Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article
Special Issue

Symmetries, Differential Equations, and Applications: Galois Bicentenary

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Research Article | Open Access

Volume 2012 |Article ID 854619 | 16 pages | https://doi.org/10.1155/2012/854619

New Jacobi Elliptic Function Solutions for the Zakharov Equations

Academic Editor: Fazal M. Mahomed
Received04 Sep 2012
Revised22 Oct 2012
Accepted23 Oct 2012
Published05 Dec 2012

Abstract

A generalized -expansion method is proposed to seek the exact solutions of nonlinear evolution equations. Being concise and straightforward, this method is applied to the Zakharov equations. As a result, some new Jacobi elliptic function solutions of the Zakharov equations are obtained. This method can also be applied to other nonlinear evolution equations in mathematical physics.

1. Introduction

In recent years, with the development of symbolic computation packages like Maple and Mathematica, searching for solutions of nonlinear differential equations directly has become more and more attractive [17]. This is because of the availability of computers symbolic system, which allows us to perform some complicated and tedious algebraic calculation and help us find new exact solutions of nonlinear differential equations.

In 2008, Wang et al. [8] introduced a new direct method called the -expansion method to look for travelling wave solutions of nonlinear evolution equations (NLEEs). The method is based on the homogeneous balance principle and linear ordinary differential equation (LODE) theory. It is assumed that the traveling wave solutions can be expressed by a polynomial in , and that satisfies a second-order LODE . The degree of the polynomial can be determined by the homogeneous balance between the highest order derivative and nonlinear terms appearing in the given NPDEs. The coefficients of the polynomial can be obtained by solving a set of algebraic equations. Many literatures have shown that the -expansion method is very effective, and many nonlinear equations have been successfully solved. Later, the further developed methods named the generalized -expansion method [9], the modified -expansion method [10], the extended -expansion method [11], the improved -expansion method [12], and the -expansion method [13] have been proposed.

As we know, when using the direct method, the choice of an appropriate auxiliary LODE is of great importance. In this paper, by introducing a new auxiliary LODE of different literature [8], we propose the generalized -expansion method, which can be used to obtain travelling wave solutions of NLEEs.

In our contribution, we will seek exact solutions of the Zakharov equations [14]: which are one of the classical models on governing the dynamics of nonlinear waves and describing the interactions between high- and low-frequency waves, where is the perturbed number density of the ion (in the low-frequency response), is the slow variation amplitude of the electric field intensity, is the thermal transportation velocity of the electron ion, and , , and are constants.

Recently, many exact solutions of (1.1)-(1.2) have been successfully obtained by using the extended tanh-expansion method, the extended hyperbolic function method, the -expansion method, the -expansion method [1319], and so on.

In this paper, we construct the exact solutions to (1.1)-(1.2) by using the generalized -expansion method. Furthermore, we show that the exact solutions are expressed by the Jacobi elliptic function.

2. The Generalized -Expansion Method

Suppose that we have a nonlinear partial differential equation (PDE) for in the form where is a polynomial in its arguments.

Step 1. By taking , , we look for traveling wave solutions of (2.1) and transform it to the ordinary differential equation (ODE)

Step 2. Suppose the solution of (2.2) can be expressed as a finite series in the form where , are constants to be determined later; is a solution of the auxiliary LODE where , , and are constants.

Step 3. Determine the parameter by balancing the highest order nonlinear term and the highest order partial derivative of in (2.2).

Step 4. Substituting (2.3) and (2.4) into (2.2), setting all the coefficients of all terms with the same powers of to zero, we obtain a system of nonlinear algebraic equations (NAEs) with respect to the parameters , , . By solving the NAEs if available, we can determine those parameters explicitly.

Step 5. Assuming that the constants , , can be obtained by solving the algebraic equations in Step 4, then substituting these constants and the known general solutions into (2.3), we can obtain the explicit solutions of (2.1) immediately.

3. Exact Solutions of the Zakharov Equations

In this section, we mainly apply the method proposed in Section 2 to seek the exact solutions of the Zakharov equations.

Since in (1.2) is a complex function and we are looking for the traveling wave solutions, thus we introduce a gauge transformation: where is a real-valued function, , , and are constants to be determined later, and and are constants. Substituting (3.1) into (1.1)-(1.2), we have

Integrating (3.2) twice with respect to , we have where and are integration constants. Substituting (3.4) into (3.3), we have

In (3.5), we assume that

Then (3.5) becomes the nonlinear ODE

According to the homogeneous balance between and in (3.7), we obtain . So we assume that can be expressed as where satisfies (2.4). By using (2.4) and (3.8), it is easily derived that

Substituting (3.8) and (3.9) into (3.7), the left-hand side of (3.7) becomes a polynomial in and . Setting their coefficients to zero yields a system of algebraic equations in , , , , and . Solving the overdetermined algebraic equations by Maple, we can obtain the following results:

Substituting (3.10) into (3.8), we obtain

Substituting (3.11) into (3.1) and (3.4), we have the following formal solution of (1.1)-(1.2): where , , and .

With the aid of the appendix [20] and from the formal solution (3.12), we get the following set of exact solutions of (1.1)-(1.2).

Case 1. Choosing , , , and and inserting them into (3.12), we obtain the Jacobi elliptic function solution of (1.1)-(1.2) where , , and .

Case 2. Choosing , , , and and inserting them into (3.12), we obtain the Jacobi elliptic function solution of (1.1)-(1.2) where , , and .

Case 3. Choosing , , , and , we obtain where , , and .

Case 4. Choosing , , , and , we obtain where , , and .

Case 5. Choosing , , , and , we obtain where , , and .

Case 6. Choosing , , , and , we obtain where , , and .

Case 7. Choosing , , , and , we obtain where , , and .

Case 8. Choosing , , , and , we obtain where , , and .

Case 9. Choosing , , , and , we obtain where , , and .

Case 10. Choosing , , , and , we obtain where , , and .

Case 11. Choosing , , , and , we obtain where , , and .

Case 12. Choosing , , , and , we obtain where , , and .

Case 13. Choosing , , , and , we obtain where , , and .

Case 14. Choosing , , , and , we obtain where , , and .

Case 15. Choosing , , , and , we obtain where , , and .

Case 16. Choosing , , , and , we obtain where , , and .

Case 17. Choosing , , , and , we obtain where , , and .

Case 18. Choosing , , , and , we obtain where , , and .

Case 19. Choosing , , , and , we obtain where , , and .

Case 20. Choosing , , , and , we obtain where , , and .

Case 21. Choosing , , , and , we obtain where , , and .

Case 22. Choosing , , , and , we obtain where , , and .

Case 23. Choosing , , , and , we obtain where , , and .

Case 24. Choosing , , , and , we obtain where , , and .

Case 25. Choosing , , , and , we obtain where , , and .

Case 26. Choosing , , , and , we obtain where , , and .

Case 27. Choosing , , , and , we obtain where , , and .

Case 28. Choosing , , , and , we obtain where , , and .

Case 29. Choosing , , , and , we obtain where , , and .

Case 30. Choosing , , , and , we obtain where , , and .

Case 31. Choosing , , , and , we obtain where , , and .

Case 32. Choosing , , , and , we obtain where , , and .

Case 33. Choosing , , , and , we obtain where , , and .

Case 34. Choosing , , , and , we obtain where , , and .

Case 35. Choosing , , , and , we obtain where , , and .

Case 36. Choosing , , , and , we obtain where , , and .

Case 37. Choosing , , , and , we obtain where , , and .

4. Conclusions

In this paper, by using the generalized -expansion method, we have successfully obtained some exact solutions of Jacobi elliptic function form of the Zakharov equations. When the modulus of the Jacobi elliptic function or 1, the corresponding solitary wave solutions and trigonometric function solutions are also obtained. This work shows that the generalized -expansion method provides a very effective and powerful tool for solving nonlinear equations in mathematical physics.

Appendix

For more details see Table 1 and [20].


Case

Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
Case 9
Case 10
Case 11
Case 12
Case 13
Case 14
Case 15