Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 725061, 14 pages

http://dx.doi.org/10.1155/2012/725061

## The Two-Variable -Expansion Method for Solving the Nonlinear KdV-mKdV Equation

Mathematics Department, Faculty of Science, Zagazig University, Zagazig 44519, Egypt

Received 6 January 2012; Accepted 17 March 2012

Academic Editor: Gradimir V. Milovanović

Copyright © 2012 E. M. E. Zayed and M. A. M. Abdelaziz. 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 two-variable (, )-expansion method to construct new exact traveling wave solutions with parameters of the nonlinear ()-dimensional KdV-mKdV equation. This method can be thought of as the generalization of the well-known ()-expansion method given recently by M. Wang et al. When the parameters are replaced by special values, the well-known solitary wave solutions of this equation are rediscovered from the traveling waves. It is shown that the proposed method provides a more general powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.

#### 1. Introduction

In the recent years, investigations of exact solutions to nonlinear PDEs play an important role in the study of nonlinear physical phenomena. Many powerful methods have been presented, such as the inverse scattering transform method [1], the Hirota method [2], the truncated Painlev expansion method [3–6], the Backlund transform method [7, 8], the exp-function method [9–13], the tanh function method [14–17], the Jacobi elliptic function expansion method [18–20], the original -expansion method [21–29], the two-variable -expansion method [30], and the first integral method [31]. The key idea of the original -expansion method is that the exact solutions of nonlinear PDEs can be expressed by a polynomial in one variable in which satisfies the second ordinary differential equation , where and are constants. In this paper, we will use the two-variable -expansion method, which can be considered as an extension of the original -expansion method. The key idea of the two-variable -expansion method is that the exact traveling wave solutions of nonlinear PDEs can be expressed by a polynomial in the two variables and , in which satisfies a second-order linear ODE, namely , where and are constants. The degree of this polynomial can be determined by considering the homogeneous balance between the highest-order derivatives and nonlinear terms in the given nonlinear PDEs, while the coefficients of this polynomial can be obtained by solving a set of algebraic equations resulted from the process of using the method. Recently, Li et al. [30] have applied the two-variable -expansion method and determined the exact solutions of Zakharov equations.

The objective of this paper is to apply the two-variable -expansion method to find the exact traveling wave solutions of the following nonlinear (1+1)-dimensional KdV-mKdV equation: where and are nonzero constants. This equation may describe the wave propagation of the bound particle, sound wave, and thermal pulse. It is the most popular soliton equation and often exists in practical problems, such as fluid physics and quantum field theory. Recently, Zayed and Gepreel [23] have found the exact solutions of (1.1) using the original -expansion method.

#### 2. Description of the Two-Variable -Expansion Method

Before we describe the main steps of this method, we need the following remarks (see [30]):

*Remark 2.1. *If we consider the second-order linear ODE
and set and , then we get

*Remark 2.2. *If , then the general solution of (2.1) is
where and are arbitrary constants. Consequently, we have
where .

*Remark 2.3. *If , then the general solution of (2.1) is
and hence
where .

*Remark 2.4. *If , then the general solution of (2.1) is
and hence
Suppose we have the following NLPDEs in the form:
where is a polynomial in and its partial derivatives. In the following, we give the main steps of the two-variable -expansion method [30].

*Step 1. *The traveling wave variable
reduces (2.9) to an ODE in the form
where is a constant and is a polynomial in and its total derivatives, while .

*Step 2. *Suppose that the solutions of (2.11) can be expressed by a polynomial in the two variables and as follows:
where and are constants to be determined later.

*Step 3. *Determine the positive integer in (2.12) by using the homogeneous balance between the highest-order derivatives and the nonlinear terms in (2.11).

*Step 4. *Substituting (2.12) into (2.11) along with (2.2) and (2.4), the left-hand side of (2.11) can be covered into a polynomial in and in which the degree of is not longer than 1. Equating each coefficient of this polynomial to zero yields a system of algebraic equations that can be solved by using the Maple or Mathematica to get the values of , and where . Thus, we get the exact solutions in terms of the hyperbolic functions.

*Step 5. *Similar to Step 4, substituting (2.12) into (2.11) along with (2.2) and (2.6) for (or (2.2) and (2.8) for ), we obtain the exact solutions of (2.11) expressed by trigonometric functions (or by rational functions), respectively.

#### 3. An Application

In this section, we apply the method described in Section 2 to find the exact traveling wave solutions of the nonlinear (1+1)-dimensional KdV-mKdV Equation (1.1). To this end, we see that the traveling wave variable (2.10) permits us to convert (1.1) into the following ODE: By balancing with in (3.1), we get . Consequently, we get where and are constants to be determined later. There are three cases to be discussed as follows.

*Case 1. *Hyperbolic function solutions .

If , substituting (3.2) into (3.1) and using (2.2) and (2.4), the left-hand side of (3.1) becomes a polynomial in and . Setting the coefficients of this polynomial to zero yields a system of algebraic equations in , and as follows:

Solving the algebraic equations (3.3) by the Maple or Mathematica, we get the following results.

*Result 1. *
We have

From (2.3) and (3.2) and (3.4),we deduce the traveling wave solution of (1.1) as follows:
where

In particular, by setting , and in (3.5), we have the solitary solution
while, if , and , then we have the solitary solution

*Result 2. * We have
In this result, we deduce the traveling wave solution of (1.1) as follows:
where

In particular, by setting , and in (3.10), we have the solitary solution
while, if , and , then we have the solitary solution

*Case 2. *Trigonometric function solutions .

If , substituting (3.2) into (3.1) and using (2.2) and (2.6), we get a polynomial in and . We vanish each coefficient of this polynomial to get the following algebraic equations.

Solving the algebraic equations (3.14) by the Maple or Mathematica, we obtain the following results.

*Result 1. *We have

From (2.5), (3.2), and (3.15), we deduce the traveling wave solution of (1.1) as follows:
where

In particular, by setting , and in (3.16), we have the periodic solution
while, if , and , then we have the periodic solution

*Result 2. *We have

In this result, we deduce the traveling wave solution of (1.1) as follows:
where

In particular, by setting , and in (3.21), we have the periodic solution
while, if , and , then we have the periodic solution

*Case 3. *Rational function solutions .

If , substituting (3.2) into (3.1) and using (2.2) and (2.8), we get a polynomial in and . Setting each coefficient of this polynomial to zero yields the following algebraic equations:

Solving the algebraic equations (3.25) by the Maple or Mathematica, we obtain the following results.

*Result 1. *We have

From (2.7), (3.2) and (3.26), we deduce the traveling wave solution of (1.1) as follows:
where

*Result 2. *We have

In this result, we deduce the traveling wave solution of (1.1) as follows:
where

*Remark 3.1. *All solutions of this paper have been checked with Maple by putting them back into the original equation (1.1).

#### 4. Conclusions

In this paper, the -expansion method was employed to obtain some new as well as some known solutions of a selected nonlinear equation, namely, the (1+1)-dimensional KdV-mKdV equation. As the two parameters and take special values, we obtain the solitary wave solutions. When and in (2.1) and (2.12), the two-variable -expansion method reduces to the original -expansion method. So, the two-variable -expansion method is an extension of the original -expansion method. The proposed method in this paper is more effective and more general than the original -expansion method because it gives exact solutions in more general forms. In summary, the advantage of the two-variable -expansion method over the original -expansion method is that the solutions using the first method recover the solutions using the second one.

#### Acknowledgment

The authors wish to thank the referees for their suggestions and comments on this paper.

#### References

- 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, New York, NY,USA, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Hirota, “Exact solutions of the KdV equation for multiple collisions of solutions,”
*Physical Review Letters*, vol. 27, pp. 1192–1194, 1971. View at Google Scholar - 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 Zentralblatt MATH - 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 Publisher · 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 - 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 - M. R. Miura,
*Bäcklund Transformation*, Springer, Berlin, Germany, 1978. - C. Rogers and W. F. Shadwick, “Backlund Transformations,” pp. Academic Press–New York, NY, USA, 1982. View at Google Scholar
- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. Yusufoglu, “New solitary for the MBBM equations using Exp-function method,”
*Physics Letters A*, vol. 372, pp. 442–446, 2008. View at Google Scholar - 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 - A. Bekir, “The exp-function for Ostrovsky equation,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 10, pp. 735–739, 2009. View at Google Scholar - 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 - M. A. Abdou, “The extended tanh method and its applications for solving nonlinear physical models,”
*Applied Mathematics and Computation*, vol. 190, no. 1, pp. 988–996, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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 Zentralblatt MATH - S. Zhang and T.-c. 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 - E. Yusufoğlu and A. Bekir, “Exact solutions of coupled nonlinear Klein-Gordon equations,”
*Mathematical and Computer Modelling*, vol. 48, no. 11-12, pp. 1694–1700, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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 - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - 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 Zentralblatt MATH - M. Wang, X. Li, and J. Zhang, “The $({G}^{\text{'}}/G)$-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 - S. Zhang, J.-L. Tong, and W. Wang, “A generalized $({G}^{\text{'}}/G)$-expansion method for the mKdV equation with variable coefficients,”
*Physics Letters A*, vol. 372, pp. 2254–2257, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. M. E. Zayed and K. A. Gepreel, “The $({G}^{\text{'}}/G)$-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 - E. M. E. Zayed, “The $({G}^{\text{'}}/G)$-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 - 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 - B. Ayhan and A. Bekir, “The $({G}^{\text{'}}/G)$-expansion method for the nonlinear lattice equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 9, pp. 3490–3498, 2012. View at Google Scholar - N. A. Kudryashov, “A note on the $({G}^{\text{'}}/G)$-expansion method,”
*Applied Mathematics and Computation*, vol. 217, no. 4, pp. 1755–1758, 2010. View at Publisher · View at Google Scholar - I. Aslan, “A note on the $({G}^{\text{'}}/G)$-expansion method again,”
*Applied Mathematics and Computation*, vol. 217, no. 2, pp. 937–938, 2010. View at Publisher · View at Google Scholar - N. A. Kudryashov, “Meromorphic solutions of nonlinear ordinary differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 10, pp. 2778–2790, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L.-X. Li, E.-Q. Li, and M.-L. Wang, “The $({G}^{\text{'}}/G,1/G)$-expansion method and its application to travelling wave solutions of the Zakharov equations,”
*Applied Mathematics B*, vol. 25, no. 4, pp. 454–462, 2010. View at Publisher · View at Google Scholar - F. Tascan and A. Bekir, “Applications of the first integral method to the nonlinear evolution equations,”
*Chinese Physics B*, vol. 19, Article ID 080201, 2010. View at Google Scholar