Bifurcation Phenomena of Nonlinear Waves in a Generalized Zakharov-Kuznetsov Equation

bu 4 )u x +γu xxx +δu xyy = 0. We reveal four kinds of interesting bifurcation phenomena.The first kind is that the low-kink waves can be bifurcated from the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The second kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves, the symmetric solitary waves, and the 2-blow-up waves. The third kind is that the periodic-blow-up waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves.

For the generalized Zakharov-Kuznetsov equation where , , , and  are real constants, Song and Cai [6] got some solitary wave and kink wave solutions of (4).When  = , Zhang [7] used the new generalized algebraic method to obtain some soliton solutions, combined soliton solutions, triangular periodic solutions, Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, and rational function solutions of (4).Biswas and Zerrad [8] obtained 1-soliton solution of (4) with dual-power law nonlinearity.
When  = 0, Liu and Yan [9] obtained some common expressions and two kinds of bifurcation phenomena for nonlinear waves of (4).Meanwhile, they pointed out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively.
We obtain three types of explicit expressions of nonlinear wave solutions.Under different parameters conditions, these expressions represent symmetric and antisymmetric solitary waves, kink and anti-kink waves, symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blowup waves.Furthermore, we reveal four kinds of interesting bifurcation phenomena which are introduced in the abstract above.
This paper is organized as follows.The four kinds of interesting bifurcation phenomena are shown in Sections 2-5.A brief conclusion is given in Section 6.

Bifurcation of the Low-Kink Waves
In this section, we show that the low-kink waves can be bifurcated from the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves.

and they represent four symmetric solitary waves (see Figures 2(a)-2(c))
. In particular, when  → −(5 2 /48)+ 0, the four symmetric solitary waves become four lowkink waves (see Figure 2(d)) which were given by Song and Cai [6].This implies that one extends the previous results.For the varying process, see Figure 2.
For the varying process, see Figure 4.

Bifurcation of the 1-Blow-Up Waves
In this section, we show that the 1-blow-up waves can be bifurcated from the 2-blow-up waves, the symmetric solitary waves, and the periodic-blow-up waves.
Similar to the proof of Proposition 1, we get the results of Proposition 3.
For the varying process, see Figure 8.
Completing the integral in (44) and solving the equation for , it follows that where  3 is given in (40),  1 =  1 () is an arbitrary constant, and In (45) letting  1 = /2, we obtain the solutions  ±  as (39).
(2) Note that Thus, we have Furthermore, we get lim Hereto, we have completed the proof for Proposition 4.

Bifurcation of the Periodic-Blow-Up Waves
In this section, we show that the periodic-blow-up waves can be bifurcated from symmetric periodic waves.