New Traveling Wave Solutions by the Extended Generalized Riccati Equation Mapping Method of the -Dimensional Evolution Equation
The generalized Riccati equation mapping is extended with the basic -expansion method which is powerful and straightforward mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty-seven traveling wave solutions for the (2+1)-dimensional modified Zakharov-Kuznetsov equation by applying this method. Further, the auxiliary equation is executed with arbitrary constant coefficients and called the generalized Riccati equation. The obtained solutions including solitons and periodic solutions are illustrated through the hyperbolic functions, the trigonometric functions, and the rational functions. In addition, it is worth declaring that one of our solutions is identical for special case with already established result which verifies our other solutions. Moreover, some of obtained solutions are depicted in the figures with the aid of Maple.
The study of analytical solutions for nonlinear partial differential equations (PDEs) has become more imperative and stimulating research fields in mathematical physics, engineering sciences, and other technical arena [1–47]. In the recent past, a wide range of methods have been developed to construct traveling wave solutions of nonlinear PDEs such as, the inverse scattering method , the Backlund transformation method , the Hirota bilinear transformation method , the bifurcation method [4, 5], the Jacobi elliptic function expansion method [6–8], the Weierstrass elliptic function method , the direct algebraic method , the homotopy perturbation method [11, 12], the Exp-function method [13–17], and others [18–28].
Recently, Wang et al.  presented a widely used method, called the ()-expansion method to obtain traveling wave solutions for some nonlinear evolution equations (NLEEs). Further, in this method, the second-order linear ordinary differential equation is implemented, as an auxiliary equation, where and are constant coefficients. Afterwards, many researchers investigated many nonlinear PDEs to construct traveling wave solutions via this powerful ()-expansion method. For example, Feng et al.  applied the same method for obtaining exact solutions of the Kolmogorov-Petrovskii-Piskunov equation. In , Naher et al. concerned about this method to construct traveling wave solutions for the higher-order Caudrey-Dodd-Gibbon equation. Zayed and Al-Joudi  studied some nonlinear partial differential equations to obtain analytical solutions by using the same method whereas Gepreel  executed this method and found exact solutions of nonlinear PDEs with variable coefficients in mathematical physics. Abazari and Abazari  constructed exact solutions for the Hirota-Ramani equation by using this method. Ozis and Aslan  established some traveling wave solutions for the Kawahara type equations via the same method. Naher et al.  investigated higher dimensional nonlinear evolution equation for obtaining traveling wave solutions by applying the improved ()-expansion method. Naher and Abdullah  implemented this method to construct some new traveling wave solutions of the nonlinear reaction diffusion equation whilst they  studied the combined KdV-MKdV equation for obtaining abundant solutions via the same method. And  they executed this method to establish some new traveling wave solutions of the -dimensional modified Zakharov-Kuznetsov equation and so on.
Zhu  investigated the -dimensional Boiti-Leon-Pempinelle equation by applying the generalized Riccati equation mapping with the extended tanh-function method. In addition, is used, as an auxiliary equation and called generalized Riccati equation, where , , and are arbitrary constants. Bekir and Cevikel  concerned about the tanh-coth method combined with the Riccati equation to study nonlinear coupled equation in mathematical physics. Guo et al.  implemented the extended Riccati equation mapping method for solving the diffusion-reaction and the mKdV equation with variable coefficient whilst Li et al.  studied higher-dimensional Jimbo-Miwa equation via the generalized Riccati equation expansion method. Salas  obtained some exact solutions for the Caudrey-Dodd-Gibbon equation by applying the projective Riccati equation method whereas Naher and Abdullah  studied the modified Benjamin-Bona-Mahony equation via the generalized Riccati equation mapping with the basic ()-expansion method for constructing traveling wave solutions and so on.
Many researchers implemented various methods to investigate the -dimensional modified Zakharov-Kuznetsov equation. For instance, Khalfallah  used homogeneous balance method to establish traveling wave solutions of this equation. In , Bekir applied basic ()-expansion method for obtaining exact traveling wave solutions for the same equation. In this basic ()-expansion method, they employed second-order linear ordinary differential equation (LODE) with constant coefficients, as an auxiliary equation instead of generalized Riccati equation.
The importance of our present work is, in order to construct many new traveling wave solutions including solitons, periodic, and rational solutions, a -dimensional Modified Zakharov-Kuznetsov equation considered by applying the extended generalized Riccati equation mapping method.
2. The Extended Generalized Riccati Equation Mapping Method
Suppose the general nonlinear partial differential equation where is an unknown function, is a polynomial in , and the subscripts indicate the partial derivatives.
Step 1. Consider the traveling wave variable: where is the speed of the traveling wave. Now using (2.2), (2.1) is converted into an ordinary differential equation for : where the superscripts stand for the ordinary derivatives with respect to .
Step 2. Equation (2.3) integrates term by term one or more times according to possibility and yields constant(s) of integration. The integral constant(s) may be zero for simplicity.
Step 3. Suppose that the traveling wave solution of (2.3) can be expressed in the form [29, 40] where and , with is the solution of the generalized Riccati equation: where , , are arbitrary constants and .
Step 4. To decide the positive integer , consider the homogeneous balance between the nonlinear terms and the highest order derivatives appearing in (2.3).
Step 5. Substitute (2.4) along with (2.5) into the (2.3), then collect all the coefficients with the same order, the left hand side of (2.3) converts into polynomials in and . Then equating each coefficient of the polynomials to zero and yield a set of algebraic equations for , , , , and .
Step 6. Solve the system of algebraic equations which are found in Step 5 with the aid of algebraic software Maple to obtain values for and then, substitute obtained values in (2.4) along with (2.5) with the value of , we obtain exact solutions of (2.1).
In the following, we have twenty seven solutions including four different families of (2.5).
Family 1. When and or , the solutions of (2.5) are: where and are two nonzero real constants. where .
Family 2. When and or , the solutions of (2.5) are: where and are two nonzero real constants and satisfy . where .
Family 3. When and , the solution (2.5) becomes: where is an arbitrary constant.
Family 4. when and , the solution of (2.5) becomes: where is an arbitrary constant.
3. Applications of the Method
In this section, we have constructed new traveling wave solutions for the -dimensional modified Zakharov-Kuznetsov equation by using the method.
3.1. The -Dimensional Modified Zakharov-Kuznetsov Equation
We consider the -dimensional Modified Zakharov-Kuznetsov equation followed by Bekir  Now, we use the wave transformation (2.2) into the (3.1), which yields: Equation (3.2) is integrable, therefore, integrating with respect once yields: where is an integral constant which 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:
By substituting (3.5) into (3.3), the left hand side is converted into polynomials in and . Setting each coefficient of these resulted polynomials to zero, we obtain a set of algebraic equations for , , , , , , and (algebraic equations are not shown, for simplicity). Solving the system of algebraic equations with the help of algebraic software Maple, we obtain
Family 5. The soliton and soliton-like solutions of (3.1) (when and or ) are: where , and , , are arbitrary constants. where , , and are two nonzero real constants.
Family 6. The periodic form solutions of (3.1) (when and or ) are: where , and , , are arbitrary constants. where , , and are two nonzero real constants and satisfies .
Family 7. The soliton and soliton-like solutions of (3.1) (when and ) are: where is an arbitrary constant, .
Family 8. The rational function solution (when and ) is: where is an arbitrary constant and .
4. Results and Discussion
It is significant to mention that one of our solutions is coincided for some special case with already published results which are presented in Table 1. Furthermore, some of newly constructed solutions are illustrated in Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.
As in Table 1, we have newly constructed traveling wave solutions to which are not being stated in the earlier literature.
4.1. Graphical Depictions of Newly Obtained Traveling Wave Solutions
In this paper, we have investigated the -dimensional modified Zakharov-Kuznetsov equation via the extended generalized Riccati equation mapping method. Twenty seven exact traveling wave solutions are constructed including solitons and periodic wave solutions by applying this powerful method. In addition, newly obtained solutions are depicted in terms of the hyperbolic, the trigonometric, and the rational functional form. The obtained solutions reveal that this method is a promising mathematical tool because it can establish a variety of new solutions of dissimilar physical structures if compared with existing methods. The correctness of newly constructed solutions is verified to be compared with already published results. Consequently, nonlinear evolution equations which regularly arise in many scientific real-time application fields can be studied by applying the extended generalized Riccati equation mapping method.
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.
C. Rogers and W. F. Shadwick, Backlund Transformations and Their Applications, Academic Press, New York, NY, USA, 1982.View at: Zentralblatt MATH
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.View at: Google Scholar
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.View at: Publisher Site | Google Scholar | Zentralblatt MATH
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.View at: Google Scholar
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.View at: Google Scholar
K. A. Gepreel, “Exact solutions for nonlinear PDEs with the variable coefficients in mathematical physics,” Journal of Information and Computing Science, vol. 6, no. 1, pp. 003–014, 2011.View at: Google Scholar
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.View at: Google Scholar
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.View at: Google Scholar
A. Bekir and A. C. Cevikel, “The tanh-coth method combined with the Riccati equation for solving nonlinear coupled equation in mathematical physics,” Journal of King Saud University, vol. 23, pp. 127–132, 2011.View at: Google Scholar
H. Naher and F. A. Abdullah, “The modified Benjamin-Bona-Mahony equation via the extended generalized Riccati equation mapping method,” Applied Mathematical Sciences, vol. 6, no. 111, pp. 5495–5512, 2012.View at: Google Scholar