Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 972519 | 11 pages | https://doi.org/10.1155/2014/972519

Exact Solutions for -Dimensional Nonlinear Fokas Equation Using Extended F-Expansion Method and Its Variant

Academic Editor: Kim M. Liew
Received06 Feb 2014
Revised23 Mar 2014
Accepted24 Mar 2014
Published17 Apr 2014

Abstract

The construction of exact solution for higher-dimensional nonlinear equation plays an important role in knowing some facts that are not simply understood through common observations. In our work, -dimensional nonlinear Fokas equation, which is an important physical model, is discussed by using the extended F-expansion method and its variant. And some new exact solutions expressed by Jacobi elliptic function, Weierstrass elliptic function, hyperbolic function, and trigonometric function are obtained. The related results are enriched.

1. Introduction

It has recently become more interesting to obtain exact solutions of nonlinear partial differential equations. These equations are mathematical models of complex physical phenomena that arise in engineering, applied mathematics, chemistry, biology, mechanics, physics, and so forth. Thus, the investigation of the traveling wave solutions to nonlinear evolution equations (NLEEs) plays an important role in mathematical physics. A lot of physical models have supported a wide variety of solitary wave solutions.

In the recent years, much efforts have been spent on this task and many significant methods have been established such as inverse scattering transform [1], Backlund and Darboux transform [2], Hirota [3], homogeneous balance method [4], symmetry reductions method [58], Jacobi elliptic function method [9], tanh-function method [10], exp-function method [1113], simple equation method [14], the meshless methods [1520], -expansion method [2123], F-expansion method [24, 25], improved F-expansion method [26, 27], and extended F-expansion method [28].

Wang and Li [25] developed a new algebraic method, belonging to the simplest equation method [2932], to seek more new solutions of NLEEs that can be expressed as polynomial in an elementary function which satisfies a more general subequation than other subequations like Riccati equation, auxiliary ordinary equation, elliptic equation, and generalized Riccati equation. The Fans method not only gives a unified formation to construct various traveling wave solutions but also provides a guideline to classify the various types of traveling wave solutions according to five parameters. An extended F-expansion method is proposed by Yomba in 2005 by giving more solutions of the general subequation. Using the new method, exact solutions of many NLEEs are successfully obtained [28].

The higher-dimensional integrable model is one of the important problems in mathematical physics which can be obtained from several lower-dimensional integrable equations by extending Lax pairs to higher dimensions. Fokas [33] extended the integrable Kadomtsev-Petviashvili and Davey-Stewartson equation to present a new -dimensional (4-dimensional space and one-dimensional time) nonlinear wave equation which is given by Due to important applications of higher-dimensional equations in real world problems, it is necessary to investigate its analytic solutions. Therefore, (1) is studied by many authors. In [34], Yang and Yan investigated symmetries of (1) including point symmetries and the potential symmetries. In [35], Lee et al. discussed the exact solutions of (1) by modified tanh-coth method, extended Jacobi elliptic function method, and the exp-function method. In [36], Kim obtained a group of exact solutions using -expansion method.

In [37], we have successfully applied the extended F-expansion method on a higher-order wave equation of KdV type. In this work, we apply this method and its variant on -dimensional nonlinear Fokas equation for obtaining new exact traveling solutions. Compared with [37], the subequation is discussed more in detail. Besides the four cases discussed in [37], the other two cases of the subequation are also considered, that is, and , so we can obtain richer results. In addition, in present paper two forms of solution are adopted. They are (4) and (5). Equation (4) is the same with the form in [37]. However, (5) is a new form, by which some new solutions can be obtained. The details can be found in Section 3.

The organization of the paper is as follows: in Section 2, a brief description of the extended F-expansion for finding traveling wave solutions of nonlinear equations is given. In Section 3, we will study (1) by the extended F-expansion methods. Finally conclusions are given in Section 4.

2. Description of the Extended F-Expansion Methods

Based on F-expansion method, the main procedures of the extended F-expansion method are as follows [22].

Step 1. Consider a general nonlinear PDE in the form Using ,  , we can rewrite (2) as the following nonlinear ODE: where the prime denotes differentiation with respect to .

Step 2. Suppose that the solution of ODE (3) can be written as follows: or where , are constants to be determined later, is a positive integer that is given by the homogeneous balance principle, and satisfies the following equation: where , , , , are constant.

Step 3. Substituting (4) or (5) along with (6) into (3) and then setting all the coefficients of of the resulting system to zero yields a set of over-determined nonlinear algebraic equations for , , .

Step 4. Assuming that the constants , , can be obtained by solving the algebraic equations in Step 3 and then substituting these constants and the solutions of (6), depending on the special conditions chosen for the , , , , and into (4) (or (5)), we can obtain the explicit solutions of (2) immediately.

3. Exact Solutions of (1)

Making a transformation with , (1) can be reduced to the following ODE: where , , , , and are nonzero constants. Balancing and in (7), we obtain which gives .

Suppose that (7) owns the solutions in the form

Substituting (8) and (6) into (7) and then setting all the coefficients of of the resulting system to zero, we can obtain the following results.

3.1.

In this situation, we obtain the following set of nontrivial solutions: where , , , and are arbitrary constants and , , , and are nonzero constants.

Substituting (9) into (8), we obtain, respectively, the following formal solution of (1): where and , , , , and are determined in (9).

When , the general elliptic equation (6) is reduced to the auxiliary ordinary equation

The solutions of (13) are given in Table 1. Combining (10)–(12) with Table 1, many exact solutions of (1) can be obtained. For simplicity, we just give out one case in Table 1; the other cases can be discussed similarly.


Case

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

When , , , the solution of (13) is or . Substituting them into (10)–(12), we can obtain the following Jacobi Elliptic function solutions of (1).

Form (10), one has where .

When , , solution (14) becomes where .

When , , solution (15) becomes where .

From (11), we have where .

When , , solution (18) becomes where .

From (12), we have When , , solution (21) becomes where .

When , , solution (22) becomes (17).

Remark 1. Considering the form of (5), we can suppose that (7) owns the solutions in the form
Substituting (24) and (13) into (7) and then setting all the coefficients of of the resulting system to zero, we can obtain the following result: where , , and are arbitrary constants, , and , , , and are nonzero constants.
Substituting (25) into (24), we obtain the following formal solution of (1):
Combining (26) with Table 1, many exact solutions of (1) can be obtained. For simplicity, we just give out one case in Table 1.
When ,  ,  , the solution of (13) is or . Substituting them into (10)–(12), we can obtain the following Jacobi Elliptic function solutions of (1): where .
When , , , and , solution (27) becomes where .
Some typical wave figures are given as follows. (14) is an unbounded periodic wave solution, as shown in Figure 1(a). As increases, the period also increases gradually, as shown in Figure 1(b). When , it becomes unbounded solitary wave solution, as shown in Figure 1(c). (18) is a smooth periodic wave solution, as shown in Figure 2(a). As increases, the period also increases gradually, as shown in Figure 2(b). When , it becomes a solitary wave solution, as shown in Figure 2(c). The figures of other solutions are similar, so we do not give them out.

3.2.

In this situation, we have the following result: where , , , and are arbitrary constants.

Substituting (30) into (8), we obtain the following formal solution of (1): where and , , , , and are determined in (30).

When , the general elliptic equation (6) is reduced to the auxiliary ordinary equation

Combining (31) with solutions of (32), many exact solutions of (1) can be obtained. The process is similar to the case of ; we omit it.

3.3.

In this situation, we have the following result: where , , and are arbitrary constants.

Substituting (33) into (8), we obtain the following formal solution of (1): where and , , , , and are determined in (33).

When , the general elliptic equation (6) is reduced to the auxiliary ordinary equation

If , , the solutions of (35) are given by the following: where is arbitrary constant.

Substituting (36) into (34), we obtain the following hyperbolic function solutions of (1): where .

Remark 2. Considering the form of (5), we can suppose that (7) owns the solutions in the form
Substituting (38) and (35) into (7) and then setting all the coefficients of of the resulting system to zero, we can obtain the following result: where , , and are arbitrary constants, , and , , , and are nonzero constants.
The solutions of (35) are given in Table 2.
Substituting (39) into (38), we obtain the following formal solution of (1): where and , , , , and are determined in (39).
Combining (40) with Table 2, many exact solutions of (1) can be obtained. For simplicity, we do not list them carefully. Using case 2 in Table 2, we can obtain the following type of wave figures. Figures 3(a) and 3(b) are kink waves and Figure 3(c) is new type of singular wave. It is interesting that they are dependent on .


Case Case

1 2

3 , 4 ,

5 , 6 ,

7 , 8 ,

9 , 10 ,

11 , 12 ,

13 14 ,

3.4. ,  ,  ,  ,  

In this case, there exists three parameters , , and such that Equation (41) is satisfied only if the following relations hold:

Equation (33) is the general Riccati equation. The solutions of (41) are listed in [12]. There are 24 group solutions named , , which we do not list for simplicity.

Substituting (41) and (8) into (7) and then setting all the coefficients of of the resulting system to zero, we can obtain the following results: where , , and are arbitrary constants, and where and are arbitrary constants.

Substituting (43)–(45) into (8), we obtain, respectively, the following formal solutions of (1): where , and where .

Substituting solutions of (41) , into (46)–(47), we can obtain a lot of solutions of (1). We just give one example.

When and , . Substituting into (46) and (51), we have where .

When , , . Substituting into (47), we have where .

3.5. ,  ,  ,  ,  

In this case, there exists three parameters , , and such that Equation (50) is satisfied only if the following relations hold: The following constraint should exist between , , and parameters:

Therefore, we can discuss the solution of (1) similarly as 3.4 under the condition (52). Here, we omit it.

3.6. ,  

In this situation, we have the following result: Substituting (53) into (8), we obtain the following formal solution of (1): where and , , , , and are determined in (53).

When , the general elliptic equation (6) is reduced to the auxiliary ordinary equation The solution of (55) is the Weierstrass elliptic doubly periodic type solution is as follows: Substituting (56) into (54), the solution of (1) is where and