Abstract and Applied Analysis

Volume 2014, Article ID 259190, 10 pages

http://dx.doi.org/10.1155/2014/259190

## The Generalized Projective Riccati Equations Method for Solving Nonlinear Evolution Equations in Mathematical Physics

Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt

Received 22 March 2014; Accepted 11 August 2014; Published 6 November 2014

Academic Editor: Luiz Duarte

Copyright © 2014 E. M. E. Zayed and K. A. E. Alurrfi. 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 generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.

#### 1. Introduction

In the recent years, investigations of exact solutions to nonlinear partial differential equations (NPDEs) play an important role in the study of nonlinear physical phenomena. Nonlinear wave phenomena appear in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics, and geochemistry. To obtain traveling wave solutions, many powerful methods have been presented, such as the inverse scattering method [1], the tanh-function method [2–8], the Hirota bilinear transform method [9], the truncated Painleve expansion method [10–13], the Backlund transform method [14, 15], the Exp-function method [16–20], the Jacobi elliptic function expansion method [21–23], the generalized Riccati equations method [24–26], the -expansion method [27–33], and the -expansion method [34–36]. Conte and Musette [37] presented an indirect method to seek more solitary wave solutions of some NPDEs that can be expressed as polynomials in two elementary functions which satisfy a projective Riccati equation [38]. Using this method, many solitary wave solutions of many NPDEs are found [38, 39]. Recently, Yan [40] developed further Conte and Musette’s method by introducing more generalized projective Riccati equations.

In this paper, we will use the generalized projective Riccati equations method to construct exact solutions for the following three nonlinear evolution equations with higher-order nonlinear terms:

(i) the nonlinear Pochhammer-Chree equation [41]: where , , and are constants and ,

(ii) the nonlinear Burgers equation [42]: where and are constants;

(iii) the nonlinear generalized Zakharov-Kuznetsov equation [43]: where , , and are nonzero real constants.

Zuo [32] has applied the extended -expansion method and determined the exact solutions of (1), and Hayek [33] has found the exact solutions of (2) using another form of the extended -expansion method, while Zhang [44] has discussed (3) using an algebraic method to find some of its exact solutions. The rest of this paper is organized as follows. In Section 2, we give the description of the generalized projective Riccati equations method. In Section 3, we apply this method to solve (1)–(3). In Section 4, physical explanations of some obtained results are obtained. In Section 5, some conclusions are given.

#### 2. Description of the Generalized Projective Riccati Equations Method

Consider we have the following NPDE: where is a polynomial in and its partial derivatives, in which the highest-order derivatives and nonlinear terms are involved. In the following, we give the main steps of this method.

*Step 1. *We use the wave transformation
where is a constant, to reduce (4) to the following ODE:
where is a polynomial in and its total derivatives, such that .

*Step 2. *We assume that (6) has the formal solution
where , , and are constants to be determined later. The functions and satisfy the ODEs
where
where and are nonzero constants.

If , (6) has the formal solution
where satisfies the ODE

*Step 3. *We determine the positive integer in (7) by using the homogeneous balance between the highest-order derivatives and the nonlinear terms in (6). In some nonlinear equations the balance number is not a positive integer. In this case, we make the following transformations.

(a) When , where is a fraction in the lowest terms, we let
and then we substitute (12) into (6) to get a new equation in the new function with a positive integer balance number.

(b) When is a negative number, we let
and then we substitute(13) into (6) to get a new equation in the new function with a positive integer balance number.

*Step 4. *Substitute (7) along with (8)-(9) into (6) or (10) along with (11) into (6). Collect all terms of the same order of (or , ). Setting each coefficient to zero yields a set of algebraic equations which can be solved to find the values of , , , , , and .

*Step 5. *It is well known [24] that (8) admits the following solutions.

(i) If , ,

(ii) If , ,

(iii) If ,
where is nonzero constant.

*Step 6. *Substituting the values of , , , , , and as well as the solutions (14)–(16) into (7), we obtain the exact solutions of (4).

We close this section with the remark that without loss of generality we take (similarly the case can be done which is omitted here for simplicity).

#### 3. Applications

In this section, we will apply the proposed method described in Section 2 to find the exact traveling wave solutions of the nonlinear equations (1)–(3).

*Example 1 (the nonlinear Pochhammer-Chree equation (1)). *In this example, we find the exact solutions of (1). To this end, we see that the traveling wave variable (5) permits us to convert (1) into the following ODE:
Integrating (17) twice with respect to and vanishing the constants of integration, we get
By balancing with in (18) we get . According to Step 3, we use the transformation
where is a new function of . Substituting (19) into (18), we get the new ODE
Balancing with in (20), we get . Consequently, we get
where , , and are constants to be determined later.

Substituting (21) into (20) and using (8)-(9) with , the left-hand side of (20) becomes a polynomial in and . Setting the coefficients of this polynomial to be zero yields the following system of algebraic equations:
On solving the above algebraic equations using the Maple or Mathematica, we get the following results.

*Case 1. *We have
From (14), (19), (21), and (23), we deduce the following exact solutions:
where .

*Case 2. *We have
In this case, we deduce the following exact solutions:
where .

*Case 3. *We have
In this case, we deduce the following exact solutions:
where .

*Example 2 (the nonlinear Burgers equation (2)). *In this example, we study the Burgers equation with power-law nonlinearity (2). To this end, we see that the traveling wave variable (4) permits us to convert (2) into the following ODE:
Integrating (32) once with respect and setting the constant of integration to be zero yield
By balancing with in (33) we get . According to Step 3, we use the transformation
where is a new function of . Substituting (34) into (33), we get the new ODE
Balancing with in (35), we get . Consequently, we get
where , , and are constants to be determined later.

Substituting (36) into (35) and using (8)-(9) with , the left-hand side of (35) becomes a polynomial in and . Setting the coefficients of this polynomial to be zero yields the following system of algebraic equations:
On solving the above algebraic equations using the Maple or Mathematica, we get the following results.

*Case 1. *We have
From (14), (34), (36), and (38), we deduce the following exact solutions:
where .

*Case 2. *We have
In this case, we deduce the following exact solutions:
where .

*Example 3 (the nonlinear generalized Zakharov-Kuznetsov equation (3)). *In this example, we study the generalized Zakharov-Kuznetsov equation with power-law nonlinearity (3). To this end, we use the traveling wave variable
where and are nonzero constants, to reduce (3) to the following ODE:
By balancing with in (45) we get . According to Step 3, we use the transformation
where is a new function of . Substituting (46) into (45), we get the new ODE
Balancing with in (47), we get . Consequently, we get
where , , and are constants to be determined later.

Substituting (48) into (47) and using (8)-(9) with , the left-hand side of (47) becomes a polynomial in and . Setting the coefficients of this polynomial to be zero yields a system of algebraic equations in , , , , and , which can be solved using the Maple or Mathematica; we get the following results.

*Case 1. *We have
From (14), (46), (48), and (49), we deduce the following exact solutions:
where , , /.

*Case 2. *We have
In this case, we have the exact solutions
where , , /.

*Case 3. *We have
In this case, we have the exact solutions
where , , /.

*Case 4. *We have
In this case, we have the exact solutions
where , , /.

*Case 5. *We have
In this case, we have the exact solutions
where , , .

#### 4. Physical Explanations of Some Obtained Solutions

In this section, we have presented some graphs of the obtained solutions constructed by taking suitable values of involved unknown parameters to visualize the underlying mechanism of the original equation. Using mathematical software Maple, three-dimensional plots of some obtained exact solutions have been shown in Figures 1, 2, 3, 4, 5, 6, 7, and 8.

##### 4.1. The Nonlinear Pochhammer-Chree Equation (1)

The obtained solutions for this equation are hyperbolic. From these explicit results it is easy to say that the solution (24) is a bell shaped soliton solution; (25) is a singular bell shaped soliton solution; (27) is a kink shaped soliton solution; (28) is a singular kink shaped soliton solution; (30) is a bell-kink shaped soliton solution; and (31) is a singular bell-kink shaped soliton solution.

##### 4.2. The Nonlinear Burgers Equation (2)

From the obtained solutions for the nonlinear Burgers equation (2) we observe that the solution (39) is a bell-kink shaped soliton solution; (40) is a singular bell-kink shaped soliton solution; (42) is a kink shaped soliton solution; and (43) is a singular kink shaped soliton solution.

##### 4.3. The Generalized Nonlinear Zakharov-Kuznetsov Equation (3)

From the obtained solutions for the generalized nonlinear Zakharov-Kuznetsov equation (3) we can easily conclude that the solution (50) is a bell shaped soliton solution; (51) is a singular bell shaped soliton solution; (53), (56), and (62) are bell-kink shaped solitons solutions; (54), (57), and (63) are singular bell-kink shaped solitons solutions; (59) is a kink shaped soliton solution; and (60) is a singular kink shaped soliton solution.

#### 5. Conclusions

The generalized projective Riccati equations method is used in this paper to obtain some new exact solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation, and the generalized Zakharov-Kuznetsov equation. On comparing our results in this paper with the well-known results obtained in [32, 33, 41–44] we deduce that our results are different and new and are not published elsewhere. The proposed method of this paper is effective and can be applied to many other nonlinear equations in mathematical physics.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### References

- M. J. Ablowitz and P. A. Clarkson,
*Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform*, vol. 149 of*New York, NY, USA*, Cambridge University Press, 1991. - E. M. E. Zayed and M. A. M. Abdelaziz, “Exact solutions for the nonlinear Schrödinger equation with variable coefficients using the generalized extended tanh-function, the sine-cosine and the exp-function methods,”
*Applied Mathematics and Computation*, vol. 218, no. 5, pp. 2259–2268, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - E. M. E. Zayed, H. A. Zedan, and K. A. Gepreel, “Group analysis and modified extended tanh-function to find the invariant solutions and soliton solutions for nonlinear Euler equations,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 5, no. 3, pp. 221–234, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. M. Wazwaz, “The extended tanh method for abundant solitary wave solutions of nonlinear wave equations,”
*Applied Mathematics and Computation*, vol. 187, no. 2, pp. 1131–1142, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - E. M. E. Zayed and M. A. M. Abdelaziz, “The tanh function method using a generalized wave transformation for nonlinear equations,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 11, no. 8, pp. 595–601, 2010. View at Google Scholar · View at Scopus - S. A. El-Wakil and M. A. Abdou, “New exact travelling wave solutions using modified extended tanh-function method,”
*Chaos, Solitons & Fractals*, vol. 31, no. 4, pp. 840–852, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - E. Fan, “Extended tanh-function method and its applications to nonlinear equations,”
*Physics Letters A*, vol. 277, no. 4-5, pp. 212–218, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Zhang and T. Xia, “A further improved tanh-function method exactly solving the (2 + 1)-dimensional dispersive long wave equations,”
*Applied Mathematics E-Notes*, vol. 8, pp. 58–66, 2008. View at Google Scholar · View at MathSciNet · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Scopus - J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,”
*Journal of Mathematical Physics*, vol. 24, no. 3, pp. 522–526, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - N. A. Kudryashov, “Exact soliton solutions of a generalized evolution equation of wave dynamics,”
*Journal of Applied Mathematics and Mechanics*, vol. 52, pp. 361–365, 1988. View at Google Scholar - N. A. Kudryashov, “Exact solutions of the generalized Kuramoto-Sivashinsky equation,”
*Physics Letters A*, vol. 147, no. 5-6, pp. 287–291, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - N. A. Kudryashov, “On types of nonlinear nonintegrable equations with exact solutions,”
*Physics Letters. A*, vol. 155, no. 4-5, pp. 269–275, 1991. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. R. Miura,
*Backlund Transformation*, Springer, Berlin, Germany, 1978. - C. Rogers and W. F. Shadwick,
*Backlund Transformations and Their Applications*, Academic Press, New York, NY, USA, 1982. View at MathSciNet - J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,”
*Chaos, Solitons & Fractals*, vol. 30, no. 3, pp. 700–708, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - E. Yusufoglu, “New solitonary solutions for the MBBM equations using Exp-function method,”
*Physics Letters A*, vol. 372, no. 4, pp. 442–446, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. Zhang, “Application of Exp-function method to high-dimensional nonlinear evolution equation,”
*Chaos, Solitons and Fractals*, vol. 38, no. 1, pp. 270–276, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - A. Bekir, “The exp-function method for Ostrovsky equation,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 10, no. 6, pp. 735–739, 2009. View at Google Scholar · View at Scopus - A. Bekir, “Application of the exp-function method for nonlinear differential-difference equations,”
*Applied Mathematics and Computation*, vol. 215, no. 11, pp. 4049–4053, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. Chen and Q. Wang, “Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1+1)-dimensional dispersive long wave equation,”
*Chaos, Solitons and Fractals*, vol. 24, no. 3, pp. 745–757, 2005. View at Publisher · View at Google Scholar · View at Scopus - S. Liu, Z. Fu, 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. Lü, “Jacobi elliptic function solutions for two variant Boussinesq equations,”
*Chaos, Solitons and Fractals*, vol. 24, no. 5, pp. 1373–1385, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - B. Li and Y. Chen, “Nonlinear partial differential equations solved by projective riccati equations ansatz,”
*Zeitschrift für Naturforschung*, vol. 58, no. 9-10, pp. 511–519, 2003. View at Google Scholar · View at Scopus - B. Li and Y. Chen, “On exact solutions of the nonlinear Schrödinger equations in optical fiber,”
*Chaos, Solitons and Fractals*, vol. 21, no. 1, pp. 241–247, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. Lu, B. Hong, and L. Tian, “New explicit exact solutions for the generalized coupled Hirota-Satsuma KdV system,”
*Computers & Mathematics with Applications*, vol. 53, no. 8, pp. 1181–1190, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Wang, X. Li, and J. Zhang, “The $\left({G}^{\prime}/G\right)$-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,”
*Physics Letters A*, vol. 372, no. 4, pp. 417–423, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Zhang, J. L. Tong, and W. Wang, “A generalized (G′/G)-expansion method for the mKdV equation with variable coefficients,”
*Physics Letters A*, vol. 372, pp. 2254–2257, 2008. View at Google Scholar - E. M. E. Zayed and K. A. Gepreel, “The $\left({G}^{\text{'}}/G\right)$-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics,”
*Journal of Mathematical Physics*, vol. 50, no. 1, Article ID 013502, 12 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - E. M. E. Zayed, “The
*G*′/*G*-expansion method and its applications to some nonlinear evolution equations in the mathematical physics,”*Journal of Applied Mathematics and Computing*, vol. 30, no. 1-2, pp. 89–103, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. Bekir, “Application of the $({G}^{\text{'}}/G)$-expansion method for nonlinear evolution equations,”
*Physics Letters A*, vol. 372, no. 19, pp. 3400–3406, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J.-M. Zuo, “Application of the extended $(G\text{'}/G)$-expansion method to solve the Pochhammer-Chree equations,”
*Applied Mathematics and Computation*, vol. 217, no. 1, pp. 376–383, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Hayek, “Constructing of exact solutions to the KdV and Burgers equations with power-law nonlinearity by the extended $\left({G}^{\text{'}}/G\right)$-expansion method,”
*Applied Mathematics and Computation*, vol. 217, no. 1, pp. 212–221, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - L. Li, E. Li, and M. Wang, “The (G′/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations,”
*Applied Mathematics*, vol. 25, no. 4, pp. 454–462, 2010. View at Publisher · View at Google Scholar · View at Scopus - E. M. E. Zayed and M. A. M. Abdelaziz, “The two-variable (
*G*′/*G*, 1/*G*)-expansion method for solving the nonlinear KDV-mKDV Equation,”*Mathematical Problems in Engineering*, vol. 2012, Article ID 725061, 14 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - E. M. E. Zayed, S. A. H. Ibrahim, and M. A. M. Abdelaziz, “Traveling wave solutions of the nonlinear $(3+1)$-dimensional Kadomtsev-Petviashvili equation using the two variables $({G}^{\text{'}}/G,1/G)$-expansion method,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 560531, 8 pages, 2012. View at Publisher · View at Google Scholar - R. Conte and M. Musette, “Link between solitary waves and projective Riccati equations,”
*Journal of Physics A: Mathematical and General*, vol. 25, no. 21, pp. 5609–5623, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - T. C. Bountis, V. Papageorgiou, and P. Winternitz, “On the integrability of systems of nonlinear ordinary differential equations with superposition principles,”
*Journal of Mathematical Physics*, vol. 27, no. 5, pp. 1215–1224, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - G. X. Zhang, Z. B. Li, and Y. S. Duan, “Exact solitary wave solutions of nonlinear wave equations,”
*Science in China A*, vol. 44, no. 3, pp. 396–401, 2001. View at Publisher · View at Google Scholar · View at Scopus - Z. Y. Yan, “Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres,”
*Chaos, Solitons and Fractals*, vol. 16, no. 5, pp. 759–766, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. M. Wazwaz, “The tanh-coth and the sine-cosine methods for kinks, solitons, and periodic solutions for the Pochhammer-Chree equations,”
*Applied Mathematics and Computation*, vol. 195, no. 1, pp. 24–33, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A.-M. Wazwaz, “Tavelling wave solutions of generalized forms of Burgers, Burgers-KdV and Burgers-Huxley equations,”
*Applied Mathematics and Computation*, vol. 169, no. 1, pp. 639–656, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - N. Hongsit, M. A. Allen, and G. Rowlands, “Growth rate of transverse instabilities of solitary pulse solutions to a family of modified Zakharov-Kuznetsov equations,”
*Physics Letters A*, vol. 372, no. 14, pp. 2420–2422, 2008. View at Publisher · View at Google Scholar · View at Scopus - L. Zhang, “Travelling wave solutions for the generalized Zakharov-Kuznetsov equation with higher-order nonlinear terms,”
*Applied Mathematics and Computation*, vol. 208, no. 1, pp. 144–155, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus