# Some Explicit Expressions and Interesting Bifurcation Phenomena for Nonlinear Waves in Generalized Zakharov Equations

Authors: Shaoyong Li; Rui Liu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Using bifurcation method of dynamical systems, we investigate the nonlinear waves for the generalized Zakharov equationsutt-cs2uxx=β(|E|2)xx,iEt+αExx-δ1uE+δ2|E|2E+δ3|E|4E=0, where α,β,δ1,δ2,δ3, and cs are real parameters, E=E(x,t) is a complex function, and u=u(x,t) is a real function. We obtain the following results. (i) Three types of explicit expressions of nonlinear waves are obtained, that is, the fractional expressions, the trigonometric expressions, and the exp-function expressions. (ii) Under different parameter conditions, these expressions represent symmetric and antisymmetric solitary waves, kink and antikink waves, symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blow-up waves. We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) Five kinds of interesting bifurcation phenomena are revealed. The first kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up and 2-blow-up waves. The second kind is that the 2-blow-up waves can be bifurcated from the periodic-blow-up waves. The third kind is that the symmetric solitary waves can be bifurcated from the symmetric periodic waves. The fourth 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. The fifth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves. We also show that the exp-function expressions include some results given by pioneers.


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## Body

## 1. Introduction

Since the exact solutions to nonlinear wave equations help to understand the characteristics of nonlinear equations, seeking exact solutions of nonlinear equations is an important subject. For this purpose, there have been many methods such as the Jacobi elliptic function method [1, 2], F-expansion method [3, 4], and (G′/G)-expansion method [5, 6].Recently, the bifurcation method of dynamical systems [7–9] has been introduced to study the nonlinear partial differential equations. Up to now, the method is widely used in literatures such as [10–16].In this paper, we consider the generalized Zakharov equations [17], which read as
(1)utt-cs2uxx=β(|E|2)xx,iEt+αExx-δ1uE+δ2|E|2E+δ3|E|4E=0,
where α,β,δ1,δ2,δ3, and cs are real parameters. E=E(x,t) is a complex function which represents the envelop of the electric field, and u=u(x,t) is a real function which represents the plasma density measured from its equilibrium value. Huang and Zhang [17] used Fan's direct algebraic method to obtain some exact travelling wave solutions of (1) as follows:
(2)ua0±=βc2-cs2(φa0±)2,Ea0±=φa0±ei(γx-ωt),ub0±=βc2-cs2(φb0±)2,Eb0±=φb0±ei(γx-ωt),uc0±=βc2-cs2(φc0±)2,Ec0±=φc0±ei(γx-ωt),ud0±=βc2-cs2(φd0±)2,Ed0±=φd0±ei(γx-ωt),ue0±=βc2-cs2(φe0±)2,Ee0±=φe0±ei(γx-ωt),
where γandw are two constants and
(3)φa0±=±-12p3q-9q2-48prcos(2-pξ),(4)φb0±=±2pq(-1+tanh(pξ)),(5)φc0±=±2pq(-1-tanh(pξ)),(6)φd0±=±12p-3q+9q2-48prcosh(2pξ),(7)φe0±=±2pq(-1-coth(pξ)),(8)ξ=x-ct,(9)p=αγ2-ωα,(10)q=βδ1α(c2-cs2)-δ2α,(11)r=-δ3α,(12)c=2αγ.
When α=1,β=-1,δ1=-2,δ2=2λ,δ3=0,andcs=1, (1) reduce to the equations
(13)utt-uxx+(|E|2)xx=0,iEt+Exx+2uE+2λ|E|2E=0.
El-Wakil et al. [18] used the extended Jacobi elliptic function expansion method to obtain some Jacobi elliptic function expression solutions of (13).Whenλ=0, (13) reduce to the equations
(14)utt-uxx+(|E|2)xx=0,iEt+Exx+2uE=0.
By multisymplectic numerical method, Wang [19] proved the preservation of discrete normal conservation law of (14) theoretically and investigated the propagation and collision behaviors of the solitary waves numerically. There are also many other researchers studying (1) or its special case; for more information, one can see [20–24].In this paper, we investigate the nonlinear waves and the bifurcation phenomena of (1). Coincidentally, under some transformations, (1) reduce to a planar system (54) which is similar to the planar system obtained by Feng and Li [16]. Many exact explicit parametric representations of solitary waves, kink and antikink waves, and periodic waves were obtained in [16], and their work is very important for the ϕ6 model. In order to find the travelling wave solutions of (1), here we consider (1) by using the bifurcation method mentioned above; firstly, we obtain three types of explicit nonlinear wave solutions, this is, the fractional expressions, the trigonometric expressions, and the exp-function expressions. Secondly, we point out that these expressions represent symmetric and antisymmetric solitary waves, kink and antikink waves, symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blow-up waves under different parameter conditions. Thirdly, we reveal five kinds of interesting bifurcation phenomena mentioned in the abstract above.The remainder of this paper is organized as follows. In Section2, we give some notations and state our main results. In Section 3, we give derivations for our results. A brief conclusion is given in Section 4.


## 2. Main Results

In this section, we state our main results. To relate conveniently, let us give some notations which will be used in the latter statement and the derivations.Letli(i=1,2,…,7) represent the following seven curves:
(15)l1:p=q24r(r>0,q<0),(16)l2:p=3q216r(r>0,q<0),(17)l3:p=0(r>0,q<0),(18)l4:p=0(r>0,q>0),(19)l5:p=0(r<0,q>0),(20)l6:p=3q216r(r<0,q>0),(21)l7:p=q24r(r<0,q<0).LetAi(i=1,2,…,12) represent the regions surrounded by the curves li(i=1,2,…,7) and the coordinate axes (see Figure 1).The locations of the regionsAi(i=1,2,…,12) and curves li(i=1,2,…,7).
(a)
r
>
0
(b)
r
<
0Let(22)u(x,t)=βc2-cs2φ2(ξ),(23)E(x,t)=φ(ξ)ei(γx-ωt),(24)Δ=q2-4pr,(25)φ1=-q+Δ2r,(26)φ2=-q-Δ2r,(27)H(φ,y)=h,
where H(φ,y) is the first integral which will be given later and h is the integral constant.In order to search for the solutions of (1) and studing the bifurcation phenomena, we only need to get the solution φ(ξ) according to (22) and (23). For convenience, throughout the following work we only discuss the solution φ(ξ). Now let us state our main results in the following Propositions 1, 2, and 3.


### 2.1. When the OrbitΓ Is Defined by H(φ,y)=H(0,0)

Proposition 1.
(i) Forp=0, (1) have two fractional nonlinear wave solutions
(28)ua±=βc2-cs2(φa±)2,Ea±=φa±ei(γx-ωt),
where
(29)φa±=±6q3q2ξ2-4r.
If (q,p)∈l4, then φa± are 1-blow-up waves (refer to Figure 2(d)). If (q,p)∈l3, then φa± are 2-blow-up waves (refer to Figure 3(d)). If (q,p)∈l5, then φa± are symmetric solitary waves (refer to Figure 4(d)).
(ii) Forp<0, (1) have two nonlinear wave solutions
(30)ub±=βc2-cs2(φb±)2,Eb±=φb±ei(γx-ωt),
where
(31)φb±=±-12p3q+9q2-48prcos(2-pξ).
These solutions have the following properties and wave shapes.
(1) When(q,p)∈A1 or A2, then φa0± and φb± are periodic-blow-up waves (refer to Figure 2(a) or Figure 3(a)). Specially, in region A1 when p→0-0, the periodic-blow-up waves φa0± become 1-blow-up waves φa±, and for the varying process, see Figure 2, while the periodic-blow-up waves φb± become a trivial wave φ=0. In region A2 when p→0-0, the periodic-blow-up waves φb± become 2-blow-up waves φa±, and for the varying process, see Figure 3.
(2) When(q,p)∈A12, then φa0± and φb± are symmetric periodic waves (refer to Figure 5(a)). If p→0-0, the symmetric periodic waves φa0± become symmetric solitary waves φa±, and for the varying process, see Figure 4. If p→3q2/16r+0, the symmetric periodic waves φa0± and φb± become two trivial waves φ=±-(3q/4r), and for the varying process, see Figure 5.
(iii) Forp>0, (1) have four nonlinear wave solutions
(32)uc±=βc2-cs2(φc±)2,Ec±=φc±ei(γx-ωt),ud±=βc2-cs2(φd±)2,Ed±=φd±ei(γx-ωt),
where
(33)φc±=±4λpλ2e2pξ+(λ0/12)e-2pξ-λq,φd±=±4λpλ2e-2pξ+(λ0/12)e2pξ-λq,λ is a nonzero arbitrary real constant and λ0=3q2-16pr. Corresponding to λ>0 or λ<0, these solutions have the following properties and wave shapes.
(1) For the case ofλ>0, there are four properties as follows.(1)a
When(q,p)∈l2, that is,r>0,q<0, andr=3q2/16p, thenφc±andφd±become(34)φc1±=±4pλe2pξ-q,φd1±=±4pλe-2pξ-q,which represent four low-kink waves (refer to Figure6(d) or Figure 7(d)). Specially, letλ=-q>0, thenφc1±andφd1±becomeφb0±andφc0±.(1)b
If(q,p)belongs to one ofA3,A7,A8, andλ≠λ0/12, thenφc±≠φd±, and they represent four symmetric solitary waves (refer to Figure6(a)). Specially, when(q,p)∈A3andp→3q2/16r-0, then the four symmetric solitary wavesφc±andφd±become the four low-kink wavesφc1±andφd1±, and for the varying process, see Figure6.(1)c
If(q,p)belongs to one ofA4,A5,A6,l1, andλ≠λ0/12, thenφc±≠φd±, and they represent four 1-blow-up waves (refer to Figure7(a)). Specially, when(q,p)∈A4andp→3q2/16r+0, then the four 1-blow-up wavesφc±andφd±become the four low-kink wavesφc1±andφd1±, and for the varying process, see Figure7.(1)d
If(q,p)belongs to one ofA3,A7,A8, andλ=λ0/12, thenφc±=φd±=φd0±. Specially, when(q,p)∈A3andp→3q2/16r-0, thenφd0±tend to two trivial solutionsφ=±-(3q/4r).(The 1-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures ofφa0± when (q,p)∈A1 and p→0-0, where r=1, q=4 and (a) p=0-10-1, (b) p=0-10-2, (c) p=0-10-3, and (d) p=0-10-4.
(a)
(b)
(c)
(d)(The 2-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures ofφb± when (q,p)∈A2 and p→0-0, where r=1,  q=-4 and (a) p=0-1, (b) p=0-0.5, (c) p=0-0.1, and (d) p=0-0.01.
(a)
(b)
(c)
(d)(The symmetric solitary waves are bifurcated from the symmetric periodic waves.) The varying process for the figures ofφa0± when (q,p)∈A12 and p→0-0, where r=-1, q=4 and (a) p=0-0.5, (b) p=0-0.1, (c) p=0-0.01, and (d) p=0-0.001.
(a)
(b)
(c)
(d)(The symmetric periodic waves become two trivial waves.) The varying process for the figures ofφa0± and φb± when (q,p)∈A12 and p→3q2/16r+0, where r=-1,  q=4,andl6:p=3q2/16r=-3 and (a) p=-3+10-1, (b) p=-3+10-2, (c) p=-3+10-3, and (d) p=-3+10-6.
(a)
(b)
(c)
(d)(The four low-kink waves are bifurcated from four symmetric solitary waves.) The varying process for the figures ofφc± and φd± when (q,p)∈A3,λ>0,λ≠λ0/12 and p→3q2/16r-0, where r=1,q=-4,λ=4,andl2:p=3q2/16r=3 and (a) p=3-10-1, (b) p=3-10-3, (c) p=3-10-7, and (d) p=3-10-9.
(a)
(b)
(c)
(d)(The four low-kink waves are bifurcated from four 1-blow-up waves.) The varying process for the figures ofφc± and φd± when (q,p)∈A4,λ>0,λ≠λ0/12, and p→3q2/16r+0, where r=1,q=-4,λ=4,andl2:p=3q2/16r=3 and (a) p=3+10-1, (b) p=3+10-3, (c) p=3+10-6, and (d) p=3+10-9.
(a)
(b)
(c)
(d)(2)For the case ofλ<0, there are three properties as follows.(2)a
If(q,p)∈l2, that is,r>0,q<0, andr=3q2/16p, thenφc±andφd±becomeφc1±andφd1±which represent four 1-blow-up waves (refer to Figure8(d)). Specially, letλ=q<0, thenφc1±andφd1±become the hyperbolic 1-blow-up wave solutionsφe0±and(35)φd2±=±2pq(-1+coth(pξ)).(2)b
If(q,p)∈A3andλ≠-λ0/12, thenφc±≠φd±, and they represent four 2-blow-up waves (refer to Figure8(a)). Specially, whenp→3q2/16r-0, then the four 2-blow-up waves become four 1-blow-up wavesφc1±andφd1±, and for the varying process, see Figure8.(2)c
If(q,p)∈A3andλ=-λ0/12, thenφc±=φd±=φcd±of forms(36)φcd±=±12p-3q-9q2-48prcosh(2pξ),which represent hyperbolic blow-up waves. Whenp→3q2/16r-0, thenφcd±tend to two trivial solutionsφ=±-(3q/4r).(The four 2-blow-up waves become four 1-blow-up waves.) The varying process for the figures ofφc± and φd± when (q,p)∈A3,λ<0,λ≠-λ0/12, and p→3q2/16r-0, where r=1, q=-4,λ=-4,andl2:p=3q2/16r=3 and (a) p=3-10-1, (b) p=3-10-3, (c) p=3-10-6, and (d) p=3-10-9.
(a)
(b)
(c)
(d)


### 2.2. When the OrbitΓ Is Defined by H(φ,y)=H(φ1,0)

Proposition 2.
If(q,p) belongs to one of the regions A1,A2,A3,A4,A11,andA12 or curves l1,l2,l3,l6, and l7, then (1) have four nonlinear wave solutions
(37)ue±=βc2-cs2(φe±)2,Ee±=φe±ei(γx-ωt),uf±=βc2-cs2(φf±)2,Ef±=φf±ei(γx-ωt),
where
(38)φe±=±((Δ-q)/2r)(2Δ+q-2μreθξ)(2Δ+q)2+4μr(4Δ-q)eθξ+4μ2r2e2θξ,φf±=±((Δ-q)/2r)(2Δ+q-2μre-θξ)(2Δ+q)2+4μr(4Δ-q)e-θξ+4μ2r2e-2θξ,μ is a nonzero arbitrary real constant, Δ is given in (24), and
(39)θ=1rΔ(Δ-q).
Let
(40)μ0=q+2Δ2r.
About μ0, one has the following fact:
(41)μ0{>0for(q,p)∈A1,A2,A3,l3,=0for(q,p)∈l2,<0for(q,p)∈A4,A11,A12,l1,l6,l7.
Corresponding to μ>0 and μ<0, these solutions have the following properties and wave shapes.
(
1
o
)   For the case of μ>0, there are six properties as follows.(1o)a
If(q,p)∈l2, that is, r>0,q<0, and r=3q2/16p, then φe± and φf± become
(42)φe1∓=∓4μp16pe-2pξ-μq,(43)φf1∓=∓4μp16pe2pξ-μq,which represent four low-kink waves (refer to Figure9(d)). Specially, let μ=-(16p/q)>0, then φe1∓=φc0∓ and φf1∓=φb0∓.(1o)b
If(q,p)∈A1,A2,A3, and μ≠|μ0|, then φe±≠φf±, and they represent four tall-kink waves (refer to Figure 9(a)). Specially, when (q,p)∈A3 and p→3q2/16r-0, then φe± and φf± become φe1∓ and φf1∓, and for the varying process, see Figure 9.(1o)c
If(q,p)∈A4,A11,A12,l6, and μ≠|μ0|, then φe±≠φf±. When (q,p)∈A4, they represent four antisymmetric solitary waves (refer to Figure 10(a)). When (q,p)∈A11,A12,andl6, they represent four symmetric solitary waves (refer to Figure 11(a)). Specially, when (q,p)∈A4, if p→3q2/16r+0, then φe± and φf± become φe1∓ and φf1∓, and for the varying process, see Figure 10. When (q,p)∈A11 and p→q2/4r+0, then φe± and φf± tend to two trivial solutions φ=±-(q/2r), and for the varying process, see Figure 11.(1o)d
If(q,p)∈A3 and μ=|μ0|, then φe±=φf∓=φg∓ of forms
(44)φg∓=∓(1/r)(Δ-q)(2Δ+q)sinh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ),which represent two tall-kink waves and tend to a trivial waveφ=0 when p→3q2/16r-0.(1o)e
If(q,p)∈A4 and μ=|μ0|, then φe±=φf±=φh∓ of forms
(45)φh∓=∓(1/r)(Δ-q)(2Δ+q)cosh((θ/2)ξ)q-4Δ+(q+2Δ)cosh(θξ),which represent two antisymmetric solitary waves, tend to a trivial waveφ=0 when p→3q2/16r+0, and tend to two trivial waves φ=±-(q/2r) when p→q2/4r-0.(1o)f
If(q,p)∈A11,A12,l6 and μ=|μ0|, then φe±=φf±=φh± which represent two symmetric solitary waves. Specially, when (q,p)∈A11 and p→q2/4r-0, then φh± tend to two trivial waves φ=±-(q/2r).
(
2
o
)   For the case of μ<0, there are five properties as follows.(2o)a
If(q,p)∈l2, then φe1± and φf1± represent four 1-blow-up waves (refer to Figure 12(d)). Specially, let μ=16p/q<0, then φe1±=φe0± and φf±=φd2∓.(2o)b
If(q,p)∈A1,A2,A3,A4 and μ≠-|μ0|, then φe±≠φf±, and they represent four pairs of 1-blow-up waves (refer to Figure 12(a)). In particular, when (q,p)∈A4, if p→3q2/16r+0, then the four pairs of 1-blow-up waves become two pairs of 1-blow-up waves, and the varying process is displayed in Figure 12. If p→q2/4r-0, then the four pairs of 1-blow-up waves become two trivial waves φ=±-(q/2r).(2o)c
If(q,p)∈A11,A12,l6 and μ≠-|μ0|, then φe±≠φf±, and they represent four tall-kink waves.(2o)d
Ifμ=-|μ0|, when (q,p)∈A3, then φe±=φf±=φh∓ which represent two pairs of 1-blow-up waves. When (q,p)∈A4, then φe±=φf∓=φg∓ which represent two pairs of 1-blow-up waves.(2o)e
If(q,p)∈A11,A12,l6 and μ=-|μ0|, then φe±=φf∓=φg∓ which represent two tall-kink waves.(The four low-kink waves are bifurcated from the four tall-kink waves.) The varying process for the figures ofφe± and φf± when μ>0,μ≠|μ0|,(q,p)∈A3, and p→3q2/16r-0, where r=1,q=-4,l2:p=3q2/16r=3,andμ=4 and (a) p=3-0.5, (b) p=3-10-2, (c) p=3-10-4, and (d) p=3-10-6.
(a)
(b)
(c)
(d)(The four low-kink waves are bifurcated from the four antisymmetric solitary waves.) The varying process for the figures ofφe± and φf± when μ>0,μ≠|μ0|,(q,p)∈A4, and p→3q2/16r+0, where r=1,q=-4,l2:p=3q2/16r=3,andμ=4 and (a) p=3+10-1, (b) p=3+10-3, (c) p=3+10-5, and (d) p=3+10-7.
(a)
(b)
(c)
(d)(The four symmetric solitary waves become two trivial waves.) The varying process for the figures ofφe± and φf± when μ>0,μ≠|μ0|,(q,p)∈A11, and p→q2/4r+0, where r=-1, q=4,l7:p=q2/4r=-4,andμ=1 and (a) p=-3.5, (b) p=-4+10-1, (c) p=-4+10-2, and (d) p=-4+10-5.
(a)
(b)
(c)
(d)(The two pairs of 1-blow-up waves are bifurcated from the four pairs of 1-blow-up waves.) The varying process for the figures ofφe± and φf± when μ<0,μ≠-|μ0|,(q,p)∈A4, and p→3q2/16r+0, where r=1,q=-4,l2:p=3q2/16r=3,andμ=-4 and (a) p=3+10-2, (b) p=3+10-4, (c) p=3+10-6, and (d) p=3+10-9.
(a)
(b)
(c)
(d)


### 2.3. When the OrbitΓ Is Defined by H(φ,y)=H(φ2,0)

Proposition 3.
(i) If(q,p) belongs to one of A3,A4,A11, and l2, then (1) have four nonlinear wave solutions
(46)ui±=βc2-cs2(φi±)2,Ei±=φi±ei(γx-ωt),uj±=βc2-cs2(φj±)2,Ej±=φj±ei(γx-ωt),
where
(47)φi±=±(q+Δ)(2Δ-q)r(q+4Δ+(q-2Δ)cos(ηξ))cos(η2ξ),(48)φj±=±(q+Δ)(2Δ-q)r(q+4Δ-(q-2Δ)cos(ηξ))sin(η2ξ),(49)η=1r(4pr-q(q+Δ)).
When(q,p)∈A3,A4, or l2, then φi± and φj± represent periodic blow-up wave solutions (refer to Figure 13(a)). When (q,p)∈A11, then φi± and φj± represent periodic wave solutions (refer to Figure 14(a)).
(ii) If(q,p)∈l1 or l7, then (1) have two fractional wave solutions
(50)uk±=βc2-cs2(φk±)2,Ek±=φk±ei(γx-ωt),
where
(51)φk±=±-q2rqξq2ξ2-12r.
When(q,p)∈l1,  φk± represent two 1-blow-up waves (refer to Figure 13(d)). When (q,p)∈l7,  φk± represent two tall-kink waves (refer to Figure 14(d)).
(iii) If(q,p)∈A4 and p→q2/4r-0, then the periodic blow-up waves φi± tend to two trivial solutions φ=±-(q/2r), while φj± tend to the 1-blow-up waves φk±, and the varying process is displayed in Figure 13. If (q,p)∈A11 and p→q2/4r+0, then the periodic waves φi± tend to two trivial solutions φ=±-(q/2r), while φj± tend to the tall-kink waves φk±, and for the varying process, see Figure 14.(The 1-blow-up waves are bifurcated from the periodic blow-up waves.) The varying process for the figures ofφj± when (q,p)∈A4 and p→q2/4r-0, where r=1,q=-4,andl1:p=q2/4r=4 and (a) p=4-10-1, (b) p=4-10-2, (c) p=4-10-4, and (d) p=4-10-6.
(a)
(b)
(c)
(d)(The two tall-kink waves are bifurcated form two periodic waves.) The varying process for the figures ofφj± when (q,p)∈A11 and p→q2/4r+0, where r=-1,q=4,andl7:p=q2/4r=-4 and (a) p=-3.5, (b) p=-4+10-1, (c) p=-4+10-3, and (d) p=-4+10-5.
(a)
(b)
(c)
(d)


## 2.1. When the OrbitΓ Is Defined by H(φ,y)=H(0,0)

Proposition 1.
(i) Forp=0, (1) have two fractional nonlinear wave solutions
(28)ua±=βc2-cs2(φa±)2,Ea±=φa±ei(γx-ωt),
where
(29)φa±=±6q3q2ξ2-4r.
If (q,p)∈l4, then φa± are 1-blow-up waves (refer to Figure 2(d)). If (q,p)∈l3, then φa± are 2-blow-up waves (refer to Figure 3(d)). If (q,p)∈l5, then φa± are symmetric solitary waves (refer to Figure 4(d)).
(ii) Forp<0, (1) have two nonlinear wave solutions
(30)ub±=βc2-cs2(φb±)2,Eb±=φb±ei(γx-ωt),
where
(31)φb±=±-12p3q+9q2-48prcos(2-pξ).
These solutions have the following properties and wave shapes.
(1) When(q,p)∈A1 or A2, then φa0± and φb± are periodic-blow-up waves (refer to Figure 2(a) or Figure 3(a)). Specially, in region A1 when p→0-0, the periodic-blow-up waves φa0± become 1-blow-up waves φa±, and for the varying process, see Figure 2, while the periodic-blow-up waves φb± become a trivial wave φ=0. In region A2 when p→0-0, the periodic-blow-up waves φb± become 2-blow-up waves φa±, and for the varying process, see Figure 3.
(2) When(q,p)∈A12, then φa0± and φb± are symmetric periodic waves (refer to Figure 5(a)). If p→0-0, the symmetric periodic waves φa0± become symmetric solitary waves φa±, and for the varying process, see Figure 4. If p→3q2/16r+0, the symmetric periodic waves φa0± and φb± become two trivial waves φ=±-(3q/4r), and for the varying process, see Figure 5.
(iii) Forp>0, (1) have four nonlinear wave solutions
(32)uc±=βc2-cs2(φc±)2,Ec±=φc±ei(γx-ωt),ud±=βc2-cs2(φd±)2,Ed±=φd±ei(γx-ωt),
where
(33)φc±=±4λpλ2e2pξ+(λ0/12)e-2pξ-λq,φd±=±4λpλ2e-2pξ+(λ0/12)e2pξ-λq,λ is a nonzero arbitrary real constant and λ0=3q2-16pr. Corresponding to λ>0 or λ<0, these solutions have the following properties and wave shapes.
(1) For the case ofλ>0, there are four properties as follows.(1)a
When(q,p)∈l2, that is,r>0,q<0, andr=3q2/16p, thenφc±andφd±become(34)φc1±=±4pλe2pξ-q,φd1±=±4pλe-2pξ-q,which represent four low-kink waves (refer to Figure6(d) or Figure 7(d)). Specially, letλ=-q>0, thenφc1±andφd1±becomeφb0±andφc0±.(1)b
If(q,p)belongs to one ofA3,A7,A8, andλ≠λ0/12, thenφc±≠φd±, and they represent four symmetric solitary waves (refer to Figure6(a)). Specially, when(q,p)∈A3andp→3q2/16r-0, then the four symmetric solitary wavesφc±andφd±become the four low-kink wavesφc1±andφd1±, and for the varying process, see Figure6.(1)c
If(q,p)belongs to one ofA4,A5,A6,l1, andλ≠λ0/12, thenφc±≠φd±, and they represent four 1-blow-up waves (refer to Figure7(a)). Specially, when(q,p)∈A4andp→3q2/16r+0, then the four 1-blow-up wavesφc±andφd±become the four low-kink wavesφc1±andφd1±, and for the varying process, see Figure7.(1)d
If(q,p)belongs to one ofA3,A7,A8, andλ=λ0/12, thenφc±=φd±=φd0±. Specially, when(q,p)∈A3andp→3q2/16r-0, thenφd0±tend to two trivial solutionsφ=±-(3q/4r).(The 1-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures ofφa0± when (q,p)∈A1 and p→0-0, where r=1, q=4 and (a) p=0-10-1, (b) p=0-10-2, (c) p=0-10-3, and (d) p=0-10-4.
(a)
(b)
(c)
(d)(The 2-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures ofφb± when (q,p)∈A2 and p→0-0, where r=1,  q=-4 and (a) p=0-1, (b) p=0-0.5, (c) p=0-0.1, and (d) p=0-0.01.
(a)
(b)
(c)
(d)(The symmetric solitary waves are bifurcated from the symmetric periodic waves.) The varying process for the figures ofφa0± when (q,p)∈A12 and p→0-0, where r=-1, q=4 and (a) p=0-0.5, (b) p=0-0.1, (c) p=0-0.01, and (d) p=0-0.001.
(a)
(b)
(c)
(d)(The symmetric periodic waves become two trivial waves.) The varying process for the figures ofφa0± and φb± when (q,p)∈A12 and p→3q2/16r+0, where r=-1,  q=4,andl6:p=3q2/16r=-3 and (a) p=-3+10-1, (b) p=-3+10-2, (c) p=-3+10-3, and (d) p=-3+10-6.
(a)
(b)
(c)
(d)(The four low-kink waves are bifurcated from four symmetric solitary waves.) The varying process for the figures ofφc± and φd± when (q,p)∈A3,λ>0,λ≠λ0/12 and p→3q2/16r-0, where r=1,q=-4,λ=4,andl2:p=3q2/16r=3 and (a) p=3-10-1, (b) p=3-10-3, (c) p=3-10-7, and (d) p=3-10-9.
(a)
(b)
(c)
(d)(The four low-kink waves are bifurcated from four 1-blow-up waves.) The varying process for the figures ofφc± and φd± when (q,p)∈A4,λ>0,λ≠λ0/12, and p→3q2/16r+0, where r=1,q=-4,λ=4,andl2:p=3q2/16r=3 and (a) p=3+10-1, (b) p=3+10-3, (c) p=3+10-6, and (d) p=3+10-9.
(a)
(b)
(c)
(d)(2)For the case ofλ<0, there are three properties as follows.(2)a
If(q,p)∈l2, that is,r>0,q<0, andr=3q2/16p, thenφc±andφd±becomeφc1±andφd1±which represent four 1-blow-up waves (refer to Figure8(d)). Specially, letλ=q<0, thenφc1±andφd1±become the hyperbolic 1-blow-up wave solutionsφe0±and(35)φd2±=±2pq(-1+coth(pξ)).(2)b
If(q,p)∈A3andλ≠-λ0/12, thenφc±≠φd±, and they represent four 2-blow-up waves (refer to Figure8(a)). Specially, whenp→3q2/16r-0, then the four 2-blow-up waves become four 1-blow-up wavesφc1±andφd1±, and for the varying process, see Figure8.(2)c
If(q,p)∈A3andλ=-λ0/12, thenφc±=φd±=φcd±of forms(36)φcd±=±12p-3q-9q2-48prcosh(2pξ),which represent hyperbolic blow-up waves. Whenp→3q2/16r-0, thenφcd±tend to two trivial solutionsφ=±-(3q/4r).(The four 2-blow-up waves become four 1-blow-up waves.) The varying process for the figures ofφc± and φd± when (q,p)∈A3,λ<0,λ≠-λ0/12, and p→3q2/16r-0, where r=1, q=-4,λ=-4,andl2:p=3q2/16r=3 and (a) p=3-10-1, (b) p=3-10-3, (c) p=3-10-6, and (d) p=3-10-9.
(a)
(b)
(c)
(d)


## 2.2. When the OrbitΓ Is Defined by H(φ,y)=H(φ1,0)

Proposition 2.
If(q,p) belongs to one of the regions A1,A2,A3,A4,A11,andA12 or curves l1,l2,l3,l6, and l7, then (1) have four nonlinear wave solutions
(37)ue±=βc2-cs2(φe±)2,Ee±=φe±ei(γx-ωt),uf±=βc2-cs2(φf±)2,Ef±=φf±ei(γx-ωt),
where
(38)φe±=±((Δ-q)/2r)(2Δ+q-2μreθξ)(2Δ+q)2+4μr(4Δ-q)eθξ+4μ2r2e2θξ,φf±=±((Δ-q)/2r)(2Δ+q-2μre-θξ)(2Δ+q)2+4μr(4Δ-q)e-θξ+4μ2r2e-2θξ,μ is a nonzero arbitrary real constant, Δ is given in (24), and
(39)θ=1rΔ(Δ-q).
Let
(40)μ0=q+2Δ2r.
About μ0, one has the following fact:
(41)μ0{>0for(q,p)∈A1,A2,A3,l3,=0for(q,p)∈l2,<0for(q,p)∈A4,A11,A12,l1,l6,l7.
Corresponding to μ>0 and μ<0, these solutions have the following properties and wave shapes.
(
1
o
)   For the case of μ>0, there are six properties as follows.(1o)a
If(q,p)∈l2, that is, r>0,q<0, and r=3q2/16p, then φe± and φf± become
(42)φe1∓=∓4μp16pe-2pξ-μq,(43)φf1∓=∓4μp16pe2pξ-μq,which represent four low-kink waves (refer to Figure9(d)). Specially, let μ=-(16p/q)>0, then φe1∓=φc0∓ and φf1∓=φb0∓.(1o)b
If(q,p)∈A1,A2,A3, and μ≠|μ0|, then φe±≠φf±, and they represent four tall-kink waves (refer to Figure 9(a)). Specially, when (q,p)∈A3 and p→3q2/16r-0, then φe± and φf± become φe1∓ and φf1∓, and for the varying process, see Figure 9.(1o)c
If(q,p)∈A4,A11,A12,l6, and μ≠|μ0|, then φe±≠φf±. When (q,p)∈A4, they represent four antisymmetric solitary waves (refer to Figure 10(a)). When (q,p)∈A11,A12,andl6, they represent four symmetric solitary waves (refer to Figure 11(a)). Specially, when (q,p)∈A4, if p→3q2/16r+0, then φe± and φf± become φe1∓ and φf1∓, and for the varying process, see Figure 10. When (q,p)∈A11 and p→q2/4r+0, then φe± and φf± tend to two trivial solutions φ=±-(q/2r), and for the varying process, see Figure 11.(1o)d
If(q,p)∈A3 and μ=|μ0|, then φe±=φf∓=φg∓ of forms
(44)φg∓=∓(1/r)(Δ-q)(2Δ+q)sinh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ),which represent two tall-kink waves and tend to a trivial waveφ=0 when p→3q2/16r-0.(1o)e
If(q,p)∈A4 and μ=|μ0|, then φe±=φf±=φh∓ of forms
(45)φh∓=∓(1/r)(Δ-q)(2Δ+q)cosh((θ/2)ξ)q-4Δ+(q+2Δ)cosh(θξ),which represent two antisymmetric solitary waves, tend to a trivial waveφ=0 when p→3q2/16r+0, and tend to two trivial waves φ=±-(q/2r) when p→q2/4r-0.(1o)f
If(q,p)∈A11,A12,l6 and μ=|μ0|, then φe±=φf±=φh± which represent two symmetric solitary waves. Specially, when (q,p)∈A11 and p→q2/4r-0, then φh± tend to two trivial waves φ=±-(q/2r).
(
2
o
)   For the case of μ<0, there are five properties as follows.(2o)a
If(q,p)∈l2, then φe1± and φf1± represent four 1-blow-up waves (refer to Figure 12(d)). Specially, let μ=16p/q<0, then φe1±=φe0± and φf±=φd2∓.(2o)b
If(q,p)∈A1,A2,A3,A4 and μ≠-|μ0|, then φe±≠φf±, and they represent four pairs of 1-blow-up waves (refer to Figure 12(a)). In particular, when (q,p)∈A4, if p→3q2/16r+0, then the four pairs of 1-blow-up waves become two pairs of 1-blow-up waves, and the varying process is displayed in Figure 12. If p→q2/4r-0, then the four pairs of 1-blow-up waves become two trivial waves φ=±-(q/2r).(2o)c
If(q,p)∈A11,A12,l6 and μ≠-|μ0|, then φe±≠φf±, and they represent four tall-kink waves.(2o)d
Ifμ=-|μ0|, when (q,p)∈A3, then φe±=φf±=φh∓ which represent two pairs of 1-blow-up waves. When (q,p)∈A4, then φe±=φf∓=φg∓ which represent two pairs of 1-blow-up waves.(2o)e
If(q,p)∈A11,A12,l6 and μ=-|μ0|, then φe±=φf∓=φg∓ which represent two tall-kink waves.(The four low-kink waves are bifurcated from the four tall-kink waves.) The varying process for the figures ofφe± and φf± when μ>0,μ≠|μ0|,(q,p)∈A3, and p→3q2/16r-0, where r=1,q=-4,l2:p=3q2/16r=3,andμ=4 and (a) p=3-0.5, (b) p=3-10-2, (c) p=3-10-4, and (d) p=3-10-6.
(a)
(b)
(c)
(d)(The four low-kink waves are bifurcated from the four antisymmetric solitary waves.) The varying process for the figures ofφe± and φf± when μ>0,μ≠|μ0|,(q,p)∈A4, and p→3q2/16r+0, where r=1,q=-4,l2:p=3q2/16r=3,andμ=4 and (a) p=3+10-1, (b) p=3+10-3, (c) p=3+10-5, and (d) p=3+10-7.
(a)
(b)
(c)
(d)(The four symmetric solitary waves become two trivial waves.) The varying process for the figures ofφe± and φf± when μ>0,μ≠|μ0|,(q,p)∈A11, and p→q2/4r+0, where r=-1, q=4,l7:p=q2/4r=-4,andμ=1 and (a) p=-3.5, (b) p=-4+10-1, (c) p=-4+10-2, and (d) p=-4+10-5.
(a)
(b)
(c)
(d)(The two pairs of 1-blow-up waves are bifurcated from the four pairs of 1-blow-up waves.) The varying process for the figures ofφe± and φf± when μ<0,μ≠-|μ0|,(q,p)∈A4, and p→3q2/16r+0, where r=1,q=-4,l2:p=3q2/16r=3,andμ=-4 and (a) p=3+10-2, (b) p=3+10-4, (c) p=3+10-6, and (d) p=3+10-9.
(a)
(b)
(c)
(d)


## 2.3. When the OrbitΓ Is Defined by H(φ,y)=H(φ2,0)

Proposition 3.
(i) If(q,p) belongs to one of A3,A4,A11, and l2, then (1) have four nonlinear wave solutions
(46)ui±=βc2-cs2(φi±)2,Ei±=φi±ei(γx-ωt),uj±=βc2-cs2(φj±)2,Ej±=φj±ei(γx-ωt),
where
(47)φi±=±(q+Δ)(2Δ-q)r(q+4Δ+(q-2Δ)cos(ηξ))cos(η2ξ),(48)φj±=±(q+Δ)(2Δ-q)r(q+4Δ-(q-2Δ)cos(ηξ))sin(η2ξ),(49)η=1r(4pr-q(q+Δ)).
When(q,p)∈A3,A4, or l2, then φi± and φj± represent periodic blow-up wave solutions (refer to Figure 13(a)). When (q,p)∈A11, then φi± and φj± represent periodic wave solutions (refer to Figure 14(a)).
(ii) If(q,p)∈l1 or l7, then (1) have two fractional wave solutions
(50)uk±=βc2-cs2(φk±)2,Ek±=φk±ei(γx-ωt),
where
(51)φk±=±-q2rqξq2ξ2-12r.
When(q,p)∈l1,  φk± represent two 1-blow-up waves (refer to Figure 13(d)). When (q,p)∈l7,  φk± represent two tall-kink waves (refer to Figure 14(d)).
(iii) If(q,p)∈A4 and p→q2/4r-0, then the periodic blow-up waves φi± tend to two trivial solutions φ=±-(q/2r), while φj± tend to the 1-blow-up waves φk±, and the varying process is displayed in Figure 13. If (q,p)∈A11 and p→q2/4r+0, then the periodic waves φi± tend to two trivial solutions φ=±-(q/2r), while φj± tend to the tall-kink waves φk±, and for the varying process, see Figure 14.(The 1-blow-up waves are bifurcated from the periodic blow-up waves.) The varying process for the figures ofφj± when (q,p)∈A4 and p→q2/4r-0, where r=1,q=-4,andl1:p=q2/4r=4 and (a) p=4-10-1, (b) p=4-10-2, (c) p=4-10-4, and (d) p=4-10-6.
(a)
(b)
(c)
(d)(The two tall-kink waves are bifurcated form two periodic waves.) The varying process for the figures ofφj± when (q,p)∈A11 and p→q2/4r+0, where r=-1,q=4,andl7:p=q2/4r=-4 and (a) p=-3.5, (b) p=-4+10-1, (c) p=-4+10-3, and (d) p=-4+10-5.
(a)
(b)
(c)
(d)


## 3. The Derivations of Main Results

To derive our results, substituting (23) and u(x,t)=u(ξ) with ξ=x-ct into (1), it follows that
(52)(c2-cs2)u′′=β(φ2)′′,αφ′′+i(2αγ-c)φ′+(w-αγ2)φ-δ1uφ+δ2φ3+δ3φ5=0.Integrating the first equation of (52) twice with respect to ξ and taking the integral constants to zero, we get (22). Substituting (22) into the second equation of (52) and letting c=2αγ, it follows that
(53)φ′′=pφ+qφ3+rφ5.Form (53) we obtain the planar system
(54)dφdξ=y,dydξ=pφ+qφ3+rφ5,
with the first integral
(55)H(φ,y)=y2-(pφ2+q2φ4+r3φ6)=h.According to the qualitative theory, we obtain the bifurcation phase portraits of system (54) as Figures 15 and 16. Using the information given by Figures 15 and 16, we give derivations to Propositions 1, 2, and 3, respectively.Figure 15
The bifurcation phase portraits of system (54) for r>0.Figure 16
The bifurcation phase portraits of system (54) for r<0.


### 3.1. The Derivations to Proposition1

When the orbitΓ is defined by H(φ,y)=H(0,0), from (55) we obtain
(56)y=±pφ2+q2φ4+r3φ6.Substituting (56) into the first equation of (54) and integrating it, we have
(57)∫lφ‍dsps2+(q/2)s4+(r/3)s6=ξ,
where l is an arbitrary constant or ±∞.Forp=0, letting l=-(3q/2r) or ±∞ and completing the above integral, it follows that
(58)φ=±6q3q2ξ2-4r,
which yields φa± as (29).Forp<0, completing (57) and solving for φ, it follows that
(59)φ=±-12p3q-9q2-48prsin(λ+2-pξ),
where λ=λ(l) is an arbitrary real constant. Let λ=±(π/2), respectively, and we obtain the solutions φa0± and φb± as (3) and (31).When(q,p)∈A1, that is, r>0,q>0,andp<0, let
(60)f(p)=3q-9q2-48prcos(2-pξ),g(p)=3q+9q2-48prcos(2-pξ).Thus, we have(61)limp→0-0φa0±=limp→0-0±-12pf(p)=±limp→0-0-12f′(p)=±(limp→0-0(-4rcos(2-pξ)(4rcos(2-pξ)-4rcos(2-pξ))-129q2-48pr)(3q2-16pr)ξ2)-129q2-48pr×(sin(2-pξ)/2-pξ)-4rcos(2-pξ)(4rcos(2-pξ)-4rcos(2-pξ))-129q2-48pr)(3q2-16pr)ξ2)-129q2-48pr)limp→0-0(29q2-48pr)1/2=±6q3q2ξ2-4r=φa±(see(29)),limp→0-0φb±=limp→0-0±-12pg(p)=±limp→0-0-12pg(p)=0.When(q,p)∈A2, that is, r>0,q<0,andp<0, similarly we get
(62)limp→0-0φb±=φa±.When(q,p)∈A12, that is, r<0,q>0,andp<0, if p→3q2/16r+0, then f(p) and g(p)→0.Thus,(63)limp→0-0φa0±(orφb±)=±-3q4r.Forp>0, completing (57) and solving for φ, it follows that
(64)φ=±4λpe2pξλ2e4pξ+(q2/4-4pr/3)-λqe2pξ,
where λ=λ(l) is an arbitrary real constant.Note that ifφ(ξ) is a solution of (53), so is φ(-ξ). Thus, from (64) we obtain the solutions φc± and φd± as in (33).For the case ofλ>0. When (q,p)∈l2, that is, r>0,q<0 and r=3q2/16p. In (33) letting r=3q2/16p, we get (34). Furthermore, in (34) letting λ=-q>0, it follows that
(65)φc1±=±4p-q(e2pξ+1)=±4pe-pξ-q(epξ+e-pξ)=±2p-q(1-epξ-e-pξepξ+e-pξ)=±2p-q(1-tanh(pξ))=±2pq(-1+tanh(pξ))=φb0±(see(4)),φd1±=±4p-q(e-2pξ+1)=±4pepξ-q(epξ+e-pξ)=±2p-q(1+epξ-e-pξepξ+e-pξ)=±2p-q(1+tanh(pξ))=±2pq(-1-tanh(pξ))=φc0±(see(5)).Whenλ=λ0/12, by (33) it follows that
(66)φc±=φd±=±24p-6q+9q2-48pr(epξ+e-pξ)=±12p-3q+9q2-48prcosh(2pξ)=φd0±(see(6)).For the case ofλ<0, similarly we can obtain (7), (35), and (36), and here we omit the process. Hereto, we have completed the derivations for Proposition 1.


### 3.2. The Derivations to Proposition2

When the orbitΓ is defined by H(φ,y)=H(φ1,0), from (55) we obtain
(67)y=±pφ2+q2φ4+r3φ6+H(φ1,0)=±r3(φ12-φ2)2(φ2+μ0),
where φ1 and μ0 are given in (25) and (40). Substituting (67) into the first equation of (54) and integrating it, we have
(68)∫mφ‍ds(r/3)(φ12-s2)2(s2+μ0)=ξ,
where m is an arbitrary constant or ±∞.Completing the above integral and solving forφ, it follows that
(69)φ=±φ12-4a1μeθξμ2e2θξ-2μb1eθξ+b12-4a1,
where θ is given in (39), μ=μ(m) is an arbitrary constant, and
(70)a1=3Δ(Δ-q)4r2,b1=q-4Δ2r.Similar to the derivations forφc± and φd±, we get φe± and φf± (see (38)) from (69).For the case ofμ>0, when (q,p)∈l2, that is, r>0,q<0,andr=3q2/16p, then θ=2p and q+2Δ=0. From (38) and (41), it is easy to check that φe± and φf± become φe1∓ and φf1∓ (see (42)), respectively. Furthermore, in (42) letting μ=-(16p/q), it follows that
(71)φe1∓=∓-4pq(1+e-2pξ)=∓2pq(-1-tanh(pξ))=φc0∓(see(5)),φf1∓=∓-4pq(1+e2pξ)=∓2pq(-1+tanh(pξ))=φb0∓(see(4)).If(q,p)∈A3 and μ=|μ0|, that is, r>0,μ=μ0>0, and 2Δ+q>0, we have(72)φe±=±(Δ-q)/2r(2Δ+q-2μreθξ)(2Δ+q)2+4μr(4Δ-q)eθξ+4μ2r2e2θξ=±(Δ-q)/2r(2Δ+q)(1-eθξ)(q+2Δ)2+2(q+2Δ)(4Δ-q)eθξ+(q+2Δ)2e2θξ=±(1/2r)(Δ-q)(2Δ+q)(1-eθξ)q+2Δ+2(4Δ-q)eθξ+(q+2Δ)e2θξ=±(1/2r)(Δ-q)(2Δ+q)(e-θξ/2-eθξ/2)2(4Δ-q)+(q+2Δ)(e-θξ+eθξ)=∓(1/r)(Δ-q)(2Δ+q)sinh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ)=φg∓(see(44)),φf±=±(Δ-q)/2r(2Δ+q-2μre-θξ)(2Δ+q)2+4μr(4Δ-q)e-θξ+4μ2r2e-2θξ=±(Δ-q)/2r(2Δ+q)(1-e-θξ)(q+2Δ)2+2(q+2Δ)(4Δ-q)e-θξ+(q+2Δ)2e-2θξ=±(1/2r)(Δ-q)(2Δ+q)(1-e-θξ)q+2Δ+2(4Δ-q)e-θξ+(q+2Δ)e-2θξ=±(1/2r)(Δ-q)(2Δ+q)(eθξ/2-e-θξ/2)2(4Δ-q)+(q+2Δ)(e-θξ+eθξ)=±(1/r)(Δ-q)(2Δ+q)sinh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ)=φg±(see(44)).If(q,p)∈A4 and μ=|μ0|, that is, r>0,μ=-μ0>0, and 2Δ+q<0, we have (73)φe±=±(Δ-q)/2r(2Δ+q-2μreθξ)(2Δ+q)2+4μr(4Δ-q)eθξ+4μ2r2e2θξ=±(Δ-q)/2r(2Δ+q)(1+eθξ)(q+2Δ)2+2(q+2Δ)(4Δ-q)eθξ+(q+2Δ)2e2θξ=∓(1/2r)(Δ-q)(2Δ+q)(1+eθξ)q+2Δ+2(4Δ-q)eθξ+(q+2Δ)e2θξ=∓(1/2r)(Δ-q)(2Δ+q)(e-θξ/2+eθξ/2)2(4Δ-q)+(q+2Δ)(e-θξ+eθξ)=∓(1/r)(Δ-q)(2Δ+q)cosh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ)=φh∓(see(45)),φf±=±(Δ-q)/2r(2Δ+q-2μre-θξ)(2Δ+q)2+4μr(4Δ-q)e-θξ+4μ2r2e-2θξ=±(Δ-q)/2r(2Δ+q)(1+e-θξ)(q+2Δ)2+2(q+2Δ)(4Δ-q)e-θξ+(q+2Δ)2e-2θξ=∓(1/2r)(Δ-q)(2Δ+q)(1+e-θξ)q+2Δ+2(4Δ-q)e-θξ+(q+2Δ)e-2θξ=∓(1/2r)(Δ-q)(2Δ+q)(e-θξ/2+eθξ/2)2(4Δ-q)+(q+2Δ)(e-θξ+eθξ)=∓(1/r)(Δ-q)(2Δ+q)cosh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ)=φh∓(see(45)).If(q,p)∈A11,A12,l6 and μ=|μ0|, that is, r<0,μ=-μ0>0, and 2Δ+q>0, we have(74)φe±=±(Δ-q)/2r(2Δ+q-2μreθξ)(2Δ+q)2+4μr(4Δ-q)eθξ+4μ2r2e2θξ=±(Δ-q)/2r(2Δ+q)(1+eθξ)(q+2Δ)2+2(q+2Δ)(4Δ-q)eθξ+(q+2Δ)2e2θξ=±(1/2r)(Δ-q)(2Δ+q)(1+eθξ)q+2Δ+2(4Δ-q)eθξ+(q+2Δ)e2θξ=±(1/2r)(Δ-q)(2Δ+q)(e-θξ/2+eθξ/2)2(4Δ-q)+(q+2Δ)(e-θξ+eθξ)=±(1/r)(Δ-q)(2Δ+q)cosh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ)=φh±(see(45)),φf±=±(Δ-q)/2r(2Δ+q-2μre-θξ)(2Δ+q)2+4μr(4Δ-q)e-θξ+4μ2r2e-2θξ=±(Δ-q)/2r(2Δ+q)(1+e-θξ)(q+2Δ)2+2(q+2Δ)(4Δ-q)e-θξ+(q+2Δ)2e-2θξ=±(1/2r)(Δ-q)(2Δ+q)(1+e-θξ)q+2Δ+2(4Δ-q)e-θξ+(q+2Δ)e-2θξ=±(1/2r)(Δ-q)(2Δ+q)(e-θξ/2+eθξ/2)2(4Δ-q)+(q+2Δ)(e-θξ+eθξ)=±(1/r)(Δ-q)(2Δ+q)cosh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ)=φh±(see(45)).For the case ofμ<0, similarly we can obtain the relations of the solutions φe±,φf±,φg±, and φh±, and here we omit the process. Hereto, we have completed the derivations for Proposition 2.


### 3.3. The Derivations to Proposition3

When the orbitΓ is defined by H(φ,y)=H(φ2,0), firstly, if (q,p) belongs to one of A3,A4,A11, and l2, then from (55) we obtain
(75)y=±pφ2+q2φ4+r3φ6+H(φ2,0)=±r3(φ2-φ22)2(φ2+δ0),
where φ2 is given in (16) and δ0=(q-2Δ)/2r. Substituting (75) into the first equation of (54) and integrating it, we have
(76)∫nφ‍ds(r/3)(s2-φ22)2(s2+δ0)=ξ,
where n is an arbitrary constant or ±∞.Completing the above integral and solving forφ, it follows that
(77)φ=±φ22-2a2b2+δ0sin(δ+ηξ),
where η is given in (49),  δ=δ(n) is an arbitrary constant, and
(78)a2=3Δ(Δ+q)4r2,b2=-q+4Δ2r.In (77) letting δ=±(π/2), respectively, we obtain the solutions φi± and φj± as (47) and (48).Secondly, if(q,p)∈l1 or l7, that is, Δ=0 and φ12=-δ0=-(q/2r), thus from (75), we obtain
(79)y=±r3(φ2+q2r)3.Similarly, we have(80)φ=±-q2rqξq2ξ2-12r,
which yields φk± as in (51).Whenp→q2/4r, it follows that
(81)Δ=q2-4pr→0,η=1r(4pr-q(q+Δ))=-Δ(Δ+q)r→0,cos(ηξ)=1-η2ξ22+η4ξ44!+⋯=1+Δ(Δ+q)2rξ2+o(Δ2).Thus, we have(82)limp→(q2/4r)φi±=±limp→q2/4r(q+Δ)(2Δ-q)r(q+4Δ+(q-2Δ)cos(ηξ))cos(η2ξ)=±-q22rq=±-q2r,limp→q2/4rφj±=limp→q2/4r±(q+Δ)(2Δ-q)r(q+4Δ-(q-2Δ)cos(ηξ))sin(η2ξ)=limp→q2/4r±(q+Δ)(2Δ-q)(1-cos(ηξ))2r(q+4Δ-(q-2Δ)cos(ηξ))=limp→q2/4r±((-(q+Δ)(2Δ-q)Δ(Δ+q)2rξ2+o(Δ)2Δ(Δ+q)2r)×(2r[(Δ2)(1+Δ(Δ+q)2rξ2)q+4Δ-(q-2Δ)×(1+Δ(Δ+q)2rξ2)+o(Δ2)(1+Δ(Δ+q)2rξ2)]Δ(Δ+q)2r)-1)1/2=limp→q2/4r±((-(Δ+q)2(2Δ-q)2rξ2+o(Δ))×(+o(Δ)(1+Δ(Δ+q)2rξ2)])-12r[6-(q-2Δ)(1+Δ(Δ+q)2rξ2)+o(Δ)(1+Δ(Δ+q)2rξ2)](+o(Δ)(1+Δ(Δ+q)2rξ2)])-12r[6-(q-2Δ)(1+Δ(Δ+q)2rξ2))-1(1+Δ(Δ+q)2rξ2))1/2=±(q3/2r)ξ22r(6-(q2/2r)ξ2)=±-q2rqξq2ξ2-12r=φk±(see(51)).Hereto, we have completed the derivations for our main results.


## 3.1. The Derivations to Proposition1

When the orbitΓ is defined by H(φ,y)=H(0,0), from (55) we obtain
(56)y=±pφ2+q2φ4+r3φ6.Substituting (56) into the first equation of (54) and integrating it, we have
(57)∫lφ‍dsps2+(q/2)s4+(r/3)s6=ξ,
where l is an arbitrary constant or ±∞.Forp=0, letting l=-(3q/2r) or ±∞ and completing the above integral, it follows that
(58)φ=±6q3q2ξ2-4r,
which yields φa± as (29).Forp<0, completing (57) and solving for φ, it follows that
(59)φ=±-12p3q-9q2-48prsin(λ+2-pξ),
where λ=λ(l) is an arbitrary real constant. Let λ=±(π/2), respectively, and we obtain the solutions φa0± and φb± as (3) and (31).When(q,p)∈A1, that is, r>0,q>0,andp<0, let
(60)f(p)=3q-9q2-48prcos(2-pξ),g(p)=3q+9q2-48prcos(2-pξ).Thus, we have(61)limp→0-0φa0±=limp→0-0±-12pf(p)=±limp→0-0-12f′(p)=±(limp→0-0(-4rcos(2-pξ)(4rcos(2-pξ)-4rcos(2-pξ))-129q2-48pr)(3q2-16pr)ξ2)-129q2-48pr×(sin(2-pξ)/2-pξ)-4rcos(2-pξ)(4rcos(2-pξ)-4rcos(2-pξ))-129q2-48pr)(3q2-16pr)ξ2)-129q2-48pr)limp→0-0(29q2-48pr)1/2=±6q3q2ξ2-4r=φa±(see(29)),limp→0-0φb±=limp→0-0±-12pg(p)=±limp→0-0-12pg(p)=0.When(q,p)∈A2, that is, r>0,q<0,andp<0, similarly we get
(62)limp→0-0φb±=φa±.When(q,p)∈A12, that is, r<0,q>0,andp<0, if p→3q2/16r+0, then f(p) and g(p)→0.Thus,(63)limp→0-0φa0±(orφb±)=±-3q4r.Forp>0, completing (57) and solving for φ, it follows that
(64)φ=±4λpe2pξλ2e4pξ+(q2/4-4pr/3)-λqe2pξ,
where λ=λ(l) is an arbitrary real constant.Note that ifφ(ξ) is a solution of (53), so is φ(-ξ). Thus, from (64) we obtain the solutions φc± and φd± as in (33).For the case ofλ>0. When (q,p)∈l2, that is, r>0,q<0 and r=3q2/16p. In (33) letting r=3q2/16p, we get (34). Furthermore, in (34) letting λ=-q>0, it follows that
(65)φc1±=±4p-q(e2pξ+1)=±4pe-pξ-q(epξ+e-pξ)=±2p-q(1-epξ-e-pξepξ+e-pξ)=±2p-q(1-tanh(pξ))=±2pq(-1+tanh(pξ))=φb0±(see(4)),φd1±=±4p-q(e-2pξ+1)=±4pepξ-q(epξ+e-pξ)=±2p-q(1+epξ-e-pξepξ+e-pξ)=±2p-q(1+tanh(pξ))=±2pq(-1-tanh(pξ))=φc0±(see(5)).Whenλ=λ0/12, by (33) it follows that
(66)φc±=φd±=±24p-6q+9q2-48pr(epξ+e-pξ)=±12p-3q+9q2-48prcosh(2pξ)=φd0±(see(6)).For the case ofλ<0, similarly we can obtain (7), (35), and (36), and here we omit the process. Hereto, we have completed the derivations for Proposition 1.


## 3.2. The Derivations to Proposition2

When the orbitΓ is defined by H(φ,y)=H(φ1,0), from (55) we obtain
(67)y=±pφ2+q2φ4+r3φ6+H(φ1,0)=±r3(φ12-φ2)2(φ2+μ0),
where φ1 and μ0 are given in (25) and (40). Substituting (67) into the first equation of (54) and integrating it, we have
(68)∫mφ‍ds(r/3)(φ12-s2)2(s2+μ0)=ξ,
where m is an arbitrary constant or ±∞.Completing the above integral and solving forφ, it follows that
(69)φ=±φ12-4a1μeθξμ2e2θξ-2μb1eθξ+b12-4a1,
where θ is given in (39), μ=μ(m) is an arbitrary constant, and
(70)a1=3Δ(Δ-q)4r2,b1=q-4Δ2r.Similar to the derivations forφc± and φd±, we get φe± and φf± (see (38)) from (69).For the case ofμ>0, when (q,p)∈l2, that is, r>0,q<0,andr=3q2/16p, then θ=2p and q+2Δ=0. From (38) and (41), it is easy to check that φe± and φf± become φe1∓ and φf1∓ (see (42)), respectively. Furthermore, in (42) letting μ=-(16p/q), it follows that
(71)φe1∓=∓-4pq(1+e-2pξ)=∓2pq(-1-tanh(pξ))=φc0∓(see(5)),φf1∓=∓-4pq(1+e2pξ)=∓2pq(-1+tanh(pξ))=φb0∓(see(4)).If(q,p)∈A3 and μ=|μ0|, that is, r>0,μ=μ0>0, and 2Δ+q>0, we have(72)φe±=±(Δ-q)/2r(2Δ+q-2μreθξ)(2Δ+q)2+4μr(4Δ-q)eθξ+4μ2r2e2θξ=±(Δ-q)/2r(2Δ+q)(1-eθξ)(q+2Δ)2+2(q+2Δ)(4Δ-q)eθξ+(q+2Δ)2e2θξ=±(1/2r)(Δ-q)(2Δ+q)(1-eθξ)q+2Δ+2(4Δ-q)eθξ+(q+2Δ)e2θξ=±(1/2r)(Δ-q)(2Δ+q)(e-θξ/2-eθξ/2)2(4Δ-q)+(q+2Δ)(e-θξ+eθξ)=∓(1/r)(Δ-q)(2Δ+q)sinh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ)=φg∓(see(44)),φf±=±(Δ-q)/2r(2Δ+q-2μre-θξ)(2Δ+q)2+4μr(4Δ-q)e-θξ+4μ2r2e-2θξ=±(Δ-q)/2r(2Δ+q)(1-e-θξ)(q+2Δ)2+2(q+2Δ)(4Δ-q)e-θξ+(q+2Δ)2e-2θξ=±(1/2r)(Δ-q)(2Δ+q)(1-e-θξ)q+2Δ+2(4Δ-q)e-θξ+(q+2Δ)e-2θξ=±(1/2r)(Δ-q)(2Δ+q)(eθξ/2-e-θξ/2)2(4Δ-q)+(q+2Δ)(e-θξ+eθξ)=±(1/r)(Δ-q)(2Δ+q)sinh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ)=φg±(see(44)).If(q,p)∈A4 and μ=|μ0|, that is, r>0,μ=-μ0>0, and 2Δ+q<0, we have (73)φe±=±(Δ-q)/2r(2Δ+q-2μreθξ)(2Δ+q)2+4μr(4Δ-q)eθξ+4μ2r2e2θξ=±(Δ-q)/2r(2Δ+q)(1+eθξ)(q+2Δ)2+2(q+2Δ)(4Δ-q)eθξ+(q+2Δ)2e2θξ=∓(1/2r)(Δ-q)(2Δ+q)(1+eθξ)q+2Δ+2(4Δ-q)eθξ+(q+2Δ)e2θξ=∓(1/2r)(Δ-q)(2Δ+q)(e-θξ/2+eθξ/2)2(4Δ-q)+(q+2Δ)(e-θξ+eθξ)=∓(1/r)(Δ-q)(2Δ+q)cosh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ)=φh∓(see(45)),φf±=±(Δ-q)/2r(2Δ+q-2μre-θξ)(2Δ+q)2+4μr(4Δ-q)e-θξ+4μ2r2e-2θξ=±(Δ-q)/2r(2Δ+q)(1+e-θξ)(q+2Δ)2+2(q+2Δ)(4Δ-q)e-θξ+(q+2Δ)2e-2θξ=∓(1/2r)(Δ-q)(2Δ+q)(1+e-θξ)q+2Δ+2(4Δ-q)e-θξ+(q+2Δ)e-2θξ=∓(1/2r)(Δ-q)(2Δ+q)(e-θξ/2+eθξ/2)2(4Δ-q)+(q+2Δ)(e-θξ+eθξ)=∓(1/r)(Δ-q)(2Δ+q)cosh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ)=φh∓(see(45)).If(q,p)∈A11,A12,l6 and μ=|μ0|, that is, r<0,μ=-μ0>0, and 2Δ+q>0, we have(74)φe±=±(Δ-q)/2r(2Δ+q-2μreθξ)(2Δ+q)2+4μr(4Δ-q)eθξ+4μ2r2e2θξ=±(Δ-q)/2r(2Δ+q)(1+eθξ)(q+2Δ)2+2(q+2Δ)(4Δ-q)eθξ+(q+2Δ)2e2θξ=±(1/2r)(Δ-q)(2Δ+q)(1+eθξ)q+2Δ+2(4Δ-q)eθξ+(q+2Δ)e2θξ=±(1/2r)(Δ-q)(2Δ+q)(e-θξ/2+eθξ/2)2(4Δ-q)+(q+2Δ)(e-θξ+eθξ)=±(1/r)(Δ-q)(2Δ+q)cosh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ)=φh±(see(45)),φf±=±(Δ-q)/2r(2Δ+q-2μre-θξ)(2Δ+q)2+4μr(4Δ-q)e-θξ+4μ2r2e-2θξ=±(Δ-q)/2r(2Δ+q)(1+e-θξ)(q+2Δ)2+2(q+2Δ)(4Δ-q)e-θξ+(q+2Δ)2e-2θξ=±(1/2r)(Δ-q)(2Δ+q)(1+e-θξ)q+2Δ+2(4Δ-q)e-θξ+(q+2Δ)e-2θξ=±(1/2r)(Δ-q)(2Δ+q)(e-θξ/2+eθξ/2)2(4Δ-q)+(q+2Δ)(e-θξ+eθξ)=±(1/r)(Δ-q)(2Δ+q)cosh((θ/2)ξ)-q+4Δ+(q+2Δ)cosh(θξ)=φh±(see(45)).For the case ofμ<0, similarly we can obtain the relations of the solutions φe±,φf±,φg±, and φh±, and here we omit the process. Hereto, we have completed the derivations for Proposition 2.


## 3.3. The Derivations to Proposition3

When the orbitΓ is defined by H(φ,y)=H(φ2,0), firstly, if (q,p) belongs to one of A3,A4,A11, and l2, then from (55) we obtain
(75)y=±pφ2+q2φ4+r3φ6+H(φ2,0)=±r3(φ2-φ22)2(φ2+δ0),
where φ2 is given in (16) and δ0=(q-2Δ)/2r. Substituting (75) into the first equation of (54) and integrating it, we have
(76)∫nφ‍ds(r/3)(s2-φ22)2(s2+δ0)=ξ,
where n is an arbitrary constant or ±∞.Completing the above integral and solving forφ, it follows that
(77)φ=±φ22-2a2b2+δ0sin(δ+ηξ),
where η is given in (49),  δ=δ(n) is an arbitrary constant, and
(78)a2=3Δ(Δ+q)4r2,b2=-q+4Δ2r.In (77) letting δ=±(π/2), respectively, we obtain the solutions φi± and φj± as (47) and (48).Secondly, if(q,p)∈l1 or l7, that is, Δ=0 and φ12=-δ0=-(q/2r), thus from (75), we obtain
(79)y=±r3(φ2+q2r)3.Similarly, we have(80)φ=±-q2rqξq2ξ2-12r,
which yields φk± as in (51).Whenp→q2/4r, it follows that
(81)Δ=q2-4pr→0,η=1r(4pr-q(q+Δ))=-Δ(Δ+q)r→0,cos(ηξ)=1-η2ξ22+η4ξ44!+⋯=1+Δ(Δ+q)2rξ2+o(Δ2).Thus, we have(82)limp→(q2/4r)φi±=±limp→q2/4r(q+Δ)(2Δ-q)r(q+4Δ+(q-2Δ)cos(ηξ))cos(η2ξ)=±-q22rq=±-q2r,limp→q2/4rφj±=limp→q2/4r±(q+Δ)(2Δ-q)r(q+4Δ-(q-2Δ)cos(ηξ))sin(η2ξ)=limp→q2/4r±(q+Δ)(2Δ-q)(1-cos(ηξ))2r(q+4Δ-(q-2Δ)cos(ηξ))=limp→q2/4r±((-(q+Δ)(2Δ-q)Δ(Δ+q)2rξ2+o(Δ)2Δ(Δ+q)2r)×(2r[(Δ2)(1+Δ(Δ+q)2rξ2)q+4Δ-(q-2Δ)×(1+Δ(Δ+q)2rξ2)+o(Δ2)(1+Δ(Δ+q)2rξ2)]Δ(Δ+q)2r)-1)1/2=limp→q2/4r±((-(Δ+q)2(2Δ-q)2rξ2+o(Δ))×(+o(Δ)(1+Δ(Δ+q)2rξ2)])-12r[6-(q-2Δ)(1+Δ(Δ+q)2rξ2)+o(Δ)(1+Δ(Δ+q)2rξ2)](+o(Δ)(1+Δ(Δ+q)2rξ2)])-12r[6-(q-2Δ)(1+Δ(Δ+q)2rξ2))-1(1+Δ(Δ+q)2rξ2))1/2=±(q3/2r)ξ22r(6-(q2/2r)ξ2)=±-q2rqξq2ξ2-12r=φk±(see(51)).Hereto, we have completed the derivations for our main results.


## 4. Conclusions

In this paper, we have investigated the explicit expressions of the nonlinear waves and bifurcation phenomena in (1). Firstly, we obtained three types of explicit nonlinear wave solutions. The first type is the fractional expressions φa± and φk± (see (29) and (51)). The second type is the trigonometric expressions φi± and φj± (see (47) and (48)). The third type is the exp-function expressions φc±, φd±, φe±, and φf± (see (33) and (38)).Secondly, we revealed five kinds of interesting bifurcation phenomena in (1). The first phenomena is that the 1-blow-up waves can be bifurcated from the periodic-blow-up (see Figure 2 or Figure 13) and 2-blow-up waves (see Figure 8). The second phenomena is that the 2-blow-up waves can be bifurcated from the periodic-blow-up waves (see Figure 3). The third kind is that the symmetric solitary waves can be bifurcated from the symmetric periodic waves (see Figure 4). The fourth kind is that the low-kink waves can be bifurcated from the symmetric solitary waves (see Figure 6), the 1-blow-up waves (see Figure 7), the tall-kink waves (see Figure 9), and the antisymmetric solitary waves (see Figure 10). The fifth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves (see Figure 14).Thirdly, we showed that some previous results are our special cases. For instants,φb0±, φc0±, and φe0± are included in φc±, φd±, φe±, and φf± (see (4), (5), (7), (33), and (38)). φd0± are included in φc± and φd± (see (6) and (33)).Finally, we employed the software Mathematica to check the correctness of these solutions. For example, the commands forua and Ea are as follows:w=αγ2,p=(αγ2-w)/α,q=βδ1/α(c2-cs2)-δ2/α,r=-δ3/α,c=2αγ,ξ=x-ct,φa=6q/(3q2ξ2-4r),ua=(β/(c2-cs2))φa2,Ea=φaExp[i(γx-wt)],Simplify[D[ua,{t,2}]-cs2D[ua,{x,2}]-βD[φa2,{x,2}]]Simplify[iD[Ea,t]+αD[Ea,{x,2}]-δ1uaEa+δ2φa2Ea+δ3φa4Ea]00



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