended or proposed to deal with nonlinear problems arise in mechanics. The family of numerical methods includes finite difference [21], finite element [22], the finite volume [23–25], and some related extensions which gives numerical solutions of the problems. The class of methods including Adomian decomposition, differential transform [11], variation iteration [10], Homotopy perturbation, Homotopy analysis etc., which give solutions in terms of series contains different drawbacks including calculation of polynomials, multipliers, perturbing, and sometimes diverged for the problems contains the higher nonlinearity factor [19, 20, 24–26]. In the spectral methods, convergence insurance and accuracy level achievement is really high but it got some issues in terms of time and cost. Several studies have been carried out to study these nonlinear problems using soliton methods including Exp-function, , , expansion expansion, etc., and their extensions. Herein, we have included some very recent studies related to the topic. An efficient scheme was first time developed by Wang et al. [27] to explore the exact solution of evolution problem arising in the mathematical physics. Bakir [28] further extended this scheme for the nonlinear problem to investigate its abundant travelling wave solution. The obtained novel solution are of hyperbolic, trigonometric, and rational function forms. Zhang et al. [29] modified this classical (*G*’/*G*)-expansion scheme and introduced its extended version. This extended version of classical (*G*’/*G*)-expansion was applied successfully to obtain the analytical solutions of the Nizhink–Novikov–Vesselov problem. Hamid et al. [17] obtained relatively novel lump kink, lump, and *N*-soliton solutions for (2 + 1)-Kadomtsev–Petviashvili (KPE) problem using a multiple exp-function based approach. Usman and Mohyud-Din [30] proposed new soliton methods to seek the behavior of seventh order Lax and Kaup Kuperschmidt equations and based on the methods generalized solitary wave solutions have been constructed. Kayum et al. [31] reported solitary wave solutions of nonlinear coupled evolutionary models to seek the electric signals and electric transmission lines and in telegraph lines. The analysis is graphically illustrated through contour plots and it is found that the MSE method is well-suited, easy, and influential mathematical tool to extract the soliton solutions. Hyder and Barakat [32] proposed an improved general Kudryashov method (KM) using the concept of a new kind of general auxiliary equation. The compatibility of the method is tested for a class of nonlinear evolutionary problems and obtained solutions are graphically plotted while categorized as periodic and solitary wave solutions [26, 33–40].

Bulut et al. [41] proposed a sine-Gordon methods and used to examine the nonlinear models including, the cubic Boussinesq equation, Zakharov–Kuznetsov modified equal width equation and the modified regularized long wave equation. The solutions have been graphically plotted using the Wolfram Mathematica software while a detailed physical evolution of results is made. Tang et al. [42] reported interaction based soliton results for BLMP and evolutionary equations via a direct approach. It is further stated that a lump seemed from the one side of the soliton wave and dispersed gradually from this side of the soliton wave, after approaching the extreme departure at time zero, the lump gradually treads upon the further side of the soliton wave and last swallowed by the other side. Yıldırım and Yaşar [43] fruitfully applied a multiexp-function based soliton method to analyze the one, two, and three-soliton category of solutions for nonlinear evolution equation Sawada–Kotera (SK) equations. Li and Chen [44] proposed a deep learning based scheme to convalesce the inherent nonlinear dynamics from spatiotemporal data straightly. The numerical tests are performed on some third-order nonlinear evolutionary problems for example, Sharma–Tasso–Olver, KdV–Burgers, modified KdV, and Korteweg–de Vries (KdV) equations. The readers are referred to see more knowledge on the topic in references [34, 45–60]. It is concluded after the above comprehensive literature review is that soliton solution attain significance important to understand the physical model like Zakharov–Kuznetsov model, Korteweg–de Vries model, Sharma–Tasso–Olver, KdV–Burgers model etc., as mentioned in above. In this context, an inclusive study of soliton solution of problem (1) is missing in the literature and needs to be covered to understand the physical behavior of problem (1) in a better way.

The concern of the current study is to extract the soliton solutions of the evolutionary type models including Klein–Gordon, Sine-Gordon, Boussinesq equations that are not reported yet in the above literature survey. It is important to highlight that the soliton solutions of nonlinear evolutionary type of models specifically Boussinesq equations, Klein–Gordon equations and sine-Gordon equations that appear in mathematical physics. However, the travelling wave solutions has been fetched while we proposed an eminent conventional expansion method. As it is noted through the literature that the said scheme got a lot of advantages like simple to apply, less computational cost and rational trigonometric solutions. By means of this proposed soliton scheme, firstly we altered the nonlinear partial differential equation (NL PDEs) under study into a nonlinear ordinary (NL ODEs) problem by means of some suitable transformation. Then, using this scheme the obtained nonlinear ordinary differential equation easily converted into some polynomials Physical behavior of the obtained solution is presented in detail by varying the parameters under study. We also presented a solid relation among the basic ideas of the anticipated method and the current literature is also offered. It is observed that the current scheme is more reliable to build the novel travelling wave solutions of the discussed nonlinear physical problems which is not described before. The work is ordered as the literature survey, and a brief introduction is reported in Section 1. In Section 2, the detailed mathematical methodology is explained. Section 3 is devoted to performing a detailed analysis, results, and physical discussion of the proposed model. Sections 4 and 5 are, respectively, devoted to conclusions and references.

#### 2. Methodology

In this section, we will discuss the methodology of the classical expansion method for investigating the travelling wave solutions of the nonlinear evolution equations under discussion [61, 62]. From the last two decades, this approach is widely used to investigate the travelling wave solution of nonlinear problems of complex nature such as, KdV, modified KdV, Zakharov–Kuzentsov Benjamin Bona Mahony model, Schrodinger problem, Burger model, Nizhnik–Novikov–Vesselov model of (2 + 1) dimensions etc. The discussed expansion method is very simple and has the following step to investigate the soliton solution: *Step 1*. First, we assume the given nonlinear multidimensional partial differential equation which is of the following form: Here in above, is a polynomial in its influences. In order to convert the discussed nonlinear multidimension partial differential problem into nonlinear order differential problem, we consider the following transformation: where is denotes the wave speed. By means of this transformation the discussed nonlinear model reduces to the following ordinary differential equation as follows: and confine the universal solution on travelling wave solutions. Here, is the polynomials of ordinary derivative w.r.t. . In this methodology, we consider that the solution of the problem under study can be conveyed by a polynomial of , which is given as follows: In the above equation, are the unknown parameters that need to be calculated, and the function present in the trial solution must fulfils the following auxiliary equation which is second order linear ordinary differential equation: where and are some arbitrary constants. The General solution of the auxiliary equation is actually depends on the selection of and . There are two main and well-known general solutions of auxiliary equation which are given as follows: where and are the arbitrary constants and it follows, that Here, the prime represents the derivative with respective to *Step 2*. Now, we have to find the value of parameter , for this we will use the homogenous balancing principle that is balancing the highest nonlinear term or terms with the highest-order derivative present in the reduced problem (5). *Step 3*. Substitute the trial solution along with the value of computed , and the above-computed derivatives of into the reduced ordinary differential problem, which yields the polynomial in . Later on, collect the terms like powers of . *Step 4*. System of nonlinear algebraic equations attained by equating the coefficient of like power of . *Step 5*. To solve the system of nonlinear algebraic equations achieved in the last step for unknown parameters using any mathematical software. *Step 6*. After solving the system of nonlinear equations we obtained the multiple solution sets. For each solution set, the travelling wave solution is attained of the problem under study after incorporating it with the solution of nonlinear auxiliary equation into the trial solution.

#### 3. Applications of (*G*′/*G*)-Expansion Method

This section concerns the applications of different cases of the problem given in equation (1) by means of the proposed expansion method. The work is further divided into different cases that will help readers to understand and seek the behavior more easily. *Case 1*. Boussinesq equation. Consider the following fourth-order nonlinear Boussinesq partial differential equation as follows: It is noted that the above-mentioned Boussinesq model is the completely integrable. Furthermore, when the problem under discussion is known as the bad Boussinesq equation or ill posed classical problem, while when then the model under discussion is called the good Boussinesq equation or well-posed problem. In order to find the travelling wave solution of this problem, we first consider the following transformation: By means of this transformation the problem (6) takes the following form: The simplified form of this problem is given bellow after integrating it as follows: According to the traditional expansion scheme the trial solution for solving the discussed problem is given as follows: Here, the parameter can be investigated by using the homogenous balancing principle, that is balancing the highest nonlinear term and the highest-order derivative as, or Therefore, the reduced trial solution after inserting the values of calculated in above is given as follows: After incorporating the reduced trial solution (14) into the nonlinear ordinary differential equation given in (12), we obtained the algebraic equation or polynomial having different powers of as follows: Equating the coefficients of like powers of on both sides, we obtained the system of nonlinear algebraic equations. After solving this system of nonlinear algebraic equation we obtained the following solution sets: The multiple travelling wave solution of the considered model is obtained after incorporating the above-given solution sets along with and the solution of auxiliary equation into the reduced trial solution (14). *Case 2*. Klein–Gordon equation. Consider the following nonlinear Klein–Gordon partial differential equation as follows: It is also known as the relativistic wave equation that is related to the well-known Klein–Gordon equation. In order to find the closed form solution, we first consider the following transformation: By means of this transformation the discussed problem takes the following form: According to the classical expansion scheme the trial solution for solving the discussed problem is given as follows: Here, the parameter can be investigated by using the homogenous balancing principle, that is balancing the highest nonlinear term and the highest-order derivative as, or Therefore, the reduced trial solution after inserting the values of calculated in above is given as follows: After incorporating the reduced trial solution (21) into the nonlinear ordinary differential equation given in (19), we obtained the algebraic equation or polynomial having different powers of as follows: Equating the coefficients of like powers of on both sides, we obtained the system of nonlinear algebraic equations. After solving this system of the nonlinear algebraic equation we obtained the following solution sets: The multiple travelling wave solution of the considered model is obtained after incorporating the above-given solution sets along with and the solution of auxiliary equation into the reduced trial solution. *Case 3*. Sine-Gordon equation. Consider the following nonlinear Sine-Gordon partial differential equation as follows: This problem increased its reputation when it offered kink and anti-kink solutions through the collisional performances of the solitons. It is noted that the first arrival of this problem is not in the wave equations; however, in differential geometry of surfaces through Gaussian curvature . It is important to mentioned that the sine-Gordon model arising in many scientific fields like promulgation of fluxons in Josephson junctions among two superconductors, solid state physics, gesture of rigid pendulum committed to a strained wire, metals dislocation and nonlinear optics. To investigate the travelling wave solution of the discussed nonlinear sine-Gordon equation. First, we introduced the following transformation: This implies that From the above-given expression, we have By means of the above-given transformation, the discussed problem is converted to the following simplest form as follows: Now, consider the following transformation: By means of this transformation the sine-Gordon problem takes the following form: According to the classical expansion scheme the trial solution for solving the discussed problem is given as follows: Here, the parameter can be investigated by using the homogenous balancing principle, that is balancing the highest nonlinear term and the highest-order derivative as, or Therefore, the reduced trial solution after inserting the values of calculated in above is given as follows: After incorporating the reduced trial solution (21) into the nonlinear ordinary differential equation given in above, we obtained the algebraic equation or polynomial having different powers of as follows: Equating the coefficients of like powers of on both sides, we obtained the system of nonlinear algebraic equations. After solving this system of nonlinear algebraic equation we obtained the following solution sets: The multiple travelling wave solution of the considered model is obtained after incorporating the above solution sets along with and the solution of auxiliary equation into the reduced trial solution.

#### 4. Results and Discussion

In the last section, we studied in detail about the travelling wave solution of nonlinear evolution problems including bad and good Boussinesq, Klein–Gordon and Sine-Gordon problems by means of (*G*′/*G*)-expansion method. This section is contracted to discuss the physical behavior of the obtained solution using the classical (*G*′/*G*)-expansion scheme. Maple code is developed for the discussed problems using the proposed algorithm and then simulate various simulation to the physical behavior of the considered problem on core(TM) i7-1165G7 with 2.80 GHz processor and 16 GB RAM.

Figure 1 is plotted to examine the behavior of the good Boussinesq model for various values of under discussed parameters and time. It is noted that the solution are in the form of soliton with single hump and singular which is travelling as time increases gradually. It is also noted that as *A*_{2} decreases the amplitude of the solutions decreases and when *A*_{1} < *A*_{2} the behavior of the solutions lies in the fourth coordinate of the Cartesian plan. Figure 2 show that graphical illustration of the bad Boussinesq model. In that figures we can analyze that there is insignificant effect of the time on the travelling solution solutions and the singular region also increases in this case as compare to the good Boussinesq equation.

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In order to demonstrate the behavior of Klein–Gordon equation, Figure 3 is plotted. Here, we can analyze the same phenomena which we obtained before that is it form a soliton with single hump and singular which is travelling as time increases gradually. But here the wave is travelled very slowly as enhancing the time. Insignificant impact of when is obtained here. But when the clear difference is obtained from the previous behavior. In this case all the soliton solution lies in the first coordinate. Figure 4, contains the physical behavior the nonlinear Sine-Gordon model under the variation of different parameters. It is observed that, it also demonstrate the similar phenomena as we obtained in case of Klein–Gordon equation. That is, soliton is obtained with single hump and singular which is travelling as time increases gradually. Again, the minor influence of when is attained. However, when the clear change is attained from the preceding behavior. Table 1 is constructed to show the comparison of the obtained results with the exsiting [3]. It is worthy to point out that some solution at particular values of parameter matches the solution present in the literature [3], that shows the effectiveness of the suggested scheme and the solution against the rest of the solution sets are novel and first time investigated.

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#### 5. Conclusions

The concern of the current study is to extract the soliton solutions of the evolutionary type models including Klein–Gordon, Sine-Gordon, Boussinesq equation. The soliton solutions of nonlinear evolutionary type of models specifically Boussinesq equations, Klein–Gordon equations and sine-Gordon equations that appear in mathematical physics. However, the travelling wave solutions has been fetched while we proposed an eminent conventional (*G*′/*G*)-expansion method. The key findings are as follows:(i)All solutions are in the form of soliton with single hump and singular which is travelling as time increases gradually(ii)As *A*_{2} decreases the amplitude of the solutions decreases and when *A*_{1} < *A*_{2} the behavior of the solutions lies in the fourth coordinate of the Cartesian plan(iii)The beauty of the proposed scheme is that it is easy to implement, reliable, concise, and efficient and the method provides some remarkably specific type of solutions(iv)The proposed scheme is hard to implement for variable order fractional differential equations arising in mathematical physics(v)This scheme does not produce link-lump and multiple solution of nonlinear problems(vi)In future, the presented method can be extended for fractional-order and variable-order physical model of physical nature(vii)Moreover, a more exact solution can be found using the appropriate choice of trial function

#### Data Availability

The datasets used and/or analyzed during the current study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest.

#### Acknowledgments

The author give thanks for language polishing service from EMATE at http://www.emate.ac.cn/en/home.html.