Journal of Applied Mathematics

Volume 2012 (2012), Article ID 438928, 20 pages

doi:10.1155/2012/438928

Research Article

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

Hasibun Naher and Farah Aini Abdullah

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

Received 24 April 2012; Accepted 22 May 2012

Academic Editor: Shiping Lu

Copyright © 2012 Hasibun Naher and Farah Aini Abdullah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

we apply the improved -expansion method for constructing abundant new exact traveling wave solutions of the (2+1)-dimensional Modified Zakharov-Kuznetsov equation. In addition, together with is employed in this method, where , and 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 -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 -expansion method, and is implemented as traveling wave solutions, where . 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 -expansion method. They employed the method as traveling wave solutions, where either or may be zero, but both and cannot be zero at a time. Afterwards, many researchers studied different nonlinear partial differential equations to construct traveling wave solutions by using this improved -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 -expansionmethod to construct traveling wave solutions for the same equation. In this basic -expansion method, they utilized , where , as traveling wave solutions, instead of , where either or may be zero, but both and 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 -expansion method.

2. Description of the Improved -Expansion Method

Consider the general nonlinear partial differential equation

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

The main steps of the improved -expansion method [51] are as follows.

Step 1. Consider the traveling wave variable

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

where the superscripts indicate the ordinary derivatives with respect to .

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]

with satisfing the second-order linear ODE

where , and are constants.

Step 4. To determine the integer , 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 obtained in Step 4. Equate the coefficients of and then set each coefficient to zero and obtain a set of algebraic equations for , and .

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 and . Then, substitute the obtained values in (2.4) along with (2.5) with the value of 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 -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]:

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

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

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

Taking the homogeneous balance between and in (3.3), we obtain .

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

where , and 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 . According to Step 5, collecting all terms with the same power of , and 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 the algebraic equations with the help of algebraic software Maple, we obtain three different values.

Case 1. We have

where and are free parameters.

Case 2. We have

where and are free parameters.

Case 3. We have

where and 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 , we obtain

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

For example,

(i)if but , we obtain

(ii)if but , we obtain

(iii)if , we obtain

Family 2 (trigonometric function solutions). When , we obtain

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

For example,

(iv) if but , we obtain

(v) if but , we obtain

(vi) if , we obtain

Family 3 (rational function solution). When , we obtain

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 but ),

where ,

where ,

where .

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 but ),

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 ),

where  ,

where  ,

where  .

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 but ),

where ,

where ,

where .

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 but ),

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 ),

where ,

where ,

where .

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 ),

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 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 and solution equation (4.9) from Bekir [41] (from Section 4) becomes:

    .
(i) If and , solution

     becomes: .

(ii) If   and solution equation (4.9) from Bekir [41] (from Section 4) becomes:

      .
(ii) If and , solution

       becomes: .

(iii) If and solution equation (4.9) from Bekir [41] (from Section 4) becomes: .
(iii) If and , solution

      becomes: .

(iv) If and solution equation (4.9) from Bekir [41] (from Section 4) becomes:

    .
(iv) If and , solution

      becomes: .

(v) If and solution equation (4.9) from Bekir [41] (from Section 4) becomes:

    .
(v) If , and

      , solution becomes:

      .

(vi) If and solution equation (4.9) from Bekir [41] (from Section 4) becomes:

     .
(vi) If , and     

      , solution becomes:

      .

(vii) If and solution equation (4.9) from Bekir [41] (from Section 4) becomes:

      .
(vii) If , and

      , solution becomes:

      .

Figure 1: Solitons solution for .

Figure 2: Solitons solution for .

Figure 3: Solitons solution for .

Figure 4: Periodic solution for .

Figure 5: Periodic solution for .

Figure 6: Periodic solution for .

Figure 7: Solitons solution for .

Figure 8: Solitons solution for .

Figure 9: Solitons solution for .

Figure 10: Solitons solution for .

Figure 11: Solitons solution for .

Figure 12: Solitons solution for .

Figure 13: Solitons solution for .

Figure 14: Periodic solution for .

Beyond Table 1, we obtain new exact traveling wave solutions ,, and , 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 -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 -expansion method is more effective and more general than the basic -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

C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, vol. 161 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1982.

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1991.

M. L. Wang, Y. B. Zhou, and Z. B. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,” Physics Letters Section A, vol. 216, no. 1–5, pp. 67–75, 1996.

R. Hirota, “Exact solution of the korteweg-de vries equation for multiple collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971.

S. Liu, Z. Fu, S. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69–74, 2001.

Z. I. A. Al-Muhiameed and E. A.-B. Abdel-Salam, “Generalized Jacobi elliptic function solution to a class of nonlinear Schrödinger-type equations,” Mathematical Problems in Engineering, vol. 2011, Article ID 575679, 11 pages, 2011.

K. A. Gepreel and A. R. Shehata, “Jacobi elliptic solutions for nonlinear differential difference equations in mathematical physics,” Journal of Applied Mathematics, vol. 2012, Article ID 710375, 15 pages, 2012.

Z. Yan and H. Zhang, “New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in shallow water,” Physics Letters A, vol. 285, no. 5-6, pp. 355–362, 2001.

W. Malfliet, “Solitary wave solutions of nonlinear wave equations,” American Journal of Physics, vol. 60, no. 7, pp. 650–654, 1992.

A.-M. Wazwaz, “The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1467–1475, 2007.

A. Bekir and A. C. Cevikel, “Solitary wave solutions of two nonlinear physical models by tanh-coth method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1804–1809, 2009.

M. Wang and X. Li, “Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1257–1268, 2005.

M. A. Abdou, “The extended F-expansion method and its application for a class of nonlinear evolution equations,” Chaos, Solitons and Fractals, vol. 31, no. 1, pp. 95–104, 2007.

A. A. Soliman and H. A. Abdo, “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, vol. 7, no. 3, pp. 274–282, 2009.

A. H. Salas and C. A. Gómez, “Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation,” Mathematical Problems in Engineering, vol. 2010, Article ID 194329, 14 pages, 2010.

J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700–708, 2006.

H. Naher, F. A. Abdullah, and M. Ali Akbar, “New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the exp-function method,” Journal of Applied Mathematics, vol. 2012, Article ID 575387, 14 pages, 2012.

S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Exp-function method for traveling wave solutions of modified Zakharov-Kuznetsov equation,” Journal of King Saud University, vol. 22, no. 4, pp. 213–216, 2010.

H. Naher, F. A. Abdullah, and M. A. Akbar, “The exp-function method for new exact solutions of the nonlinear partial differential equations,” International Journal of the Physical Sciences, vol. 6, no. 29, pp. 6706–6716, 2011.

N. A. Khan, M. Jamil, and A. Ara, “Approximate solutions to time-fractional Schrodinger equation via homotopy analysis method,” ISRN Mathematical Physics, vol. 2012, Article ID 197068, 11 pages, 2012.

A. Yıldırım and Z. Pınar, “Application of the exp-function method for solving nonlinear reaction-diffusion equations arising in mathematical biology,” Computers & Mathematics with Applications, vol. 60, no. 7, pp. 1873–1880, 2010.

I. Aslan, “Application of the exp-function method to nonlinear lattice differential equations for multi-wave and rational solutions,” Mathematical Methods in the Applied Sciences, vol. 34, no. 14, pp. 1707–1710, 2011.

A. Bekir and A. Boz, “Exact solutions for nonlinear evolution equations using Exp-function method,” Physics Letters A, vol. 372, no. 10, pp. 1619–1625, 2008.

C. H. Che Hussin and A. Kiliçman, “On the solutions of nonlinear higher-order boundary value problems by using differential transformation method and Adomian decomposition method,” Mathematical Problems in Engineering, vol. 2011, Article ID 724927, 19 pages, 2011.

Y. M. Chen, G. Meng, J. K. Liu, and J. P. Jing, “Homotopy analysis method for nonlinear dynamical system of an electrostatically actuated microcantilever,” Mathematical Problems in Engineering, vol. 2011, Article ID 173459, 14 pages, 2011.

X. Wu, W. Rui, and X. Hong, “Exact travelling wave solutions of explicit type, implicit type and parametric type for K(m,n)equation,” Journal of Applied Mathematics, vol. 2012, Article ID 236875, 23 pages, 2012.

R. Zhang, “Bifurcation analysis for a kind of nonlinear finance system with delayed feedback and its application to control of chaos,” Journal of Applied Mathematics, vol. 2012, Article ID 316390, 18 pages, 2012.

A. M. Wazwaz, “A new (2+1)-dimensional Korteweg-de-Vries equation and its extension to a new (3+1)-dimensional Kadomtsev-Petviashvili equation,” Physica Scripta, vol. 84, Article ID 035010, 2011.

M. Noor, K. Noor, A. Waheed, and E. A. Al-Said, “An efficient method for solving system of third-order nonlinear boundary value problems,” Mathematical Problems in Engineering, vol. 2011, Article ID 250184, 14 pages, 2011.

N. Hua, T. Li-Xin, and Y. Hong-Xing, “On the positive almost periodic solutions of a class of nonlinear Lotka-Volterra type system with feedback control,” Journal of Applied Mathematics, vol. 2012, Article ID 135075, 15 pages, 2012.

Q. Liu and R. Xu, “Periodic solutions of a cohen-grossberg-type BAM neural networks with distributed delays and impulses,” Journal of Applied Mathematics, vol. 2012, Article ID 643418, 17 pages, 2012.

B. I. Yun, “An iteration method generating analytical solutions for Blasius problem,” Journal of Applied Mathematics, vol. 2011, Article ID 925649, 8 pages, 2011.

A. H. Bhrawy, A. S. Alofi, and S. I. El-Soubhy, “An extension of the Legendre-Galerkin Method for solving sixth-order differential equations with variable polynomial coefficients,” Mathematical Problems in Engineering, vol. 2012, Article ID 896575, 13 pages, 2012.

Z. I. A. Al-Muhiameed and E. A. B. Abdel-Salam, “Generalized hyperbolic function solution to a class of nonlinear Schrodinger-type equations,” Journal of Applied Mathematics, vol. 2012, Article ID 365348, 15 pages, 2012.

L. Zhengrong, J. Tianpei, Q. Peng, and X. Qinfeng, “Trigonometric function periodic wave solutions and their limit forms for the KdV-like and the PC-like equations,” Mathematical Problems in Engineering, vol. 2011, Article ID 810217, 23 pages, 2011.

A. Biswas and E. Zerrad, “1-soliton solution of the Zakharov-Kuznetsov equation with dual-power law nonlinearity,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 9-10, pp. 3574–3577, 2009.

A. Biswas, “1-soliton solution of the generalized Zakharov-Kuznetsov equation with nonlinear dispersion and time-dependent coefficients,” Physics Letters A, vol. 373, no. 33, pp. 2931–2934, 2009.

A. Biswas and E. Zerrad, “Solitary wave solution of the Zakharov-Kuznetsov equation in plasmas with power law nonlinearity,” Nonlinear Analysis. Real World Applications, vol. 11, no. 4, pp. 3272–3274, 2010.

E. V. Krishnan and A. Biswas, “Solutions to the Zakharov-Kuznetsov equation with higher order nonlinearity by mapping and Ansatz methods,” Physics of Wave Phenomena, vol. 18, no. 4, pp. 256–261, 2010.

Y. Li and C. Wang, “Three positive periodic solutions to nonlinear neutral functional differential equations with parameters on variable time scales,” Journal of Applied Mathematics, vol. 2012, Article ID 516476, 28 pages, 2012.

A. Bekir, “Application of the -expansion method for nonlinear evolution equations,” Physics Letters A, vol. 372, no. 19, pp. 3400–3406, 2008.

M. Wang, X. Li, and J. Zhang, “The -method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008.

H. Naher, F. A. Abdullah, and M. A. Akbar, “The -expansion method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon equation,” Mathematical Problems in Engineering, vol. 2011, Article ID 218216, 11 pages, 2011.

R. Abazari and R. Abazari, “Hyperbolic, trigonometric and rational function solutions of Hirota-Ramani equation via -expansion method,” Mathematical Problems in Engineering, vol. 2011, Article ID 424801, 11 pages, 2011.

E. M. E. Zayed and S. Al-Joudi, “Applications of an extended -expansion method to find exact solutions of nonlinear PDEs in mathematical physics,” Mathematical Problems in Engineering, vol. 2010, Article ID 768573, 19 pages, 2010.

E. M. E. Zayed, “Traveling wave solutions for higher dimensional nonlinear evolution equations using the -expansion method,” Journal of Applied Mathematics & Informatics, vol. 28, no. 1-2, pp. 383–395, 2010.

A. Jabbari, H. Kheiri, and A. Bekir, “Exact solutions of the coupled Higgs equation and the Maccari system using He's semi-inverse method and -expansion method,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2177–2186, 2011.

T. Öziş and I. Aslan, “Application of the -expansion method to Kawahara type equations using symbolic computation,” Applied Mathematics and Computation, vol. 216, no. 8, pp. 2360–2365, 2010.

K. A. Gepreel, “Exact complexiton soliton solutions for nonlinear partial differential equations,” International Mathematical Forum, vol. 6, no. 25–28, pp. 1261–1272, 2011.

A. Borhanifar and A. Z. Moghanlu, “Application of the -expansion method for the Zhiber-Shabat equation and other related equations,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2109–2116, 2011.

J. Zhang, F. Jiang, and X. Zhao, “An improved -expansion method for solving nonlinear evolution equations,” International Journal of Computer Mathematics, vol. 87, no. 8, pp. 1716–1725, 2010.

Y.-M. Zhao, Y.-J. Yang, and W. Li, “Application of the improved -expansion method for the variant Boussinesq equations,” Applied Mathematical Sciences, vol. 5, no. 57–60, pp. 2855–2861, 2011.

T. A. Nofel, M. Sayed, Y. S. Hamad, and S. K. Elagan, “The improved -expansion method for solving the fifth-order KdV equation,” Annals of Fuzzy Mathematics and Informatics, vol. 3, no. 1, pp. 9–17, 2011.

Y. S. Hamad, M. Sayed, S. K. Elagan, and E. R. El-Zahar, “The improved -expansion method for solving (3 + 1)-dimensional potential-YTSF equation,” Journal of Modern Methods in Numerical Mathematics, vol. 2, no. 1-2, pp. 32–38, 2011.

H. Naher, F. A. Abdullah, and M. A. Akbar, “New traveling wave solutions of the higher dimensional nonlinear evolution equation by the improved -expansion method,” World Applied Sciences Journal, vol. 16, no. 1, pp. 11–21, 2012.

H. Naher and F. A. Abdullah, “Some new traveling wave solutions of the nonlinear reaction diffusion equation by using the improved -expansion method,” Mathematical Problems in Engineering, vol. 2012, Article ID 871724, 17 pages, 2012.

H. Naher and F. A. Abdullah, “Some new solutions of the combined KdV-MKdV equation by using the improved -expansion method,” World Applied Sciences Journal, vol. 16, no. 11, pp. 1559–1570, 2012.

M. Khalfallah, “New exact traveling wave solutions of the (2 + 1) dimensional Zakharov-Kuznetsov (ZK) equation,” Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 15, no. 2, pp. 35–43, 2007.