we apply the improved (G^′/G)-expansion method for constructing abundant new exact traveling wave solutions of the (2+1)-dimensional Modified Zakharov-Kuznetsov equation. In addition, G^″ + λG^′ + μG = 0 together with is employed in this method, where p_q (q = 0, ±1, ±2, …, ±w), λ and μ are constants. Moreover, the obtained solutions including solitons and periodic solutions are described by three different families. Also, it is noteworthy to mention out that, some of our solutions are coincided with already published results, if parameters taken particular values. Furthermore, the graphical presentations are demonstrated for some of newly obtained solutions.

1. Introduction

The investigations of traveling wave solutions for nonlinear evolution equations (NLEEs) play an outstanding role in analysing nonlinear physical phenomena. In the recent past, a wide range of methods have been presented to establish analytical solutions for nonlinear partial differential equations (PDEs), such as the Backlund transformation method [1], the inverse scattering method [2], the homogeneous balance method [3], the Hirota bilinear transformation method [4], the Jacobi elliptic function expansion method [5–7], the generalized Riccati equation method [8], the tanh-coth method [9–11], the F-expansion method [12, 13], the direct algebraic method [14], the Cole-Hopf transformation method [15], the Exp-function method [16–23], the Adomian decomposition method [24], the homotopy analysis method [25], the bifurcation method [26, 27], and others [28–40].

Wang et al. [42] presented the basic (G′/G)-expansion method, and is implemented as traveling wave solutions, where a_m ≠ 0. Later on, nonlinear partial differential equations are investigated to construct traveling wave solutions via this method [41, 43–50]. More recently, Zhang et al. [51] extended this method and called it the improved (G′/G)-expansion method. They employed the method as traveling wave solutions, where either p_−w or p_w may be zero, but both p_−w and p_w cannot be zero at a time. Afterwards, many researchers studied different nonlinear partial differential equations to construct traveling wave solutions by using this improved (G′/G)-expansion method. For example, Zhao et al. [52] executed the same method to establish exact solutions of the variant Boussinesq equations. Nofel et al. [53] constructed traveling wave solutions for the fifth-order KdV equation by using this method. Hamad et al. [54] studied higher-dimensional potential YTSF equation to obtain analytical solutions via the same method. Naher et al. [55] applied this powerful method to construct traveling wave solutions of the higher-dimensional modified KdV-Zakharov-Kuznetsev equation. In [56], Naher and Abdullah were concerned about the same method to obtain exact solutions for the nonlinear reaction diffusion equation whilst in [57] they investigated the combined KdV-MKdV equation for constructing traveling wave solutions by applying this method and so on.

Many researchers studied the (2+1)-dimensional modified Zakharov-Kuznetsov equation by using different methods. For instance, Khalfallah [58] implemented homogeneous balance method to investigate this equation for obtaining exact solutions. In [41], Bekir employed the basic (G′/G)-expansionmethod to construct traveling wave solutions for the same equation. In this basic (G′/G)-expansion method, they utilized , where a_m ≠ 0, as traveling wave solutions, instead of , where either p_−w or p_w may be zero, but both p_−w and p_w cannot be zero at a time.

The significance of this work is that the (2+1)-dimensional modified Zakharov-Kuznetsov equation is considered to construct many new exact traveling wave solutions including solitons, periodic, and rational solutions by applying the improved (G′/G)-expansion method.

2. Description of the Improved (G^′/G)-Expansion Method

Consider the general nonlinear partial differential equation

(2.1)

where u = u(x, y, t) is an unknown function, A is a polynomial in u(x, y, t), and the subscripts stand for the partial derivatives.

The main steps of the improved (G′/G)-expansion method [51] are as follows.

Step 1. Consider the traveling wave variable

(2.2)

where C is the wave speed. Now using (2.2), (2.1) is converted into an ordinary differential equation for b(α)

(2.3)

where the superscripts indicate the ordinary derivatives with respect to α.

Step 2. According to possibility, (2.3) can be integrated term by term one or more times, yielding constant(s) of integration. The integral constant may be zero, for simplicity.

Step 3. Suppose that the traveling wave solution of (2.3) can be expressed in the form [51]

(2.4)

with G = G(α) satisfing the second-order linear ODE

(2.5)

where p_q (q = 0, ±1, ±2, …, ±w), λ, and μ are constants.

Step 4. To determine the integer w, substitute (2.4) along with (2.5) into (2.3) and then take the homogeneous balance between the highest-order nonlinear terms and the highest-order derivatives appearing in (2.3).

Step 5. Substitute (2.4) and (2.5) into (2.3) with the value of w obtained in Step 4. Equate the coefficients of (G′/G)^r, (r = 0, ±1, ±2, …) and then set each coefficient to zero and obtain a set of algebraic equations for p_q (q = 0, ±1, ±2, …, ±w), C, λ, and μ.

Step 6. By solving the system of algebraic equations which are obtained in Step 5 with the aid of algebraic software Maple, and we obtain values for p_q (q = 0, ±1, ±2, …, ±w), C, λ and μ. Then, substitute the obtained values in (2.4) along with (2.5) with the value of w to obtain the traveling wave solutions of (2.1).

3. Applications of the Method

In this section, we have studied the (2+1)-dimensional modified Zakharov-Kuznetsov equation to construct new exact traveling wave solutions including solitons, periodic solutions, and rational solutions via the improved (G′/G)-expansion method.

3.1. The (2+1)-Dimensional Modified Zakharov-Kuznetsov Equation

We consider the (2+1)-dimensional Modified Zakharov-Kuznetsov equation followed by Bekir [41]:

(3.1)

Now, we use the wave transformation equation (2.2) into (3.1), which yields

(3.2)

Equation (3.2) is integrable; therefore, integrating with respect to α once yields:

(3.3)

where H is an integral constant that is to be determined later.

Taking the homogeneous balance between b³ and b^′′ in (3.3), we obtain w = 1.

Therefore, the solution of (3.3) is of the form

(3.4)

where p₋₁, p₀, and p₁ are constants to be determined.

Substituting (3.4) together with (2.5) into (3.3), the left-hand side of (3.3) is converted into a polynomial of (G′/G)^r, (r = 0, ±1, ±2, …). According to Step 5, collecting all terms with the same power of (G′/G), and then setting each coefficient of the resulted polynomial to zero, yields a set of algebraic equations (for simplicity, which are not presented) for p₋₁, p₀, p₁, C, H, λ, and μ.

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

Case 1. We have

(3.5)

where λ and μ are free parameters.

Case 2. We have

(3.6)

where λ and μ are free parameters.

Case 3. We have

(3.7)

where λ and μ are free parameters.

Substituting the general solution equation (2.5) into (3.4), we obtain three different families of traveling wave solutions of(3.3)

Family 1 (hyperbolic function solutions). When λ² − 4μ > 0, we obtain

(3.8)

If U and V take particular values, various known solutions can be rediscovered.

For example,

(i)
if U = 0 but V ≠ 0, we obtain
(3.9)
(ii)
if V = 0 but U ≠ 0, we obtain
(3.10)
(iii)
if U ≠ 0, U > V, we obtain
(3.11)

Family 2 (trigonometric function solutions). When λ² − 4μ < 0, we obtain

(3.12)

If U and V take particular values, various known solutions can be rediscovered.

For example,

(iv)
if U = 0 but V ≠ 0, we obtain
(3.13)
(v)
if V = 0 but U ≠ 0, we obtain
(3.14)
(vi)
if U ≠ 0, U > V, we obtain
(3.15)

Family 3 (rational function solution). When λ² − 4μ = 0, we obtain

(3.16)

Family 4 (hyperbolic function solutions). Substituting (3.5), (3.6), and (3.7) together with the general solution equation (2.5) into (3.4) yields the hyperbolic function solution equation (3.8), then using (3.9), our traveling wave solutions become, respectively (if U = 0 but V ≠ 0),

(3.17)

where α = x + y + (λ² − 4μ)t,

(3.18)

where α = x + y + (λ² − 4μ)t,

(3.19)

where α = x + y + (λ² + 8μ)t.

Again, substituting (3.5), (3.6), and (3.7) together with the general solution equation (2.5) into (3.4), we obtain the hyperbolic function solution equation (3.8), then using (3.10), we obtain the following exact solutions, respectively (if V = 0 but U ≠ 0),

(3.20)

(3.21)

(3.22)

Moreover, substituting (3.5), (3.6), and (3.7) together with the general solution equation (2.5) into (3.4) yields the hyperbolic function solution equation (3.8), and then using (3.11), our obtained wave solutions become, respectively (if U ≠ 0, U > V),

(3.23)

where α₀ = tanh⁻¹ (V/U),

(3.24)

where α₀ = tanh⁻¹ (V/U),

(3.25)

where α₀ = tanh⁻¹ (V/U).

Family 5 (trigonometric function solutions). Substituting (3.5), (3.6), and (3.7) together with the general solution equation (2.5) into (3.4) yields the trigonometric function solution equation (3.12), and then using (3.13), we obtain the following solutions, respectively (if U = 0 but V ≠ 0),

(3.26)

where α = x + y − (4μ − λ²)t,

(3.27)

where α = x + y − (4μ − λ²)t,

(3.28)

where α = x + y − (−8μ − λ²)t.

Also, substituting (3.5), (3.6), and (3.7) together with the general solution equation (2.5) into (3.4) yields the trigonometric function solution equation (3.12), and then using (3.14), our solutions become, respectively (if V = 0 but U ≠ 0),

(3.29)

(3.30)

(3.31)

Furthermore, substituting (3.5), (3.6), and (3.7) together with the general solution equation (2.5) into (3.4) yields the trigonometric function solution equation (3.12), and then using (3.15), our obtained traveling wave solutions, become respectively (if U ≠ 0, U > V),

(3.32)

where α₀ = tan⁻¹ (V/U),

(3.33)

where α₀ = tan⁻¹ (V/U),

(3.34)

where α₀ = tan⁻¹ (V/U).

Family 6 (rational function solutions). Substituting (3.5), (3.6), and (3.7) together with the general solution equation (2.5) into (3.4), we obtain the rational function solution equation (3.16), and our wave solutions become, respectively (if λ² − 4μ = 0),

(3.35)

where α = x + y + (λ² − 4μ)t,

(3.36)

where α = x + y + (λ² − 4μ)t,

(3.37)

where α = x + y + (λ² + 8μ)t.

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 Table 1. Moreover, some of the newly obtained exact traveling wave solutions are described in Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, and 14.

Table 1. Comparison between Bekir [41] solutions and Newly obtained solutions.

Bekir [41] solutions	New solutions
(i) If C₁ = 0, C₂ ≠ 0, μ = 2 and λ = 3 solution equation (4.9) from Bekir [41] (from Section 4) becomes: .	(i) If μ = 2, λ = 3 and b₁(α) = u_1,2(ξ), solution b₁(α) becomes: .

(ii) If C₁ ≠ 0, C₂ = 0, μ = 2 and λ = 3 solution equation (4.9) from Bekir [41] (from Section 4) becomes: .	(ii) If μ = 2, λ = 3 and b₄(α) = u_1,2(ξ), solution b₄(α) becomes: .

(iii) If C₁ = 0, C₂ ≠ 0, μ = 2 and λ = 2 solution equation (4.9) from Bekir [41] (from Section 4) becomes: .	(iii) If μ = 2, λ = 2 and b₁₀(α) = u_3,4(ξ), solution b₁₀(α) becomes: .

(iv) If C₁ ≠ 0, C₂ = 0, μ = 2 and λ = 2 solution equation (4.9) from Bekir [41] (from Section 4) becomes: .	(iv) If μ = 2, λ = 2 and b₁₃(α) = u_3,4(ξ), solution b₁₃(α) becomes: .

(v) If C₁ = 0, C₂ = 1 and λ² − 4μ = 0 solution equation (4.9) from Bekir [41] (from Section 4) becomes: .	(v) If U = 0, V = 1, λ² − 4μ = 0 and b₁₉(α) = u_5,6(ξ), solution b₁₉(α) becomes: .

(vi) If C₁ = 1, C₂ = 1/2 and λ² − 4μ = 0 solution equation (4.9) from Bekir [41] (from Section 4) becomes: .	(vi) If U = 1, V = 1/2, λ² − 4μ = 0 and b₁₉(α) = u_5,6(ξ), solution b₁₉(α) becomes: .

(vii) If C₁ = 1, C₂ = −1/2 and λ² − 4μ = 0 solution equation (4.9) from Bekir [41] (from Section 4) becomes: .	(vii) If U = 1, V = −1/2, λ² − 4μ = 0 and b₁₉(α) = u_5,6(ξ), solution b₁₉(α) becomes: .

Details are in the caption following the image — **Figure 1**
Open in figure viewer PowerPoint

Solitons solution for λ = 3, μ = 2.

Beyond Table 1, we obtain new exact traveling wave solutions b₂, b₃, b₅, b₆, b₇, b₈, b₉, b₁₁, b₁₂, b₁₄, b₁₅, b₁₆, b₁₇, b₁₈, b₂₀, and b₂₁, which are not established in the previous literature.

4.1. Graphical Representations of the Solutions

The graphical presentations of some solutions are depicted in Figures 1–14 with the aid of commercial software Maple.

5. Conclusions

In this paper, the improved (G′/G)-expansion method is implemented to investigate the nonlinear partial differential equation, namely, the (2+1)-dimensional modified Zakharov-Kuznetsov equation. We have constructed abundant exact traveling wave solutions including solitons, periodic, and rational solutions. Moreover, it is worth stating that some of the newly obtained solutions are identical with already-published results, for special case. The obtained solutions show that the improved (G′/G)-expansion method is more effective and more general than the basic (G′/G)-expansion method, because it gives many new solutions. Consequently, this simple and powerful method can be more successfully applied to study nonlinear partial differential equations, which frequently arise in engineering sciences, mathematical physics, and other scientific real-time application fields.

Acknowledgments

This paper is supported by the USM short term Grant (Ref. no. 304/PMATHS/6310072) and the authors would like to express their thanks to the School of Mathematical Sciences, USM for providing related research facilities. The authors are also grateful to the referees for their valuable comments and suggestions.

References

1 Rogers C. and Shadwick W. F., Bäcklund Transformations and Their Applications, 1982, 161, Academic Press, New York, NY, USA, Mathematics in Science and Engineering, 658491.
Google Scholar
2 Ablowitz M. J. and Clarkson P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering, 1991, 149, Cambridge University Press, Cambridge, UK, London Mathematical Society Lecture Note Series, https://doi.org/10.1017/CBO9780511623998, 1149378.
10.1017/CBO9780511623998
Google Scholar
3 Wang M. L., Zhou Y. B., and Li Z. B., Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters Section A. (1996) 216, no. 1–5, 67–75, 2-s2.0-30244524644.
10.1016/0375-9601(96)00283-6
CAS Web of Science® Google Scholar
4 Hirota R., Exact solution of the korteweg-de vries equation for multiple collisions of solitons, Physical Review Letters. (1971) 27, no. 18, 1192–1194, 2-s2.0-35949040541, https://doi.org/10.1103/PhysRevLett.27.1192.
10.1103/PhysRevLett.27.1192
Web of Science® Google Scholar
5 Liu S., Fu Z., Liu S., and Zhao Q., Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Physics Letters A. (2001) 289, no. 1-2, 69–74, https://doi.org/10.1016/S0375%2D9601(01)00580%2D1, 1862082.
10.1016/S0375-9601(01)00580-1
CAS Web of Science® Google Scholar
6 Al-Muhiameed Z. I. A. and Abdel-Salam E. A.-B., Generalized Jacobi elliptic function solution to a class of nonlinear Schrödinger-type equations, Mathematical Problems in Engineering. (2011) 2011, 11, 575679, https://doi.org/10.1155/2011/575679, 2786629.
10.1155/2011/575679
Web of Science® Google Scholar
7 Gepreel K. A. and Shehata A. R., Jacobi elliptic solutions for nonlinear differential difference equations in mathematical physics, Journal of Applied Mathematics. (2012) 2012, 15, https://doi.org/10.1155/2012/710375, 710375.
10.1155/2012/710375
Google Scholar
8 Yan Z. and Zhang H., New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water, Physics Letters A. (2001) 285, no. 5-6, 355–362, https://doi.org/10.1016/S0375%2D9601(01)00376%2D0, 1851579.
10.1016/S0375-9601(01)00376-0
CAS Web of Science® Google Scholar
9 Malfliet W., Solitary wave solutions of nonlinear wave equations, American Journal of Physics. (1992) 60, no. 7, 650–654, https://doi.org/10.1119/1.17120, 1174588.
10.1119/1.17120
Web of Science® Google Scholar
10 Wazwaz A.-M., The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations, Applied Mathematics and Computation. (2007) 188, no. 2, 1467–1475, https://doi.org/10.1016/j.amc.2006.11.013, 2335643.
10.1016/j.amc.2006.11.013
Web of Science® Google Scholar
11 Bekir A. and Cevikel A. C., Solitary wave solutions of two nonlinear physical models by tanh-coth method, Communications in Nonlinear Science and Numerical Simulation. (2009) 14, no. 5, 1804–1809, 2-s2.0-56049127915, https://doi.org/10.1016/j.cnsns.2008.07.004.
10.1016/j.cnsns.2008.07.004
Web of Science® Google Scholar
12 Wang M. and Li X., Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation, Chaos, Solitons and Fractals. (2005) 24, no. 5, 1257–1268, https://doi.org/10.1016/j.chaos.2004.09.044, 2123272.
10.1016/j.chaos.2004.09.044
Web of Science® Google Scholar
13 Abdou M. A., The extended F-expansion method and its application for a class of nonlinear evolution equations, Chaos, Solitons and Fractals. (2007) 31, no. 1, 95–104, https://doi.org/10.1016/j.chaos.2005.09.030, 2263268.
10.1016/j.chaos.2005.09.030
Web of Science® Google Scholar
14 Soliman A. A. and Abdo H. A., New exact solutions of nonlinear variants of the RLW, and PHI-four and Boussinesq equations based on modified extended direct algebraic method, International Journal of Nonlinear Science. (2009) 7, no. 3, 274–282, 2538454.
Google Scholar
15 Salas A. H. and Gómez C. A., Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation, Mathematical Problems in Engineering. (2010) 2010, 14, 194329, https://doi.org/10.1155/2010/194329, 2606914.
10.1155/2010/194329
Web of Science® Google Scholar
16 He J.-H. and Wu X.-H., Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals. (2006) 30, no. 3, 700–708, https://doi.org/10.1016/j.chaos.2006.03.020, 2238695.
10.1016/j.chaos.2006.03.020
Web of Science® Google Scholar
17 Naher H., Abdullah F. A., and Ali Akbar M., New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the exp-function method, Journal of Applied Mathematics. (2012) 2012, 14, 575387, https://doi.org/10.1155/2012/575387, 2872358.
10.1155/2012/575387
Google Scholar
18 Mohyud-Din S. T., Noor M. A., and Noor K. I., Exp-function method for traveling wave solutions of modified Zakharov-Kuznetsov equation, Journal of King Saud University. (2010) 22, no. 4, 213–216, 2-s2.0-77952378676, https://doi.org/10.1016/j.jksus.2010.04.015.
10.1016/j.jksus.2010.04.015
Google Scholar
19 Naher H., Abdullah F. A., and Akbar M. A., The exp-function method for new exact solutions of the nonlinear partial differential equations, International Journal of the Physical Sciences. (2011) 6, no. 29, 6706–6716.
Google Scholar
20 Khan N. A., Jamil M., and Ara A., Approximate solutions to time-fractional Schrodinger equation via homotopy analysis method, ISRN Mathematical Physics. (2012) 2012, 11, https://doi.org/10.5402/2012/197068, 197068.
10.5402/2012/197068
Google Scholar
21 Yıldırım A. and Pınar Z., Application of the exp-function method for solving nonlinear reaction-diffusion equations arising in mathematical biology, Computers & Mathematics with Applications. (2010) 60, no. 7, 1873–1880, https://doi.org/10.1016/j.camwa.2010.07.020, 2719706.
10.1016/j.camwa.2010.07.020
Web of Science® Google Scholar
22 Aslan I., Application of the exp-function method to nonlinear lattice differential equations for multi-wave and rational solutions, Mathematical Methods in the Applied Sciences. (2011) 34, no. 14, 1707–1710, https://doi.org/10.1002/mma.1476, 2833823.
10.1002/mma.1417
Web of Science® Google Scholar
23 Bekir A. and Boz A., Exact solutions for nonlinear evolution equations using Exp-function method, Physics Letters A. (2008) 372, no. 10, 1619–1625, https://doi.org/10.1016/j.physleta.2007.10.018, 2396241.
10.1016/j.physleta.2007.10.018
CAS Web of Science® Google Scholar
24 Che Hussin C. H. and Kiliçman A., On the solutions of nonlinear higher-order boundary value problems by using differential transformation method and Adomian decomposition method, Mathematical Problems in Engineering. (2011) 2011, 19, 724927, https://doi.org/10.1155/2011/724927, 2786623.
10.1155/2011/724927
Web of Science® Google Scholar
25 Chen Y. M., Meng G., Liu J. K., and Jing J. P., Homotopy analysis method for nonlinear dynamical system of an electrostatically actuated microcantilever, Mathematical Problems in Engineering. (2011) 2011, 14, https://doi.org/10.1155/2011/173459, 173459.
10.1155/2011/173459
Web of Science® Google Scholar
26 Wu X., Rui W., and Hong X., Exact travelling wave solutions of explicit type, implicit type and parametric type for K(m,n)equation, Journal of Applied Mathematics. (2012) 2012, 23, https://doi.org/10.1155/2012/236875, 236875.
10.1155/2012/236875
Google Scholar
27 Zhang R., Bifurcation analysis for a kind of nonlinear finance system with delayed feedback and its application to control of chaos, Journal of Applied Mathematics. (2012) 2012, 18, https://doi.org/10.1155/2012/316390, 316390.
10.1155/2012/316390
Google Scholar
28 Wazwaz A. M., A new (2+1)-dimensional Korteweg-de-Vries equation and its extension to a new (3+1)-dimensional Kadomtsev-Petviashvili equation, Physica Scripta. (2011) 84, 035010, https://doi.org/10.1088/0031%2D8949/84/03/035010.
10.1088/0031-8949/84/03/035010
Web of Science® Google Scholar
29 Noor M., Noor K., Waheed A., and Al-Said E. A., An efficient method for solving system of third-order nonlinear boundary value problems, Mathematical Problems in Engineering. (2011) 2011, 14, 250184, https://doi.org/10.1155/2011/250184, 2799873.
10.1155/2011/250184
Web of Science® Google Scholar
30 Hua N., Li-Xin T., and Hong-Xing Y., On the positive almost periodic solutions of a class of nonlinear Lotka-Volterra type system with feedback control, Journal of Applied Mathematics. (2012) 2012, 15, https://doi.org/10.1155/2012/135075, 135075.
10.1155/2012/135075
Google Scholar
31 Liu Q. and Xu R., Periodic solutions of a cohen-grossberg-type BAM neural networks with distributed delays and impulses, Journal of Applied Mathematics. (2012) 2012, 17, https://doi.org/10.1155/2012/643418, 643418.
10.1155/2012/643418
Google Scholar
32 Yun B. I., An iteration method generating analytical solutions for Blasius problem, Journal of Applied Mathematics. (2011) 2011, 8, 925649, https://doi.org/10.1155/2011/925649, 2822401.
10.1155/2011/925649
Google Scholar
33 Bhrawy A. H., Alofi A. S., and El-Soubhy S. I., An extension of the Legendre-Galerkin Method for solving sixth-order differential equations with variable polynomial coefficients, Mathematical Problems in Engineering. (2012) 2012, 13, https://doi.org/10.1155/2012/896575, 896575.
10.1155/2012/896575
Web of Science® Google Scholar
34 Al-Muhiameed Z. I. A. and Abdel-Salam E. A. B., Generalized hyperbolic function solution to a class of nonlinear Schrodinger-type equations, Journal of Applied Mathematics. (2012) 2012, 15, https://doi.org/10.1155/2012/365348, 365348.
10.1155/2012/265348
Google Scholar
35 Zhengrong L., Tianpei J., Peng Q., and Qinfeng X., Trigonometric function periodic wave solutions and their limit forms for the KdV-like and the PC-like equations, Mathematical Problems in Engineering. (2011) 2011, 23, 810217, https://doi.org/10.1155/2011/810217, 2823612.
10.1155/2011/810217
Web of Science® Google Scholar
36 Biswas A. and Zerrad E., 1-soliton solution of the Zakharov-Kuznetsov equation with dual-power law nonlinearity, Communications in Nonlinear Science and Numerical Simulation. (2009) 14, no. 9-10, 3574–3577, 2-s2.0-63449088138, https://doi.org/10.1016/j.cnsns.2008.10.004.
10.1016/j.cnsns.2008.10.004
Web of Science® Google Scholar
37 Biswas A., 1-soliton solution of the generalized Zakharov-Kuznetsov equation with nonlinear dispersion and time-dependent coefficients, Physics Letters A. (2009) 373, no. 33, 2931–2934, https://doi.org/10.1016/j.physleta.2009.06.029, 2555468.
10.1016/j.physleta.2009.06.029
CAS Web of Science® Google Scholar
38 Biswas A. and Zerrad E., Solitary wave solution of the Zakharov-Kuznetsov equation in plasmas with power law nonlinearity, Nonlinear Analysis. Real World Applications. (2010) 11, no. 4, 3272–3274, https://doi.org/10.1016/j.nonrwa.2009.08.007, 2661987.
10.1016/j.nonrwa.2009.08.007
Web of Science® Google Scholar
39 Krishnan E. V. and Biswas A., Solutions to the Zakharov-Kuznetsov equation with higher order nonlinearity by mapping and Ansatz methods, Physics of Wave Phenomena. (2010) 18, no. 4, 256–261, 2-s2.0-78650038597, https://doi.org/10.3103/S1541308X10040059.
10.3103/S1541308X10040059
Web of Science® Google Scholar
40 Li Y. and Wang C., Three positive periodic solutions to nonlinear neutral functional differential equations with parameters on variable time scales, Journal of Applied Mathematics. (2012) 2012, 28, https://doi.org/10.1155/2012/516476, 516476.
10.1155/2012/516476
Google Scholar
41 Bekir A., Application of the (G^′/G)-expansion method for nonlinear evolution equations, Physics Letters A. (2008) 372, no. 19, 3400–3406, https://doi.org/10.1016/j.physleta.2008.01.057, 2414708.
10.1016/j.physleta.2008.01.057
CAS Web of Science® Google Scholar
42 Wang M., Li X., and Zhang J., The (G^′/G)-method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A. (2008) 372, no. 4, 417–423, https://doi.org/10.1016/j.amc.2010.12.071, 2770205.
10.1016/j.physleta.2007.07.051
CAS Web of Science® Google Scholar
43 Naher H., Abdullah F. A., and Akbar M. A., The (G^′/G)-expansion method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon equation, Mathematical Problems in Engineering. (2011) 2011, 11, 218216.
10.1155/2011/218216
Web of Science® Google Scholar
44 Abazari R. and Abazari R., Hyperbolic, trigonometric and rational function solutions of Hirota-Ramani equation via (G^′/G)-expansion method, Mathematical Problems in Engineering. (2011) 2011, 11, https://doi.org/10.1155/2011/424801, 424801.
10.1155/2011/424801
Web of Science® Google Scholar
45 Zayed E. M. E. and Al-Joudi S., Applications of an extended (G^′/G)-expansion method to find exact solutions of nonlinear PDEs in mathematical physics, Mathematical Problems in Engineering. (2010) 2010, 19, 768573, https://doi.org/10.1155/2010/768573, 2674244.
10.1155/2010/768573
Web of Science® Google Scholar
46 Zayed E. M. E., Traveling wave solutions for higher dimensional nonlinear evolution equations using the (G’/G)-expansion method, Journal of Applied Mathematics & Informatics. (2010) 28, no. 1-2, 383–395.
Google Scholar
47 Jabbari A., Kheiri H., and Bekir A., Exact solutions of the coupled Higgs equation and the Maccari system using He′s semi-inverse method and (G^′/G)-expansion method, Computers & Mathematics with Applications. (2011) 62, no. 5, 2177–2186, https://doi.org/10.1016/j.camwa.2011.07.003, 2831678.
10.1016/j.camwa.2011.07.003
Web of Science® Google Scholar
48 Öziş T. and Aslan I., Application of the (G^′/G)-expansion method to Kawahara type equations using symbolic computation, Applied Mathematics and Computation. (2010) 216, no. 8, 2360–2365, https://doi.org/10.1016/j.amc.2010.03.081, 2647109.
10.1016/j.amc.2010.03.081
Web of Science® Google Scholar
49 Gepreel K. A., Exact complexiton soliton solutions for nonlinear partial differential equations, International Mathematical Forum. (2011) 6, no. 25–28, 1261–1272, 2820222.
Google Scholar
50 Borhanifar A. and Moghanlu A. Z., Application of the (G^′/G)-expansion method for the Zhiber-Shabat equation and other related equations, Mathematical and Computer Modelling. (2011) 54, no. 9-10, 2109–2116, https://doi.org/10.1016/j.mcm.2011.05.020, 2834615.
10.1016/j.mcm.2011.05.020
Google Scholar
51 Zhang J., Jiang F., and Zhao X., An improved (G^′/G)-expansion method for solving nonlinear evolution equations, International Journal of Computer Mathematics. (2010) 87, no. 8, 1716–1725, https://doi.org/10.1080/00207160802450166, 2665745.
10.1080/00207160802450166
Web of Science® Google Scholar
52 Zhao Y.-M., Yang Y.-J., and Li W., Application of the improved (G^′/G)-expansion method for the variant Boussinesq equations, Applied Mathematical Sciences. (2011) 5, no. 57–60, 2855–2861, 2852276.
Google Scholar
53 Nofel T. A., Sayed M., Hamad Y. S., and Elagan S. K., The improved (G^′/G)-expansion method for solving the fifth-order KdV equation, Annals of Fuzzy Mathematics and Informatics. (2011) 3, no. 1, 9–17.
Google Scholar
54 Hamad Y. S., Sayed M., Elagan S. K., and El-Zahar E. R., The improved (G^′/G)-expansion method for solving (3 + 1)-dimensional potential-YTSF equation, Journal of Modern Methods in Numerical Mathematics. (2011) 2, no. 1-2, 32–38.
10.20454/jmmnm.2011.78
Google Scholar
55 Naher H., Abdullah F. A., and Akbar M. A., New traveling wave solutions of the higher dimensional nonlinear evolution equation by the improved (G^′/G)-expansion method, World Applied Sciences Journal. (2012) 16, no. 1, 11–21.
Google Scholar
56 Naher H. and Abdullah F. A., Some new traveling wave solutions of the nonlinear reaction diffusion equation by using the improved (G^′/G)-expansion method, Mathematical Problems in Engineering. (2012) 2012, 17, https://doi.org/10.1155/2012/871724, 871724.
10.1155/2012/871724
Web of Science® Google Scholar
57 Naher H. and Abdullah F. A., Some new solutions of the combined KdV-MKdV equation by using the improved (G^′/G)-expansion method, World Applied Sciences Journal. (2012) 16, no. 11, 1559–1570.
Google Scholar
58 Khalfallah M., New exact traveling wave solutions of the (2 + 1) dimensional Zakharov-Kuznetsov (ZK) equation, Analele Stiintifice ale Universitatii Ovidius Constanta. (2007) 15, no. 2, 35–43, 2397271.
Google Scholar

Citing Literature

All articles

The Improved (G’/G)-Expansion Method for the (2+1)-Dimensional Modified Zakharov-Kuznetsov Equation

Abstract

1. Introduction

2. Description of the Improved (G^′/G)-Expansion Method

3. Applications of the Method

3.1. The (2+1)-Dimensional Modified Zakharov-Kuznetsov Equation

4. Results and Discussion

4.1. Graphical Representations of the Solutions

5. Conclusions

Acknowledgments

References

Citing Literature

Figures

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

The Improved (G’/G)-Expansion Method for the (2+1)-Dimensional Modified Zakharov-Kuznetsov Equation

Abstract

1. Introduction

2. Description of the Improved (G′/G)-Expansion Method

3. Applications of the Method

3.1. The (2+1)-Dimensional Modified Zakharov-Kuznetsov Equation

4. Results and Discussion

4.1. Graphical Representations of the Solutions

5. Conclusions

Acknowledgments

References

Citing Literature

Figures

References

Related

Information

2. Description of the Improved (G^′/G)-Expansion Method