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

Authors: Yun Wu; Zhengrong Liu
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

We study the bifurcation phenomena of nonlinear waves described by a generalized Zakharov-Kuznetsov
equationut+au2+bu4ux+γuxxx+δuxyy=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.


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## Body

## 1. Introduction and Preliminary

Zakharov-Kuznetsov (Z-K) equation [1],
(1)ut+auux+(∇2u)x=0,
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
(2)ut+(au+bu2)ux+γuxxx+δuxyy=0,
Yan and Liu [4] gave some polynomial solutions, triangular function solutions and elliptic periodic solutions, of (2) via a direct symmetry method.Whena=0, b=1, and γ=δ=1, equation (2) reduces to
(3)ut+u2ux+uxxx+uxyy=0;
Bekir [5] used the (G′/G)-expansion method to obtain three types of traveling wave solutions of (3).For the generalized Zakharov-Kuznetsov equation(4)ut+(au2+bu4)ux+γuxxx+δuxyy=0,
where a, b, γ, and δ are real constants, Song and Cai [6] got some solitary wave and kink wave solutions of (4).Whenγ=δ, 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.Whenδ=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.In order to investigate the bifurcation phenomena of (4), letting c>0 be wave speed and substituting u=φ(ξ) with ξ=x+y-ct into (4), it follows that
(5)-cφ′+aφ2φ′+bφ4φ′+γφ′′′+δφ′′′=0.
Integrating (5), we get
(6)-cφ+a3φ2+b5φ5+γφ′′+δφ′′=0.
Setting φ′=ψ; yields the following planar system:
(7)φ′=ψ,ψ′=cφ-(a/3)φ3-(b/5)φ5γ+δ.
Obviously, system (7) is a Hamiltonian system with Hamiltonian function
(8)H(φ,ψ)=(γ+δ)ψ2-cφ2+a6φ4+b15φ6=h,
where h is the integral constant.Let(9)φ1=-16b(5a+Δ),(10)φ2=-16b(5a-Δ),(11)Δ=25a2+180bc.Ona-b parametric plane, let li(i=1,2,3,4) represent the following four curves:
(12)l1:b=0(a>0),l2:b=-16a275c,l3:b=-2a29c,l4:b=0(a<0).
Let Ai(i=1,2,…,6) represent the regions surrounded by li(i=1,2,3,4) and the coordinate axes (see Figure 1).Figure 1
The locations of the regionsAi(i=1,2,…,6) and curves li(i=1,2,3,4).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 Sections2–5. 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.
Forab≠0, γ+δ>0, and H(φ,ψ)=H(0,0), (4) has four nonlinear wave solutions as follows:
(13)ua±=±4α1λλ2e-τ1ξ-2λβ1+(β12-4α1)eτ1ξ,ub±=±4α1λλ2eτ1ξ-2λβ1+(β12-4α1)e-τ1ξ,
where
(14)α1=-15cb,(15)β1=5a2b,(16)τ1=2cγ+δ,
and λ≠0 is an arbitrary real constant. For λ>0, one has the following results and bifurcation phenomena.(1)
Ifλ≠β12-4α1and(a,b)∈A2, thenua±≠ub±,
and they represent four symmetric solitary waves (see Figures2(a)–2(c)). In particular, whenb→-(5a2/48c)+0, the four symmetric solitary waves become four low-kink waves (see Figure2(d))(17)ua0±=±12c(-bλ/5)e-2(c/(γ+δ))ξ+a,(18)ub0±=±12c-(bλ/5)e2(c/(γ+δ))ξ+a,which were given by Song and Cai [6]. This implies that one extends the previous results. For the varying process, see Figure 2.(2)
Ifλ≠β12-4α1and(a,b)belongs to any one of the regionsA3,l3,A4, andA5, thenua±≠ub±and they represent four 1-blow-up waves (see Figures3(a)–3(c)). In particular, when(a,b)∈A3andb→-(5a2/48c)-0, the four 1-blow-up waves become four low-kink waves with the expressionsua0±andub0±. For the varying process, see Figure3.(3)
If(a,b)∈A2andλ=β12-4α1, thenua±=ub±equal to the hyperbolic solitary wave solutions(19)uab±=±-12c(25a2+240bc/5)cosh[2c/(γ+δ)ξ]-a,which were given by Song and Cai [6]. This implies that one extends the previous results. Whenb→-(5a2/48c)+0,
uab±tend to two trivial solutionsu=±12c/a.Four low-kink waves are bifurcated from four symmetric solitary waves. The varying process for the figures ofua± and ub± when λ>0, λ≠β12-4α1, (a,b)∈A2, and b→-(5a2/48c)+0, where λ=50, γ=δ=c=a=1, and (a) b=-(5a2/48c)+10-2, (b) b=-(5a2/48c)+10-4, (c) b=-(5a2/48c)+10-6, and (d) b=-(5a2/48c)+10-9.
(a)
(b)
(c)
(d)Four low-kink waves are bifurcated from four 1-blow-up waves. The varying process for the figures ofua± and ub± when λ>0, λ≠β12-4α1, (a,b)∈A3, and b→-(5a2/48c)-0, where λ=50, γ=δ=c=a=1, and (a) b=-(5a2/48c)-10-2, (b) b=-(5a2/48c)-10-5, (c) b=-(5a2/48c)-10-7, and (d) b=-(5a2/48c)-10-9.
(a)
(b)
(c)
(d)Proof.
In (8), letting h=H(0,0), it follows that
(20)ψ=±1γ+δ(cφ2-a6φ4-b15φ6).
Substituting (20) into dφ/dξ=ψ and integrating it, we have
(21)∫νφ‍ds(s2/(γ+δ))(c-(a/6)s2-(b/15)s4)=ξ,
where ν is an arbitrary constant.
Completing the integral above and solving the equation forφ, it follows that
(22)φ=±4α1λeτ1ξλ2e2τ1ξ-2λβ1eτ1ξ+(β12-4α1),
where λ=λ(ν) is an arbitrary real number.
Note that ifu=φ(ξ) is a solution of (4), so is u=φ(-ξ). Therefore, from (22) we obtain the solutions ua± and ub± as (13).
In (13) letting b→-5a2/48c, then β12-4α1→0, and we get (17) and (18). From (13), (17) and (18), we get results (1) and (2) of Proposition 1.
Whenλ=β12-4α1, via (13) it follows that
(23)ua±=ub±=±4α1β12-4α1(e-τ1ξ+eτ1ξ)-2β1=±2α1β12-4α1cosh(τ1ξ)-β1=uab±(see(19)),
which is result (3) of Proposition 1.


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

Proposition 2.
Ifγ+δ>0, H(φ,ψ)=H(φ1,0), and (a,b) belongs to one of the regions A2, A3, l2, and l3, then (4) has four real nonlinear wave solutions as follows:
(24)uc±=±-(5a+Δ)/6b(2Δ-5a+6bηeτ2ξ)(5a-2Δ)2+36b2η2e2τ2ξ-12b(5a+4Δ)ηeτ2ξ,ud±=±-(5a+Δ)/6b(2Δ-5a+6bηe-τ2ξ)(5a-2Δ)2+36b2η2e-2τ2ξ-12b(5a+4Δ)ηe-τ2ξ,
where η≠0 is an arbitrary real constant, Δ is given in (11), and
(25)τ2=-Δ(Δ+5a)45b(γ+δ).
Letting(26)μ0=5a-2Δ6b,
corresponding to η>0, one has the following results and bifurcation phenomena.(1)
If(a,b)∈A2andη≠|μ0|, thenuc±≠ud±,
and they represent four tall-kink waves (see Figures4(a)–4(c)). Whenb→-(5a2/48c)+0, the four tall-kink waves become(27)uc0∓=∓12cηaη+48ce-2(c/(γ+δ))ξ,(28)ud0∓=∓12cηaη+48ce2(c/(γ+δ))ξ,which represent four low-kink waves (see Figure4(d)). For the varying process, see Figure 4.(2)
If(a,b)∈A3andη≠|μ0|, thenuc±≠ud±,
and they represent four antisymmetry solitary waves (see Figures5(a)–5(c)). Whenb→-(5a2/36c)+0, the four antisymmetry solitary waves become two trivial wavesu=±6c/a. In particular, whenb→-(5a2/48c)-0, the four antisymmetry solitary waves become four low-kink waves with the expressionsuc0±andud0±(see Figure5(d)). For the varying process, see Figure 5.(3)
If(a,b)∈A2andη=|μ0|, thenuc±=ud∓=ucd∓of forms(29)ucd∓=∓(5a+Δ)(5a-2Δ)3b[(2Δ-5a)cosh(τ2ξ)+5a+4Δ]sinh(τ22ξ),which represent two tall-kink waves and tend to a trivial waveu=0whenb→-(5a2/48c)+0.(4)
If(a,b)∈A3andη=|μ0|, thenuc±=ud±=ucd*∓of forms(30)ucd*∓=∓(5a+Δ)(2Δ-5a)3b[(5a-2Δ)cosh(τ2ξ)+5a+4Δ]cosh(τ22ξ),which represent two antisymmetric solitary waves and tend to the trivial waveu=0whenb→-(5a2/48c)-0and tend tou=±12c/awhenb→-(5a2/36c)+0.Four low-kink waves are bifurcated from four tall-kink waves. The varying process for the figures ofuc± and ud± when η>0,  η≠|μ0|, (a,b)∈A2, and b→-(5a2/48c)+0, where η=10, δ=γ=c=a=1, and (a) b=-(5a2/48c)+10-2, (b) b=-(5a2/48c)+10-3, (c) b=-(5a2/48c)+10-5, and (d) b=-(5a2/48c)+10-7.
(a)
(b)
(c)
(d)Four low-kink waves are bifurcated from four antisymmetric solitary waves. The varying process for the figures ofuc± and ud± when η>0, η≠|μ0|, (a,b)∈A3, and b→-(5a2/48c)-0, where η=10, δ=γ=c=a=1, and (a) b=-(5a2/48c)-10-2, (b) b=-(5a2/48c)-10-3, (c) b=-(5a2/48c)-10-5, and (d) b=-(5a2/48c)-10-7.
(a)
(b)
(c)
(d)Proof.
In (8), letting h=H(φ1,0), it follows that
(31)ψ=±-b15(γ+δ)(φ12-φ2)2(φ2+μ0),
where φ1 and μ0 are given in (9) and (26), respectively. Substituting (31) into dφ/dξ=ψ and integrating it, we have
(32)∫pφ‍ds-(b/15(γ+δ))(φ12-φ2)2(φ2+μ0)=ξ,
where p is an arbitrary constant.
Completing the integral above and solving the equation forφ, it follows that
(33)φ=±φ12-4α2ηeτ2ξη2e2τ2ξ-2ηβ2eτ2ξ+(β22-4α2),
where τ2 is given in (25), η=η(p) is an arbitrary real number, and
(34)α2=Δ(Δ+5a)12b2,β2=4Δ+5a6b.
Similarly, if u=φ(ξ) is a solution of (4), so is u=φ(-ξ). Substituting (34) into (33), we get uc± and ud± (see (24)).
Whenb→-5a2/48c, it follows that
(35)τ2⟶2cγ+δ,5a-2Δ⟶0.
From (24), it is easy to check that uc± and ud± become uc0± and ud0± (see (27) and (28)).
If(a,b)∈A2, then 2Δ-5a>0 and η=|μ0|=(5a-2Δ)/6b, and we have
(36)uc±=±-(5a+Δ)/6b(2Δ-5a+6bηeτ2ξ)(5a-2Δ)2+36b2η2e2τ2ξ-12b(5a+4Δ)ηeτ2ξ=±-(5a+Δ)/6b(2Δ-5a)(1-eτ2ξ)(5a-2Δ)2+(5a-2Δ)2e2τ2ξ-2(5a+4Δ)(5a-2Δ)eτ2ξ=∓2-5a+Δ6b·2Δ-5a2(2Δ-5a)cosh(τ2ξ)+2(4Δ+5a)sinh(τ22ξ)=∓(5a+Δ)(5a-2Δ)3b[(2Δ-5a)cosh(τ2ξ)+5a+4Δ]sinh(τ22ξ)=ucd∓(see(29)).
Similarly, we have ud±=ucd±.
If(a,b)∈A3, then 5a-2Δ>0 and η=|μ0|=(2Δ-5a)/6b, and we have
(37)uc±=±-(5a+Δ)/6b(2Δ-5a+6bηeτ2ξ)(5a-2Δ)2+36b2η2e2τ2ξ-12b(5a+4Δ)ηeτ2ξ=±-(5a+Δ)/6b(2Δ-5a)(eτ2ξ+1)(5a-2Δ)2+(5a-2Δ)2e2τ2ξ+2(5a+4Δ)(5a-2Δ)eτ2ξ=∓-5a+Δ6b·5a-2Δ(eτ2ξ/2+e-τ2ξ/2)(5a-2Δ)(eτ2ξ/2+e-τ2ξ/2)+2(4Δ+5a)=∓2-5a+Δ6b·5a-2Δ2(5a-2Δ)cosh(τ2ξ)+2(4Δ+5a)cosh(τ22ξ)=∓(5a+Δ)(2Δ-5a)3b[(5a-2Δ)cosh(τ2ξ)+5a+4Δ]cosh(τ22ξ)=ucd*∓(see(30)).
Similarly, we have ud±=ucd*∓.
Hereto, we have completed the proof for Proposition2.


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

Proposition 1.
Forab≠0, γ+δ>0, and H(φ,ψ)=H(0,0), (4) has four nonlinear wave solutions as follows:
(13)ua±=±4α1λλ2e-τ1ξ-2λβ1+(β12-4α1)eτ1ξ,ub±=±4α1λλ2eτ1ξ-2λβ1+(β12-4α1)e-τ1ξ,
where
(14)α1=-15cb,(15)β1=5a2b,(16)τ1=2cγ+δ,
and λ≠0 is an arbitrary real constant. For λ>0, one has the following results and bifurcation phenomena.(1)
Ifλ≠β12-4α1and(a,b)∈A2, thenua±≠ub±,
and they represent four symmetric solitary waves (see Figures2(a)–2(c)). In particular, whenb→-(5a2/48c)+0, the four symmetric solitary waves become four low-kink waves (see Figure2(d))(17)ua0±=±12c(-bλ/5)e-2(c/(γ+δ))ξ+a,(18)ub0±=±12c-(bλ/5)e2(c/(γ+δ))ξ+a,which were given by Song and Cai [6]. This implies that one extends the previous results. For the varying process, see Figure 2.(2)
Ifλ≠β12-4α1and(a,b)belongs to any one of the regionsA3,l3,A4, andA5, thenua±≠ub±and they represent four 1-blow-up waves (see Figures3(a)–3(c)). In particular, when(a,b)∈A3andb→-(5a2/48c)-0, the four 1-blow-up waves become four low-kink waves with the expressionsua0±andub0±. For the varying process, see Figure3.(3)
If(a,b)∈A2andλ=β12-4α1, thenua±=ub±equal to the hyperbolic solitary wave solutions(19)uab±=±-12c(25a2+240bc/5)cosh[2c/(γ+δ)ξ]-a,which were given by Song and Cai [6]. This implies that one extends the previous results. Whenb→-(5a2/48c)+0,
uab±tend to two trivial solutionsu=±12c/a.Four low-kink waves are bifurcated from four symmetric solitary waves. The varying process for the figures ofua± and ub± when λ>0, λ≠β12-4α1, (a,b)∈A2, and b→-(5a2/48c)+0, where λ=50, γ=δ=c=a=1, and (a) b=-(5a2/48c)+10-2, (b) b=-(5a2/48c)+10-4, (c) b=-(5a2/48c)+10-6, and (d) b=-(5a2/48c)+10-9.
(a)
(b)
(c)
(d)Four low-kink waves are bifurcated from four 1-blow-up waves. The varying process for the figures ofua± and ub± when λ>0, λ≠β12-4α1, (a,b)∈A3, and b→-(5a2/48c)-0, where λ=50, γ=δ=c=a=1, and (a) b=-(5a2/48c)-10-2, (b) b=-(5a2/48c)-10-5, (c) b=-(5a2/48c)-10-7, and (d) b=-(5a2/48c)-10-9.
(a)
(b)
(c)
(d)Proof.
In (8), letting h=H(0,0), it follows that
(20)ψ=±1γ+δ(cφ2-a6φ4-b15φ6).
Substituting (20) into dφ/dξ=ψ and integrating it, we have
(21)∫νφ‍ds(s2/(γ+δ))(c-(a/6)s2-(b/15)s4)=ξ,
where ν is an arbitrary constant.
Completing the integral above and solving the equation forφ, it follows that
(22)φ=±4α1λeτ1ξλ2e2τ1ξ-2λβ1eτ1ξ+(β12-4α1),
where λ=λ(ν) is an arbitrary real number.
Note that ifu=φ(ξ) is a solution of (4), so is u=φ(-ξ). Therefore, from (22) we obtain the solutions ua± and ub± as (13).
In (13) letting b→-5a2/48c, then β12-4α1→0, and we get (17) and (18). From (13), (17) and (18), we get results (1) and (2) of Proposition 1.
Whenλ=β12-4α1, via (13) it follows that
(23)ua±=ub±=±4α1β12-4α1(e-τ1ξ+eτ1ξ)-2β1=±2α1β12-4α1cosh(τ1ξ)-β1=uab±(see(19)),
which is result (3) of Proposition 1.


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

Proposition 2.
Ifγ+δ>0, H(φ,ψ)=H(φ1,0), and (a,b) belongs to one of the regions A2, A3, l2, and l3, then (4) has four real nonlinear wave solutions as follows:
(24)uc±=±-(5a+Δ)/6b(2Δ-5a+6bηeτ2ξ)(5a-2Δ)2+36b2η2e2τ2ξ-12b(5a+4Δ)ηeτ2ξ,ud±=±-(5a+Δ)/6b(2Δ-5a+6bηe-τ2ξ)(5a-2Δ)2+36b2η2e-2τ2ξ-12b(5a+4Δ)ηe-τ2ξ,
where η≠0 is an arbitrary real constant, Δ is given in (11), and
(25)τ2=-Δ(Δ+5a)45b(γ+δ).
Letting(26)μ0=5a-2Δ6b,
corresponding to η>0, one has the following results and bifurcation phenomena.(1)
If(a,b)∈A2andη≠|μ0|, thenuc±≠ud±,
and they represent four tall-kink waves (see Figures4(a)–4(c)). Whenb→-(5a2/48c)+0, the four tall-kink waves become(27)uc0∓=∓12cηaη+48ce-2(c/(γ+δ))ξ,(28)ud0∓=∓12cηaη+48ce2(c/(γ+δ))ξ,which represent four low-kink waves (see Figure4(d)). For the varying process, see Figure 4.(2)
If(a,b)∈A3andη≠|μ0|, thenuc±≠ud±,
and they represent four antisymmetry solitary waves (see Figures5(a)–5(c)). Whenb→-(5a2/36c)+0, the four antisymmetry solitary waves become two trivial wavesu=±6c/a. In particular, whenb→-(5a2/48c)-0, the four antisymmetry solitary waves become four low-kink waves with the expressionsuc0±andud0±(see Figure5(d)). For the varying process, see Figure 5.(3)
If(a,b)∈A2andη=|μ0|, thenuc±=ud∓=ucd∓of forms(29)ucd∓=∓(5a+Δ)(5a-2Δ)3b[(2Δ-5a)cosh(τ2ξ)+5a+4Δ]sinh(τ22ξ),which represent two tall-kink waves and tend to a trivial waveu=0whenb→-(5a2/48c)+0.(4)
If(a,b)∈A3andη=|μ0|, thenuc±=ud±=ucd*∓of forms(30)ucd*∓=∓(5a+Δ)(2Δ-5a)3b[(5a-2Δ)cosh(τ2ξ)+5a+4Δ]cosh(τ22ξ),which represent two antisymmetric solitary waves and tend to the trivial waveu=0whenb→-(5a2/48c)-0and tend tou=±12c/awhenb→-(5a2/36c)+0.Four low-kink waves are bifurcated from four tall-kink waves. The varying process for the figures ofuc± and ud± when η>0,  η≠|μ0|, (a,b)∈A2, and b→-(5a2/48c)+0, where η=10, δ=γ=c=a=1, and (a) b=-(5a2/48c)+10-2, (b) b=-(5a2/48c)+10-3, (c) b=-(5a2/48c)+10-5, and (d) b=-(5a2/48c)+10-7.
(a)
(b)
(c)
(d)Four low-kink waves are bifurcated from four antisymmetric solitary waves. The varying process for the figures ofuc± and ud± when η>0, η≠|μ0|, (a,b)∈A3, and b→-(5a2/48c)-0, where η=10, δ=γ=c=a=1, and (a) b=-(5a2/48c)-10-2, (b) b=-(5a2/48c)-10-3, (c) b=-(5a2/48c)-10-5, and (d) b=-(5a2/48c)-10-7.
(a)
(b)
(c)
(d)Proof.
In (8), letting h=H(φ1,0), it follows that
(31)ψ=±-b15(γ+δ)(φ12-φ2)2(φ2+μ0),
where φ1 and μ0 are given in (9) and (26), respectively. Substituting (31) into dφ/dξ=ψ and integrating it, we have
(32)∫pφ‍ds-(b/15(γ+δ))(φ12-φ2)2(φ2+μ0)=ξ,
where p is an arbitrary constant.
Completing the integral above and solving the equation forφ, it follows that
(33)φ=±φ12-4α2ηeτ2ξη2e2τ2ξ-2ηβ2eτ2ξ+(β22-4α2),
where τ2 is given in (25), η=η(p) is an arbitrary real number, and
(34)α2=Δ(Δ+5a)12b2,β2=4Δ+5a6b.
Similarly, if u=φ(ξ) is a solution of (4), so is u=φ(-ξ). Substituting (34) into (33), we get uc± and ud± (see (24)).
Whenb→-5a2/48c, it follows that
(35)τ2⟶2cγ+δ,5a-2Δ⟶0.
From (24), it is easy to check that uc± and ud± become uc0± and ud0± (see (27) and (28)).
If(a,b)∈A2, then 2Δ-5a>0 and η=|μ0|=(5a-2Δ)/6b, and we have
(36)uc±=±-(5a+Δ)/6b(2Δ-5a+6bηeτ2ξ)(5a-2Δ)2+36b2η2e2τ2ξ-12b(5a+4Δ)ηeτ2ξ=±-(5a+Δ)/6b(2Δ-5a)(1-eτ2ξ)(5a-2Δ)2+(5a-2Δ)2e2τ2ξ-2(5a+4Δ)(5a-2Δ)eτ2ξ=∓2-5a+Δ6b·2Δ-5a2(2Δ-5a)cosh(τ2ξ)+2(4Δ+5a)sinh(τ22ξ)=∓(5a+Δ)(5a-2Δ)3b[(2Δ-5a)cosh(τ2ξ)+5a+4Δ]sinh(τ22ξ)=ucd∓(see(29)).
Similarly, we have ud±=ucd±.
If(a,b)∈A3, then 5a-2Δ>0 and η=|μ0|=(2Δ-5a)/6b, and we have
(37)uc±=±-(5a+Δ)/6b(2Δ-5a+6bηeτ2ξ)(5a-2Δ)2+36b2η2e2τ2ξ-12b(5a+4Δ)ηeτ2ξ=±-(5a+Δ)/6b(2Δ-5a)(eτ2ξ+1)(5a-2Δ)2+(5a-2Δ)2e2τ2ξ+2(5a+4Δ)(5a-2Δ)eτ2ξ=∓-5a+Δ6b·5a-2Δ(eτ2ξ/2+e-τ2ξ/2)(5a-2Δ)(eτ2ξ/2+e-τ2ξ/2)+2(4Δ+5a)=∓2-5a+Δ6b·5a-2Δ2(5a-2Δ)cosh(τ2ξ)+2(4Δ+5a)cosh(τ22ξ)=∓(5a+Δ)(2Δ-5a)3b[(5a-2Δ)cosh(τ2ξ)+5a+4Δ]cosh(τ22ξ)=ucd*∓(see(30)).
Similarly, we have ud±=ucd*∓.
Hereto, we have completed the proof for Proposition2.


## 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 λ<0, one has the following results and bifurcation phenomena.(1)
If(a,b)∈A2andλ≠-β12-4α1, thenua±≠ub±,
and they represent four 2-blow-up waves. Whenb→-(5a2/48c)+0,ua±,
andub±,
respectively, becomeua0±andub0±(see (17) and (18)) which represent four 1-blow-up waves (see Figure 6(d)). For the varying process, see Figure 6.(2)
Ifλ=-β12-4α1, thenua±=ub±and become(38)uab*±=±12c(25a2+240bc/5)cosh[2c/(γ+δ)ξ]+a,which were given by Song and Cai [6]. This implies that one extends the previous results.
When(a,b)∈A2, uab*± represent hyperbolic blow-up waves. Specially, when b→-(5a2/48c)+0, uab*± tend to two trivial solutions u=±12c/a.
When(a,b) belongs to any one of the regions A1, A6, uab*± represent two symmetric solitary waves. In particular, when (a,b)∈A6 and b→0+0,  uab*± become two 1-blow-up waves. For the varying process, see Figure 7.Four 2-blow-up waves become four 1-blow-up waves. The varying process for the figures ofua± and ub± when λ<0, λ≠-β12-4α1, (a,b)∈A2, and b→-(5a2/48c)+0, where λ=-50, γ=δ=c=a=1, and (a) b=-(5a2/48c)+10-2, (b) b=-(5a2/48c)+10-4, (c) b=-(5a2/48c)+10-5, and (d) b=-(5a2/48c)+10-7.
(a)
(b)
(c)
(d)Two 1-blow-up waves are bifurcated from two symmetric solitary waves. The varying process for the figures ofuab*± when λ=-β12-4α1, (a,b)∈A6, and b→0+0, where λ=a=-1, γ=δ=c=1, and (a) b=0+5×10-1, (b) b=0+10-2, (c) b=0+10-4, and (d) b=0+10-7.
(a)
(b)
(c)
(d)Similar to the proof of Proposition1, we get the results of Proposition 3.


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

Proposition 4.
Underγ+δ>0 and H(φ,ψ)=H(φ2,0), one has the following results and bifurcation phenomena.(1)
If(a,b)belongs to one of the regionsA2,A3,
andl2, then (4) has two periodic-blow-up wave solutions(39)ue±=±(5a-Δ)(5a+2Δ)3b[(5a+2Δ)cos(τ3ξ)+4Δ-5a]sin(τ32ξ),where(40)τ3=Δ(Δ-5a)45b(γ+δ).(2)
If(a,b)∈A3andb→-(5a2/36c)+0, the periodic-blow-up wave solutionsue±become two fractional wave solutions(41)ue0±=±6cξa[cξ2-3(γ+δ)],which represent two 1-blow-up waves (see Figure8(d)). For the varying process, see Figure 8.The 1-blow-up waves are bifurcated from the periodic-blow-up waves. The varying process for the figures ofue± when b→-(5a2/36c)+0, where γ=δ=c=a=1 and (a) b=-(5a2/36c)+10-1, (b) b=-(5a2/36c)+10-2, (c) b=-(5a2/36c)+10-3, and (d) b=-(5a2/36c)+10-4.
(a)
(b)
(c)
(d)Proof.
(1) In (8), letting h=H(φ2,0), it follows that
(42)ψ=±-b15(γ+δ)(φ2-φ22)2(φ2+μ1),
where φ2 is given in (10), and
(43)μ1=5a+2Δ6b.
Substituting (42) into dφ/dξ=ψ and integrating it, we have
(44)∫qφ‍ds-(b/15(γ+δ))(φ2-φ22)2(φ2+μ1)=ξ,
where q is an arbitrary constant.
Completing the integral in (44) and solving the equation for φ, it follows that
(45)φ=±φ22-2α3μ1sin(τ3ξ+θ1)+β3,
where τ3 is given in (40), θ1=θ1(q) is an arbitrary constant, and
(46)α3=Δ(Δ-5a)12b2,β3=4Δ-5a6b.
In (45) letting  θ1=π/2, we obtain the solutions ue± as (39).
(2) Note that(47)cos(τ3ξ)=1-τ32ξ22+τ34ξ44!+⋯=1-Δ(Δ+5a)ξ290b(γ+δ)+O(Δ2),(48)limb→-(5a2/36c)+0Δ=limb→-(5a2/36c)+025a2+180bc=0.
Thus, we have(49)ue±=±(5a+Δ)(5a-2Δ)[1-cos(τ3ξ)]6b[(5a-2Δ)cos(τ3ξ)-5a-4Δ]=±(1/90b(γ+δ))(5a+Δ)(5a-2Δ)Δ(Δ+5a)ξ2+O(Δ2)6b[(5a-2Δ)(1-(Δ(5a+Δ)ξ2)/90b(γ+δ))-5a-4Δ+O(Δ2)]=±(5a+Δ)2(5a-2Δ)ξ2+O(Δ)6b[-540b(γ+δ)-(5a-2Δ)(5a+Δ)ξ2+O(Δ)].
Furthermore, we get(50)limb→-(5a2/36c)+0ue±=limb→-(5a2/36c)+0±(5a+Δ)2(5a-2Δ)ξ2+O(Δ)6b[-540b(γ+δ)-(5a-2Δ)(5a+Δ)ξ2+O(Δ)]=±25a2×5aξ26×(-5a2/36c)[-540×(-5a2/36c)(γ+δ)-25a2ξ2]=±6cξa[cξ2-3(γ+δ)]=ue0±(see(41)).Hereto, we have completed the proof for Proposition 4.


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

Proposition 3.
In (13), corresponding to λ<0, one has the following results and bifurcation phenomena.(1)
If(a,b)∈A2andλ≠-β12-4α1, thenua±≠ub±,
and they represent four 2-blow-up waves. Whenb→-(5a2/48c)+0,ua±,
andub±,
respectively, becomeua0±andub0±(see (17) and (18)) which represent four 1-blow-up waves (see Figure 6(d)). For the varying process, see Figure 6.(2)
Ifλ=-β12-4α1, thenua±=ub±and become(38)uab*±=±12c(25a2+240bc/5)cosh[2c/(γ+δ)ξ]+a,which were given by Song and Cai [6]. This implies that one extends the previous results.
When(a,b)∈A2, uab*± represent hyperbolic blow-up waves. Specially, when b→-(5a2/48c)+0, uab*± tend to two trivial solutions u=±12c/a.
When(a,b) belongs to any one of the regions A1, A6, uab*± represent two symmetric solitary waves. In particular, when (a,b)∈A6 and b→0+0,  uab*± become two 1-blow-up waves. For the varying process, see Figure 7.Four 2-blow-up waves become four 1-blow-up waves. The varying process for the figures ofua± and ub± when λ<0, λ≠-β12-4α1, (a,b)∈A2, and b→-(5a2/48c)+0, where λ=-50, γ=δ=c=a=1, and (a) b=-(5a2/48c)+10-2, (b) b=-(5a2/48c)+10-4, (c) b=-(5a2/48c)+10-5, and (d) b=-(5a2/48c)+10-7.
(a)
(b)
(c)
(d)Two 1-blow-up waves are bifurcated from two symmetric solitary waves. The varying process for the figures ofuab*± when λ=-β12-4α1, (a,b)∈A6, and b→0+0, where λ=a=-1, γ=δ=c=1, and (a) b=0+5×10-1, (b) b=0+10-2, (c) b=0+10-4, and (d) b=0+10-7.
(a)
(b)
(c)
(d)Similar to the proof of Proposition1, we get the results of Proposition 3.


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

Proposition 4.
Underγ+δ>0 and H(φ,ψ)=H(φ2,0), one has the following results and bifurcation phenomena.(1)
If(a,b)belongs to one of the regionsA2,A3,
andl2, then (4) has two periodic-blow-up wave solutions(39)ue±=±(5a-Δ)(5a+2Δ)3b[(5a+2Δ)cos(τ3ξ)+4Δ-5a]sin(τ32ξ),where(40)τ3=Δ(Δ-5a)45b(γ+δ).(2)
If(a,b)∈A3andb→-(5a2/36c)+0, the periodic-blow-up wave solutionsue±become two fractional wave solutions(41)ue0±=±6cξa[cξ2-3(γ+δ)],which represent two 1-blow-up waves (see Figure8(d)). For the varying process, see Figure 8.The 1-blow-up waves are bifurcated from the periodic-blow-up waves. The varying process for the figures ofue± when b→-(5a2/36c)+0, where γ=δ=c=a=1 and (a) b=-(5a2/36c)+10-1, (b) b=-(5a2/36c)+10-2, (c) b=-(5a2/36c)+10-3, and (d) b=-(5a2/36c)+10-4.
(a)
(b)
(c)
(d)Proof.
(1) In (8), letting h=H(φ2,0), it follows that
(42)ψ=±-b15(γ+δ)(φ2-φ22)2(φ2+μ1),
where φ2 is given in (10), and
(43)μ1=5a+2Δ6b.
Substituting (42) into dφ/dξ=ψ and integrating it, we have
(44)∫qφ‍ds-(b/15(γ+δ))(φ2-φ22)2(φ2+μ1)=ξ,
where q is an arbitrary constant.
Completing the integral in (44) and solving the equation for φ, it follows that
(45)φ=±φ22-2α3μ1sin(τ3ξ+θ1)+β3,
where τ3 is given in (40), θ1=θ1(q) is an arbitrary constant, and
(46)α3=Δ(Δ-5a)12b2,β3=4Δ-5a6b.
In (45) letting  θ1=π/2, we obtain the solutions ue± as (39).
(2) Note that(47)cos(τ3ξ)=1-τ32ξ22+τ34ξ44!+⋯=1-Δ(Δ+5a)ξ290b(γ+δ)+O(Δ2),(48)limb→-(5a2/36c)+0Δ=limb→-(5a2/36c)+025a2+180bc=0.
Thus, we have(49)ue±=±(5a+Δ)(5a-2Δ)[1-cos(τ3ξ)]6b[(5a-2Δ)cos(τ3ξ)-5a-4Δ]=±(1/90b(γ+δ))(5a+Δ)(5a-2Δ)Δ(Δ+5a)ξ2+O(Δ2)6b[(5a-2Δ)(1-(Δ(5a+Δ)ξ2)/90b(γ+δ))-5a-4Δ+O(Δ2)]=±(5a+Δ)2(5a-2Δ)ξ2+O(Δ)6b[-540b(γ+δ)-(5a-2Δ)(5a+Δ)ξ2+O(Δ)].
Furthermore, we get(50)limb→-(5a2/36c)+0ue±=limb→-(5a2/36c)+0±(5a+Δ)2(5a-2Δ)ξ2+O(Δ)6b[-540b(γ+δ)-(5a-2Δ)(5a+Δ)ξ2+O(Δ)]=±25a2×5aξ26×(-5a2/36c)[-540×(-5a2/36c)(γ+δ)-25a2ξ2]=±6cξa[cξ2-3(γ+δ)]=ue0±(see(41)).Hereto, 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.
Ifab≠0, γ+δ<0, and H(φ,ψ)=H(0,0), (4) has two nonlinear wave solutions
(51)uf±=±-2α1β12-4α1cos(τ4ξ)+β1,
where
(52)τ4=2-cγ+δ.
One has the following results and bifurcation phenomena.(1)
If(a,b)belongs to any one of the regionsA1andA6, thenuf±represent periodic-blow-up waves.(2)
If(a,b)∈A2, thenuf±represent periodic waves. In particular, whenb→0-0, the periodic waves become periodic-blow-up waves as follows:(53)uf0±=±12ca[1-cos(τ4ξ)].For the varying process, see Figure9.
When
b
→
-
(
5
a
2
/
48
c
)
+
0
, the periodic wave tends to two trivial waves
u
=
±
12
c
/
a
. For the varying process, see Figure10.The periodic-blow-up waves are bifurcated from the symmetric periodic waves. The varying process for the figures ofuf± when (a,b)∈A2 and b→0-0, where c=a=1, γ=δ=-1, and (a) b=0-10-1, (b) b=0-10-2, (c) b=0-10-4, and (d) b=0-10-7.
(a)
(b)
(c)
(d)The periodic waves become the trivial waves. The varying process for the figures ofuf± when (a,b)∈A2 and b→-(5a2/48c)+0, where c=a=1, γ=δ=-1, and (a) b=-(5a2/48c)+10-1, (b) b=-(5a2/48c)+10-2, (c) b=-(5a2/48c)+10-3, and (d) b=-(5a2/48c)+10-7.
(a)
(b)
(c)
(d)Proof.
Completing the integral in (21) and solving the equation for φ, it follows that
(54)φ=±-2α1β12-4α1sin(τ4ξ+θ2)+β1,
where τ4 is given in (52) and θ2=θ2(ν) is an arbitrary constant.
In (54) letting θ2=π/2, we obtain the solutions uf± as (51).
From (14) and (15), we have
(55)β12-4α1=240bc+25a24b2=-240bc+25a22b.
Lettingb→0-0, then
(56)limb→0-0uf±=limb→0-0±-2α1β12-4α1cos(τ4ξ)+β2=limb→0-0±60c-240bc+25a2cos(τ4ξ)+5a=±12ca[1-cos(τ4ξ)]=uf0±(see(53)).
Hereto, we have completed the proof for Proposition5.


## 4.1. Bifurcation from Periodic Waves

Proposition 5.
Ifab≠0, γ+δ<0, and H(φ,ψ)=H(0,0), (4) has two nonlinear wave solutions
(51)uf±=±-2α1β12-4α1cos(τ4ξ)+β1,
where
(52)τ4=2-cγ+δ.
One has the following results and bifurcation phenomena.(1)
If(a,b)belongs to any one of the regionsA1andA6, thenuf±represent periodic-blow-up waves.(2)
If(a,b)∈A2, thenuf±represent periodic waves. In particular, whenb→0-0, the periodic waves become periodic-blow-up waves as follows:(53)uf0±=±12ca[1-cos(τ4ξ)].For the varying process, see Figure9.
When
b
→
-
(
5
a
2
/
48
c
)
+
0
, the periodic wave tends to two trivial waves
u
=
±
12
c
/
a
. For the varying process, see Figure10.The periodic-blow-up waves are bifurcated from the symmetric periodic waves. The varying process for the figures ofuf± when (a,b)∈A2 and b→0-0, where c=a=1, γ=δ=-1, and (a) b=0-10-1, (b) b=0-10-2, (c) b=0-10-4, and (d) b=0-10-7.
(a)
(b)
(c)
(d)The periodic waves become the trivial waves. The varying process for the figures ofuf± when (a,b)∈A2 and b→-(5a2/48c)+0, where c=a=1, γ=δ=-1, and (a) b=-(5a2/48c)+10-1, (b) b=-(5a2/48c)+10-2, (c) b=-(5a2/48c)+10-3, and (d) b=-(5a2/48c)+10-7.
(a)
(b)
(c)
(d)Proof.
Completing the integral in (21) and solving the equation for φ, it follows that
(54)φ=±-2α1β12-4α1sin(τ4ξ+θ2)+β1,
where τ4 is given in (52) and θ2=θ2(ν) is an arbitrary constant.
In (54) letting θ2=π/2, we obtain the solutions uf± as (51).
From (14) and (15), we have
(55)β12-4α1=240bc+25a24b2=-240bc+25a22b.
Lettingb→0-0, then
(56)limb→0-0uf±=limb→0-0±-2α1β12-4α1cos(τ4ξ)+β2=limb→0-0±60c-240bc+25a2cos(τ4ξ)+5a=±12ca[1-cos(τ4ξ)]=uf0±(see(53)).
Hereto, we have completed the proof for Proposition5.


## 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.
Underγ+δ<0 and H(φ,ψ)=H(φ1,0), one has the following results and bifurcation phenomena.(1)
If(a,b)belongs to the regionA3, then (4) has two periodic wave solutions(57)ug±=(5a+Δ)(5a-2Δ)3b[(5a-2Δ)cos(τ5ξ)-5a-4Δ]sin(τ52ξ),where(58)τ5=Δ(Δ+5a)45b(γ+δ).(2)
If(a,b)∈A3andb→-(5a2/36c)+0, the periodic wave solutionsug±tend to two fractional wave solutionsug0±which have the expressions asue0±(see (41)) and represent two tall-kink waves (see Figure 11(d)). For the varying process, see Figure 11.Two tall-kink waves are bifurcated form the symmetric periodic waves. The varying process for the figures ofug± when (a,b)∈A3 and b→-(5a2/36c)+0, where c=a=1, δ=γ=-1, and (a) b=-(5a2/36c)+10-2, (b) b=-(5a2/36c)+10-3, (c) b=-(5a2/36c)+10-4, and (d) b=-(5a2/36c)+10-7.
(a)
(b)
(c)
(d)Proof.
Completing the integral in (31) and solving the equation for φ, it follows that
(59)φ=±φ12-2α2μ0sin(τ5ξ+θ3)-β2,
where τ5 is given in (58) and θ3=θ3(p) is an arbitrary constant.
In (59) letting θ3=π/2, we obtain the solutions ug± as (57).
Similar to the proof of Proposition4, we get the results of Proposition 6.
Besides these bifurcation phenomena above, there is another bifurcation phenomenon as follows.Proposition 7.
Ifγ+δ<0 and (a,b) belongs to one of the regions A2, A3, and l2, then (4) has four symmetric solitary wave solutions (see Figures 12(a)–12(c)) as follows:
(60)uh±=±(Δ-5a)/6b(6bωeτ6ξ-5a-2Δ)36b2ω2e2τ6ξ-12b(5a-4Δ)ωeτ6ξ+(5a+2Δ)2,ui±=±(Δ-5a)/6b(6bωe-τ6ξ-5a-2Δ)36b2ω2e-2τ6ξ-12b(5a-4Δ)ωe-τ6ξ+(5a+2Δ)2,
where
(61)τ6=Δ(5a-Δ)45b(γ+δ).
In particular, when (a,b)∈A3 and b→-(5a2/36c)+0, uh± and ui± tend to two trivial solutions u=±12c/a. For the varying process, see Figure 12.Four symmetric solitary waves become two trivial waves. The varying process for the figures ofuh± and ui± when b→-(5a2/36c)+0, where c=a=ω=1, γ=δ=-1, and (a) b=-(5a2/36c)+10-2, (b) b=-(5a2/36c)+10-3, (c) b=-(5a2/36c)+10-4, and (d) b=-(5a2/36c)+10-6.
(a)
(b)
(c)
(d)Proof.
Completing the integral in (44) and solving the equation for φ, it follows that
(62)φ=φ22-4α3ωeτ6ξω2e2τ6ξ+2ωβ3eτ6ξ+(β32-4α3),
where α3, β3, and τ6 are given in (46) and (61) and ω=ω(q) is an arbitrary constant.
Similar to the derivations foruc± and ud±, we get uh± and ui± (see (60)) from (62).
Hereto, we have completed the proofs for all propositions.


## 5.1. Bifurcation from Symmetric Periodic Waves

Proposition 6.
Underγ+δ<0 and H(φ,ψ)=H(φ1,0), one has the following results and bifurcation phenomena.(1)
If(a,b)belongs to the regionA3, then (4) has two periodic wave solutions(57)ug±=(5a+Δ)(5a-2Δ)3b[(5a-2Δ)cos(τ5ξ)-5a-4Δ]sin(τ52ξ),where(58)τ5=Δ(Δ+5a)45b(γ+δ).(2)
If(a,b)∈A3andb→-(5a2/36c)+0, the periodic wave solutionsug±tend to two fractional wave solutionsug0±which have the expressions asue0±(see (41)) and represent two tall-kink waves (see Figure 11(d)). For the varying process, see Figure 11.Two tall-kink waves are bifurcated form the symmetric periodic waves. The varying process for the figures ofug± when (a,b)∈A3 and b→-(5a2/36c)+0, where c=a=1, δ=γ=-1, and (a) b=-(5a2/36c)+10-2, (b) b=-(5a2/36c)+10-3, (c) b=-(5a2/36c)+10-4, and (d) b=-(5a2/36c)+10-7.
(a)
(b)
(c)
(d)Proof.
Completing the integral in (31) and solving the equation for φ, it follows that
(59)φ=±φ12-2α2μ0sin(τ5ξ+θ3)-β2,
where τ5 is given in (58) and θ3=θ3(p) is an arbitrary constant.
In (59) letting θ3=π/2, we obtain the solutions ug± as (57).
Similar to the proof of Proposition4, we get the results of Proposition 6.
Besides these bifurcation phenomena above, there is another bifurcation phenomenon as follows.Proposition 7.
Ifγ+δ<0 and (a,b) belongs to one of the regions A2, A3, and l2, then (4) has four symmetric solitary wave solutions (see Figures 12(a)–12(c)) as follows:
(60)uh±=±(Δ-5a)/6b(6bωeτ6ξ-5a-2Δ)36b2ω2e2τ6ξ-12b(5a-4Δ)ωeτ6ξ+(5a+2Δ)2,ui±=±(Δ-5a)/6b(6bωe-τ6ξ-5a-2Δ)36b2ω2e-2τ6ξ-12b(5a-4Δ)ωe-τ6ξ+(5a+2Δ)2,
where
(61)τ6=Δ(5a-Δ)45b(γ+δ).
In particular, when (a,b)∈A3 and b→-(5a2/36c)+0, uh± and ui± tend to two trivial solutions u=±12c/a. For the varying process, see Figure 12.Four symmetric solitary waves become two trivial waves. The varying process for the figures ofuh± and ui± when b→-(5a2/36c)+0, where c=a=ω=1, γ=δ=-1, and (a) b=-(5a2/36c)+10-2, (b) b=-(5a2/36c)+10-3, (c) b=-(5a2/36c)+10-4, and (d) b=-(5a2/36c)+10-6.
(a)
(b)
(c)
(d)Proof.
Completing the integral in (44) and solving the equation for φ, it follows that
(62)φ=φ22-4α3ωeτ6ξω2e2τ6ξ+2ωβ3eτ6ξ+(β32-4α3),
where α3, β3, and τ6 are given in (46) and (61) and ω=ω(q) is an arbitrary constant.
Similar to the derivations foruc± and ud±, we get uh± and ui± (see (60)) from (62).
Hereto, we have completed the proofs for all propositions.


## 6. Conclusion

In this paper, we have studied the bifurcation behavior of the nonlinear waves in a generalized Z-K equation. Firstly, we obtained three types of explicit nonlinear wave solutions. The first type is the exp-function expressionsua±, ub±, uc±, ud±, uh±, and ui± (see (13), (24), and (60)). The second type is the trigonometric expressions ue±, uf±, and ug± (see (39), (51), and (57)). The third type is the fractional expressions ue0± (see (41)). Furthermore, four kinds of interesting bifurcation phenomena have been revealed. The first kind is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves (see Propositions 1 and 2). The second kind is 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 (see Propositions 3 and 4). The third kind is that the periodic-blow-up waves can be bifurcated from the symmetric periodic waves (see Proposition 5). The fourth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves (see Proposition 6). Some previous results are our some special cases (see (17), (19), and (38)).



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