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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 582532, 9 pages
http://dx.doi.org/10.1155/2013/582532
Research Article

Infinitely Many Elliptic Solutions to a Simple Equation and Applications

Institute of Applied Mathematics and Engineering Computations, Hangzhou Dianzi University, Zhejiang, Hangzhou 310018, China

Received 4 June 2013; Accepted 4 July 2013

Academic Editor: Pavel Kurasov

Copyright © 2013 Long Wei and Yang Wang. 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

Based on auxiliary equation method and Bäcklund transformations, we present an idea to find infinitely many Weierstrass and Jacobi elliptic function solutions to some nonlinear problems. First, we give some nonlinear iterated formulae of solutions and some elliptic function solutions to a simple auxiliary equation, which results in infinitely many Weierstrass and Jacobi elliptic function solutions of the simple equation. Then applying auxiliary equation method to some nonlinear problems and combining the results with exact solutions of the auxiliary equation, we obtain infinitely many elliptic function solutions to the corresponding nonlinear problems. The employed approach is powerful and can be also applied to solve other nonlinear differential equations.

1. Introduction

Partial differential equations (PDEs) describe various nonlinear phenomena in natural and applied sciences such as fluid dynamics, plasma physics, solid state physics, optical fibers, acoustics, mechanics, biology, and mathematical finance. Their solution spaces are infinite dimensional and contain diverse solution structures. It is of significant importance to solve nonlinear PDEs from both theoretical and practical points of view. In some decades, many authors are devoted to investigate the exact solutions to nonlinear PDEs. With the help of exact solutions, when they exist, the mechanism of complicated physical phenomena and dynamical processes modeled by these nonlinear PDEs can be better understood. They can also help to analyze the stability of these solutions and to check numerical analysis for these nonlinear PDEs.

In recent years, reducing PDEs into ordinary differential equations (ODEs) has been proved to be a successful idea to generate exact solutions of nonlinear wave equations. Many approaches to exact solutions in the literature follow such an idea, which contain tanh-function method [14], sech-function method [5, 6], homogeneous balance method [7, 8], extended tanh-function method [9], Jacobi elliptic function method [10], exp-function method [11], -expansion method [12], and transformed rational function method [13]. The tanh-function method and the expansion methods are special cases of general Frobenius’ idea [14] or the transformed rational function method [13]. Recently, the multiple exp-function was also used to present three waves in (3 + 1) dimension [15]. There is also a generalized theory of the Bell polynomials [16], which describe generalized bilinear differential equations [17]. Among those, auxiliary equation methods are widely used to find exact solutions (especially elliptic solutions) of PDEs by some authors (see [1, 3, 1824], etc.). But in many literatures, the authors constructed exact solutions of PDEs making use of finite solutions of the auxiliary equations employed and have not obtained infinitely many solutions, especially Weierstrass and Jacobi elliptic solutions.

In this work, we will seek for nonlinear iterated formulae of solutions to a simple auxiliary equation. Simultaneously, in order to obtain infinitely many new elliptic solutions of some nonlinear PDEs, we will also give some Weierstrass and Jacobi elliptic function solutions of the simple auxiliary equation. Based on these facts, we can construct infinitely many Weierstrass and Jacobi elliptic function solutions of destination equations. The obtained results are new and introduce useful analysis for equations of identical nonlinearities.

This paper is organized as follows: in Section 2, we provide some nonlinear iterated formulae of solutions and some new elliptic solutions to a simple equation. In Section 3, we give some examples to demonstrate the application of our approach. The remainder of this paper is a discussion.

2. Preliminaries

In this section, we try to find some nonlinear iterated formulae of solutions and infinitely many Weierstrass and Jacobi elliptic function solutions to the auxiliary equation We will solve some NPDEs by the exact solutions of the auxiliary equation; consequently we can obtain infinitely many Weierstrass and Jacobi elliptic function solutions and periodic and soliton solutions of destination equations.

2.1. Nonlinear Iterated Formulae of Solutions to (1)

In order to find infinitely many elliptic solutions to (1), we assume that .

By the Bäcklund transformations and a modified truncation approach, we can obtain the following nonlinear iterated formulae of solutions; that is, if is a solution to (1), then so is : where with with with with

Above, we have given finitely many nonlinear iterated formulae of solutions to (1). In fact, it is possible to obtain infinitely many ones by substituting one into another repeatedly. Here, it should be pointed out that the previous iterative procedure can be done finite times for some formulae but infinite times for some other ones. For example, substituting (3) into (2) we can obtain a new nonlinear iterated formula: but repeating the procedure again will result in (3) itself; substituting into results in ; substituting into results in and so on. But for solutions and so forth, the iterative procedure are infinite For instance, substituting or into and iterating the results by can generate infinitely many iterated formulae of solutions.

2.2. Infinitely Many Weierstrass Elliptic Function Solutions to (1)

Let be the Weierstrass elliptic function satisfying where , are real parameters and called invariants [25], and denotes the first-order derivative. It follows from (21) that the second-order derivative is From the last previous two equations, we will give some Weierstrass elliptic function solutions to (1), among which there are many new solutions different from those in [26] and some latter references. In fact, in [26] the author considered the following auxiliary equation: but this equation can be converted into (1) by the transformation , with , , and .

From (21) and (22), we have the following Weierstrass elliptic function solutions to (1).

(i) When , , then orwhere .

(ii) When , , then where .

(iii) When and , then with , , and ; with , , and ; or with , , , and , where , and are two arbitrary constants.

Note that in any pair of solutions, the latter can be obtained from the former through formula (2). Indeed, combining solutions (24)–(31) with the nonlinear iterated formulae (2)–(20) and their infinitely many derivatives can yield infinitely many Weierstrass elliptic function solutions of (1).

2.3. Infinitely Many Jacobi Elliptic Function Solutions to (1)

Assume that , , and are arbitrary constants, and is the elliptic modulus; then (1) has the following Jacobi elliptic solutions.

(I) ,  ,  , Indeed, the former two solutions and the latter two ones can transform each other by formula (2). In what follows, we omit the solutions that can be generated by formula (2).

(II) ,  ,  ,

(III) ,  ,  ,

(IV) ,  ,  ,

(V) ,  ,  ,

(VI) ,  ,  ,

(VII) ,  ,  ,

(VIII) ,  ,  ,

(IX) ,  ,  ,

(X) ,  ,  ,

(XI) ,  ,  ,

(XII) ,  ,  ,

Combining solutions (32)–(43) with the nonlinear iterated formulae (2)–(20) and their derivatives can give infinitely many Jacobi elliptic function solutions of (1), and these solutions can be degenerated to solutions with hyperbolic or trigonometric functions as long as letting go to 1 or 0.

3. Infinitely Many Elliptic Solutions of Some Nonlinear PDEs

Now we apply auxiliary equation methods to some nonlinear PDEs and obtain their infinitely many elliptic function solutions.

Example 1. First we consider the Kadomtsev-Petviashvili (KP) equation which was introduced in 1970 [27] and is of both physical and mathematical interests. By using the wave variable and integrating twice, the KP equation (44) can be converted to the ODE if we let the integration constants be zeros.
Assume that the solutions of (45) have the following form: where satisfies (1), () and () are constants to be determinated. Substituting (46) into (45) and equating corresponding coefficients of to zero yield an algebraic system of , , , , and . Solving this system with the aid of Maple, we obtain the following solutions of (45): where satisfies (1), with , and , , , , and are arbitrary constants. where satisfies (1), with , , , , , are arbitrary constants.
Now substituting the solutions ’s of (1) into (47) and (48), we obtain immediately infinitely many Weierstrass and Jacobi elliptic solutions of (44), and these elliptic solutions can be degenerated to solutions with hyperbolic or trigonometric functions as long as letting go to 1 or 0.
For instance, from (48), (6), (9), and (24) (or (35)) we can obtain infinitely many Weierstrass or Jacobi elliptic solutions of (44) as follows: (i) For Weierstrass elliptic function solutions, with where , , and are arbitrary constants.(ii) For Jacobi elliptic function solutions, with where , , and are arbitrary constants and is the elliptic modulus. If tends to or , we can also obtain multiple soliton or periodic solutions.

Example 2. Next we investigate infinitely many elliptic function solutions to a negative order KdV equation which actually comes from the negative KdV hierarchy and can be transformed to the Camassa-Holm equation through a gauge transform (see [28]). The negative order KdV equation (54) will be transformed to the ODE upon using wave variable and integrating once, where is a integration constant to be determinated. Similar to the procedure of Example 1, one can get solutions of (55) as follows, where satisfies (1), with , , , , , are arbitrary constants.
Similarly, substituting the solutions ’s of (1) into (56), we obtain infinitely many Weierstrass and Jacobi elliptic solutions of (54), among which including hyperbolic and trigonometric functions solutions as long as letting go to 1 or 0.
For instance, from (56), (3), (9) and (27) (or (34)) we can obtain infinitely many Weierstrass or Jacobi elliptic solutions of (54) as follows (i) for Weierstrass elliptic function solutions, with where , and are arbitrary constants;(ii) for Jacobi elliptic function solutions, with where , , are arbitrary constants and is the elliptic modulus. If tends to 1 or 0, we can also obtain multiple soliton or periodic solutions.

Example 3. We consider the two-dimensional Davey-Stewartson equation This equation is completely integrable and often used to describe the long time evolution of a two-dimensional wave packet [29, 30]. In recent years, various methods such as inverse scattering, Darboux transformation, variable separation, and Bäcklund transformation have been used to solve the equation, respectively. In this work, we try to deal with (62) by new idea and give it infinitely many Weierstrass and Jacobi elliptic-like solutions.
For our purpose, we introduce the following transformations: Then system (62) is reduced to the following system of ordinary differential equations: Integrating the second equation in (64) and neglecting the constants of integration we find that Substituting (65) into the first equation of the system leads to where prime denotes differentiation with respect to . Further multiplying (66) by and integrating it, we obtain that which is regrouped as follows: where is an arbitrary constant of integration.
In order to write (68) in a simple form that we want, let us introduce the following transformation: Then (68) can be written as (1) with
Now from the solutions ’s of (1) and taking into account (63), (65), and (69), we obtain infinitely many Weierstrass and Jacobi elliptic-like solutions to the two-dimensional Davey-Stewartson equation (62) as follows: where , ; taking account of (70), , , , , , and satisfy the relations such that (1) has an elliptic solution.
For instance, from (6), (9), and (24) (or (35)) we can obtain infinitely many Weierstrass or Jacobi elliptic-like solutions to the two-dimensional Davey-Stewartson equation (62) as follows: (i) For Weierstrass elliptic function solutions, with where , , , , , and are arbitrary real constants.(ii) For Jacobi elliptic function solutions, with where , , , and are arbitrary constants and is the elliptic modulus. If tends to 1 or 0, we can also obtain multiple soliton or periodic solutions.

4. Discussion

In this paper, we have obtained some nonlinear iterated formulae of solutions to the simple equation (1) and its infinitely many Weierstrass and Jacobi elliptic function solutions. Based on this fact, we can obtain infinitely many Weierstrass and Jacobi elliptic function solutions of some nonlinear equations. Our results suggest that the proposed method is reliable and effective to find infinitely many elliptic solutions to some nonlinear problems. The method is concise, and it can also be applied to other kinds of nonlinear equations in mathematical physics.

Acknowledgments

The first author is partly supported by Zhejiang Provincial Natural Science Foundation of China under Grant no. LY12A01003 and Subjects Research and Development Foundation of Hangzhou Dianzi University under Grant no. ZX100204004-6. The second author is partly supported by NSFC under Grant no. 11101111.

References

  1. E. J. Parkes and B. R. Duffy, “An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations,” Computer Physics Communications, vol. 98, no. 3, pp. 288–300, 1996. View at Publisher · View at Google Scholar · View at Scopus
  2. W. Malfliet and W. Hereman, “The tanh method. I. Exact solutions of nonlinear evolution and wave equations,” Physica Scripta, vol. 54, no. 6, pp. 563–568, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. 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
  4. A.-M. Wazwaz, “The tanh method for traveling wave solutions of nonlinear equations,” Applied Mathematics and Computation, vol. 154, no. 3, pp. 713–723, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. W. X. Ma, “Travelling wave solutions to a seventh order generalized KdV equation,” Physics Letters A, vol. 180, no. 3, pp. 221–224, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  6. E. J. Parkes, Z. Zhu, B. R. Duffy, and H. C. Huang, “Sech-polynomial travelling solitary-wave solutions of odd-order generalized KdV equations,” Physics Letters A, vol. 248, no. 2–4, pp. 219–224, 1998. View at Scopus
  7. 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 MathSciNet
  8. 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 Scopus
  9. 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 · View at MathSciNet
  10. 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
  11. J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 700–708, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Liu and K. Yang, “The extended F-expansion method and exact solutions of nonlinear PDEs,” Chaos, Solitons & Fractals, vol. 22, no. 1, pp. 111–121, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  13. W.-X. Ma and J.-H. Lee, “A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation,” Chaos, Solitons & Fractals, vol. 42, no. 3, pp. 1356–1363, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. W.-X. Ma, H. Wu, and J. He, “Partial differential equations possessing Frobenius integrable decompositions,” Physics Letters A, vol. 364, no. 1, pp. 29–32, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. W.-X. Ma and Z. Zhu, “Solving the (3+1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm,” Applied Mathematics and Computation, vol. 218, no. 24, pp. 11871–11879, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  16. W. X. Ma, “Bilinear equations, Bell polynomials and linear superposition principle,” Journal of Physics: Conference Series, vol. 411, Article ID 012021, 2013. View at Publisher · View at Google Scholar
  17. W. X. Ma, “Generalized bilinear differential equations,” Studies in Nonlinear Sciences, vol. 2, pp. 140–144, 2011.
  18. L. Wei, “A function transformation method and exact solutions to a generalized sinh-Gordon equation,” Computers & Mathematics with Applications, vol. 60, no. 11, pp. 3003–3011, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. L. Wei, “Exact solutions to a combined sinh-cosh-Gordon equation,” Communications in Theoretical Physics, vol. 54, no. 4, pp. 599–602, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. V. Demina and N. A. Kudryashov, “From Laurent series to exact meromorphic solutions: the Kawahara equation,” Physics Letters A, vol. 374, no. 39, pp. 4023–4029, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. M. V. Demina and N. A. Kudryashov, “On elliptic solutions of nonlinear ordinary differential equations,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9849–9853, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. M. V. Demina and N. A. Kudryashov, “Explicit expressions for meromorphic solutions of autonomous nonlinear ordinary differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1127–1134, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. 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
  24. N. A. Kudryashov and D. I. Sinelshchikov, “Exact solutions of the Swift-Hohenberg equation with dispersion,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 26–34, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. D. F. Lawden, Elliptic Functions and Applications, vol. 80 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1989. View at MathSciNet
  26. Z. Yan, “An improved algebra method and its applications in nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 21, no. 4, pp. 1013–1021, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. B. B. Kadomtsev and V. I. Petviashvili, “On the stability of solitary waves in weakyly dispersive media,” Soviet Physics. Doklady, vol. 15, no. 16, pp. 539–541, 1970.
  28. Z. Qiao and J. Li, “Negative-order KdV equation with both solitons and kink wave solutions,” EPL, vol. 94, no. 5, Article ID 50003, pp. 1–5, 2011. View at Publisher · View at Google Scholar · View at Scopus
  29. Y. Hase and J. Satsuma, “An N-soliton solution for the nonlinear Schrödinger equation coupled to the Boussinesq equation,” Journal of the Physical Society of Japan, vol. 57, no. 3, pp. 679–682, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  30. E. Fan and J. Zhang, “Applications of the Jacobi elliptic function method to special-type nonlinear equations,” Physics Letters A, vol. 305, no. 6, pp. 383–392, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet