About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 896748, 21 pages
http://dx.doi.org/10.1155/2012/896748
Research Article

A New Extended Jacobi Elliptic Function Expansion Method and Its Application to the Generalized Shallow Water Wave Equation

1Department of Mathematics, North University of China, Taiyuan 030051, China
2Software School, North University of China, Taiyuan 030051, China
3School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received 25 October 2012; Accepted 19 November 2012

Academic Editor: Renat Zhdanov

Copyright © 2012 Yafeng Xiao 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

With the aid of symbolic computation, a new extended Jacobi elliptic function expansion method is presented by means of a new ansatz, in which periodic solutions of nonlinear evolution equations, which can be expressed as a finite Laurent series of some 12 Jacobi elliptic functions, are very effective to uniformly construct more new exact periodic solutions in terms of Jacobi elliptic function solutions of nonlinear partial differential equations. As an application of the method, we choose the generalized shallow water wave (GSWW) equation to illustrate the method. As a result, we can successfully obtain more new solutions. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.

1. Introduction

In recent years, the nonlinear partial differential equations (NPDEs) are widely used to describe many important phenomena and dynamic processes in physics, mechanics, chemistry, biology, and so forth. With the development of soliton theory, There has been a great amount of activities aiming to find methods for exact solutions of nonlinear differential equations, such as Bäcklund transformation, Darboux transformation, Cole-Hopf transformation, similarity reduction method, variable separation approach, Exp-function method, homogeneous balance method, varied tanh methods, and varied Jacobi elliptic function methods [121].

Among those, the direct ansätz method [821] provides a straightforward and effective algorithm to obtain such particular solutions for a large number of nonlinear partial differential equations, in which the starting point is the ansätz that the solution sought is expressible as a finite series of special function, such as tanh function, sech function, tan function, sec function, sine-cosine function, Weierstrass elliptic function, theta function, and Jacobi elliptic function.

In this paper, a new Jacobi elliptic function expansion method is presented by means of a new general ansätz and is more powerful to uniformly construct more new exact doubly-periodic solutions in terms of Jacobi elliptic functions of nonlinear partial differential equations. The algorithm and its application are demonstrated later.

This paper is organized as follows. In Section 2, we summarize the extended Jacobi elliptic function expansion method. In Section 3, we apply the extended method to the generalized shallow water wave (GSWW) equation and bring out more new solutions. Conclusions will be presented in finally.

2. Summary of the Extended Jacobi Elliptic Function Expansion Method

In the following, we would like to outline the main steps of our extended method.

Step 1. For a given nonlinear partial differential equation with some physical fields in three variables , , , by using the wave transformation where , , and are constants to be determined later. Then the nonlinear partial differential equation (2.1) is reduced to a nonlinear ordinary differential equation (ODE):

Step 2. We introduce some new ansätz in terms of finite Jacobi elliptic function expansion in the following forms:(1) expansion:(2) expansion: (3) expansion: where , , are Jacobi elliptic sine function, the Jacobi elliptic cosine function, and the Jacobi elliptic function of the third kind, which are periodic and possess the following properties:

(1) properties of triangular functions

(2) derivatives of the Jacobi elliptic functions where is a modulus. The Jacobi-Glaisher functions for elliptic function can be found in [22, 23]. It is necessary to point out that above combinations only require solving the recurrent coefficient relation or derivative relation for the terms of polynomial for computation closed. Therefore, other Jacobi elliptic functions can be chosen to combine as new ansätz. For simplicity, we omit them here.

Step 3. In order to obtain the value of in (2.4a)–(2.4c), we define the degree of as , which gives rise to the degree of other expressions as Therefore, we can get the value of in (2.4a)–(2.4c). If is a nonnegative integer, then we first make the transformation .

Step 4. Respectively substitute three cases of (2.4a)–(2.4c) into (2.3) along with (2.5) and (2.6) and then respectively set all coefficients of to be zero to get an overdetermined system of nonlinear algebraic equations with respect to , , , , , .

Step 5. By use of the Maple software package “Charsets” by Dongming Wang, which is based on the Wu-elimination method [24], solving the overdetermined algebraic equations, we would end up with the explicit expressions for , , , , , . In this way, we can get periodic solutions with Jacobi elliptic function.
Since degenerate respectively as in the following form:

(1) solitary wave solutions:

(2) triangular function formal solutions:

So the new extended Jacobi elliptic function expansion method can obtain more new solutions, which contain solitary wave solutions, singular solitary solutions, and triangular function solutions. These new solutions cannot be obtained by other Jacobi elliptic function methods, such as Fu et al. [13], Fan [14], Yan [1517], and Chen et al. [1821].

3. Exact Solutions of the GSWW Equation

The Boussinesq approximation theory [25] for generalized classical shallow water wave leads to the GSWW equation: where and are nonzero constants. Using in [26], (3.1) can be simplified to which has recently attracted many attentions from researchers under the following cases.

(1) For the case when , the following AKNS-SWW equation: was discussed by Ablowitz et al. [27].

(2) For the case when , the following equation: was discussed by Hirota and Satsuma [28].

The GSWW equation (3.1) in potential form was studied by Clarkson and Mansfield [26, 29] who gave a complete catalog of classical and nonclassical symmetry reductions. The necessary conditions of Painlevé tests were given by Weiss et al. [30] and the complete integrability of (3.1) for the case when and was established by Ablowitz et al. [31]. It has been proven by Hietarinta [32] that the GSWW equation (3.2) can be expressed in Hirota’s bilinear form and when , (3.2) can be reduced to (3.3) or to (3.4) when . In other words, both (3.3) and (3.4) are solvable by using Hirota’s bilinear method [33]. The -soliton solutions for (3.3) and (3.4) had also been found using this technique [28]. Although there is no scaling transformation that can reduce (3.3) to (3.4), the classical methods of Lie, the nonclassical method of Bluman and Cole, and the direct method of Clarkson and Kruskal can be applied to solve the GSWW equation (3.2) to obtain some kinds of symmetry reductions [34, 35]. Recently, Elwakil found a lot of exact solutions by using the modified extended tanh-function method [36]. Yang and Hon employed a rational expansion to generalize Fan’s method for exact travelling wave solutions for the generalized shallow water wave (GSWW) equation [37].

According to the above method, to seek travelling wave solutions of (3.1), we make the following transformation: where and are constants to be determined later. Thus (3.1) becomes Equation (3.6) is integrated at once yielding For the following, without loss of generality, we can set the integration constant equal to zero.

Now we consider the system (3.7) in the above three cases, so that (2.4a)–(2.4c). According to Step 2 and Step 3 in Section 2, by balancing the nonlinear term with the highest-order differential term in (3.7), we suppose that (3.7) has the following formal solutions.

3.1. Expansion

Now we consider the ansätz (2.4a). For (3.7), the ansätz (2.4a) becomes where , , , , , , and are constants to be determined later.

With the aid of Maple, substituting (3.8) along with (2.5) and (2.6) into (3.7), yields a set of algebraic equations for . Setting the coefficients of these terms to be zero yields a set of overdetermined algebraic equations with respect to , , , , , , , and .

By use of the Maple software package “Charsets” by Dongming Wang, which is based on the Wu-elimination method [24], solving the overdetermined algebraic equations, we get the following results.

Case 1.

Case 2.

Case 3.

Case 4.

Case 5.

Case 6.
From (3.8) and Cases 16, we obtain the following solutions for (3.1).

Family 1. From (3.9), we obtain the following expansion solutions for the GSWW equation as follows: where and and are arbitrary constants.

Family 2. From (3.10), we obtain the following expansion solutions for the GSWW equation as follows: where and and are arbitrary constants.

Family 3. From (3.11), we obtain the following expansion solutions for the GSWW equation as follows: where and and are arbitrary constants.

Family 4. From (3.12), we obtain the following expansion solutions for the GSWW equation as follows: where and and are arbitrary constants.

Family 5. From (3.13), we obtain the following expansion solutions for the GSWW equation as follows: where and and are arbitrary constants.

Family 6. From (3.14), we obtain the following expansion solutions for the GSWW equation as follows: where and and are arbitrary constants.

Remark 3.1. Here we find that the modulus of the Jacobi elliptic functions has relations with and in Cases 16. To further analyze their relations, we take the solutions (3.9), (3.11), and (3.13) as samples by three figures (see Figures 1, 2, and 3).

896748.fig.001
Figure 1: Figure of in (3.9).
896748.fig.002
Figure 2: Figure of in (3.11).
896748.fig.003
Figure 3: Figure of in (3.13).
3.2.   Expansion

Now we consider the ansätz (2.4b). For (3.7), the ansätz (2.4b) becomes where , , , , , , , and are constants to be determined later. Following the same steps in Section 3.1, we can obtain the following expansion solutions.

Family 7. where and and are arbitrary constants.

Family 8. where and and are arbitrary constants.

Family 9. where and and are arbitrary constants.

Family 10. where and and are arbitrary constants.

Family 11. where and and are arbitrary constants.

Family 12. where and and are arbitrary constants.

3.3. Expansion

Now we consider the ansätz (2.4c). For (3.7), the ansätz (2.4c) becomes where , , , , , , , and are constants to be determined later. Following the same steps in Section 3.1, we can obtain the following expansion solutions.

Family 13. where and and are arbitrary constants.

Family 14. where and and are arbitrary constants.

Family 15. where and and are arbitrary constants.

Family 16. where and and are arbitrary constants.

Family 17. where and and are arbitrary constants.

Family 18. where and and are arbitrary constants.

Remark 3.2. The solutions obtained here, to our knowledge, are all new families of periodic solution of the GSWW equation.

Remark 3.3. In order to understand the significance of these solutions in Families 1 and 18, here we take the solutions (3.15), (3.17), (3.19), (3.22), (3.24), (3.26), (3.29), (3.31), and (3.33) as samples to further analyze their properties by some figures (see Figures 4, 5, 6, 7,8, 9, 10, 11, and 12).

896748.fig.004
Figure 4: The solution of GSWW equation with , , , .
896748.fig.005
Figure 5: The solution of GSWW equation with , , , .
896748.fig.006
Figure 6: The solution of GSWW equation with , , , .
896748.fig.007
Figure 7: The solution of GSWW equation with , , , .
896748.fig.008
Figure 8: The solution of GSWW equation with , , , .
896748.fig.009
Figure 9: The solution of GSWW equation with , , , .
896748.fig.0010
Figure 10: The solution of GSWW equation with , , , .
896748.fig.0011
Figure 11: The solution of GSWW equation with , , , .
896748.fig.0012
Figure 12: The solution of GSWW equation with , , , .

4. Conclusion and More General Form

In short, we have presented the new extended Jacobi elliptic function expansion method. The GSWW equation is chosen to illustrate the method such that many families of new Jacobi elliptic function solutions are obtained. When the modulus or 0, some of these obtained solutions degenerate as solitary solutions or trigonometric function solutions. The algorithm can be applied to many nonlinear differential equations in mathematical physics. Other types of solutions of (2.1) need to be studied further.

It is easy to see that above-mentioned method is only applied to these nonlinear ODEs with constant coefficients or nonlinear partial differential equations, which can be reduced to the corresponding nonlinear ODEs with constant coefficients by using some transformations; otherwise, the method will not work. In order to overcome the disadvantage of the method, we change the method into a general form as follows.

If we do not reduce (2.1) to a nonlinear ODE (2.3) with constant coefficients, then we directly assume that (2.1) has the following solutions:(1) expansion:(2) expansion: (3) expansion: where , , (), and are functions to be determined later. Substituting (4.1a), (4.1b), and (4.1c) with (2.5) and (2.6) into (2.1) yields a set of nonlinear partial differential equations with respect to these unknown functions , , (), and , respectively. If we can solve these functions from the obtained set of nonlinear partial differential equations, then we may obtain more types of doubly periodic solutions of (2.1). The more general form is better applied to nonlinear differential equations with variable coefficients. Particularly, if we only deduce the conclusions that all these function , , () are all constants and is of the form (, constants), then the obtained results are the same as the ones found by using the method presented in Section 2. About the applications of the more general form, we will give some examples in future.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

Many thanks are due to suggestions from the referees and support from the National Natural Science Foundation of China (Grant no. 61171179) and the Young Scientists Fund of the National Natural Science Foundation of China (Grant no. 10901145).

References

  1. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149, Cambridge University Press, Cambridge, UK, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer, Berlin, Germany, 1991. View at Zentralblatt MATH
  3. M. Wadati, “Wave propagation in nonlinear lattice—I,” Journal of the Physical Society of Japan, vol. 38, pp. 673–680, 1975.
  4. P. A. Clarkson and M. D. Kruskal, “New similarity reductions of the Boussinesq equation,” Journal of Mathematical Physics, vol. 30, no. 10, pp. 2201–2213, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. S.-Y. Lou and J. Lu, “Special solutions from the variable separation approach: the Davey-Stewartson equation,” Journal of Physics A, vol. 29, no. 14, pp. 4209–4215, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J.-H. He and M. A. Abdou, “New periodic solutions for nonlinear evolution equations using Exp-function method,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1421–1429, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. M. L. Wang, “Application of a homogeneous balance method to exact solutions of nonlinear equations inmathematical physics,” Physics Letters A, vol. 216, pp. 67–75, 1996.
  8. A.-M. Wazwaz, “The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 1002–1014, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. A.-M. Wazwaz, “The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1467–1475, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. 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
  11. 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, pp. 288–300, 1996.
  12. Z. Yan, “New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations,” Physics Letters A, vol. 292, no. 1-2, pp. 100–106, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. 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
  14. E. Fan, “Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics,” Chaos, Solitons and Fractals, vol. 16, no. 5, pp. 819–839, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Z. Yan, “Abundant families of Jacobi elliptic function solutions of the (2 + 1)-dimensional integrable Davey-Stewartson-type equation via a new method,” Chaos, Solitons and Fractals, vol. 18, no. 2, pp. 299–309, 2003. View at Publisher · View at Google Scholar
  16. Z. Yan, “The extended Jacobian elliptic function expansion method and its application in the generalized Hirota-Satsuma coupled KdV system,” Chaos, Solitons and Fractals, vol. 15, no. 3, pp. 575–583, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. Z. Yan, “Extended Jacobian elliptic function algorithm with symbolic computation to construct new doubly-periodic solutions of nonlinear differential equations,” Computer Physics Communications, vol. 148, no. 1, pp. 30–42, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. 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
  19. Y. Chen, Q. Wang, and B. Li, “Elliptic equation rational expansion method and new exact travelling solutions for Whitham-Broer-Kaup equations,” Chaos, Solitons and Fractals, vol. 26, no. 1, pp. 231–246, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. Q. Wang, Y. Chen, and H. Zhang, “An extended Jacobi elliptic function rational expansion method and its application to (2 + 1)-dimensional dispersive long wave equation,” Physics Letters A, vol. 340, no. 5-6, pp. 411–426, 2005. View at Publisher · View at Google Scholar
  21. Y. Chen and Q. Wang, “A new elliptic equation rational expansion method and its application to the shallow long wave approximate equations,” Applied Mathematics and Computation, vol. 173, no. 2, pp. 1163–1182, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. K. Chandrasekharan, Elliptic Function, Springer, Berlin, Germany, 1978.
  23. P. Du Val, Elliptic Functions and Elliptic Curves, Cambridge University Press, London, UK, 1973.
  24. W. J. Wu, Algorithm and Computation, Springer, Berlin, Germany, 1994.
  25. A. Espinosa and J. Fujioka, “Hydrodynamic foundation and Painlevé analysis of Hirota-Satsuma-type equations,” Journal of the Physical Society of Japan, vol. 63, no. 4, pp. 1289–1294, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. P. A. Clarkson and E. L. Mansfield, “On a shallow water wave equation,” Nonlinearity, vol. 7, no. 3, pp. 975–1000, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse scattering transform-Fourier analysis for nonlinear problems,” vol. 53, no. 4, pp. 249–315, 1974. View at Zentralblatt MATH
  28. R. Hirota and J. Satsuma, “N-soliton solutions of model equations for shallow water waves,” Journal of the Physical Society of Japan, vol. 40, no. 2, pp. 611–612, 1976.
  29. P. A. Clarkson and E. L. Mansfield, “Symmetry reductions and exact solutions of shallow water wave equations,” Acta Mathematicae Applicatae Sinica, vol. 39, no. 1–3, pp. 245–276, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. 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
  31. M. J. Ablowitz, A. Ramani, and H. Segur, “A connection between nonlinear evolution equations and ordinary differential equations of p-type—I,” Journal of Mathematical Physics, vol. 21, no. 4, pp. 715–721, 1980. View at Publisher · View at Google Scholar
  32. J. Hietarinta, “Hirota's bilinear method and partial integrability,” in Partially Integrable Evolution Equations in Physics, R. Conte and N. Bocarra, Eds., vol. 310 of NATO ASI Series C: Mathematical and Physical Sciences, pp. 459–478, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. View at Zentralblatt MATH
  33. R. Hirota, “Direct methods in soliton theory,” in Solitons, R. K. Bullough and P. J. Caudrey, Eds., Springer, Berlin, Germany, 1980.
  34. G. W. Bluman and J. D. Cole, “The general similarity solution of the heat equation,” Journal of Mathematics and Mechanics, vol. 18, pp. 1025–1042, 1969.
  35. P. A. Clarkson and M. D. Kruskal, “New similarity reductions of the Boussinesq equation,” Journal of Mathematical Physics, vol. 30, no. 10, pp. 2201–2213, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. S. A. Elwakil, S. K. El-Labany, M. A. Zahran, and R. Sabry, “Modified extended tanh-function method and its applications to nonlinear equations,” Applied Mathematics and Computation, vol. 161, no. 2, pp. 403–412, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  37. Z. Yang and Y. C. Hon, “A rational generalization of Fan's method and its application to generalized shallow water wave equations,” Applied Mathematics and Computation, vol. 216, no. 7, pp. 1984–1995, 2010. View at Publisher · View at Google Scholar