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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 438928, 20 pages
http://dx.doi.org/10.1155/2012/438928
Research Article

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

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

Received 24 April 2012; Accepted 22 May 2012

Academic Editor: ShipingΒ Lu

Copyright Β© 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 (πΊξ…ž/𝐺)-expansion method for constructing abundant new exact traveling wave solutions of the (2+1)-dimensional Modified Zakharov-Kuznetsov equation. In addition, πΊξ…žξ…ž+πœ†πΊξ…ž+πœ‡πΊ=0 together with βˆ‘π‘(𝛼)=π‘€π‘ž=βˆ’π‘€π‘π‘ž(πΊξ…ž/𝐺)π‘ž is employed in this method, where π‘π‘ž(π‘ž=0,Β±1,Β±2,…,±𝑀), πœ† and πœ‡ 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 βˆ‘π‘’(πœ‰)=π‘šπ‘–=0π‘Žπ‘–(𝐺′/𝐺)𝑖 is implemented as traveling wave solutions, where π‘Žπ‘šβ‰ 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 βˆ‘π‘(𝛼)=π‘€π‘ž=βˆ’π‘€π‘π‘ž(𝐺′/𝐺)π‘ž method as traveling wave solutions, where either π‘βˆ’π‘€ or 𝑝𝑀 may be zero, but both π‘βˆ’π‘€ and 𝑝𝑀 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 βˆ‘π‘’(πœ‰)=π‘šπ‘–=0π‘Žπ‘–(𝐺′/𝐺)𝑖, where π‘Žπ‘šβ‰ 0, as traveling wave solutions, instead of βˆ‘π‘(𝛼)=π‘€π‘ž=βˆ’π‘€π‘π‘ž(𝐺′/𝐺)π‘ž, where either π‘βˆ’π‘€ or 𝑝𝑀 may be zero, but both π‘βˆ’π‘€ and 𝑝𝑀 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 (πΊξ…ž/𝐺)-Expansion Method

Consider the general nonlinear partial differential equation 𝐴𝑒,𝑒𝑑,𝑒π‘₯,𝑒𝑦,𝑒π‘₯𝑑,𝑒𝑦𝑑,𝑒π‘₯𝑦,𝑒𝑑𝑑,𝑒π‘₯π‘₯,𝑒𝑦𝑦,…=0,(2.1) where 𝑒=𝑒(π‘₯,𝑦,𝑑) is an unknown function, 𝐴 is a polynomial in 𝑒(π‘₯,𝑦,𝑑), 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 𝑒(π‘₯,𝑦,𝑑)=𝑏(𝛼),𝛼=π‘₯+π‘¦βˆ’πΆπ‘‘,(2.2) where 𝐢 is the wave speed. Now using (2.2), (2.1) is converted into an ordinary differential equation for 𝑏(𝛼)𝐡𝑏,𝑏′,π‘ξ…žξ…ž,π‘ξ…žξ…žξ…žξ€Έ,…=0,(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] 𝑏(𝛼)=π‘€ξ“π‘ž=βˆ’π‘€π‘π‘žξ‚΅πΊξ…žπΊξ‚Άπ‘ž,(2.4) with 𝐺=𝐺(𝛼) satisfing the second-order linear ODE πΊξ…žξ…ž+πœ†πΊβ€²+πœ‡πΊ=0,(2.5) where π‘π‘ž(π‘ž=0,Β±1,Β±2,…,±𝑀),πœ†, and πœ‡ are constants.

Step 4. To determine the integer 𝑀, 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 𝑀 obtained in Step 4. Equate the coefficients of (𝐺′/𝐺)π‘Ÿ,(π‘Ÿ=0,Β±1,Β±2,…) and then set each coefficient to zero and obtain a set of algebraic equations for π‘π‘ž(π‘ž=0,Β±1,Β±2,…,±𝑀),𝐢,πœ†, and πœ‡.

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 π‘π‘ž(π‘ž=0,Β±1,Β±2,…,±𝑀),𝐢,πœ† and πœ‡. Then, substitute the obtained values in (2.4) along with (2.5) with the value of 𝑀 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]: 𝑒𝑑+𝑒2𝑒π‘₯+𝑒π‘₯π‘₯π‘₯+𝑒π‘₯𝑦𝑦=0.(3.1) Now, we use the wave transformation equation (2.2) into (3.1), which yields βˆ’πΆπ‘β€²+𝑏2𝑏′+2π‘ξ…žξ…žξ…ž=0.(3.2)

Equation (3.2) is integrable; therefore, integrating with respect to 𝛼 once yields: 1π»βˆ’πΆπ‘+3𝑏3+2π‘ξ…žξ…ž=0,(3.3) where 𝐻 is an integral constant that is to be determined later.

Taking the homogeneous balance between 𝑏3 and π‘ξ…žξ…ž in (3.3), we obtain 𝑀=1.

Therefore, the solution of (3.3) is of the form 𝑏(𝛼)=π‘βˆ’1(𝐺′/𝐺)βˆ’1+𝑝0+𝑝1(𝐺′/𝐺),(3.4) where π‘βˆ’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)π‘Ÿ,(π‘Ÿ=0,Β±1,Β±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 π‘βˆ’1,𝑝0,𝑝1,𝐢,𝐻,πœ†, and πœ‡.

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

Case 1. We have π‘βˆ’1=0,𝑝0√=Β±πœ†π‘–3,𝑝1√=Β±2𝑖3,𝐢=4πœ‡βˆ’πœ†2,𝐻=0,(3.5) where πœ† and πœ‡ are free parameters.

Case 2. We have π‘βˆ’1√=Β±2πœ‡π‘–3,𝑝0√=Β±πœ†π‘–3,𝑝1=0,𝐢=4πœ‡βˆ’πœ†2,𝐻=0,(3.6) where πœ† and πœ‡ are free parameters.

Case 3. We have π‘βˆ’1√=Β±2πœ‡π‘–3,𝑝0√=Β±πœ†π‘–3,𝑝1√=Β±2𝑖3,𝐢=βˆ’8πœ‡βˆ’πœ†2√,𝐻=Β±8πœ†πœ‡π‘–3,(3.7) where πœ† and πœ‡ 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 πœ†2βˆ’4πœ‡>0, we obtain 𝑏(𝛼)=π‘βˆ’1ξƒ©βˆ’πœ†2+12ξ”πœ†2βˆšβˆ’4πœ‡π‘ˆsinh(1/2)πœ†2βˆšβˆ’4πœ‡π›Ό+𝑉cosh(1/2)πœ†2βˆ’4πœ‡π›Όβˆšπ‘ˆcosh(1/2)πœ†2βˆšβˆ’4πœ‡π›Ό+𝑉sinh(1/2)πœ†2ξƒͺβˆ’4πœ‡π›Όβˆ’1+𝑝0+𝑝1ξƒ©βˆ’πœ†2+12ξ”πœ†2βˆšβˆ’4πœ‡π‘ˆsinh(1/2)πœ†2βˆšβˆ’4πœ‡π›Ό+𝑉cosh(1/2)πœ†2βˆ’4πœ‡π›Όβˆšπ‘ˆcosh(1/2)πœ†2βˆšβˆ’4πœ‡π›Ό+𝑉sinh(1/2)πœ†2ξƒͺ.βˆ’4πœ‡π›Ό(3.8) If π‘ˆ and 𝑉 take particular values, various known solutions can be rediscovered.
For example,(i)if π‘ˆ=0 but 𝑉≠0, we obtain 𝑏(𝛼)=π‘βˆ’1ξ‚΅βˆ’πœ†2+12ξ”πœ†21βˆ’4πœ‡coth2ξ”πœ†2ξ‚Άβˆ’4πœ‡π›Όβˆ’1+𝑝0+𝑝1ξ‚΅βˆ’πœ†2+12ξ”πœ†21βˆ’4πœ‡coth2ξ”πœ†2ξ‚Ά,βˆ’4πœ‡π›Ό(3.9)(ii)if 𝑉=0 but π‘ˆβ‰ 0, we obtain 𝑏(𝛼)=π‘βˆ’1ξ‚΅βˆ’πœ†2+12ξ”πœ†21βˆ’4πœ‡tanh2ξ”πœ†2ξ‚Άβˆ’4πœ‡π›Όβˆ’1+𝑝0+𝑝1ξ‚΅βˆ’πœ†2+12ξ”πœ†21βˆ’4πœ‡tanh2ξ”πœ†2ξ‚Ά,βˆ’4πœ‡π›Ό(3.10)(iii)if π‘ˆβ‰ 0,π‘ˆ>𝑉, we obtain 𝑏(𝛼)=π‘βˆ’1ξ‚΅βˆ’πœ†2+12ξ”πœ†2ξ‚΅1βˆ’4πœ‡tanh2ξ”πœ†2βˆ’4πœ‡π›Ό+𝛼0ξ‚Άξ‚Άβˆ’1+𝑝0+𝑝1ξ‚΅βˆ’πœ†2+12ξ”πœ†2ξ‚΅1βˆ’4πœ‡tanh2ξ”πœ†2βˆ’4πœ‡π›Ό+𝛼0.ξ‚Άξ‚Ά(3.11)

Family 2 (trigonometric function solutions). When πœ†2βˆ’4πœ‡<0, we obtain 𝑏(𝛼)=π‘βˆ’1ξƒ©βˆ’πœ†2+124πœ‡βˆ’πœ†2βˆšβˆ’π‘ˆsin(1/2)4πœ‡βˆ’πœ†2βˆšπ›Ό+𝑉cos(1/2)4πœ‡βˆ’πœ†2π›Όβˆšπ‘ˆcos(1/2)4πœ‡βˆ’πœ†2βˆšπ›Ό+𝑉sin(1/2)4πœ‡βˆ’πœ†2𝛼ξƒͺβˆ’1+𝑝0+𝑝1ξƒ©βˆ’πœ†2+124πœ‡βˆ’πœ†2βˆšβˆ’π‘ˆsin(1/2)4πœ‡βˆ’πœ†2βˆšπ›Ό+𝑉cos(1/2)4πœ‡βˆ’πœ†2π›Όβˆšπ‘ˆcos(1/2)4πœ‡βˆ’πœ†2βˆšπ›Ό+𝑉sin(1/2)4πœ‡βˆ’πœ†2𝛼ξƒͺ.(3.12) If π‘ˆ and 𝑉 take particular values, various known solutions can be rediscovered.
For example,(iv) if π‘ˆ=0 but 𝑉≠0, we obtain 𝑏(𝛼)=π‘βˆ’1ξ‚΅βˆ’πœ†2+124πœ‡βˆ’πœ†2cot124πœ‡βˆ’πœ†2π›Όξ‚Άβˆ’1+𝑝0+𝑝1ξ‚΅βˆ’πœ†2+124πœ‡βˆ’πœ†2cot124πœ‡βˆ’πœ†2𝛼,(3.13)(v) if 𝑉=0 but π‘ˆβ‰ 0, we obtain 𝑏(𝛼)=π‘βˆ’1ξ‚΅βˆ’πœ†2βˆ’124πœ‡βˆ’πœ†21tan24πœ‡βˆ’πœ†2π›Όξ‚Άβˆ’1+𝑝0+𝑝1ξ‚΅βˆ’πœ†2βˆ’124πœ‡βˆ’πœ†21tan24πœ‡βˆ’πœ†2𝛼,(3.14)(vi) if π‘ˆβ‰ 0,π‘ˆ>𝑉, we obtain 𝑏(𝛼)=π‘βˆ’1ξ‚΅βˆ’πœ†2+124πœ‡βˆ’πœ†2ξ‚΅1tan24πœ‡βˆ’πœ†2π›Όβˆ’π›Ό0ξ‚Άξ‚Άβˆ’1+𝑝0+𝑝1ξ‚΅βˆ’πœ†2+124πœ‡βˆ’πœ†2ξ‚΅1tan24πœ‡βˆ’πœ†2π›Όβˆ’π›Ό0.ξ‚Άξ‚Ά(3.15)

Family 3 (rational function solution). When πœ†2βˆ’4πœ‡=0, we obtain 𝑏(𝛼)=π‘βˆ’1ξ‚€βˆ’πœ†2+π‘‰ξ‚π‘ˆ+π‘‰π›Όβˆ’1+𝑝0+𝑝1ξ‚€βˆ’πœ†2+𝑉.π‘ˆ+𝑉𝛼(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 π‘ˆ=0 but 𝑉≠0), 𝑏1(𝛼)=±𝑖3ξ€·πœ†2ξ€Έ1βˆ’4πœ‡coth2ξ”πœ†2βˆ’4πœ‡π›Ό,(3.17) where 𝛼=π‘₯+𝑦+(πœ†2βˆ’4πœ‡)𝑑, 𝑏2√(𝛼)=±𝑖3βŽ›βŽœβŽœβŽξƒ©2πœ‡βˆ’πœ†2+βˆšπœ†2βˆ’4πœ‡21coth2ξ”πœ†2ξƒͺβˆ’4πœ‡π›Όβˆ’1⎞⎟⎟⎠,+πœ†(3.18) where 𝛼=π‘₯+𝑦+(πœ†2βˆ’4πœ‡)𝑑, 𝑏3√(𝛼)=±𝑖3βŽ›βŽœβŽœβŽξƒ©2πœ‡βˆ’πœ†2+βˆšπœ†2βˆ’4πœ‡21coth2ξ”πœ†2ξƒͺβˆ’4πœ‡π›Όβˆ’1+ξ”πœ†21βˆ’4πœ‡coth2ξ”πœ†2⎞⎟⎟⎠,βˆ’4πœ‡π›Ό(3.19) where 𝛼=π‘₯+𝑦+(πœ†2+8πœ‡)𝑑.
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 π‘ˆβ‰ 0), 𝑏4(𝛼)=±𝑖3ξ€·πœ†2ξ€Έ1βˆ’4πœ‡tanh2ξ”πœ†2βˆ’4πœ‡π›Ό,(3.20)𝑏5(βˆšπ›Ό)=±𝑖3βŽ›βŽœβŽœβŽξƒ©2πœ‡βˆ’πœ†2+βˆšπœ†2βˆ’4πœ‡21tanh2ξ”πœ†2ξƒͺβˆ’4πœ‡π›Όβˆ’1⎞⎟⎟⎠,+πœ†(3.21)𝑏6√(𝛼)=±𝑖3βŽ›βŽœβŽœβŽξƒ©2πœ‡βˆ’πœ†2+βˆšπœ†2βˆ’4πœ‡21tanh2ξ”πœ†2ξƒͺβˆ’4πœ‡π›Όβˆ’1+ξ”πœ†21βˆ’4πœ‡tanh2ξ”πœ†2⎞⎟⎟⎠.βˆ’4πœ‡π›Ό(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 π‘ˆβ‰ 0,π‘ˆ>𝑉), 𝑏7(𝛼)=±𝑖3ξ€·πœ†2ξ€Έξ‚΅1βˆ’4πœ‡tanh2ξ”πœ†2βˆ’4πœ‡π›Ό+𝛼0ξ‚Ά,(3.23) where 𝛼0= tanhβˆ’1(𝑉/π‘ˆ), 𝑏8√(𝛼)=±𝑖3βŽ›βŽœβŽœβŽξƒ©2πœ‡βˆ’πœ†2+βˆšπœ†2βˆ’4πœ‡2ξ‚΅1tanh2ξ”πœ†2βˆ’4πœ‡π›Ό+𝛼0ξ‚Άξƒͺβˆ’1⎞⎟⎟⎠,+πœ†(3.24) where 𝛼0= tanhβˆ’1(𝑉/π‘ˆ), 𝑏9√(𝛼)=±𝑖3βŽ›βŽœβŽœβŽξƒ©2πœ‡βˆ’πœ†2+βˆšπœ†2βˆ’4πœ‡2ξ‚΅1tanh2ξ”πœ†2βˆ’4πœ‡π›Ό+𝛼0ξ‚Άξƒͺβˆ’1+ξ”πœ†2ξ‚΅1βˆ’4πœ‡tanh2ξ”πœ†2βˆ’4πœ‡π›Ό+𝛼0ξ‚ΆβŽžβŽŸβŽŸβŽ ,(3.25) where 𝛼0= tanhβˆ’1(𝑉/π‘ˆ).

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 π‘ˆ=0 but 𝑉≠0), 𝑏10(𝛼)=±𝑖3ξ€·4πœ‡βˆ’πœ†2ξ€Έcot124πœ‡βˆ’πœ†2𝛼,(3.26) where 𝛼=π‘₯+π‘¦βˆ’(4πœ‡βˆ’πœ†2)𝑑, 𝑏11√(𝛼)=±𝑖3βŽ›βŽœβŽœβŽξƒ©2πœ‡βˆ’πœ†2+√4πœ‡βˆ’πœ†22cot124πœ‡βˆ’πœ†2𝛼ξƒͺβˆ’1⎞⎟⎟⎠,+πœ†(3.27) where 𝛼=π‘₯+π‘¦βˆ’(4πœ‡βˆ’πœ†2)𝑑, 𝑏12√(𝛼)=±𝑖3βŽ›βŽœβŽœβŽξƒ©2πœ‡βˆ’πœ†2+√4πœ‡βˆ’πœ†22cot124πœ‡βˆ’πœ†2𝛼ξƒͺβˆ’1+4πœ‡βˆ’πœ†2cot124πœ‡βˆ’πœ†2π›ΌβŽžβŽŸβŽŸβŽ ,(3.28) where 𝛼=π‘₯+π‘¦βˆ’(βˆ’8πœ‡βˆ’πœ†2)𝑑.

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 π‘ˆβ‰ 0), 𝑏13(𝛼)=βˆ“π‘–3ξ€·4πœ‡βˆ’πœ†2ξ€Έ1tan24πœ‡βˆ’πœ†2𝛼,(3.29)𝑏14(βˆšπ›Ό)=±𝑖3βŽ›βŽœβŽœβŽξƒ©2πœ‡βˆ’πœ†2βˆ’βˆš4πœ‡βˆ’πœ†221tan24πœ‡βˆ’πœ†2𝛼ξƒͺβˆ’1⎞⎟⎟⎠,+πœ†(3.30)𝑏15√(𝛼)=±𝑖3βŽ›βŽœβŽœβŽξƒ©2πœ‡βˆ’πœ†2βˆ’βˆš4πœ‡βˆ’πœ†221tan24πœ‡βˆ’πœ†2𝛼ξƒͺβˆ’1βˆ’ξ”4πœ‡βˆ’πœ†21tan24πœ‡βˆ’πœ†2π›ΌβŽžβŽŸβŽŸβŽ .(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 π‘ˆβ‰ 0,π‘ˆ>𝑉), 𝑏16(𝛼)=±𝑖3ξ€·4πœ‡βˆ’πœ†2ξ€Έξ‚΅1tan24πœ‡βˆ’πœ†2π›Όβˆ’π›Ό0ξ‚Ά,(3.32) where 𝛼0=tanβˆ’1(𝑉/π‘ˆ), 𝑏17√(𝛼)=±𝑖3βŽ›βŽœβŽœβŽξƒ©2πœ‡βˆ’πœ†2+√4πœ‡βˆ’πœ†22ξ‚΅1tan24πœ‡βˆ’πœ†2π›Όβˆ’π›Ό0ξ‚Άξƒͺβˆ’1⎞⎟⎟⎠,+πœ†(3.33) where 𝛼0=tanβˆ’1(𝑉/π‘ˆ), 𝑏18√(𝛼)=±𝑖3βŽ›βŽœβŽœβŽξƒ©2πœ‡βˆ’πœ†2+√4πœ‡βˆ’πœ†22ξ‚΅1tan24πœ‡βˆ’πœ†2π›Όβˆ’π›Ό0ξ‚Άξƒͺβˆ’1+4πœ‡βˆ’πœ†2ξ‚΅1tan24πœ‡βˆ’πœ†2π›Όβˆ’π›Ό0ξ‚ΆβŽžβŽŸβŽŸβŽ ,(3.34) where 𝛼0=tanβˆ’1(𝑉/π‘ˆ).

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 πœ†2βˆ’4πœ‡=0), 𝑏19√(𝛼)=±𝑖32𝑉,(π‘ˆ+𝑉𝛼)(3.35) where 𝛼=π‘₯+𝑦+(πœ†2βˆ’4πœ‡)𝑑, 𝑏20√(𝛼)=±𝑖3ξ‚΅ξ‚€2πœ‡βˆ’πœ†2+π‘‰ξ‚π‘ˆ+π‘‰π›Όβˆ’1ξ‚Ά,+πœ†(3.36) where 𝛼=π‘₯+𝑦+(πœ†2βˆ’4πœ‡)𝑑, 𝑏21√(𝛼)=±𝑖3ξ‚΅ξ‚€2πœ‡βˆ’πœ†2+π‘‰ξ‚π‘ˆ+π‘‰π›Όβˆ’1+2𝑉,π‘ˆ+𝑉𝛼(3.37) where 𝛼=π‘₯+𝑦+(πœ†2+8πœ‡)𝑑.

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.

tab1
Table 1: Comparison between Bekir [41] solutions and Newly obtained solutions.
438928.fig.001
Figure 1: Solitons solution for πœ†=3,πœ‡=2.
438928.fig.002
Figure 2: Solitons solution for πœ†=2,πœ‡=0.5.
438928.fig.003
Figure 3: Solitons solution for πœ†=6,πœ‡=8.
438928.fig.004
Figure 4: Periodic solution for πœ†=5,πœ‡=6.
438928.fig.005
Figure 5: Periodic solution for πœ†=5,πœ‡=4.
438928.fig.006
Figure 6: Periodic solution for πœ†=3,πœ‡=2.
438928.fig.007
Figure 7: Solitons solution for πœ†=2,πœ‡=0.5.
438928.fig.008
Figure 8: Solitons solution for πœ†=1,πœ‡=1.
438928.fig.009
Figure 9: Solitons solution for πœ†=7,πœ‡=14.
438928.fig.0010
Figure 10: Solitons solution for πœ†=6,πœ‡=10.
438928.fig.0011
Figure 11: Solitons solution for πœ†=4,πœ‡=5.
438928.fig.0012
Figure 12: Solitons solution for πœ†=3,πœ‡=3.
438928.fig.0013
Figure 13: Solitons solution for πœ†=3,πœ‡=4.
438928.fig.0014
Figure 14: Periodic solution for πœ†=0.5,πœ‡=1.

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.

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