Research Article | Open Access

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

**Academic Editor:**K. S. Govinder

#### 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], BÃ¤cklund 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 [10â€“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 â€‰where (2) When , , and , (7) has a periodic peakon wave solution â€‰where (3) When , , and , (7) has two smooth periodic wave solutions â€‰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)**

*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)**

**(a)**

**(b)**

**(c)**

**(d)**

**(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 2â€“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)**

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) , ,**

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