1. Introduction

In recent years, various methods have been introduced to generate analytical solutions of the nonlinear evolution equations (NLEEs) in mathematical physics, engineering sciences and other real world problems. For example, the inverse scattering method ^[1], the homogeneous balance method ^[2], the generalized Riccati equation method ^{[3, 4, 5, 6]}, the F-expansion method ^[7], the tanh-coth method ^[8], the Exp-function method [9-11]^[9] and others ^{[12, 13, 14]}.

Firstly, the -expansion method is proposed by Wang et al. ^[15] in 2008. They applied this method to some nonlinear partial differential equations and is used as the travelling wave solution where Subsequently, many researchers applied this method to study various nonlinear partial differential equations ^{[16, 17, 18]}.

Recently, -expansion method is extended by Zhang et al. ^[19] and they presented the solution form as , where either or may be zero, but both or cannot be zero at a time. Consequently, a diverse group of scientists investigated a different class of nonlinear PDEs by using this effective and straightforward method for establishing many new travelling wave solutions. For example, Hamad et al. ^[20] studied higher dimensional potential YTSF equation to construct analytical solutions via this method. Nofel et al. ^[21] implemented same method to the fifth-order KdV equation for obtaining travelling wave solutions whereas Naher et al. ^[22] investigated compound KdV-Burgers equation.

Naher and Abdullah ^[23] executed the same method to construct travelling wave solutions of the nonlinear reaction diffusion equation whilst they ^[24] studied the (2+1)-dimensional modified Zakharov-Kuznetsov equation for constructing abundant new travelling wave solutions by applying this method. Again, Naher and Abdullah ^[25] constructed analytical solutions of the (2+1)-dimensional breaking soliton equation and Naher et al. ^[26] investigated the higher dimensional Jimbo-Miwa equation via the same method for obtaining some new solutions. Furthermore, in Ref. ^[27] Naher and Abdullah applied this method to the higher dimensional Kadomstev-Petviashvili equation to establish some new analytical solutions and so on.

In this article, auxiliary equation intends to be used to generate analytical solutions of the SMCH equation. In addition, some new solutions coincide with previous results which are available in the open literature. Also, some new solutions are displayed in the figures.

2. Description of the Method

Consider the general nonlinear partial differential equation:

(1)

where is an unknown function, is a polynomial in and the subscripts stand for the partial derivatives.

The main steps of the method ^[19] are as follows:

Step 1. Suppose, combining the real variables and by a new variable :

(2)

where is the speed of the travelling wave. Now using transformation of (2) in equation (1) obtain to the following ordinary differential equation for

(3)

where the superscripts indicate the ordinary derivatives with respect to .

Step 2. According to possibility, equation (3) can be integrated term by term one or more times, then we can obtain constant(s) of integration. For simplicity, we can consider zero for integral constant.

Step 3. Suppose that the travelling wave solution of (3) can be expressed in the following form:

(4)

with satisfies the following second order linear ODE:

(5)

where and are constants to be determined later.

Step 4. To determine the positive integer , for this reason need to take the homogeneous balance between highest order nonlinear terms and highest order derivatives of equation (3).

Step 5. Substituting equation (4) and equation (5) into equation (3) with the value of which obtained in Step 4, then obtained polynomials in and setting each coefficient of the resulted polynomials to zero, it follows a set of algebraic equations for and

Step 6. Solving the system of algebraic equations which are obtained in step 5 with the aid of algebraic software Maple and then, we can obtain values for and Furthermore, substituting the values of and in equation (4) along with the general solution of (5) which is well known, then obtaining some new travelling wave solutions of equation (1).

3. Application of the Method

In this section, we have investigated the SMCH equation by construction some new analytical solutions which including solitons solutions and periodic solutions via the improved -expansion method.

In this work, we choose the SMCH equation:

(6)

where and

Details of CH and MCH equation can be found in references ^{[28, 29, 30, 31, 32]}.

Now, we use the wave transformation equation (2) into equation (6), which yields:

(7)

where the superscripts stand for the derivatives with respect to

Equation (7) is integrable, therefore, integrating with respect to once yields:

(8)

where is an integral constant which is to be determined later.

Balancing with in equation (8), we obtain So, the solution of (8) is the form:

(9)

where and are constants to be determined.

Substituting equation (9) together with equation(5) into (8), the left-hand side of (8) is converted into a polynomial of According to Step 5, collecting all terms with the same power of Then, setting each coefficient of the resulted polynomial to zero, yields a set of algebraic equations (for simplicity, which are not presented) for and

Solving the system of obtained algebraic equations with the help of algebraic software Maple, we obtain three different values.

Case 1:

(10)

where are free parameters and

Case 2:

(11)

where are free parameters and

Case 3:

(12)

where are free parameters and

Substituting the general solution equation (5) into (9), we obtain three different families of travelling wave solutions of (8):

Family 1: Hyperbolic function solutions:

When we obtain

(13)

If and are taken particular values, various known solutions can be rediscovered.

Family 2: Trigonometric function solutions:

When we obtain

(14)

If and are taken particular values, various known solutions can be rediscovered.

Family 3: Rational function solution:

When we obtain

(15)

Substituting expression of (10), (11) and (12) in (9), account into the general solution (5), then yields the hyperbolic function solution for equation (13), with corresponds to the conditions , :

where

where

where

Substituting expression of (10), (11) and (12) in (9), account into the general solution (5), then yields the trigonometric function solution for equation (14), which are traveling wave solutions corresponds to the conditions , :

Substituting expression of (10), (11) and (12) in (9), account into the general solution of (5) then yields the rational function solution for equation (15), we obtain following traveling wave solutions corresponds to the condition :

where

where

where

4. Results and Discussion

It is worth declaring that some of our obtained solutions are in good agreement with already published results which are presented in the following table. Moreover, some of obtained traveling wave solutions are described from the Figure 1 to Figure 8.

Figure 1. Periodic solution for

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Figure 2. Periodic solution for

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Table

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Beyond this table, we obtain new exact travelling wave solutions and which are not being established in the previous literature.

The graphical presentations of some new solutions are depicted in the figures with the aid of commercial software Maple:

Figure 3. Solitons solution for

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Figure 4. Solitons solution for

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Figure 5. Solitons solution for

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Figure 6. Periodic solution for

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Figure 7. Solitons solution for

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Figure 8. Periodic solution for

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5. Conclusions

In this article, the linear ODE is used with the improved -expansion method to construct some new travelling wave solutions of the SMCH equation. The new obtained solutions are presented through the hyperbolic function, the trigonometric function and the rational functions form. If parameters take particular values some of the obtained solutions coincide with published results which describes in result and discussion section and verify that our obtained all solutions are correct. In addition, some figures are illustrated which examined new obtained solutions.

References

[1]	Ablowitz, M.J. & Clarkson, P.A. 1991. Solitons, nonlinear evolution equations and inverse scattering transform. Press, .
	In article		CrossRef PubMed
[2]	Wang, M.L., Zhou, Y.B. & Li, Z.B. 1996. Application of homogeneous balance method to exact solutions of nonlinear equations in mathematical physics. Pyss. Lett. A 216: 67-75.
	In article		CrossRef
[3]	Yan, Z. & Zhang, H.Q. 2001. New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water. Phy. Lett. A 285: 355-362.
	In article		CrossRef
[4]	Naher, H., Abdullah, F. A., & Mohyud-Din, S. T. (2013). Extended generalized Riccati equation mapping method for the fifth-order Sawada-Kotera equation. AIP Advances, 3(5), 052104.
	In article		CrossRef
[5]	Naher, H., & Abdullah, F. A. (2012). New Traveling Wave Solutions by the Extended Generalized Riccati Equation Mapping Method of the-Dimensional Evolution Equation. Journal of Applied Mathematics, 2012.
	In article
[6]	Naher, H., & Abdullah, F. A. (2012). The modified Benjamin-Bona-Mahony equation via the extended generalized Riccati equation mapping method. Applied Mathematical Sciences, 6(111), 5495-5512.
	In article
[7]	Wang, M.L. & Li, X.Z. 2005. Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos, Solitons and Fractals, 24: 1257-1268.
	In article		CrossRef
[8]	Malfliet, W. 1992. Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60: 650-654.
	In article		CrossRef
[9]	He, J.H. & Wu, X.H. 2006. Exp-function method for nonlinear wave equations. Chaos Solitons and Fractals. 30: 700-708.
	In article		CrossRef
[10]	Naher, H., Abdullah, F.A. & Akbar, M.A. 2011. The exp-function method for new exact solutions of the nonlinear partial differential equations. International Journal of the Physical Sciences. 6(29): 6706-6716.
	In article
[11]	Naher, H., Abdullah, F.A. & Akbar, M.A. 2012. New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the Exp-function method. Journal of Applied Mathematics. Article ID: 575387, 14 pages.
	In article
[12]	Maurya, V.N., Gandel, Y. V., Dushkin, V.D. 2014. The Approximate Method for Solving the Boundary Integral Equations of the Problem of Wave Scattering by Superconducting Lattice. American Journal of Applied Mathematics and Statistics, 2 (6A): 13-19.
	In article		CrossRef
[13]	Daga, A., Pradhan, V. 2014. Variational Homotopy Perturbation Method for the Nonlinear Generalized Regularized Long Wave Equation. American Journal of Applied Mathematics and Statistics, 2 (4): 231-234.
	In article		CrossRef
[14]	Patel, K. K., Mehta, M. N., Singh, T. R. 2014. Application of Homotopy Analysis Method in One-Dimensional Instability Phenomenon Arising in Inclined Porous Media. American Journal of Applied Mathematics and Statistics, 2 (3): 106-114.
	In article		CrossRef
[15]	Wang, M., Li, X. & Zhang, J. 2008. The (G’/G)--expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Physics Letters A 372: 417-423.
	In article		CrossRef
[16]	Feng, J., Li, W. & Wan, Q. 2011. Using (G’/G)-expansion method to seek traveling wave solution of Kolmogorov-Petrovskii-Piskunov equation. Applied Mathematics and Computation 217: 5860-5865.
	In article		CrossRef
[17]	Naher, H., Abdullah, F.A. & Akbar, M.A. 2011. The (G’/G) -expansion method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon equation. Mathematical Problems in Engineering. Article ID: 218216 11 pages.
	In article
[18]	Naher, H., & Abdullah, F. A. (2012). The Basic (G'/G) -expansion method for the fourth order Boussinesq equation. Applied Mathematics, 3(10).
	In article		CrossRef
[19]	Zhang, J., Jiang, F. & Zhao, X. 2010. An improved (G’/G) -expansion method for solving nonlinear evolution equations, International Journal of Computer Mathematics, 87(8): 1716-1725.
	In article		CrossRef
[20]	Hamad, Y.S., Sayed, M., Elagan, S.K. & El-Zahar, E.R. 2011. The improved (G’/G) -expansion method for solving (3+1)-dimensional potential-YTSF equation, Journal of Modern Methods in Numerical Mathematics 2(1-2): 32-38.
	In article
[21]	Nofel, T.A., Sayed, M. Hamad, Y.S & Elagan, S.K. 2011. The improved (G’/G) -expansion method for solving the fifth-order KdV equation, Annals of Fuzzy Mathematics and Informatics 3(1): 9-17.
	In article
[22]	Naher, H., Abdullah, F. A., & Bekir, A. (2012). Abundant traveling wave solutions of the compound KdV-Burgers equation via the improved (G′/G)-expansion method. AIP Advances, 2(4), 042163.
	In article		CrossRef
[23]	Naher, H. & Abdullah, F.A. 2012. Some new traveling wave solutions of the nonlinear reaction diffusion equation by using the improved (G’/G) -expansion method. Mathematical Problems in Engineering. Article ID: 871724, 17 pages.
	In article
[24]	Naher, H. & Abdullah, F.A. 2012. The improved (G’/G) -expansion method for the (2+1)-dimensional Modified Zakharov-Kuznetsov equation. Journal of Applied Mathematics, Article ID: 438928, 20 pages.
	In article
[25]	Naher, H., Abdullah, F. A. 2014. The improved (G'/G)-expansion method to the (2+1)-dimensional breaking soliton equation. Journal of Computational Analysis & Applications, 16(2), 220-235.
	In article
[26]	Naher, H., Abdullah, F. A., Rashid, A. 2014. Some New Solutions of the (3+1)-dimensional Jimbo-Miwa equation via the Improved (G'/G)-expansion method. Journal of Computational Analysis & Applications, 17(2) 287-296.
	In article
[27]	Naher, H., Abdullah, F. A. 2013. The Improved (G’/G)-expansion method to the (3+1)-dimensional Kadomstev-Petviashvili equation. American Journal of Applied Mathematics and Statistics, 1 (4) 64-70.
	In article		CrossRef
[28]	Camassa, R. & Holm, D. 1993. An integrable shallow water equation with peaked solitons. Physical Review Letters 71(11): 1661-1664.
	In article		CrossRef PubMed
[29]	Tian, L. & Song, X. 2004. New peaked solitary wave solutions of the generalized Camassa-Holm equation. Chaos Solitons and Fractals 19: 621-637.
	In article		CrossRef
[30]	Wazwaz, A.M. 2005. New compact and noncompact solutions for variants of a modified Camassa-Holm equation. Applied Mathematics and Computation 163(3): 1165-1179.
	In article		CrossRef
[31]	Lenells, J. 2005. Traveling wave solutions of the Camassa-Holm equation, J. Diff. Eq. 217: 393-430.
	In article		CrossRef
[32]	Liu, X., Tian, L. & Wu, Y. 2010. Application of (G’/G) -expansion method to two nonlinear evolution equations. Applied Mathematics and Computation 217: 1376-1384.
	In article		CrossRef