Journal of Applied Mathematics

Volume 2013 (2013), Article ID 156139, 14 pages

http://dx.doi.org/10.1155/2013/156139

Research Article

Some Further Results on Traveling Wave Solutions for the ZK-BBM() Equations

Shaoyong Li1,2 and Zhengrong Liu²

¹College of Mathematics and Information Sciences, Shaoguan University, Shaoguan, Guangdong 512005, China

²Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510640, China

Received 30 June 2013; Accepted 4 September 2013

Academic Editor: K. S. Govinder

Copyright © 2013 Shaoyong Li and Zhengrong Liu. 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 investigate the traveling wave solutions for the ZK-BBM() equations by using bifurcation method of dynamical systems. Firstly, for ZK-BBM(2, 2) equation, we obtain peakon wave, periodic peakon wave, and smooth periodic wave solutions and point out that the peakon wave is the limit form of the periodic peakon wave. Secondly, for ZK-BBM(3, 2) equation, we obtain some elliptic function solutions which include periodic blow-up and periodic wave. Furthermore, from the limit forms of the elliptic function solutions, we obtain some trigonometric and hyperbolic function solutions which include periodic blow-up, blow-up, and smooth solitary wave. We also show that our work extends some previous results.

1. Introduction

In recent years, many nonlinear wave equations have been derived from solid state physics, plasma physics, chemical physics, fluid mechanics, biology, and other fields. Thus, there has been considerable attention to find exact solutions of these problems. For this purpose, there have been many methods, such as inverse scattering transform method [1], Bäcklund and Darboux transforms [2, 3], Jacobi elliptic function method [4, 5], F-expansion and extended F-expansion method [6, 7], -expansion method [8, 9], and the bifurcation method of dynamical systems [10–14].

Zakharov-Kuznetsov (ZK) equation [15]

is a two-dimensional space generalization of the KdV equation. The nonintegrable ZK equation governs the behavior of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field [16, 17].

Benjamin-Bona-Mahony (BBM) equation [18]

is an alternative model to KdV equation for small-amplitude, surface waves of long wavelength in liquids, acoustic-gravity waves in compressible fluids, hydromagnetic waves in cold plasma, and acoustic waves in anharmonic crystals.

Combining the BBM equation with the sense of the ZK equation, Wazwaz [19] considered the following ZK-BBM equation:

and its generalized form

He presented a method called the extended tanh method to seek exact explicit compactons, solitons, solitary patterns, and plane periodic solutions of (3) and (4).

Wang and Tang [20] studied the following generalized ZK-BBM equations:

By using the bifurcation theory of planar dynamical systems, they gave some exact explicit traveling wave solutions and the sufficient conditions to guarantee the existence of smooth and nonsmooth traveling wave solutions.

In the present paper, we continue to study the traveling wave solutions for (5), which we denote by ZK-BBM() equations for convenience. Our results are as follows: (i) for ZK-BBM(2, 2) equation, we obtain peakon wave, periodic peakon wave, and smooth periodic wave solutions and point out that the peakon wave is the limit form of the periodic peakon wave; (ii) for ZK-BBM(3, 2) equation, we obtain some elliptic function solutions which include periodic blow-up and periodic wave. From the limit forms of the elliptic function solutions, we obtain some trigonometric and hyperbolic function solutions which include periodic blow-up, blow-up, and smooth solitary wave. We also check the correctness of these solutions by putting them back into the original equation.

This paper is organized as follows. In Section 2, we state our main results which are included in two propositions. In Sections 3 and 4, we give the derivations for the two propositions, respectively. A brief conclusion is given in Section 5.

2. Main Results and Remarks

In this section we list our main results and give some remarks. To begin with, let us recall some symbols. The symbols sn and cn denote the Jacobian elliptic functions sine amplitude and cosine amplitude . cosh , sinh , sech , and csch are the hyperbolic functions. For the sake of simplification, we only consider the case (the other case can be considered similarly). To relate conveniently, for given constant wave speed , let

Via the following two propositions we state our main results.

Proposition 1. Consider ZK-BBM(2, 2) equation

and its traveling wave equation

There are the following results.

(1) When , , and , (7) has a peakon wave solution

 where

(2) When , , and , (7) has a periodic peakon wave solution

 where

(3) When , , and , (7) has two smooth periodic wave solutions

 where

Remark 2. When , , and , the periodic peakon wave becomes the peakon wave ; the varying process is displayed in Figure 1.

(a)

(b)

(c)

(d)

Figure 1: The varying process for the periodic peakon wave tends to the peakon wave when , where , , and , and (a) ; (b) ; (c) ; (d) .

Remark 3. When , , and , the smooth periodic wave becomes

which can be found in [20]; this implies that we extend the previous result.

Proposition 4. Consider ZK-BBM(3, 2) equation

and its traveling wave equation

There are the following results.

() When , , and , (17) has two elliptic periodic blow-up solutions

where

() When , , and , (17) has two elliptic periodic blow-up solutions , and two symmetric elliptic periodic wave solutions ,

where

() When , , and , (17) has two elliptic periodic blow-up solutions

where

() When , , and , (17) has two elliptic periodic blow-up solutions

where

() When , , and , (17) has two elliptic periodic blow-up solutions , and two symmetric elliptic periodic wave solutions ,

where

() When , , and , (17) has two elliptic periodic blow-up solutions

where

Remark 5. When , , and , the periodic blow-up solutions (or ) and (or ) tend to two trigonometric periodic blow-up solutions, respectively,

The symmetric elliptic periodic wave solutions and become a trivial solution .

Remark 6. When , , and , the periodic blow-up solution (or ) tends to a hyperbolic blow-up solution

The elliptic periodic wave solution (or the elliptic periodic blow-up solution ) tends to a hyperbolic smooth solitary wave solution

For the varying process, see Figures 2, 3, and 4. The elliptic solutions and tend to a trivial solution .

(a)

(b)

(c)

(d)

Figure 2: The varying process for graphs of when , where , , and (a) ; (b) ; (c) ; (d) .

(a)

(b)

(c)

(d)

Figure 3: The varying process for graphs of when , where , , and (a) ; (b) ; (c) ; (d) .

(a)

(b)

(c)

(d)

Figure 4: The varying process for graphs of when , where , , and (a) ; (b) ; (c) ; (d) .

Remark 7. When , , and , the periodic blow-up solutions (or ) and (or ) tend to two trigonometric periodic blow-up solutions, respectively,

The symmetric elliptic periodic wave solutions and become the trivial solution .

Remark 8. When , , and , the periodic blow-up solution (or ) tends to a hyperbolic blow-up solution

The elliptic periodic wave solution (or the elliptic periodic blow-up solution ) tends to a hyperbolic smooth solitary wave solution

The varying process is similar to those in Figures 2–4. The elliptic solutions and tend to the trivial solution .

Remark 9. When , the solutions and , respectively, become

which can be found in [20]; this implies that we extend the previous results.

3. The Derivations for Proposition 1

In this section, we derive the precise expressions of the traveling wave solutions for ZK-BBM(2, 2) equation. Substituting with into (7), it follows that

Integrating (45) once, we have

where is an integral constant.

Letting , we obtain the following planar system:

Under the transformation , system (47) becomes

Clearly, system (47) and system (48) have the same first integral

where is an integral constant. Consequently, these two systems have the same topological phase portraits except for the straight line . Thus, we can understand the phase portraits of system (47) from those of system (48).

When the integral constant , (49) becomes

Solving equation , we get two roots

On the other hand, solving equation , we obtain

According to the qualitative theory, we obtain the phase portraits of system (48) as shown in Figure 5.

(a) , , 

(b) , ,

(c) , ,

(d) , ,

(e) , ,

(f) , ,

(g) , ,

(h) , ,

(i)

Figure 5: The phase portraits of system (48).

When , on plane the orbit has expression

Substituting (53) into and integrating it along the orbit , we obtain the peakon wave solution as (9).

When , on plane the orbit has expression

Substituting (54) into and integrating it along the orbit , we obtain the periodic peakon wave solution as (11), where

If , it follows that , (or , ), and . This implies that the periodic peakon wave solution tends to the peakon wave solution .

When and , on plane the orbit has expression

Substituting (56) into and integrating it along the orbit , we obtain the smooth periodic wave solutions and as (13).

Hereto, we have completed the derivations for Proposition 1.

4. The Derivations for Proposition 4

In this section, we derive the explicit elliptic function solutions and their limit forms for ZK-BBM(3, 2) equation. Similar to the derivations in Section 3, substituting with into (17) and integrating it, we have the following planar system:

Similarly, under the transformation , system (57) becomes

which has the first integral

When the integral constant , (59) becomes

Solving equation , we get three roots , , and as (23), (24), and (25). On the other hand, solving equations

we get three equilibrium points of system (58), where

According to the qualitative theory, we obtain the phase portraits of system (58) as shown in Figure 6.

(a) , , 

(b) , ,

(c) , ,

(d) , ,

(e) , ,

(f) , ,

(g) , ,

(h) , ,

(i) , ,

(j) , ,

(k) , ,

(l) , ,

Figure 6: The phase portraits of system (58).

Now using planar system (57) and the phase portraits in Figure 6, we derive the explicit expressions of solutions for the ZK-BBM(3, 2) equation respectively.

When , , and , has the expression

where and are complex numbers.

Substituting (63) into and integrating it, we have

Completing the integrals in the above two equations and noting that , we obtain and as (19).

When , , and , and have the expressions

Substituting (65) into and integrating them, we have

Completing the integrals in the above four equations and noting that , we obtain as (27).

When , , and , has the expression

where and are complex numbers.

Substituting (67) into and integrating it, we have

Completing the integrals in the above two equations and noting that , we obtain and as (29).

When , , and , has the expression

where and are complex numbers.

Substituting (69) into and integrating it, we have

Completing the integrals in the above two equations and noting that , we obtain and as (31).

When , , and , and have the expressions

Substituting (71) into and integrating them, we have

Completing the integrals in the above four equations and noting that , we obtain as (33).

When , , and , has the expression

where and are complex numbers.

Substituting (73) into and integrating it, we have

Completing the integrals in the above two equations and noting that , we obtain and as (35).

Hereto, we have finished the derivations for the solutions . In what follows, we will derive the limit forms of these solutions.

When , , and , it follows that

Thus we have

When , , and , it follows that

Thus we have

The limit forms of the other solutions can be derived similarly, so here we omit them. Hereto, we have completed the derivations for Proposition 4.

5. Conclusion

In this paper, we have investigated ZK-BBM equations. For ZK-BBM(2, 2) equation, we obtain peakon wave, periodic peakon wave, and smooth periodic wave solutions (see ). For ZK-BBM(3, 2) equation, we obtain some elliptic function solutions (see ). Furthermore, from the limit forms of these solutions, we obtain some trigonometric and hyperbolic function solutions (see Remarks 5–8 and the corresponding derivations). We also showed that some previous results are our special cases (see Remarks 3 and 9). We would like to study the ZK-BBM(, ) equations further.

Acknowledgment

This study is supported by the National Natural Science Foundation (no. 11171115).

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