About this Journal Submit a Manuscript Table of Contents
Advances in Mathematical Physics
Volume 2013 (2013), Article ID 890784, 11 pages
http://dx.doi.org/10.1155/2013/890784
Research Article

A New Method with a Different Auxiliary Equation to Obtain Solitary Wave Solutions for Nonlinear Partial Differential Equations

Department of Mathematics, Firat University, 23119 Elazig, Turkey

Received 3 February 2013; Revised 22 April 2013; Accepted 23 April 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 Bülent Kiliç and Hasan Bulut. 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

A new method with a different auxiliary equation from the Riccati equation is used for constructing exact travelling wave solutions of nonlinear partial differential equations. The main idea of this method is to take full advantage of a different auxilliary equation from the Riccati equation which has more new solutions. More new solitary solutions are obtained for the RLW Burgers and Hirota Satsuma coupled equations.

1. Introduction

In the recent years, remarkable progress has been made in the construction of the exact solutions for nonlinear partial differential equations, which have been a basic concern for both mathematicians and physicists [13].We do not attempt to characterize the general form of nonlinear dispersive wave equations [4, 5]. When an original nonlinear equation is directly calculated, the solution will preserve the actual physical characters of solutions [6]. The studies in finding exact solutions to nonlinear differential equation (NPDE), when they exist, are very important for the understanding of most nonlinear physical phenomena. There are many studies which obtain explicit solutions for nonlinear differential equations. Many explicit exact methods have been introduced in literature [721]. Some of them are generalized Miura transformation, Darboux transformation, Cole-Hopf transformation, Hirota’s dependent variable transformation, the inverse scattering transform and the Bäcklund transformation, tanh method, sine-cosine method, Painleve method, homogeneous balance method (HB), similarity reduction method, improved tanh method and so on.

In this article, the first section presents the scope of the study as an introduction. In the second section contains analyze of a new method and balance term definition. In the third section, we will obtain wave solutions of RLW Burgers and Hirota Satsuma coupled equations by using a new method. In the last section, we implement the conclusion.

2. Method and Its Applications

Let us simply describe the method [22]. Consider a given partial differential equation in two variables The fact that the solutions of many nonlinear equations can be expressed as a finite series of solutions of the auxiliary equation motivates us to seek for the solutions of (1) in the form where,,   and are the wave number and the wave speed respectively, is a positive integer that can be determined by balancing the linear term of highest order with the nonlinear term in (1), is balancing coefficient that will be defined in a new “Balance term” definition and ,  ,   are parameters to be determined. Substituting (2) into (1) yields a set of algebraic equations for ,  ,   because all coefficients of have to vanish. From these relations ,  ,   can be determined. The main idea of our method is to take full advantage of the new auxiliary equation. The desired auxiliary equation presents as following where and , , are constants.

Case 1. If , , then (3) has the solution .

Case 2. If , , then (3) has the solutions or .

Case 3. If , , then (3) has the solutions or

Remark 1. Depending on the and coefficients in the (3), it could be reached only three cases.

In the following we present a new approach to the “Balance term” definition.

Definition 2. When (1) is transformed with, , where and are real constants, we get a nonlinear ordinary differential equation for as following Let is the highest order derivative linear term and is the highest nonlinear term in (5) and is the auxiliary equation that is used to solve the nonlinear partial differential equation then the “Balance term” can be decided by the balancing the nonlinear term and the linear term with acceptances of and where is integer and is the balance coefficient that can be determined later.

Example 1. For the KdV equation with the transform , we have the ordinary differential equation as following By the balancing linear term with nonlinear term we have the equations above and the equating to If it is noticed that our new balance term is connected to . Namely our new balance term definition is connected to chosen auxiliary equation.

3. Application of the Method

Example 2. Let’s consider RLW Burgers equation with the transform , we have the following equation From the Definition 2 we have the balance term of RLW Burgers equation by using the auxiliary equation (3) for “Case 1”, is equal to 4. Therefore, we may choose the following ansatz: Substituting (11) into (10) along with (3) and using Mathematica yields a system of equations w.r.t. . Setting the coefficients of in the obtained system of equations to zero, we can deduce the following set of algebraic polynomials with the respect unknowns , , for the Case 1: From the system of (12) we have
(i)
(ii) From the Definition 2 we have the balance term of RLW Burgers equation by using the auxiliary equation (3) for “Case 2”, is equal to −4 then we have the following system of equations From the system of (15) we have
(i) or

Example 3. Let’s consider Hirota Satsuma coupled equation with the transform , , we have From the Definition 2 we have the balance term of Hirota Satsuma coupled equation by using the auxiliary equation (3) for “Case 1”, is equal to 4 for and . Therefore, we may choose the following ansatz: Substituting (20) into (19) along with (3) and using Mathematica yields a system of equations w.r.t. . Setting the coefficients of in the obtained system of equations to zero, we can deduce the following set of algebraic polynomials with the respect unknowns , , for the Case 1: From the system of (21) we have
(i)
(ii) From the Definition 2 we have the balance term of Hirota Satsuma coupled equation by using the auxiliary equation (3) for “Case 2”, is equal to −4 then we have the following system of equations From the system of (3) we have
(i) or Figures 1, 2, 3 and 4 gives to us 3D graphics for RLW Burgers and RLW Burgers and Hirota Satsuma coupled equations.

fig1
Figure 1: Graph of the solution from left to the right for (13) and (14).
fig2
Figure 2: Graph of the solution from left to the right for (15) and (16).
fig3
Figure 3: Graph of the solutions and corresponding to the value from left to the right for (22), (3), and (23).
fig4
Figure 4: Graph of the solutions and corresponding to the value from left to the right for (25) and (26).

4. Conclusion

We have presented a new method and balance term definition and used it to solve the RLW Burgers and Hirota Satsuma coupled equations. In fact, this method is readily applicable to a large variety of nonlinear PDEs. First, all the nonlinear PDEs which can be solved by the other methods can be solved by our method. Second, we used only the special solutions of (3). If we use other solutions of (3), we can obtain more travelling wave solutions. Third it is a computerizable method, which allow us to perform complicated and tedious algebraic calculation on computer and so our balance term definition is effectively useful for any to chosen auxiliary equation.

References

  1. B. R. Duffy and E. J. Parkes, “Travelling solitary wave solutions to a seventh-order generalized KdV equation,” Physics Letters A, vol. 214, no. 5-6, pp. 271–272, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  2. E. J. Parkes and B. R. Duffy, “Travelling solitary wave solutions to a compound KdV-Burgers equation,” Physics Letters A, vol. 229, no. 4, pp. 217–220, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. E. J. Parkes, “Exact solutions to the two-dimensional Korteweg-de Vries-Burgers equation,” Journal of Physics A, vol. 27, no. 13, pp. L497–L501, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäuser, Boston, Mass, USA, 1997. View at Zentralblatt MATH · View at MathSciNet
  5. A. M. Wazwaz, Partial Differential Equations: Methods and Applications, Balkema, Rotterdam, The Netherlands, 2002.
  6. W. Hereman, P. P. Banerjee, A. Korpel, G. Assanto, A. van Immerzeele, and A. Meerpoel, “Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method,” Journal of Physics A, vol. 19, no. 5, pp. 607–628, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  7. X.-B. Hu and W.-X. Ma, “Application of Hirota's bilinear formalism to the Toeplitz lattice–-some special soliton-like solutions,” Physics Letters A, vol. 293, no. 3-4, pp. 161–165, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  8. A. M. Abourabia and M. M. El Horbaty, “On solitary wave solutions for the two-dimensional nonlinear modified Kortweg-de Vries-Burger equation,” Chaos, Solitons & Fractals, vol. 29, no. 2, pp. 354–364, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. T. L. Bock and M. D. Kruskal, “A two-parameter Miura transformation of the Benjamin-Ono equation,” Physics Letters A, vol. 74, no. 3-4, pp. 173–176, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer Series in Nonlinear Dynamics, Springer, Berlin, Germany, 1991. View at MathSciNet
  11. F. Cariello and M. Tabor, “Painlevé expansions for nonintegrable evolution equations,” Physica D, vol. 39, no. 1, pp. 77–94, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  12. E. Fan and H. Zhang, “New exact solutions to a solutions to a system of coupled KdV equations,” Physics Letters A, vol. 245, no. 5, pp. 389–392, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. E. Fan, “Auto-Bäcklund transformation and similarity reductions for general variable coefficient KdV equations,” Physics Letters A, vol. 294, no. 1, pp. 26–30, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. E.-G. Fan, “Traveling wave solutions for nonlinear equations using symbolic computation,” Computers & Mathematics with Applications, vol. 43, no. 6-7, pp. 671–680, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Y. Ugurlu and D. Kaya, “Solutions of the Cahn-Hilliard equation,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3038–3045, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. I. E. Inan, “Exact solutions to the various nonlinear evolution equations,” Physics Letters A, vol. 371, no. 1-2, pp. 90–95, 2007. View at Publisher · View at Google Scholar
  17. E. Fan and H. Zhang, “A note on homogeneous balance method,” Physics Letters A, vol. 246, no. 5, pp. 403–406, 1998. View at Publisher · View at Google Scholar
  18. L. Yang, Z. Zhu, and Y. Wang, “Exact solutions of nonlinear equations,” Physics Letters A, vol. 260, no. 1-2, pp. 55–59, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. E. Fan, “Two new applications of the homogeneous balance method,” Physics Letters A, vol. 265, no. 5-6, pp. 353–357, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. L. 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 Zentralblatt MATH · View at MathSciNet
  21. 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
  22. B. Kiliç, Some methods for traveling wave solutions of the nonlinear partial differential equations and numerical analysis of these solutions [Ph.D. thesis], Firat University, 2012.