Abstract and Applied Analysis

Volume 2013 (2013), Article ID 268902, 5 pages

http://dx.doi.org/10.1155/2013/268902

## New Exact Solutions for a Generalized Double Sinh-Gordon Equation

International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa

Received 26 June 2013; Revised 15 July 2013; Accepted 15 July 2013

Academic Editor: Santanu Saha Ray

Copyright © 2013 Gabriel Magalakwe and Chaudry Masood Khalique. 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 study a generalized double sinh-Gordon equation, which has applications in various fields, such as fluid dynamics, integrable quantum field theory, and kink dynamics. We employ the Exp-function method to obtain new exact solutions for this generalized double sinh-Gordon equation. This method is important as it gives us new solutions of the generalized double sinh-Gordon equation.

#### 1. Introduction

It is well known that finding exact travelling wave solutions of nonlinear partial differential equations (NLPDEs) is useful in many scientific applications such as fluid mechanics, plasma physics, and quantum field theory. Due to these applications many researchers are investigating exact solutions of NLPDEs since they play a vital role in the study of nonlinear physical phenomena. Finding exact solutions of such NLPDEs provides us with a better understanding of the physical phenomena that these NLPDEs describe. Several techniques have been presented in the literature to find exact solutions of the NLPDEs. These include the homogeneous balance method, the Weierstrass elliptic function expansion method, the -expansion method, the -expansion method, the Exp-function method, the tanh function method, the extended tanh function method, and the Lie group method [1–10].

In this work, we study one such NLPDE, namely, the generalized double sinh-Gordon equation: which appears in many scientific applications [11–13]. It should be noted that when , , and , (1) becomes the generalized sinh-Gordon equation [14, 15]. Furthermore, if and , (1) reduces to the sinh-Gordon equation [16].

Many authors have studied the generalized double sinh-Gordon equation (1). Travelling waves solutions of (1) were obtained in [11] by using the tanh function method and the variable separable method. In [12] the method of bifurcation theory of dynamical system was used to prove the existence of periodic wave, solitary wave, kink and antikink wave, and unbounded wave solutions of (1). It should be noted that solutions obtained in [12] were different the ones obtained in [11]. Recently, solitary and periodic waves solutions of (1) were found in [13] by employing -expansion method. It is further shown in [13] that solutions obtained by using the -expansion method are more general than those given in [11], which were obtained by tanh function method.

In this paper, we employ an entirely different method, known as the Exp-function method, to obtain new exact solutions of the generalized sinh-Gordon equation (1). The paper is structured as follows. In Section 2, we obtain exact solutions of the generalized double sinh-Gordon equation (1) with the help of the Exp-function method. In Section 3 we present concluding remarks.

#### 2. Exact Solutions of (1) Using Exp-Function Method

In this section we employ the Exp-function method to solve the generalized double sinh-Gordon equation (1). This method was introduced by He and Wu [17]. The Exp-function method results in the travelling wave solution based on the assumption that the solution can be expressed in the following form: where , , , and are positive integers that can be determined and and are unknown constants. According to Exp-function method, we introduce the travelling wave substitution , where . Then (1) transforms to the nonlinear ordinary differential equation: Further, using the transformation on (3), we obtain We assume that the solution of (4) can be expressed as The values of and , and can be determined by balancing the linear term of the highest order with the highest order of nonlinear term in (4), that is, and . By straight forward calculation, we have where are coefficients only for simplicity. Balancing the highest order of Exp-function in (6), we have , which yields . Similarly, we balance the lowest order in (4) to determine values of and . We have where are coefficients only for simplicity. Balancing the lowest order of Exp-function in (7), we have , which yields . For simplicity, we first set and . then (5) reduces to

Inserting (8) into (4) and using Maple, we obtain where Equating the coefficients of in (9) to zero, we obtain a set of algebraic equations: Solving the system (11) with the help of Maple, we obtain the following three cases.

*Case 1. *We have the following:

*Case 2. *We have the following:

*Case 3. *We have the following:
where .

Substituting values from (12) into (8), we obtain As a result one of the solutions of (1) is given by where , and .

As a special case, if we choose and in (16), then we get , and obtain the solution of the generalized sinh-Gordon equation as which is the solution obtained in [14, 15].

Now substituting the values from (13) (Case 2) into (8) results in the second solution of (1) as with , , and .

The third solution of (1) is obtained by using the values from (14) (Case 3) and substituting them into (8). Consequently, it is given by where , , , and .

To construct more solutions of (1), we now set and . Then (5) reduces to Proceeding as above, we obtain the following three solutions of (1): where , , with , , and , and where , , and .

By taking , , , , and in the solution (16), we have its profile given in Figure 1.

By taking , , , , and in the solution (23), we have its profile given in Figure 2.

#### 3. Concluding Remarks

In this paper we obtained new exact solutions of the generalized double sinh-Gordon equation (1) using the Exp-function method. We presented six different solutions of (1). Earlier, the tanh function, the bifurcation, and the -expansion methods [11–13] were employed to obtain exact solutions of (1). The solutions obtained in this paper were new and were different from the ones obtained in [11–13]. By taking special values of the constants, we also retrieved the solution of the generalized sinh-Gordon equation, which was obtained in [14, 15]. The Exp-function method is very simple and straightforward method for solving nonlinear partial differential equations. Indeed this has some pronounced merit as compared to the other methods. The correctness of the solutions obtained here has been verified by substituting them back into (1).

#### Acknowledgments

Gabriel Magalakwe would like to thank SANHARP, NRF, and North-West University, Mafikeng Campus, South Africa, for their financial support.

#### References

- M. Wang, Y. Zhou, and Z. Li, “Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics,”
*Physics Letters A*, vol. 216, no. 1–5, pp. 67–75, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - Y. Chen and Z. Yan, “The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations,”
*Chaos, Solitons and Fractals*, vol. 29, no. 4, pp. 948–964, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - 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. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - M. Wang, X. Li, and J. Zhang, “The(${G}^{\prime}$,
*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 · View at Zentralblatt MATH · 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 - H. Zhi and H. Zhang, “Applications of the combined tanh function method with symmetry method to the nonlinear evolution equations,”
*Applied Mathematics and Computation*, vol. 188, no. 1, pp. 385–393, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - D.-S. Li and H.-Q. Zhang, “A further extended tanh-function method and new soliton-like solutions to the integrable Broer-Kaup (BK) equations in (2 + 1) dimensional spaces,”
*Applied Mathematics and Computation*, vol. 147, no. 2, pp. 537–545, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - G. W. Bluman and S. Kumei,
*Symmetries and Differential Equations*, vol. 81 of*Applied Mathematical Sciences*, Springer, New York, NY, USA, 1989. - P. J. Olver,
*Applications of lie Groups to Differential Equations*, vol. 107 of*Graduate Texts in Mathematics*, Springer, Berlin, Germany, 2nd edition, 1993. - N. H. Ibragimov,
*CRC Handbook of Lie Group Analysis of Differential Equations*, vol. 1–3, CRC Press, Boca Raton, Fla, USA, 1994–1996. - A. M. Wazwaz, “Exact solutions to the double sinh-Gordon equation by the tanh method and a variable separated ODE method,”
*Computers and Mathematics with Applications*, vol. 50, no. 10–12, pp. 1685–1696, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S. Tang and W. Huang, “Bifurcations of travelling wave solutions for the generalized double sinh-Gordon equation,”
*Applied Mathematics and Computation*, vol. 189, no. 2, pp. 1774–1781, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - H. Kheiri and A. Jabbari, “Exact solutions for the double sinh-Gordon and the generalized form of the double sinh-Gordon equations by (${G}^{\prime}$/
*G*)-expansion method,”*Turkish Journal of Physics*, vol. 34, no. 2, pp. 73–82, 2010. View at Publisher · View at Google Scholar · View at Scopus - A.-M. Wazwaz, “Exact solutions for the generalized sine-Gordon and the generalized sinh-Gordon equations,”
*Chaos, Solitons and Fractals*, vol. 28, no. 1, pp. 127–135, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - K. Parand, J. A. Rad, and A. Rezaei, “Application of Exp-function method for class of nonlinear PDE’s in mathematical physics,”
*Journal of Applied Mathematics & Informatics*, vol. 29, pp. 763–779, 2011. View at Google Scholar · View at Zentralblatt MATH - A.-M. Wazwaz, “The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations,”
*Applied Mathematics and Computation*, vol. 167, no. 2, pp. 1196–1210, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - 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 · View at MathSciNet · View at Scopus