Novel Soliton Solutions of Two-Mode Sawada-Kotera Equation and Its Applications

The Sawada-Kotera equations illustrate the non-linear wave phenomena in shallow water, ion-acoustic waves in plasmas, fluid dynamics, etc. In this article, the two-mode Sawada-Kotera equation (tmSKE) occurring in fluid dynamics is considered which is important model equations for shallow water waves, the capillary waves, the waves of foam density, the electro-hydro-dynamical model. The improved F-expansion and generalized exp $(-\phi (\zeta))$ -expansion methods are utilized in this model and abundant of solitary wave solutions of different kinds such as bright and dark solitons, multi-peak soliton, breather type waves, periodic solutions, and other wave results are obtained. These achieved novel solitary and other wave results have significant applications in fluid dynamics, applied sciences and engineering. By granting appropriate values to parameters, the structures of few results are presented in which many structures are novel. The graphical moments of the results are provided to signify the impact of the parameters. To explain the novelty between the present results and the previously attained results, a comparative study has been carried out. The restricted conditions are also added on solutions to avoid singularities. Furthermore, the executed techniques can be employed for further studies to explain the realistic phenomena arising in fluid dynamics correlated with any physical and engineering problems.


I. INTRODUCTION
The dynamic complexity of physical phenomena in the real world can be expressed by the changes in temporal and spatial events. The temporal and spatial changes of physical phenomena are greatest articulated by partial differential equations (PDEs). The nonlinear PDEs are utilized for expressing various physical phenomena in the real world to get an insight through qualitative and quantitative features of many models that arise in diverse fields. Nonlinear wave phenomena emerge in plasma physics, fluid mechanics, solid-state physics, dynamics of chemical, non-linear optics, population model and other fields of science and engineering [1]- [19]. The analytical solutions of non-linear PDEs play a decisive part in non-linear science as they inform The associate editor coordinating the review of this manuscript and approving it for publication was Ladislau Matekovits . us deep imminent into the physical characteristics of the model and can provide further physical information to help in other applications. In recent years, the approximate and exact solutions of non-linear PDEs have attracted more and more attention, as they are utilized to illustrate the nonlinear complex phenomena in dissimilar scientific areas. Numerous real-world problems are altered into equations mathematically by differential equations. Thus, the finding wave results of all kinds of PDEs are a major problem, such as the present direction of non-linear science, which originated from the research of chemistry, physics, material science, biology, and many more, and has a burly practical backdrop. They have significant realistic applications and theoretical study in mathematics.
As results, few solitons results in the form Kink, Kinks type of multiple soliton, periodic wave of singular kind, dark and bright solitons solutions have been conceded out for the aforementioned models.
The researcher Wazwaz [5] developed the tmSKE from the tmfKdV equation, and few multiple solitons results were determined by the simplified Hirota technique. Later on, the researchers in [23] investigated the tmfKdV model and established some Kink, bright and periodic solutions in singular form by using sine-cosine function and Kudryashov techniques. The authors in [18] were used modified Kudryashov and auxiliary equation methods, and dual wave solutions were constructed. It should be pointed out that the tmSKE is a special case of the tmfKdV equation. As far as the author is aware, although some two-mode PDEs have been extensively studied, the contributions to the above tmSKE are limited. It can be seen from the literature that there is room for further study of the tmSKE through the improved F-expansion and generalized exp(−φ(ζ ))-expansion methods, as well as the illustrating their physical explanations. The results executed by the projected methods are to be novel in the sense of methods application.
Several powerful and systematic methods (analytic, semianalytic, and numerical methods) have been developed for studying non-linear PDEs [29]- [62] This work aims to obtain solitons and other wave results of tmSKE. It is of interest to note here that the generalized exp(−φ(ζ ))-expansion method is an extended form of the exp(−φ(ζ ))-expansion method, and the improved F-expansion method is also an extended form of F-expansion method. Thus, motivated by the existing literature, a modest effort has been made in this study to construct some new dual-wave solutions to the TmSK equation via the project methods. The solutions attained by the improved F-expansion and generalized exp(−φ(ζ ))-expansion methods are to be new in the sense of methods application. The constructed results are novel and more general. To our best knowledge, these approaches are not utilized to address the early work on this equation. This paper is structured as follows. In Section 1, specifies the introduction. In Section 2, a summary of the general form of tm standard and tm SK equations are summarized. In Section 3, the review of the improved F-expansion and generalized exp(−φ(ζ ))-expansion techniques are depicted. The constructed results from the investigation are given in Section 4. In Section 5, a general discussion and graphical illustrations of some acquired solutions are presented. Finally, the conclusion and future recommendations of the article are illustrated in Section 6.

II. FORMULATION OF MATHEMATICAL MODELS A. GENERAL TYPE OF DUAL-MODE STANDARD MODEL
The general type of the two-mode or dual-mode model proposed by Korsonski [10] is as the above equation (1) is recognized from the equation of standard mode: In equation (1), the function u(x, t) is an unknown with (t, x) ∈ (−∞, ∞), and ν > 0 is velocity of the phase, β ≤ 1, γ ≤ 1, β and γ symbolize nonlinearityjand dispersion parameters respectively. The terms L ∂ 2 u ∂r∂x , r ≥ 2 and N u, u ∂u ∂x , . . . signify the terms of linear and nonlinear respectively.

B. DUAL-MODE SAWADA-KOTERA MODEL
The SKE in standard form having two non-linear terms [5] has as in above equation, the terms ∂ 5 u ∂x 5 and ∂ ∂x u 3 3 + u ∂ 2 u ∂x 2 are linear and nonlinear respectively. VOLUME 9, 2021 Merging the sense of Korsunsky [10], and follow Wazwaz [5], the tmSKE of the standard SKE precises by equation (2) is presented as Obviously, for ν = 0, the tmSKE specified through equation (3) after integrating the relevant time t has been simplified to the standard mode SKE given through equation (2).
The equation (3) illustrates the proliferation of two moving waves under the persuade of phase velocity ν, dispersion (γ ), and non-linearity (β) factors.

III. PORTRAYAL OF PROPOSED METHODS
Here, we reveal the algorithms of suggested techniques namely as improved F-expansion and generalized exp(−φ(ζ ))-expansion methods for constructing the wave results of two-mode Sawada-Kotera model. The general nonlinear PDE has as where the polynomial function G having unknown function v(x, t) with respect to a few specific independent variables x and t, that also having derivative terms of linear and nonlinear. Assuming the transformation for changing independent variables into sole variable has as where the constant k and ω are wave length and frequency. Utilizing (5), the equation (4) is converting into ODE as where U = dU dζ and F is a polynomial of U and its derivatives.

A. IMPROVED F-EXPANTION METHOD
The main steps are as 1st Step: Consider the solution of Eq.(6) has as where the constants A i , B −j , µ are real and the function F(ζ ) in equation (7) pledges the below ODE where δ 0 , δ 1 , δ 2 and δ 3 are real constants. 2nd Step: By utilizing homogeneous balance principle on Eq.(6), the positive integer N is obtained. 3rd Step: Deputizing Eq. (7) into Eq.(6) and taking the various coefficients of F i (ζ ) (µ+F(ζ )) j to zero, capitulate a system of equation. By using Mathematica, this system is solved and constant values can be achieved. After substituting constant values and solutions of Eq.(6), the wave solutions of Eq. (7) are constructed.
The main steps are as 1st Step: Assume the solution of Eq.(6) has the form as where where a, b, c are real constants. 2nd Step: Utilizing homogeneous balance principle on Eq.(6), the positive integer N is obtained. 3rd Step: By Deputizing equation (9) into (6) and polynomial obtained in e (−φ(ζ )) , and taking diverse powers of (e (−φ(ζ )) ) i to zero, capitulate a system of equation. By resolving this system and reverse substitution, we construct many exact solutions for Eq.(4).

IV. APPLICATIONS
In this part, we construct the solitons and other waves solutions of two-mode Sawada-Kotera equation by employing described methods. By employing the transformation described in Eq.(5), the Eq. (3) is converted into ODE as
3rd Family: In this family, we assume as δ 3 = 0, Set 1: See (26), as shown at the bottom of the page. Set 2: .

B. APPLICATION OF GENERALIZED EXP(−φ(ζ ))-EXPANSION METHOD
In this part, we employ generalized exp(−φ(ζ ))-expansion method on two-mode Sawada-Kotera for constructing the solitons and more waves solutions. Employing balancing principle of homogeneous on Eq. (11) and assume the wave solution as By substituting Eq.(30) into Eq.(11) and deputing the coefficients of e (−φ(ζ )) i to zero, we achieved a equations system A 0 , A 1 , A 2 , a, b, c, k, ν, ω, η, β. Mathematica 9 was utilized to resolve the equations set. We attained below families as: 1st Family: 2nd Family: 3rd Family: See (33), as shown at the bottom of the page. 4th Family: From 1st family, the different forms of solitons and other solutions of Eq.(3) are obtained as Type I: For a = 1, b = 0, c 2 − 4b > 0, (35), as shown at the bottom of the next page.

V. DISCUSSION OF RESULTS AND GRAPHICAL REPRESENTATION
The accomplished solutions are dissimilar from the results obtained by other researchers in the previous methods. The equations (8) and (10) present numerous dissimilar kinds of solutions by giving different values of parameters. It was announced earlier that the tmSKE was studied by some authors is given in Table 1.
Pedestal on the applications of these methods, the authors report some bright, dark, multi-solitons, singular periodic and kink structured results with the restricted conditions β = γ = 1. However in this article, eighteen wave solutions are constructed through the improved F-expansion method and thirty-six wave solutions are constructed through the generalized exp(−φ(ζ ))-expansion technique. The explored solutions demonstrate the dual-mode bright, dark, periodic, Kink, multi soliton and singular wave behaviors that are being classified as waves of right/left mode. Evaluated with published results [5], [18], [23], it is worth revealed that the constructed dual-wave solutions are new for the interests of applied methods. As a result, we have constructed several original results, which have not been explained before.
The Figures 1 to 4 indicate the solitons and other waves in dissimilar structures are described. In the Figure 1, by granting appropriate values to parameters, the formation of solutions (16) and (17) are revealed as: Fig(1-A) Dark solitary wave and its 2-dimensional (2D) in Fig(1-B), Fig(1-C) bright soliton and its 2D in Fig(1-D). By granting appropriate values to parameters, the formation of solutions (18) and (19) in Figure 2 are revealed as: Fig(2-A) is Multi-peak solitons and its 2D in Fig(2-B), Fig(2-C) is solitary wave of anti-Kink type and its 2D in Fig(2-D). In Figure 3, by granting appropriate values to parameters, the shape of solutions (22) and (23) are shown as: Fig(3-A) dark periodic solitary wave and its 2D in Fig(3-B), Fig(3-C) is dark soliton and its 2D in Fig(3-D). By granting appropriate values to parameters, the shape of solutions (24) and (29) in Figure 4 are shown as: Fig(4-A) Multi peak soliton of different amplitude and its 2D in Fig(4-B), Fig(4-C) periodic solitary wave and its 2D in Fig(4-D).
The Figures 5 and 6 illustrate the solitary waves in dissimilar structures are described. In the Figure 5, By granting appropriate values to parameters, the shape of solutions (35) and (37) are shown as: Fig(5-A) is bright soliton wave and its 2D in Fig(5-B), Fig(5-C) is dark solitary wave and its 2D in Fig(5-D). By granting appropriate values to parameters, the shape of solutions (44) and (45) in Figure 6 are shown as: Fig(6-A) is Kink soliton wave and its 2D in Fig(6-B), Fig(6-C) is Breather wave of strange shape and its 2D in Fig(6-D).

VI. CONCLUSION
The described methods namely, the improved F-expansion method and generalized exp(−φ(ζ ))-expansion method have been effectively employed on the tmSKE and as consequences, abundant of different kinds of solitons and other waves solutions such as bright and dark solitons, multi-peak soliton, breather type waves, periodic solutions are obtained. The two-mode equation describes the spread of moving two-waves under the influence of dispersion, nonlinearity, and phase velocity factors. The obtained novel solitons and other wave results have significant applications in fluid dynamics, applied sciences and engineering. The Sawada-Kotera equations illustrating the non-linear wave phenomena in shallow water, ion-acoustic waves in plasmas, fluid dynamics, etc., and tmSKE also arising in fluid dynamics is addressed in this article. We may say that these two-waves solutions could be useful in many physical and engineering applications, for example, they can be used as barrier waves to strengthen the transmission of different signals data. Also, if a huge amount of data is complicated to pass on to a single router, it can be dispersed on two routers. The graphical moments of few solutions are depicted that helps the engineers and scientists for understanding the physical phenomena of this model. The restricted conditions are also added on solutions to avoid singularities. To explain the novelty between the present results and the previously attained results, a comparative study has been presented. The computational work and constructed results approve the effectiveness, simplicity, and impact of described techniques. Furthermore, the described techniques can be employed to any two-mode nonlinear PDEs and other models arising in fluid dynamics correlated with any physical and engineering problems to explore novel dual-wave and other wave solutions. The fractional derivative of this two-model will also consider to obtain such types of results by utilizing the described techniques. Our future work would be intense towards investigating the new dual-wave solutions by using different analytical, semi-analytical, and numerical methods to the tmSKE and fractional tmSKE.