- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 284865, 7 pages

http://dx.doi.org/10.1155/2013/284865

## Abundant Exact Solition-Like Solutions to the Generalized Bretherton Equation with Arbitrary Constants

Department of Mathematics Sciences, Dezhou University, Dezhou 253023, China

Received 20 January 2013; Accepted 26 February 2013

Academic Editor: Abdel-Maksoud A. Soliman

Copyright © 2013 Xiuqing Yu et al. 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

The Riccati equation is employed to construct exact travelling wave solutions to the generalized Bretherton equation. Taking full advantage of the Riccati equation which has more new solutions, abundant new multiple solition-like solutions are obtained for the generalized Bretherton equation.

#### 1. Introduction

Nonlinear partial differential equations (PDE) are widely chosen to describe complex phenomena in physics sciences. Searching for exact solutions to nonlinear differential equations plays more and more important role in nonlinear science. Recently, various direct methods have been proposed, such as the tanh-function method [1, 2], the Jacobi elliptic function expansion method [3, 4], the F-expansion [5–8], sine-cosine method [9, 10], and the homogeneous balance method [11–13]. Among them, the tanh-function method is improved continuously [14–17] as one of the most effectively straightforward methods for constructing exact solutions to PDEs. In the paper an extended tanh-function method is used to solve the generalized Bretherton equation with arbitrary constants.

In [18], Bretherton introduced the partial differential equation in time and one spatial dimension as a model of a dispersive wave system to study the resonant nonlinear interaction between three liner models. The modified Bretherton equation was studied by Kudryashov [19], Kudryashov et al. [20], and Berloff and Howard [21], and its travelling wave solutions were obtained.

Our aim in this paper is to investigate multiple soliton-like solutions to the generalized Bretherton equation in [22] by using the solutions to the Riccati equation:

#### 2. Multiple Soliton-Like Solutions to the Generalized Bretherton Equation

We assume the travelling wave variable where is the speed of the travelling wave.

Making use of the travelling wave transformation (2), (1c) is converted into an ordinary differential equation (ODE) for as follows: We assume that the solutions to (3) can be expressed in the form where is a solution of the Riccati equation, where , and are constants to be determined later, and either or can be zero, but they cannot be zero together.

Substituting (4) into (3) together with (5) and considering the homogeneous balance between the highest-order derivative and the nonlinear term , we obtain . Thus the solution to (3) takes the following form: Substituting (6) with (5) into (3) and collecting all the terms of the same power of , the left-hand side of (3) is converted into another polynomial of . Setting the coefficients of to zero yields a set of algebraic equations Solving (7) with the help of the symbolic computation software Maple, we obtain the following.

*Case 1. *One has

*Case 2. *One has
where , , and are arbitrary constants, but cannot be zero.

*Case 3. *One has

*Case 4. *One has
where , and are arbitrary constants, but cannot be zero.

*Case 5. *One has

*Case 6. *One has
where , and are arbitrary constants, while cannot be zero in Cases 1 and 2 and cannot be zero in Cases 3–6. is an arbitrary element of .

Substituting ((8a), (8b), (8c), (8d), (8e), (8f)) into (6) respectively and taking advantage of solitions to (5), we can find the following solutions which contain multiple solition-like and triangular periodic solutions for the generalized Bretherton equation.

When , when ,

when ,

when , When , when ,

when ,

when ,

when ,

when ,

where , and are arbitrary elements of .

#### 3. Conclusion

In this paper, we have used solutions to the Riccati equation to solve the generalized Bretherton equation with arbitrary constants and obtained abundant new multiple solition-like and triangular periodic solutions. It is significant to observe practical denotation of the obtained solutions, so the obtained solutions involving arbitrary constants in this paper have potential applications in dispersive wave systems to research for resonant nonlinear interactions.

#### Acknowledgment

This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2010AL019).

#### References

- E. J. Parkes and B. R. Duffy, “An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations,”
*Computer Physics Communications*, vol. 98, no. 3, pp. 288–300, 1996. View at Publisher · View at Google Scholar - 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 · View at MathSciNet - 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 · View at MathSciNet - Z. Fu, S. Liu, S. Liu, and Q. Zhao, “New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations,”
*Physics Letters A*, vol. 290, no. 1-2, pp. 72–76, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Wang and Y. Zhou, “The periodic wave solutions for the Klein-Gordon-Schrödinger equations,”
*Physics Letters A*, vol. 318, no. 1-2, pp. 84–92, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Zhou, M. Wang, and T. Miao, “The periodic wave solutions and solitary wave solutions for a class of nonlinear partial differential equations,”
*Physics Letters A*, vol. 323, no. 1-2, pp. 77–88, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Wang and X. Li, “Extended $F$-expansion method and periodic wave solutions for the generalized Zakharov equations,”
*Physics Letters A*, vol. 343, no. 1-3, pp. 48–54, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - 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 - A.-M. Wazwaz, “A study on nonlinear dispersive partial differential equations of compact and noncompact solutions,”
*Applied Mathematics and Computation*, vol. 135, no. 2-3, pp. 399–409, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A.-M. Wazwaz, “A construction of compact and noncompact solutions for nonlinear dispersive equations of even order,”
*Applied Mathematics and Computation*, vol. 135, no. 2-3, pp. 411–424, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Wang, “Solitary wave solutions for variant Boussinesq equations,”
*Physics Letters A*, vol. 199, no. 3-4, pp. 169–172, 1995. View at Publisher · View at Google Scholar · View at MathSciNet - M. Wang, “Exact solutions for a compound KdV-Burgers equation,”
*Physics Letters A*, vol. 213, no. 5-6, pp. 279–287, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Wang, Y. Zhou, and Z. B. Li, “Application of a homogeneous balance method to exact solutions of nonlinear evolution equations in mathematical physics,”
*Physics Letters A*, vol. 216, pp. 67–75, 1996. View at Publisher · View at Google Scholar - C. Bai and H. Zhao, “New explicit exact solutions for the $(2+1)$-dimensional higher-order Broer-Kaup system,”
*Communications in Theoretical Physics*, vol. 41, no. 4, pp. 521–526, 2004. View at Google Scholar · View at MathSciNet - L. Zhang, X. Liu, and C. Bai, “New multiple soliton-like and periodic solutions for (2+1)-dimensional canonical generalized KP equation with variable coefficients,”
*Communications in Theoretical Physics*, vol. 46, pp. 793–798, 2006. View at Publisher · View at Google Scholar - Sirendaoreji, “Auxiliary equation method and new solutions of Klein-Gordon equations,”
*Chaos, Solitons and Fractals*, vol. 31, no. 4, pp. 943–950, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Sirendaoreji, “A new auxiliary equation and exact travelling wave solutions of nonlinear equations,”
*Physics Letters A*, vol. 356, no. 2, pp. 124–130, 2006. View at Publisher · View at Google Scholar - F. P. Bretherton, “Resonant interactions between waves. The case of discrete oscillations,”
*Journal of Fluid Mechanics*, vol. 20, pp. 457–479, 1964. View at Publisher · View at Google Scholar · View at MathSciNet - 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 - N. A. Kudryashov, D. I. Sinelshchikov, and M. V. Demina, “Exact solutions of the generalized Bretherton equation,”
*Physics Letters A*, vol. 375, no. 7, pp. 1074–1079, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. G. Berloff and L. N. Howard, “Nonlinear wave interactions in nonlinear nonintegrable systems,”
*Studies in Applied Mathematics*, vol. 100, no. 3, pp. 195–213, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Akbar, H. Norhashidah, A. Mohd, and E. M. E. Zayed, “Abundant exact traveling wave solutions of generalized Bretherton equation via improved $({G}^{\text{'}}/G)$-expansion method,”
*Communications in Theoretical Physics*, vol. 57, no. 2, pp. 173–178, 2012. View at Publisher · View at Google Scholar · View at MathSciNet