Research Article | Open Access

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

**Academic Editor:**Hagen Neidhardt

#### Abstract

We study the bifurcation phenomena of nonlinear waves described by a generalized Zakharov-Kuznetsov equation . 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.

#### 1. Introduction and Preliminary

Zakharov-Kuznetsov (Z-K) equation [1], was first derived for describing weakly nonlinear ion-acoustic wave in a strongly magnetized lossless plasma in two dimensions. The Z-K 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 [2, 3].

There are lots of research for various generalized Z-K equations [4–13]. For the Z-K equation Yan and Liu [4] gave some polynomial solutions, triangular function solutions and elliptic periodic solutions, of (2) via a direct symmetry method.

When , , and , equation (2) reduces to Bekir [5] used the -expansion method to obtain three types of traveling wave solutions of (3).

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 , 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.

In order to investigate the bifurcation phenomena of (4), letting be wave speed and substituting with into (4), it follows that Integrating (5), we get Setting ; yields the following planar system: Obviously, system (7) is a Hamiltonian system with Hamiltonian function where is the integral constant.

Let

On parametric plane, let represent the following four curves: Let represent the regions surrounded by and the coordinate axes (see Figure 1).

In this paper, we employ bifurcation method of dynamical systems [14–23] to investigate the bifurcation phenomena of nonlinear waves described by (4).

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-blow-up 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.

#### 2. 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.

##### 2.1. Bifurcation from Symmetric Solitary Waves and 1-Blow-Up Waves

Proposition 1. *For , , and , (4) has four nonlinear wave solutions as follows:
**
where
**
and is an arbitrary real constant. For , one has the following results and bifurcation phenomena.*(1)*If ** and **, then *,
* and they represent four symmetric solitary waves (see Figures 2(a)–2(c)). In particular, when **, the four symmetric solitary waves become four low-kink 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.*(2)*If ** and ** belongs to any one of the regions **, **, **, and **, then ** and they represent four 1-blow-up waves (see Figures 3(a)–3(c)). In particular, when ** and **, the four 1-blow-up waves become four low-kink waves with the expressions ** and **. For the varying process, see Figure 3.*(3)*If ** and **, then ** equal to the hyperbolic solitary wave solutions **which were given by Song and Cai [6]. This implies that one extends the previous results. When *,
* ** tend to two trivial solutions **.*

**(a)**

**(b)**

**(c)**

**(d)**

**(a)**

**(b)**

**(c)**

**(d)**

*Proof. * In (8), letting , it follows that
Substituting (20) into and integrating it, we have
where is an arbitrary constant.

Completing the integral above and solving the equation for , it follows that
where is an arbitrary real number.

Note that if is a solution of (4), so is . Therefore, from (22) we obtain the solutions and as (13).

In (13) letting , then , and we get (17) and (18). From (13), (17) and (18), we get results (1) and (2) of Proposition 1.

When , via (13) it follows that
which is result (3) of Proposition 1.

##### 2.2. Bifurcation from Tall-Kink Waves and Antisymmetric Solitary Waves

Proposition 2. * If , , and belongs to one of the regions , , , and , then (4) has four real nonlinear wave solutions as follows:
**
where is an arbitrary real constant, is given in (11), and
**Letting
**
corresponding to , one has the following results and bifurcation phenomena.*(1)*If ** and **, then *,
* and they represent four tall-kink waves (see Figures 4(a)–4(c)). When **, the four tall-kink waves become**which represent four low-kink waves (see Figure 4(d)). For the varying process, see Figure 4.*(2)*If ** and **, then *,
* and they represent four antisymmetry solitary waves (see Figures 5(a)–5(c)). When **, the four antisymmetry solitary waves become two trivial waves **. In particular, when **, the four antisymmetry solitary waves become four low-kink waves with the expressions ** and ** (see Figure 5(d)). For the varying process, see Figure 5.*(3)*If ** and **, then ** of forms **which represent two tall-kink waves and tend to a trivial wave ** when **.*(4)*If ** and **, then ** of forms **which represent two antisymmetric solitary waves and tend to the trivial wave ** when ** and tend to ** when **.*

**(a)**

**(b)**

**(c)**

**(d)**

**(a)**

**(b)**

**(c)**

**(d)**

*Proof. * In (8), letting , it follows that
where and are given in (9) and (26), respectively. Substituting (31) into and integrating it, we have
where is an arbitrary constant.

Completing the integral above and solving the equation for , it follows that
where is given in (25), is an arbitrary real number, and
Similarly, if is a solution of (4), so is . Substituting (34) into (33), we get and (see (24)).

When , it follows that
From (24), it is easy to check that and become and (see (27) and (28)).

If , then and , and we have
Similarly, we have .

If , then and , and we have
Similarly, we have .

Hereto, we have completed the proof for Proposition 2.

#### 3. 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.

##### 3.1. Bifurcation from 2-Blow-Up Waves and Symmetric Solitary Waves

Proposition 3. *In (13), corresponding to , one has the following results and bifurcation phenomena.*(1)*If ** and **, then *,
* and they represent four 2-blow-up waves. When **, *,
* and *,
* respectively, become ** and ** (see (17) and (18)) which represent four 1-blow-up waves (see Figure 6(d)). For the varying process, see Figure 6.*(2)*If **, then ** and become **which were given by Song and Cai [6]. This implies that one extends the previous results.**When , represent hyperbolic blow-up waves. Specially, when , tend to two trivial solutions .**When belongs to any one of the regions , , represent two symmetric solitary waves. In particular, when and , become two 1-blow-up waves. For the varying process, see Figure 7.*

**(a)**

**(b)**

**(c)**

**(d)**

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**(c)**

**(d)**

Similar to the proof of Proposition 1, we get the results of Proposition 3.

##### 3.2. Bifurcation from Periodic-Blow-Up Waves

Proposition 4. * Under and , one has the following results and bifurcation phenomena.*(1)*If ** belongs to one of the regions **, *,
* and **, then (4) has two periodic-blow-up wave solutions **where *(2)*If ** and **, the periodic-blow-up wave solutions ** become two fractional wave solutions **which represent two 1-blow-up waves (see Figure 8(d)). For the varying process, see Figure 8.*

**(a)**

**(b)**

**(c)**

**(d)**

*Proof. *(1) In (8), letting , it follows that
where is given in (10), and

Substituting (42) into and integrating it, we have
where is an arbitrary constant.

Completing the integral in (44) and solving the equation for , it follows that
where is given in (40), is an arbitrary constant, and

In (45) letting , we obtain the solutions as (39).

(2) Note that

Thus, we have

Furthermore, we getHereto, we have completed the proof for Proposition 4.

#### 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.

##### 4.1. Bifurcation from Periodic Waves

Proposition 5. * If , , and , (4) has two nonlinear wave solutions
**
where
**
One has the following results and bifurcation phenomena.*(1)*If ** belongs to any one of the regions ** and **, then ** represent periodic-blow-up waves.*(2)*If **, then ** represent periodic waves. In particular, when **, the periodic waves become periodic-blow-up waves as follows:**For the varying process, see Figure 9.**When **, the periodic wave tends to two trivial waves **. For the varying process, see Figure 10.*

**(a)**

**(b)**

**(c)**

**(d)**

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**(b)**

**(c)**

**(d)**

*Proof. *Completing the integral in (21) and solving the equation for , it follows that
where is given in (52) and is an arbitrary constant.

In (54) letting , we obtain the solutions as (51).

From (14) and (15), we have

Letting , then

Hereto, we have completed the proof for Proposition 5.

#### 5. Bifurcation of the Tall-Kink Waves

In this section, we show that the tall-kink waves can be bifurcated from the symmetric periodic waves.

##### 5.1. Bifurcation from Symmetric Periodic Waves

Proposition 6. * Under and , one has the following results and bifurcation phenomena.*(1)*If ** belongs to the region **, then (4) has two periodic wave solutions**where *(2)*If ** and **, the periodic wave solutions ** tend to two fractional wave solutions ** which have the expressions as ** (see (41)) and represent two tall-kink waves (see Figure 11(d)). For the varying process, see Figure 11.*

**(a)**

**(b)**

**(c)**

**(d)**

*Proof. *Completing the integral in (31) and solving the equation for , it follows that
where is given in (58) and is an arbitrary constant.

In (59) letting , we obtain the solutions as (57).

Similar to the proof of Proposition 4, we get the results of Proposition 6.

Besides these bifurcation phenomena above, there is another bifurcation phenomenon as follows.

Proposition 7. * If and belongs to one of the regions , , and , then (4) has four symmetric solitary wave solutions (see Figures 12(a)–12(c)) as follows:
*