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

x = 0 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 ZKBBM(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.

Zakharov-Kuznetsov (ZK) equation [15]   +   + (  +   )  = 0 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]   +   − ( 2 )  − b  = 0 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:   +   − (  )  + ((  )  + (  )  )  = 0. (5) 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, blowup, 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.

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  > 0 (the other case  < 0 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.
Remark 3. When ( + ) > 0,  ̸ = 1, and  → 0, the smooth periodic wave  4 () becomes which can be found in [20]; this implies that we extend the previous result.and its traveling wave equation There are the following results.

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: (48) Clearly, system (47) and system (48) have the same first integral where ℎ is an integral constant.Consequently, these two systems have the same topological phase portraits except for the straight line  = 0. Thus, we can understand the phase portraits of system (47) from those of system (48).
Hereto, we have completed the derivations for Proposition 1.