`Mathematical Problems in EngineeringVolumeΒ 2011Β (2011), Article IDΒ 810217, 23 pageshttp://dx.doi.org/10.1155/2011/810217`
Research Article

Trigonometric Function Periodic Wave Solutions and Their Limit Forms for the KdV-Like and the PC-Like Equations

Department of Mathematics, South China University of Technology, Guangzhou 510640, China

Received 15 February 2011; Accepted 26 May 2011

Copyright Β© 2011 Liu Zhengrong 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

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limit forms for the KdV-like equation , and PC-like equation , respectively. Via some special phase orbits, we obtain some new explicit periodic wave solutions which are called trigonometric function periodic wave solutions because they are expressed in terms of trigonometric functions. We also show that the trigonometric function periodic wave solutions can be obtained from the limits of elliptic function periodic wave solutions. It is very interesting that the two equations have similar periodic wave solutions. Our work extend previous some results.

1. Introduction

Many authors have investigated the KdV-like equation and the PC-like equation For example, Dey [1, 2] studied the exact Himiltonian density and the conservation laws, and gave two kink solutions for (1.1). Zhang et al. [3, 4] gave some solitary wave solutions and singular wave solutions for (1.1) by using two different methods. Yu [5] got an exact kink soliton for (1.1) by using homogeneous balance method. Grimshaw et al. [6] studied the large-amplitude solitons for (1.1). Fan [7, 8] gave some bell-shaped soliton solutions, kink-shaped soliton, and Jacobi periodic solutions for (1.1) by using algebraic method. Tang et al. [9] investigated solitary waves and their bifurcations for (1.1) by employing bifurcation method of dynamical systems. Peng [10] used the modified mapping method to get some solitary wave solutions composed of hyperbolic functions, periodic wave solutions composed of Jacobi elliptic functions, and singular wave solution composed of triangle functions for (1.1). Chow et al. [11] described the interaction between a soliton and a breather for (1.1) by using the Hirota bilinear method. Kaya and Inan [12] studied solitary wave solutions for (1.1) by using Adomian decomposition method. Yomba [13] used Fan's subequation method to construct exact traveling wave solutions composed of hyperbolic functions or Jacobi elliptic functions for (1.1).

Zhang and Ma [14] gave some explicit solitary wave solutions composed of hyperbolic functions by using solving algebraic equations for (1.2). Li and Zhang [15] used bifurcation method of dynamical system to study the bifurcation of traveling wave solutions and construct solitary wave solutions for (1.2). Kaya [16] discussed the exact and numerical solitary wave solutions by using a decomposition method for (1.2). Rafei et al. [17] gave numerical solutions by using He's method for (1.2).

Recently, many authors have presented some useful methods to deal with the problems in equations, for instance [18β30].

In this paper, we use the bifurcation method mentioned above to study the periodic wave solutions for (1.1) and (1.2). Through some special phase orbits, we obtain new expressions of periodic wave solutions which are composed of trigonometric functions sinβ or cosβ. These solutions are called trigonometric function periodic wave solutions. We also check the correctness by using the software Mathematica.

In Section 2, we will state our results for (1.1). In Section 3, we will state our results for (1.2). In Sections 4, and 5, we will give derivations for our main results. Some discussions and the orders for testing the correctness of the solutions will be given in Section 6.

2. Trigonometric Function Periodic Wave Solutions for (1.1)

In this section, we state our main results for (1.1). In order to state these results conveniently, we give some preparations. For given constant , on plane we define some lines and regions as follows.()When , we define lines and regions , as Figure 1(a).

Figure 1: The locations of the lines , and the regions , for given constant .
()When , we define lines and regions , as Figure 1(b).

Using the lines and regions in Figure 1, we narrate our results as follows.

Proposition 2.1. For arbitrary given constant , let Then, (1.1) has the following periodic wave solutions. (1)When and or , the expression of the periodic wave solution is which has the following limit forms.()When , and tends to the line , tends to the periodic blow-up solution (see Figure 2).()When , and tends to the line , tends to the periodic blow-up solution (see Figure 3).()When , or , and tends to , tends to the trivial solution .()When and , or when and , the expression of the periodic wave solution is where The solution has the following limit forms. ()When , and tends to , the tends to the peak-shaped solitary wave solution (see Figure 4).()When , and tends to , tends to the trivial solution .()When , and tends to , the tends to the periodic blow-up solution (see Figure 5).()When and , or when and , the expressions of the solution is where The solution has the following limit forms.()When , and tends to , the tends to the canyon-shaped solitary wave (see Figure 6) solution . ()When , and tends to , tends to the trivial solution . ()When , and tends to , the tends to the periodic blow-up wave solution (see Figure 3).

Figure 2: The limiting precess of when , , and tends to the line , where and .
Figure 3: The limiting precess of when , , and tends to the line , where and .
Figure 4: The limiting precess of when , , and tends to the line , where and .
Figure 5: The limiting precess of when , , and tends to the line , where and .
Figure 6: The limiting precess of when , , and tends to the line , where and .

Remark 2.2. Note that if is a solution of (1.1), then also is solution of (1.1), where is a arbitrary constant. According to this fact and the results listed in Proposition 2.1, the following nine functions also are periodic wave solutions of (1.1).
(1) When and or , the functions are
(2) When and or when and , the functions are
(3) When and , or when and , the functions are

Remark 2.3. In the given parametric regions, the solutions , , , , and are nonsingular. The solutions , , and are singular. The relationships of singular solutions and nonsingular solutions are displayed in the Proposition 2.1.

3. Trigonometric Function Periodic Wave Solutions for (1.2)

In this section, we state our main results for (1.2). For given and , on plane we define some rays and regions as follows.()When , we define curves and region as the domain surrounded by and , as the domain surrounded by and (see Figure 7(a)).

Figure 7: The locations of the rays , and the regions , for given and .
()When , we define curves and region as the domain surrounded by and , as the domain surrounded by and (see Figure 7(b)).

Using the rays and regions above, we state our results as follows.

Proposition 3.1. For given parameter and constant satisfying , let . Then, (1.2) has the following periodic wave solutions. ()When and or , the expression of the periodic wave solution is where For , the periodic wave solution has the following limit forms.()When , and tends to the ray , tends to the periodic blow-up solution The limiting process is similar to that in Figure 2.()When , and tends to the ray , tends to the periodic blow-up solution The limiting process is similar to that in Figure 3.()When , and tends to the curve , or and tends to the curve , tends to the trivial solution .(2)When and , or when and , the expression of the periodic wave solution is where The periodic wave solution has the following limit forms. ()When , , and tends to the curve , tends to the trivial solution .()When , , and tends to the curve , the tends to the canyon-shaped solitary wave solution The limiting process is similar to that in Figure 6.()When , , and tends to the ray , tends to the periodic blow-up wave solution The limiting process is similar to that in Figure 2.(3)When and , or when and , the expression of the periodic wave solution is where The periodic wave solution has the following limit forms:()When , , and tends to the curve , tends to the trivial solution .()When , , and tends to the curve , the tends to the peak-shaped solitary wave solution . The limiting process is similar to that in Figure 4.()When , , and tends to the ray , the tends to the periodic blow-up wave solution . The limiting process is similar to that in Figure 5.

Remark 3.2. Similar to the reason in Remark 2.2, the following nine functions also are periodic wave solutions of (1.2).
()When and or , the functions are () When and or when and , the functions are () When and or when and , the functions are

Remark 3.3. In the given regions, the solutions , , , , and are nonsingular. The solutions , , and are singular. The relationships of nonsingular solutions and singular solutions are displayed in Proposition 3.1.

4. The Derivation on Proposition 2.1

In order to derive the Proposition 2.1, letting be a constant and substituting with into (1.1), we have

Integrating (4.1) once and letting the integral constant be zero, it follows that

Letting , yields the following planar system:

Obviously, system (4.3) has the first integral

Let where is defined in (2.8). Then, it is easy to see that system (4.3) has three singular points , and when , two singular points and when , unique singular point when .

Let and be, respectively,

Using the qualitative analysis of dynamical systems, we obtain the bifurcation phase portraits of system (4.3) and the locations of and as Figures 8 and 9.

Figure 8: When , the bifurcation phase portraits of system (4.3) and the locations of and .
Figure 9: When , the bifurcation phase portraits of system (4.3) and the locations of and .

It is easy to test that the closed orbit passing passes . Thus, using the phase portraits in Figures 8 and 9, we derive as follows.

()When and or , the closed orbit passing the points and has expression Substituting (4.7) into , we have Integrating (4.8) along the closed orbit and noting that , we obtain the solution as (2.4).() When and or when and , the closed orbit passing the points and has expression Substituting (4.9) into , we get Along the closed orbit integrating (4.10) and noting that , we get the solution as (2.7).() When and or when and , the closed orbit passing the points and has expression Substituting (4.11) into , it follows that Similarly, along the closed orbit integrating (4.12), we obtain as (2.16). From the expressions of these solutions, we get their limit forms. This completes the derivation on Proposition 2.1.

5. The Derivation on Proposition 3.1

In this section, we give derivation on Proposition 3.1. Let with , where is a constant. Thus, (1.2) becomes

Integrating (5.1) twice and letting integral constant be zero, we get

Letting , we have the planar system

It is easy to see that system (5.3) has the first integral and three singular points , , and , where and is defined in (3.9).

Let and be, respectively,

Similarly, using the qualitative analysis of dynamical systems, we get the bifurcation phase portraits of system (5.3) and the locations of and as Figures 10 and 11.

Figure 10: When , the bifurcation phase portraits of system (5.3) and the locations of and .
Figure 11: When , the bifurcation phase portraits of system (5.3) and the locations of and .

It is easy to test that the closed orbit passing passes . Thus, using the phase portraits in Figures 10 and 11, we derive as follows.()When and or , the closed orbit passing the points and has expression Substituting (5.7) into , we have Integrating (5.8) along the closed orbit and noting that , we get the solution as (3.3). () When and , or when and , the closed orbit passing the points and has expression From and (5.9), it follows that Integrating (5.10) along the closed orbit, we get as (3.7).() When and , or when and , the closed orbit passing the points and has expression Substituting (5.11) into , we have Integrating (5.12) along the closed orbit, we obtain as (3.12). From the expressions of these solutions, we get their limiting properties. This completes the derivation on Proposition 3.1.

6. Discussions and Testing Orders

In this paper, Using the special closed orbits, we have obtained trigonometric function periodic wave solutions for (1.1) and (1.2), respectively. Their limit forms have been given. From these expressions, an interesting phenomena has been seen, that is, (1.1) and (1.2) have similar periodic wave solutions. Our work has extended previous results on periodic wave solutions.

Now, we point out that the trigonometric function periodic wave solutions can be obtained from the limits of the elliplic function periodic wave solution. For given real number , let where

Assume that , , and . It is easy to check that are real and satisfy

There are two closed orbits and (see Figure 12). The closed orbit passes the points and . The closed orbit passes the points and .

Figure 12: The locations of and when and .

On plane, the expression of is

Substituting (6.4) into and integrating it along , we have where

Solving (6.5) for and noting that , we obtain an elliptic function periodic wave solution where

Letting , it follows that , , , , and .

Therefore, in (6.7) letting , we obtain the trigonometric function periodic wave solution

Via Remark 2.2 and , further we get , and . Similarly, we can derive others trigonometric function periodic wave solutions.

We also have tested the correctness of each solution by using the software Mathematica. Here, we list two testing orders. Others testing orders are similar.()The orders for testing Simplify . () The orders for testing Simplify .

Acknowledgment

Research is supported by the National Natural Science Foundation of China (no. 10871073) and the Research Expences of Central Universities for students.

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