Role of shallow water waves generated by modified Camassa-Holm equation: A comparative analysis for traveling wave solutions


 There is no denying fact that harmonic crystals, cold plasma or liquids and compressible fluids are usually dependent of acoustic-gravity waves, acoustic waves, hydromagnetic waves, surface waves with long wavelength and few others. In this context, the exact solutions of the modified Camassa-Holm equation have been successfully constructed on the basis of comparative analysis of (G′ / G − 1 / G) and (1 / G′)-expansion methods. The (G′ / G − 1 / G) and (1 / G′)-expansion methods have been proved to be powerful and systematic tool for obtaining the analytical solutions of nonlinear partial differential equations so called modified Camassa-Holm equation. The solutions investigated via (G′ / G − 1 / G) and (1 / G′)-expansion methods have remarkably generated trigonometric, hyperbolic, complex hyperbolic and rational traveling wave solutions. For the sake of different traveling wave solutions, we depicted 3-dimensional, 2-dimensional and contour graphs subject to the specific values of the parameters involved in the governing equation. Two methods, which are important instruments in generating traveling wave solutions in mathematics, were compared both qualitatively and quantitatively. In addition, advantages and disadvantages of both methods are discussed and their advantages and disadvantages are revealed.


Comparative methods for Camassa-Holm equation . ( /G )-Expansion method
We consider two-variable general form of nonlinear partial di erential equations in the general form. Here, let u = u (x, t) = U (ξ ) , ξ = x + vt, v ≠ , where v is a constant and the velocity of the wave. After this, it can be converted into following nonlinear ordinary di erential equation for U (ξ ): The solution of Eq. (3) is assumed to have the form where a i , (i = , , , ..., m.) are constants, m is a positive integer, which is balancing term in Eq. (2), and G = G (ξ ) provides the following second order ordinary di erential equation as: where λ and µ are constants to be determined after, where A is constant. The Eq. (5a) is a solution of the Eq.

. (G /G − /G)-Expansion method
The form of nonlinear partial di erential equation containing two or more independent variables for which the solution can be explored by using G G − Gexpansion method is written as follows: (6) is converted into a nonlinear ordinary di erential equation and this equation can be generally written as: Here, Eq. (7) can be integrated to decrease the operational complexity. By the nature of G G − G -expansion method, G (ξ ) function is solution function of the second order ordinary di erential equation as where λ and µ are real constants. As, ϕ = ϕ (ξ ) = G G and ψ = ψ (ξ ) = G(ξ ) provides operational esthetic. We can write the derivatives of the functions de ned herein as; We can present the behaviors of the solution functions of Eq. (8) with respect to the condition of λ by considering the equations given by Eq. (9).
whereas c and c are arbitrary constants. By considering Eq. (10); Eq. (11) is easily written.
here c and c are arbitrary constants. By considering Eq. (12), there is following equation; here c and c are arbitrary constants. By considering Eq. (14), there is following equation; Finally, the solution of Eq. (7) in terms of ϕ and ψ polynomials is expressed as; Here, a i (i = , , ..., m) and b i (i = , . . . , m) numbers are the constants to be determined later. m is a positive equilibrium term which can be attained by comparing maximum order derivative with the maximum order nonlinear term in Eq. (7). If Eq. (16) is written in Eq. (7) along with Eqs. (9,11,13) or (15) 7) is attained and if ξ = x + vt transformation is operated in reverse order, we will obtain the desired u (x, t) traveling wave solution of Eq. (6).

Solutions of modi ed Camassa-Holm equation . (G /G − /G)-Expansion method
We consider Camassa-Holm Eq. (1). Using transmutation u = u (x, t) = U (ξ ) , ξ = x + vt and taking once the integral of Eq. (1), we get Where, vis the wave velocity. Thus, by nding the equilibrium term m = in Eq. (17), and in Eq. (16) we obtain to following form of the solution If we substitute the Eq. (18) in the Eq. (17) and the coe cients of the algebraic equation are equal to zero, we can establish the following algebraic equation systems aims with ready package program, reaching the solutions of system (19) then we obtained the following cases: replacing the values of Eq. (20) into Eq. (18) then we have the following trigonometric traveling wave solution for Eq. (1)

. ( /G )-Expansion method
We consider Eq. (1). For which using transmutation u = u (x, t) = U (ξ ) , ξ = x + vt, v ≠ , and taking once the integral of Eq. (1), we obtain where, v represents the velocity of the wave. Taking into account the Eq. (26), we nd the equilibrium term m = and in Eq. (4), we attain to following form of the solution If we substitute the Eq. (27) in the Eq. (26) and the coe cients of the algebraic equation are equal to zero, we can establish the following algebraic equation systems : va − vλ a − λ a a + a a = , − λµa a − λ a + a a + va − vλ a − λ a a + a a = , − λµa a − λ a a + a a a = , − vµ a − µ a a − λµa a + a a − λ a + a a = : − µ a a − λµa + a a = , Case I: replacing values Eq. (29) into Eq. (27) and we have the following new type complex hyperbolic traveling wave solution for Eq. (1):

Results and discussions
Shock waves of nonlinear partial di erential equations (NLPDEs) have been discussed on the basis of modeling of physical phenomena. The comparative analysis has been attained by two methods namely G G − Gexpansion and G -expansion method for Camassa-Holm equation. It has been traced out that the solutions are di erent than the existing solutions in literature. This assured that the results have disclosed new phenomenon for Shock waves based on two di erent methods. For the sake of physical aspects, it provides the opportunity to understand the dynamics of solitary waves obtained by two expansion methods their states. The solutions obtained with the G G − G -expansion method are trigonometric, hyperbolic, and rational traveling wave solutions. From comparison point of view, only hyperbolic and complex hyperbolic traveling wave solutions have been obtained via G -expansion method. The solutions obtained by both methods were found to be di erent from each other. In this case, the existence of many methods expresses the richness of the solutions of the di erential equation. G G − G -expansion method more complicated and G -expansion method is less dicult. In this case, we can determine the degree of diculty by referring to the system of Eq. (19) and (28). It was also observed that the processing time in G G − Gexpansion method was longer by using a ready package program with the same features. The excess of the number of equations in the equation system (19) is e ective on the extension of period. It has been observed that all obtained exact solutions, the G -expansion method is advantages in terms of process complexity, while G G − Gexpansion method is more advantages in terms of number of solutions.
In this study, the application of two di erent analytical methods is included, and the solutions obtained at the end of this application are important both mathematically and physically. Mathematically important is the generation of traveling wave solutions. Physically, traveling wave solutions, which play an important role in the transport of energy, will shed light on many problems. If the parameters in the traveling wave solution gain physical meaning by considering the physical properties of the problem under consideration, the obtained traveling wave solutions will be much more valuable. It was observed that the traveling wave solutions obtained by both analytical methods satisfy the modi ed Camassa-Holm equation. At the end of this observation, it can be said that the methods are reliable, useful and applicable methods for obtaining traveling wave solution. Both methods are recommended for obtaining traveling wave solution of NLPDEs in the future.

Conclusion
In this letter, as a result, trigonometric, hyperbolic, complex hyperbolic and rational traveling wave solutions of modi ed Camassa-Holm equation have successfully constructed using G G − G and G expansion methods. 3-D, 2-D and contour graphs are presented for the arbitrary values of the parameters in the solutions obtained. The solutions obtained by both methods have di erent properties and can shed light on some physical events such as di erent shallow water waves. Advantages and disadvantages of two methods discussed. In the future, it can be used to nd traveling wave solutions of many NLPDEs. Because both methods are powerful methods for obtaining traveling wave solutions of NLPDEs.
Acknowledgements: Author Asıf Yokuş is very grateful to Firat University, Elazig, Turkey, Hülya Durur is very grateful Ardahan University, Ardahan, Turkey and Kashif Ali Abro are highly thankful and grateful to Mehran university of Engineering and Technology, Jamshoro, Pakistan for generous support and facilities of this research work.

Con ict of interest:
The authors declare no con ict of interest.

Data Availability Statement:
The data that support the ndings of this study are available from the corresponding author upon reasonable request.