# New Types of Nonlinear Waves and Bifurcation Phenomena in Schamel-Korteweg-de Vries Equation

Authors: Yun Wu; Zhengrong Liu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study the nonlinear waves described by Schamel-Korteweg-de Vries equationut+au1/2+buux+δuxxx=0. Two new types of nonlinear waves called compacton-like waves and kink-like waves are displayed. Furthermore, two kinds of new bifurcation phenomena are revealed. The first phenomenon is that the kink waves can be bifurcated from five types of nonlinear waves which are the bell-shape solitary waves, the blow-up waves, the valley-shape solitary waves, the kink-like waves, and the compacton-like waves. The second phenomenon is that the periodic-blow-up wave can be bifurcated from the smooth periodic wave.


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## Body

## 1. Introduction and Preliminary

Consider the following Schamel-Korteweg-de Vries (S-KdV) equation [1, 2]:
(1)ut+(au1/2+bu)ux+δuxxx=0,
where a, b, and δ are constants.Equation (1) arises in plasma physics in the study of ion acoustic solitons when electron trapping is present and also it governs the electrostatic potential for a certain electron distribution in velocity space. Tagare and Chakraborti [1] showed that (1) has solitary wave solution by applying direct integral method. Lee and Sakthivel [3] gave some exact traveling wave solutions of (1) by using exp-function method.Whenb=0, (1) becomes the Schamel equation [4]:
(2)ut+au1/2ux+δuxxx=0.Whena=0, (1) becomes a well-known KdV equation
(3)ut+buux+δuxxx=0,
which has been studied successively by many authors (e.g., [5–8]).The concept of compacton, soliton with compact support or strict localization of solitary waves, appeared in the work of Rosenau and Hyman [9] where a genuinely nonlinear dispersive equation K(n;n) is defined by
(4)ut+a(un)x+(un)xxx=0.
They found certain solitary wave solutions which vanish identically outside a finite core region. These solutions are called compactons.Several studies have been conducted in [10–20]. The aim of these studies was to examine if other nonlinear dispersive equations may generate compactons structures.In order to investigate the nonlinear waves of (1), letting c>0 be wave speed and substituting u=φ(ξ) with ξ=x-ct into (1), it follows that
(5)-cφ′+aφ1/2φ′+bφφ′+δφ′′′=0.
Integrating (5), we get
(6)-cφ+23aφ3/2+b2φ2+δφ′′=0.
Setting φ′=y yields the following planar system:
(7)φ′=y,y′=cδφ-2a3δφ3/2-b2δφ2.
Obviously, system (7) is a Hamiltonian system with Hamiltonian function
(8)H(φ,y)=y2-cδφ2+8a15δφ5/2+b3δφ3.
If one puts
(9)f(φ)=cδφ-2a3δφ3/2-b2δφ2,Δ=4a2+18bc,
then one can see the following facts.WhenΔ>0, f(φ) has three zero points φ0, φ1, and φ2 which possess expressions
(10)φ0=0,φ1=(-2a-Δ3b)2,φ2=(-2a+Δ3b)2.
When Δ=0, f(φ) has two zero points φ0 and φ* which are denoted by
(11)φ0=0,φ*=4a29b2.
When Δ<0, f(φ) has one zero point φ0=0.Letting(φ~,0) be one of the singular points of system (7), then the characteristic values at (φ~,0) are
(12)λ=±f′(φ~).From the qualitative theory of dynamical systems, we get the following conclusions:(1)
iff′(φ~)>0, then (φ~,0) is a saddle point,(2)
iff′(φ~)<0, then (φ~,0) is a center point,(3)
iff′(φ~)=0, then (φ~,0) is a degenerate saddle point.Ona-b parametric plane, let l1, l2, and l3, respectively, represent the following three curves:
(13)l1:b=0,l2:b=-16a275c,l3:b=-2a29c.
Let Ai  (i=1,2,…,6) represent the regions surrounded by l1, l2, l3, and the coordinate axes (see Figures 1 and 2).Figure 1
The bifurcation phase portraits of system (7) when δ>0.Figure 2
The bifurcation phase portraits of system (7) when δ<0.According to the previous analysis, we obtain the bifurcation phase portraits of system (7) as in Figures 1 and 2.In this paper, we study the nonlinear waves and their bifurcations in (1) by using the bifurcation method of dynamical systems [21–23]. We point out that there are two new types of nonlinear waves, kink-like waves and compacton-like waves [24–33]. Furthermore, we reveal two kinds of new bifurcation phenomena which are introduced in the abstract.This paper is organized as follows. In Section2, we display the two new types of nonlinear waves. We show the two kinds of new bifurcation phenomena in Sections 3 and 4. A brief conclusion is given in Section 5.


## 2. Two New Types of Nonlinear Waves

In this section, we display two new types of nonlinear waves defined by (1).


### 2.1. Kink-Like Waves

Proposition 1.
(1) When the parameters satisfyδ>0, a>0, and -16a2/75c<b<0, (1) has a kink-like wave solution ua+=φ(ξ) and an antikink-like wave solution ua-=φ(ξ), respectively, which are hidden in the following equations:
(14)∫φ1/2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h1=ξ,0000000000000000000000000000000ξ∈(-ξ10,∞),(15)∫φ1/2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h1=-ξ,00000000000000000000000000000000ξ∈(-∞,ξ10),
where
(16)ξ10=∫0φ1/2‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h1
and h1=H(φ1,0).
(2)   When the parameters satisfy one of the following Cases.
Case  1.
δ
<
0, a>0, and -2a2/9c<b<0,
Case  2.
δ
<
0, and b>0,
Case  3.
δ
<
0, a>0, and b=0,
Case  4.δ<0, a>0, and b=-2a2/9c.
Equation (1) has a kink-like wave solution ub+=φ(ξ) and an antikink-like wave solution ub-=φ(ξ), respectively, which are hidden in the following equations:
(17)∫φ2/2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h2=ξ,0000000000000000000000000000000ξ∈(-ξ20,∞),∫φ2/2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h2=-ξ,00000000000000000000000000000000ξ∈(-∞,ξ20),
where
(18)ξ20=∫0φ2/2‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h2
and h2=H(φ2,0).Proof.
(1) Under the conditionδ>0, a>0, and -16a2/75c<b<0, (φ1,0) is a saddle point and on its left side there are two orbits Γ1± connecting with it (see Figure 3(a1)).
In (8), letting h1=H(φ1,0), it follows that
(19)Γ1±:y=±cδφ2-8a15δφ5/2-b3δφ3+h1(0<φ<φ1).
OnΓ1± suppose φ(0)=φ1/2. Substituting (19) into (7) and integrating them along Γ1+ and Γ1-, respectively, we get (14)–(16).
(2) Under one of Cases  1–4, (φ2,0) is a saddle point and on its left side there are two orbits Γ2± connecting with it (see Figures 3(a2)–3(a4)).
In (8), letting h2=H(φ2,0), it follows that
(20)Γ2±:y=±cδφ2-8a15δφ5/2-b3δφ3+h2(0<φ<φ2).
Similar to the proof of (1), we get the results of (2).Figure 3
The graphs of orbitsΓ1± and Γ2± when (a1) δ>0 and (a,b)∈A2, (a2) δ<0 and (a,b)∈A2 or A3 or l2, (a3) δ<0 and (a,b)∈A1 or A6 or l1(a>0), and (a4) δ<0 and (a,b)∈l3.Next, we simulate the planar graphs of the kink-like waves for those data given in Example2.Example 2 (Corresponding to Proposition1  (1)).
Givinga=1/2, b=-2/45, c=1, and δ=1, we get φ1=117.81152949 and φ(0)=φ1/2=58.90576474. Note that orbits Γ1± have expressions (19). From (19) we get y1+(0)=38.27939786 and y1-(0)=-38.27939786. These imply that Γ1+ passes point (φ(0),y1+(0)) and Γ1- passes point (φ(0),y1-(0)). Thus letting φ(0)=φ1/2 and φ′(0)=y1+(0) as the initial conditions of (6), we get the simulation of the integral curve which corresponds to Γ1+ as Figure 4(a1). Meanwhile, choosing φ(0)=φ1/2 and φ′(0)=y1-(0) as the initial conditions of (6), we get the simulation of the integral curve which corresponds to Γ1- as in Figure 4(a2).Figure 4
(Corresponding to Example2). The simulation of integral curve of (6) when a=1/2, b=-2/45, c=1, and δ=1. (a1) Initial conditions φ(0)=58.90576474 and φ′(0)=38.27939786. (a2) Initial conditions φ(0)=58.90576474 and φ′(0)=-38.27939786.


### 2.2. Compacton-Like Waves

Proposition 3.
Letφk be an initial value when parameters and initial value satisfy one of the following Cases.
Case  1. δ>0, b>0, and φk>γ1,
Case  2. δ>0, a>0, b=0, and φk>γ1,
Case  3. δ>0, a>0, -16a2/75c<b<0, and γ1<φk<φ1,
Case  4. δ<0, b>0, and 0<φk<φ2,
Case  5. δ<0, a>0, b=0, and 0<φk<φ2,
Case  6. δ<0, a<0, b≤0, and φk>0,
Case  7. δ<0, a>0, b<-2a2/9c, and φk>0,
Case  8. δ<0, a>0, b=-2a2/9c, and φk≠γ2(=φ2),
Case  9. δ<0, a>0, -2a2/9c<b<0, and φk>γ2 or 0<φk<φ2.
Equation (1) has compacton-like wave solutions uc+=φ(ξ) and uc-=φ(ξ), respectively, which are hidden in the following equations:
(21)∫φk/2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+hφk=ξ,0000000000000000000000000000000ξ∈(-ξ30,ξ40),∫φk/2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+hφk=-ξ,00000000000000000000000000000000ξ∈(-ξ40,ξ30),
where
(22)ξ30=∫0φk/2‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+hφk,(23)ξ40=2∫φk/2φk‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+hφk+ξ30,
and hφk=H(φk,0).Proof.
Under one of Cases  1–9, there is an orbitΓφk passing point (φk,0) (see Figure 5 (a1)–5(a6)).
In (8), letting hφk=H(φk,0), it follows that
(24)Γφk:y2=cδφ2-8a15δφ5/2-b3δφ3+hφk(0<φ<φk).
OnΓφk suppose φ(0)=φk/2. Substituting (24) into (7) and integrating it along Γφk, respectively, we obtain (21)–(23).Figure 5
The graphs of orbitΓφk when (a1) δ>0 and (a,b)∈A1 or A6 or l1(a>0), (a2) δ>0 and (a,b)∈A2, (a3) δ<0 and (a,b)∈A1 or A6 or l1(a>0), (a4) δ<0 and (a,b)∈A4 or A5 or l1(a<0), (a5) δ<0 and (a,b)∈l3, and (a6) δ<0 and (a,b)∈A2 or A3 or l2.Next, we simulate the planar graphs of the compacton-like waves for those data given in Example4.Example 4 (Corresponding to Proposition3 Case  (3)).
Givinga=1/2, b=-157/3000, c=1, and δ=1, we get φ1=62.46353941 and γ1=43.51656288. Note that orbit Γφk has expression (24). Letting φk=(φ1+γ1)/2, it follows that φ(0)=26.49502557. From (24) we get y(0)=±10.49025102. Thus letting φ(0)=26.49502557 and φ′(0)=10.49025102 as the initial conditions of (6), we get the simulation of the integral curve as in Figure 6(a1). Meanwhile, choosing φ(0)=26.49502557 and φ′(0)=-10.49025102 as the initial conditions of (6), we get the simulation of the integral curve as Figure 6(a2).Figure 6
(Corresponding to Example4). The simulation of integral curve of (6) when a=1, b=-157/3000, c=1, and δ=1. (a1) Initial conditions φ(0)=26.49502557 and φ′(0)=10.49025102. (a2) Initial conditions φ(0)=26.49502557 and φ′(0)=-10.49025102.


## 2.1. Kink-Like Waves

Proposition 1.
(1) When the parameters satisfyδ>0, a>0, and -16a2/75c<b<0, (1) has a kink-like wave solution ua+=φ(ξ) and an antikink-like wave solution ua-=φ(ξ), respectively, which are hidden in the following equations:
(14)∫φ1/2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h1=ξ,0000000000000000000000000000000ξ∈(-ξ10,∞),(15)∫φ1/2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h1=-ξ,00000000000000000000000000000000ξ∈(-∞,ξ10),
where
(16)ξ10=∫0φ1/2‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h1
and h1=H(φ1,0).
(2)   When the parameters satisfy one of the following Cases.
Case  1.
δ
<
0, a>0, and -2a2/9c<b<0,
Case  2.
δ
<
0, and b>0,
Case  3.
δ
<
0, a>0, and b=0,
Case  4.δ<0, a>0, and b=-2a2/9c.
Equation (1) has a kink-like wave solution ub+=φ(ξ) and an antikink-like wave solution ub-=φ(ξ), respectively, which are hidden in the following equations:
(17)∫φ2/2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h2=ξ,0000000000000000000000000000000ξ∈(-ξ20,∞),∫φ2/2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h2=-ξ,00000000000000000000000000000000ξ∈(-∞,ξ20),
where
(18)ξ20=∫0φ2/2‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h2
and h2=H(φ2,0).Proof.
(1) Under the conditionδ>0, a>0, and -16a2/75c<b<0, (φ1,0) is a saddle point and on its left side there are two orbits Γ1± connecting with it (see Figure 3(a1)).
In (8), letting h1=H(φ1,0), it follows that
(19)Γ1±:y=±cδφ2-8a15δφ5/2-b3δφ3+h1(0<φ<φ1).
OnΓ1± suppose φ(0)=φ1/2. Substituting (19) into (7) and integrating them along Γ1+ and Γ1-, respectively, we get (14)–(16).
(2) Under one of Cases  1–4, (φ2,0) is a saddle point and on its left side there are two orbits Γ2± connecting with it (see Figures 3(a2)–3(a4)).
In (8), letting h2=H(φ2,0), it follows that
(20)Γ2±:y=±cδφ2-8a15δφ5/2-b3δφ3+h2(0<φ<φ2).
Similar to the proof of (1), we get the results of (2).Figure 3
The graphs of orbitsΓ1± and Γ2± when (a1) δ>0 and (a,b)∈A2, (a2) δ<0 and (a,b)∈A2 or A3 or l2, (a3) δ<0 and (a,b)∈A1 or A6 or l1(a>0), and (a4) δ<0 and (a,b)∈l3.Next, we simulate the planar graphs of the kink-like waves for those data given in Example2.Example 2 (Corresponding to Proposition1  (1)).
Givinga=1/2, b=-2/45, c=1, and δ=1, we get φ1=117.81152949 and φ(0)=φ1/2=58.90576474. Note that orbits Γ1± have expressions (19). From (19) we get y1+(0)=38.27939786 and y1-(0)=-38.27939786. These imply that Γ1+ passes point (φ(0),y1+(0)) and Γ1- passes point (φ(0),y1-(0)). Thus letting φ(0)=φ1/2 and φ′(0)=y1+(0) as the initial conditions of (6), we get the simulation of the integral curve which corresponds to Γ1+ as Figure 4(a1). Meanwhile, choosing φ(0)=φ1/2 and φ′(0)=y1-(0) as the initial conditions of (6), we get the simulation of the integral curve which corresponds to Γ1- as in Figure 4(a2).Figure 4
(Corresponding to Example2). The simulation of integral curve of (6) when a=1/2, b=-2/45, c=1, and δ=1. (a1) Initial conditions φ(0)=58.90576474 and φ′(0)=38.27939786. (a2) Initial conditions φ(0)=58.90576474 and φ′(0)=-38.27939786.


## 2.2. Compacton-Like Waves

Proposition 3.
Letφk be an initial value when parameters and initial value satisfy one of the following Cases.
Case  1. δ>0, b>0, and φk>γ1,
Case  2. δ>0, a>0, b=0, and φk>γ1,
Case  3. δ>0, a>0, -16a2/75c<b<0, and γ1<φk<φ1,
Case  4. δ<0, b>0, and 0<φk<φ2,
Case  5. δ<0, a>0, b=0, and 0<φk<φ2,
Case  6. δ<0, a<0, b≤0, and φk>0,
Case  7. δ<0, a>0, b<-2a2/9c, and φk>0,
Case  8. δ<0, a>0, b=-2a2/9c, and φk≠γ2(=φ2),
Case  9. δ<0, a>0, -2a2/9c<b<0, and φk>γ2 or 0<φk<φ2.
Equation (1) has compacton-like wave solutions uc+=φ(ξ) and uc-=φ(ξ), respectively, which are hidden in the following equations:
(21)∫φk/2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+hφk=ξ,0000000000000000000000000000000ξ∈(-ξ30,ξ40),∫φk/2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+hφk=-ξ,00000000000000000000000000000000ξ∈(-ξ40,ξ30),
where
(22)ξ30=∫0φk/2‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+hφk,(23)ξ40=2∫φk/2φk‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+hφk+ξ30,
and hφk=H(φk,0).Proof.
Under one of Cases  1–9, there is an orbitΓφk passing point (φk,0) (see Figure 5 (a1)–5(a6)).
In (8), letting hφk=H(φk,0), it follows that
(24)Γφk:y2=cδφ2-8a15δφ5/2-b3δφ3+hφk(0<φ<φk).
OnΓφk suppose φ(0)=φk/2. Substituting (24) into (7) and integrating it along Γφk, respectively, we obtain (21)–(23).Figure 5
The graphs of orbitΓφk when (a1) δ>0 and (a,b)∈A1 or A6 or l1(a>0), (a2) δ>0 and (a,b)∈A2, (a3) δ<0 and (a,b)∈A1 or A6 or l1(a>0), (a4) δ<0 and (a,b)∈A4 or A5 or l1(a<0), (a5) δ<0 and (a,b)∈l3, and (a6) δ<0 and (a,b)∈A2 or A3 or l2.Next, we simulate the planar graphs of the compacton-like waves for those data given in Example4.Example 4 (Corresponding to Proposition3 Case  (3)).
Givinga=1/2, b=-157/3000, c=1, and δ=1, we get φ1=62.46353941 and γ1=43.51656288. Note that orbit Γφk has expression (24). Letting φk=(φ1+γ1)/2, it follows that φ(0)=26.49502557. From (24) we get y(0)=±10.49025102. Thus letting φ(0)=26.49502557 and φ′(0)=10.49025102 as the initial conditions of (6), we get the simulation of the integral curve as in Figure 6(a1). Meanwhile, choosing φ(0)=26.49502557 and φ′(0)=-10.49025102 as the initial conditions of (6), we get the simulation of the integral curve as Figure 6(a2).Figure 6
(Corresponding to Example4). The simulation of integral curve of (6) when a=1, b=-157/3000, c=1, and δ=1. (a1) Initial conditions φ(0)=26.49502557 and φ′(0)=10.49025102. (a2) Initial conditions φ(0)=26.49502557 and φ′(0)=-10.49025102.


## 3. Bifurcation of the Kink Waves

In this section, we show that the kink waves can be bifurcated from five other waves.


### 3.1. Bifurcation from Bell-Shape Solitary Waves

Proposition 5.
Forab≠0 and δ>0, (1) has two nonlinear wave solutions
(25)ud=(4αλλ2e-τ1ξ-2λβ+(β2-4α)eτ1ξ)2,ue=(4αλλ2eτ1ξ-2λβ+(β2-4α)e-τ1ξ)2,
where
(26)α=pq,β=-(p+q),τ1=12cδ,(27)p=γ1=-4a+16a2+75bc5b,(28)q=-4a-16a2+75bc5b,
and λ≠0 is an arbitrary real number. These solutions possess the following properties.(1)
Ifλ>0,a>0, andb=-16a2/75c, thenudanduebecome(29)ud*=(4αλe-τ1ξ-2β)2,ue*=(4αλeτ1ξ-2β)2,which represent a kink wave and an antikink wave.
In particular, whenα=pq=-3c/b, β=-(p+q)=8a/5b, c=-16a2/75b, λ=a0, and ξ=x-ct, ud* and ue* become
(30)u1=16a02a2(4aexp(±(2/5)-1/3bδa(x+(16a2/75b)t))-5a0b)2,
which was given by Lee and Sakthivel [3]. This implies that u1 is the special case of ud or ue.(2)
Ifλ=β2-4α, thenud=ueand become(31)ude*=(2αβ2-4αcosh(τ1ξ)-β)2.
When(a,b) belongs to one of the regions A2, A6, ude* represents a hyperbolic solitary wave.
In particular, whenα=pq=-3c/b, and β=-(p+q)=8a/5b, ude* becomes
(32)u2=(15c4a+16a2+75bccosh((1/2)c/δξ))2,
which was obtained by Tagare and Chakraborti [3]. This implies that u2 is the special case of ud or ue.(3)
Under one of the following Cases.
Case  1. λ>0, λ≠β2-4α, and (a,b) belongs to one of the regions A2, A6,
Case  2. λ<0, λ≠-β2-4α, and (a,b)∈A1, ud≠ue and they represent two bell-shape solitary waves.
In particular, when(a,b)∈A2 in Case  1 and b→0-0, ud and ue become
(33)ud∘=225c24a2(1+e-τ1ξ)4,(34)ue∘=225c24a2(1+eτ1ξ)4,
which are the solutions of the Schamel equation.
When(a,b)∈A2 in Case  1 and b→-16a2/75c+0, the two bell-shape solitary waves ud and ue become a kink wave and an antikink wave with the expressions (29). For the varying process, see Figures 7 and 8.(The kink wave is bifurcated from the bell-shape solitary wave). The varying process for the graph ofud when c>0, δ>0, a>0, λ>0 and b→-16a2/75c+0. Where c=1, δ=1, a=1, λ=50 and (a) b=-16a2/75c+10-1. (b) b=-16a2/75c+10-3. (c) b=-16a2/75c+10-4. (d) b=-16a2/75c+10-7.
(a)
(b)
(c)
(d)(The anti-kink wave is bifurcated from the bell-shape solitary wave). The varying process for the graph ofue when c>0, δ>0, a>0, λ>0 and b→-16a2/75c+0. Where c=1, δ=1, a=1, λ=50 and (a) b=-16a2/75c+10-1. (b) b=-16a2/75c+10-3. (c) b=-16a2/75c+10-4. (d) b=-16a2/75c+10-7.
(a)
(b)
(c)
(d)Proof.
In (8), letting h0=H(0,0), it follows that
(35)y=±cδφ2-8a15δφ5/2-b3δφ3.
Substituting (35) into dφ/dξ=y, we have
(36)dφdξ=±cδφ2-8a15δφ5/2-b3δφ3.
Let φ=w2, (36) becomes
(37)2wdwdξ=±cδw4-8a15δw5-b3δw6.
Integrating (37), we have
(38)∫νw‍dssc/δ-(8a/15δ)s-(b/3δ)s2=±12ξ,
where ν is an arbitrary constant.
Completing the previous integral and solving the equation forφ, it follows that
(39)φ=(4αλe∓τ1ξλ2e∓2τ1ξ-2λβe∓τ1ξ+(β2-4α))2,
where λ=λ(ν) is an arbitrary real number. From (39) we obtain the solutions ud and ue as (25).
In (25) letting b=-16a2/75c, we get (29). From (25) and (29), we get the result (1) of Proposition 5.
Whenλ=β2-4α, via (25) it follows that
(40)ud=ue=(4αλ(e-τ1ξ+eτ1ξ)-2β)2=(2αλcoshτ1ξ-β)2=ude*
(see (31)).
Thus, we get the result (2) of Proposition5.
In (38), letting ν=p (see (27)), it follows that
(41)λ=βp+2αp=-216a2+75bc5b.
Lettingb→0-0, then
(42)limb→0-0αλ=limb→0-03cb·5b216a2+75bc=limb→015c216a2+75bc=15c8a,(43)limb→0-0βλ=limb→0-0-8a5b·5b216a2+75bc=-1,(44)limb→0-0β2-4αλ2=limb→0-0(βλ)2-4αλ·1λ=1.
We have
(45)limb→0-0ud=limb→0-0(4αλλ2e-τ1ξ-2λβ+(β2-4α)eτ1ξ)2=limb→0-0(4α/λe-τ1ξ-2(β/λ)+((β2-4α)/λ2)·eτ1ξ)2=ud∘
(see (33)).
Similarly, we have(46)limb→0-0ue=ue∘
(see (34)).
From (41)–(46), we get result (3) of Proposition 5.


### 3.2. Bifurcation from Blow-Up Waves

Proposition 6.
Forab≠0 and δ>0, (1) has two nonlinear wave solutions as ud and ue. These solutions possess the following properties.
(1) Under one of the following Cases.
Case  1. λ>0, λ≠β2-4α, and (a,b) belongs to one of the regions A1, A3,
Case  2. λ<0, λ≠-β2-4α, and (a,b) belongs to any one of the regions A2, A3, and A6, ud≠ue and they represent two blow-up waves.
In particular, when(a,b)∈A3 in Case  1 and b→-16a2/75c-0, the two blow-up waves become a kink wave and an antikink wave with the expressions (29). For the varying process, see Figures 9 and 10.(The kink wave is bifurcated from the blow-up wave). The varying process for the graph ofud when c>0, δ>0, a>0, λ>0, and b→-16a2/75c-0, where c=1, δ=1, a=1, λ=1, and (a) b=-16a2/75c-10-2. (b) b=-16a2/75c-10-4. (c) b=-16a2/75c-10-5. (d) b=-16a2/75c-10-9.
(a)
(b)
(c)
(d)(The antikink wave is bifurcated from the blow-up wave). The varying process for the graph ofue when c>0, δ>0, a>0, λ>0, and b→-16a2/75c-0, where c=1, δ=1, a=1, λ=1, and (a) b=-16a2/75c-10-2. (b) b=-16a2/75c-10-4. (c) b=-16a2/75c-10-5. (d) b=-16a2/75c-10-9.
(a)
(b)
(c)
(d)Similar to the proof of Proposition5, we get the results of Proposition 6.


### 3.3. Bifurcation from Valley-Shape Solitary Waves

Proposition 7.
When the parameters satisfyδ>0 and (a,b)∈A3, (1) has two valley-shape solitary wave solutions uf=φ(ξ) and ug=φ(ξ), respectively, which are hidden in the following equations:
(47)∫φ2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h1=ξ000000000000000000000000000000(0<φ<φ1),(48)∫φ2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h1=-ξ0000000000000000000000000000000(0<φ<φ1).
In particular, when b→-16a2/75c-0, the two valley-shape solitary waves become a kink wave and an antikink wave with the expressions (29). For the varying process, see Figures 11 and 12.(The kink wave is bifurcated from the valley-shape solitary wave). The varying process for the graph ofuf when c>0, δ>0, a>0, and b→-16a2/75c-0, where c=1, δ=1, a=1/2, and (a) b=-16a2/75c-10-3. (b) b=-16a2/75c-10-7. (c) b=-16a2/75c-10-8. (d) b=-16a2/75c-10-10.
(a)
(b)
(c)
(d)(The antikink wave is bifurcated from the valley-shape solitary wave). The varying process for the graph ofug when c>0, δ>0, a>0, and b→-16a2/75c-0, where c=1, δ=1, a=1/2, and (a) b=-16a2/75c-10-3. (b) b=-16a2/75c-10-7. (c) b=-16a2/75c-10-8. (d) b=-16a2/75c-10-10.
(a)
(b)
(c)
(d)Proof.
Whenδ>0 and (a,b)∈A3, (φ1,0) is a saddle point and on its left side there is an orbit Γ3 connecting with it (see Figure 13).
In (8), letting h1=H(φ1,0), it follows that
(49)Γ3:y2=cδφ2-8a15δφ5/2-b3δφ3+h1(0<φ<φ1).
Substituting (49) into (7) and integrating it along the orbit Γ3, we get (47) and (48).
Lettingb→-16a2/75c-0, it follows that
(50)limb→-16a2/75c-0h1=limb→-16a2/75c-0H(φ1,0)=limb→-16a2/75c-0-cδφ12+8a15δφ15/2+b3δφ13=0.
Whenh1→0, completing the integrals in (47) and (48), we get the kink wave solution and the antikink wave solution as (29).
Hereto, we have completed the proof for the Proposition7.Figure 13
The graph of orbitΓ3 when δ>0 and (a,b)∈A3.


### 3.4. Bifurcation from Kink-Like Waves

Proposition 8.
When the parameters satisfyδ>0, (a,b)∈A2, and b→-16a2/75c+0, the kink-like wave and the antikink-like wave, respectively, become a kink wave and an anti-kink wave with the expressions (29).
For the varying process, see Figures14 and 15.(The kink wave is bifurcated from the kink-like wave). The varying process for the graph ofua+ when c>0, δ>0, a>0, and b→-16a2/75c+0, where c=1, δ=1, a=1/2, and (a) b=-16a2/75c+10-3. (b) b=-16a2/75c+10-4. (c) b=-16a2/75c+10-6. (d) b=-16a2/75c+10-8.
(a)
(b)
(c)
(d)(The antikink wave is bifurcated from the antikink-like wave). The varying process for the graph ofua- when c>0, δ>0, a>0, and b→-16a2/75c+0, where c=1, δ=1, a=1/2, and (a) b=-16a2/75c+10-3. (b) b=-16a2/75c+10-4. (c) b=-16a2/75c+10-6. (d) b=-16a2/75c+10-8.
(a)
(b)
(c)
(d)Proof.
Lettingb→-16a2/75c+0, it follows that h1→0 (see (50)) and
(51)limb→-16a2/75c+0ξ10=∫0φ1/2‍ds(c/δ)s2-(8a/15δ)s5/2+(16a2/225cδ)s3=limν→0∫νφ1/2‍ds(c/δ)s2-(8a/15δ)s5/2+(16a2/225cδ)s3=∞.
Whenh1→0, and ξ10→∞, completing the integrals in (17), we get the kink wave solution and the antikink wave solution as (29).
Hereto, we have completed the proof for the Proposition8.


### 3.5. Bifurcation from Compacton-Like Waves

Proposition 9.
When the parameters satisfyδ>0, (a,b)∈A2, and b→-16a2/75c+0, the two compacton-like waves become a kink wave and an anti-kink wave with the expressions (29).
For the varying process, see Figures16 and 17.(The kink wave is bifurcated from the compacton-like wave). The varying process for the graph ofuc+ when c>0, δ>0, a>0, and b→-16a2/75c+0, where c=1, δ=1, a=1/2, and (a) b=-16a2/75c+10-3. (b) b=-16a2/75c+10-4. (c) b=-16a2/75c+10-6. (d) b=-16a2/75c+10-8.
(a)
(b)
(c)
(d)(The antikink wave is bifurcated from the compacton-like wave). The varying process for the graph ofuc- when c>0, δ>0, a>0, and b→-16a2/75c+0, where c=1, δ=1, a=1/2, and (a) b=-16a2/75c+10-3. (b) b=-16a2/75c+10-4. (c) b=-16a2/75c+10-6. (d) b=-16a2/75c+10-8.
(a)
(b)
(c)
(d)Proof.
Lettingb→-16a2/75c+0, it follows that
(52)limb→-16a2/75c+0hφk=limb→-16a2/75c+0H(φk,0)=limb→-16a2/75c+0-cδφk2+8a15δφk5/2+b3δφk3=0,(53)limb→-16a2/75c+0ξ30=∫0φk/2‍ds(c/δ)s2-(8a/15δ)s5/2+(16a2/225cδ)s3=limν→0∫νφk‍ds(c/δ)s2-(8a/15δ)s5/2+(16a2/225cδ)s3=∞,(54)limb→-16a2/75c+0ξ40=limb→-16a2/75c+0(2∫φk/2φk‍(ds)((c/δ)s2-(8a/15δ)s5/200000000000000000000000000-(b/3δ)s3+hφk)-1/2)00000000000000+ξ30∫φk/2φk‍)=∞.
Whenhφk→0, ξ30→∞, and ξ40→∞, completing the integrals in (21), we get the kink wave solution and the antikink wave solution as (29).
Hereto, we have completed the proof for the Proposition9.


## 3.1. Bifurcation from Bell-Shape Solitary Waves

Proposition 5.
Forab≠0 and δ>0, (1) has two nonlinear wave solutions
(25)ud=(4αλλ2e-τ1ξ-2λβ+(β2-4α)eτ1ξ)2,ue=(4αλλ2eτ1ξ-2λβ+(β2-4α)e-τ1ξ)2,
where
(26)α=pq,β=-(p+q),τ1=12cδ,(27)p=γ1=-4a+16a2+75bc5b,(28)q=-4a-16a2+75bc5b,
and λ≠0 is an arbitrary real number. These solutions possess the following properties.(1)
Ifλ>0,a>0, andb=-16a2/75c, thenudanduebecome(29)ud*=(4αλe-τ1ξ-2β)2,ue*=(4αλeτ1ξ-2β)2,which represent a kink wave and an antikink wave.
In particular, whenα=pq=-3c/b, β=-(p+q)=8a/5b, c=-16a2/75b, λ=a0, and ξ=x-ct, ud* and ue* become
(30)u1=16a02a2(4aexp(±(2/5)-1/3bδa(x+(16a2/75b)t))-5a0b)2,
which was given by Lee and Sakthivel [3]. This implies that u1 is the special case of ud or ue.(2)
Ifλ=β2-4α, thenud=ueand become(31)ude*=(2αβ2-4αcosh(τ1ξ)-β)2.
When(a,b) belongs to one of the regions A2, A6, ude* represents a hyperbolic solitary wave.
In particular, whenα=pq=-3c/b, and β=-(p+q)=8a/5b, ude* becomes
(32)u2=(15c4a+16a2+75bccosh((1/2)c/δξ))2,
which was obtained by Tagare and Chakraborti [3]. This implies that u2 is the special case of ud or ue.(3)
Under one of the following Cases.
Case  1. λ>0, λ≠β2-4α, and (a,b) belongs to one of the regions A2, A6,
Case  2. λ<0, λ≠-β2-4α, and (a,b)∈A1, ud≠ue and they represent two bell-shape solitary waves.
In particular, when(a,b)∈A2 in Case  1 and b→0-0, ud and ue become
(33)ud∘=225c24a2(1+e-τ1ξ)4,(34)ue∘=225c24a2(1+eτ1ξ)4,
which are the solutions of the Schamel equation.
When(a,b)∈A2 in Case  1 and b→-16a2/75c+0, the two bell-shape solitary waves ud and ue become a kink wave and an antikink wave with the expressions (29). For the varying process, see Figures 7 and 8.(The kink wave is bifurcated from the bell-shape solitary wave). The varying process for the graph ofud when c>0, δ>0, a>0, λ>0 and b→-16a2/75c+0. Where c=1, δ=1, a=1, λ=50 and (a) b=-16a2/75c+10-1. (b) b=-16a2/75c+10-3. (c) b=-16a2/75c+10-4. (d) b=-16a2/75c+10-7.
(a)
(b)
(c)
(d)(The anti-kink wave is bifurcated from the bell-shape solitary wave). The varying process for the graph ofue when c>0, δ>0, a>0, λ>0 and b→-16a2/75c+0. Where c=1, δ=1, a=1, λ=50 and (a) b=-16a2/75c+10-1. (b) b=-16a2/75c+10-3. (c) b=-16a2/75c+10-4. (d) b=-16a2/75c+10-7.
(a)
(b)
(c)
(d)Proof.
In (8), letting h0=H(0,0), it follows that
(35)y=±cδφ2-8a15δφ5/2-b3δφ3.
Substituting (35) into dφ/dξ=y, we have
(36)dφdξ=±cδφ2-8a15δφ5/2-b3δφ3.
Let φ=w2, (36) becomes
(37)2wdwdξ=±cδw4-8a15δw5-b3δw6.
Integrating (37), we have
(38)∫νw‍dssc/δ-(8a/15δ)s-(b/3δ)s2=±12ξ,
where ν is an arbitrary constant.
Completing the previous integral and solving the equation forφ, it follows that
(39)φ=(4αλe∓τ1ξλ2e∓2τ1ξ-2λβe∓τ1ξ+(β2-4α))2,
where λ=λ(ν) is an arbitrary real number. From (39) we obtain the solutions ud and ue as (25).
In (25) letting b=-16a2/75c, we get (29). From (25) and (29), we get the result (1) of Proposition 5.
Whenλ=β2-4α, via (25) it follows that
(40)ud=ue=(4αλ(e-τ1ξ+eτ1ξ)-2β)2=(2αλcoshτ1ξ-β)2=ude*
(see (31)).
Thus, we get the result (2) of Proposition5.
In (38), letting ν=p (see (27)), it follows that
(41)λ=βp+2αp=-216a2+75bc5b.
Lettingb→0-0, then
(42)limb→0-0αλ=limb→0-03cb·5b216a2+75bc=limb→015c216a2+75bc=15c8a,(43)limb→0-0βλ=limb→0-0-8a5b·5b216a2+75bc=-1,(44)limb→0-0β2-4αλ2=limb→0-0(βλ)2-4αλ·1λ=1.
We have
(45)limb→0-0ud=limb→0-0(4αλλ2e-τ1ξ-2λβ+(β2-4α)eτ1ξ)2=limb→0-0(4α/λe-τ1ξ-2(β/λ)+((β2-4α)/λ2)·eτ1ξ)2=ud∘
(see (33)).
Similarly, we have(46)limb→0-0ue=ue∘
(see (34)).
From (41)–(46), we get result (3) of Proposition 5.


## 3.2. Bifurcation from Blow-Up Waves

Proposition 6.
Forab≠0 and δ>0, (1) has two nonlinear wave solutions as ud and ue. These solutions possess the following properties.
(1) Under one of the following Cases.
Case  1. λ>0, λ≠β2-4α, and (a,b) belongs to one of the regions A1, A3,
Case  2. λ<0, λ≠-β2-4α, and (a,b) belongs to any one of the regions A2, A3, and A6, ud≠ue and they represent two blow-up waves.
In particular, when(a,b)∈A3 in Case  1 and b→-16a2/75c-0, the two blow-up waves become a kink wave and an antikink wave with the expressions (29). For the varying process, see Figures 9 and 10.(The kink wave is bifurcated from the blow-up wave). The varying process for the graph ofud when c>0, δ>0, a>0, λ>0, and b→-16a2/75c-0, where c=1, δ=1, a=1, λ=1, and (a) b=-16a2/75c-10-2. (b) b=-16a2/75c-10-4. (c) b=-16a2/75c-10-5. (d) b=-16a2/75c-10-9.
(a)
(b)
(c)
(d)(The antikink wave is bifurcated from the blow-up wave). The varying process for the graph ofue when c>0, δ>0, a>0, λ>0, and b→-16a2/75c-0, where c=1, δ=1, a=1, λ=1, and (a) b=-16a2/75c-10-2. (b) b=-16a2/75c-10-4. (c) b=-16a2/75c-10-5. (d) b=-16a2/75c-10-9.
(a)
(b)
(c)
(d)Similar to the proof of Proposition5, we get the results of Proposition 6.


## 3.3. Bifurcation from Valley-Shape Solitary Waves

Proposition 7.
When the parameters satisfyδ>0 and (a,b)∈A3, (1) has two valley-shape solitary wave solutions uf=φ(ξ) and ug=φ(ξ), respectively, which are hidden in the following equations:
(47)∫φ2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h1=ξ000000000000000000000000000000(0<φ<φ1),(48)∫φ2φ‍ds(c/δ)s2-(8a/15δ)s5/2-(b/3δ)s3+h1=-ξ0000000000000000000000000000000(0<φ<φ1).
In particular, when b→-16a2/75c-0, the two valley-shape solitary waves become a kink wave and an antikink wave with the expressions (29). For the varying process, see Figures 11 and 12.(The kink wave is bifurcated from the valley-shape solitary wave). The varying process for the graph ofuf when c>0, δ>0, a>0, and b→-16a2/75c-0, where c=1, δ=1, a=1/2, and (a) b=-16a2/75c-10-3. (b) b=-16a2/75c-10-7. (c) b=-16a2/75c-10-8. (d) b=-16a2/75c-10-10.
(a)
(b)
(c)
(d)(The antikink wave is bifurcated from the valley-shape solitary wave). The varying process for the graph ofug when c>0, δ>0, a>0, and b→-16a2/75c-0, where c=1, δ=1, a=1/2, and (a) b=-16a2/75c-10-3. (b) b=-16a2/75c-10-7. (c) b=-16a2/75c-10-8. (d) b=-16a2/75c-10-10.
(a)
(b)
(c)
(d)Proof.
Whenδ>0 and (a,b)∈A3, (φ1,0) is a saddle point and on its left side there is an orbit Γ3 connecting with it (see Figure 13).
In (8), letting h1=H(φ1,0), it follows that
(49)Γ3:y2=cδφ2-8a15δφ5/2-b3δφ3+h1(0<φ<φ1).
Substituting (49) into (7) and integrating it along the orbit Γ3, we get (47) and (48).
Lettingb→-16a2/75c-0, it follows that
(50)limb→-16a2/75c-0h1=limb→-16a2/75c-0H(φ1,0)=limb→-16a2/75c-0-cδφ12+8a15δφ15/2+b3δφ13=0.
Whenh1→0, completing the integrals in (47) and (48), we get the kink wave solution and the antikink wave solution as (29).
Hereto, we have completed the proof for the Proposition7.Figure 13
The graph of orbitΓ3 when δ>0 and (a,b)∈A3.


## 3.4. Bifurcation from Kink-Like Waves

Proposition 8.
When the parameters satisfyδ>0, (a,b)∈A2, and b→-16a2/75c+0, the kink-like wave and the antikink-like wave, respectively, become a kink wave and an anti-kink wave with the expressions (29).
For the varying process, see Figures14 and 15.(The kink wave is bifurcated from the kink-like wave). The varying process for the graph ofua+ when c>0, δ>0, a>0, and b→-16a2/75c+0, where c=1, δ=1, a=1/2, and (a) b=-16a2/75c+10-3. (b) b=-16a2/75c+10-4. (c) b=-16a2/75c+10-6. (d) b=-16a2/75c+10-8.
(a)
(b)
(c)
(d)(The antikink wave is bifurcated from the antikink-like wave). The varying process for the graph ofua- when c>0, δ>0, a>0, and b→-16a2/75c+0, where c=1, δ=1, a=1/2, and (a) b=-16a2/75c+10-3. (b) b=-16a2/75c+10-4. (c) b=-16a2/75c+10-6. (d) b=-16a2/75c+10-8.
(a)
(b)
(c)
(d)Proof.
Lettingb→-16a2/75c+0, it follows that h1→0 (see (50)) and
(51)limb→-16a2/75c+0ξ10=∫0φ1/2‍ds(c/δ)s2-(8a/15δ)s5/2+(16a2/225cδ)s3=limν→0∫νφ1/2‍ds(c/δ)s2-(8a/15δ)s5/2+(16a2/225cδ)s3=∞.
Whenh1→0, and ξ10→∞, completing the integrals in (17), we get the kink wave solution and the antikink wave solution as (29).
Hereto, we have completed the proof for the Proposition8.


## 3.5. Bifurcation from Compacton-Like Waves

Proposition 9.
When the parameters satisfyδ>0, (a,b)∈A2, and b→-16a2/75c+0, the two compacton-like waves become a kink wave and an anti-kink wave with the expressions (29).
For the varying process, see Figures16 and 17.(The kink wave is bifurcated from the compacton-like wave). The varying process for the graph ofuc+ when c>0, δ>0, a>0, and b→-16a2/75c+0, where c=1, δ=1, a=1/2, and (a) b=-16a2/75c+10-3. (b) b=-16a2/75c+10-4. (c) b=-16a2/75c+10-6. (d) b=-16a2/75c+10-8.
(a)
(b)
(c)
(d)(The antikink wave is bifurcated from the compacton-like wave). The varying process for the graph ofuc- when c>0, δ>0, a>0, and b→-16a2/75c+0, where c=1, δ=1, a=1/2, and (a) b=-16a2/75c+10-3. (b) b=-16a2/75c+10-4. (c) b=-16a2/75c+10-6. (d) b=-16a2/75c+10-8.
(a)
(b)
(c)
(d)Proof.
Lettingb→-16a2/75c+0, it follows that
(52)limb→-16a2/75c+0hφk=limb→-16a2/75c+0H(φk,0)=limb→-16a2/75c+0-cδφk2+8a15δφk5/2+b3δφk3=0,(53)limb→-16a2/75c+0ξ30=∫0φk/2‍ds(c/δ)s2-(8a/15δ)s5/2+(16a2/225cδ)s3=limν→0∫νφk‍ds(c/δ)s2-(8a/15δ)s5/2+(16a2/225cδ)s3=∞,(54)limb→-16a2/75c+0ξ40=limb→-16a2/75c+0(2∫φk/2φk‍(ds)((c/δ)s2-(8a/15δ)s5/200000000000000000000000000-(b/3δ)s3+hφk)-1/2)00000000000000+ξ30∫φk/2φk‍)=∞.
Whenhφk→0, ξ30→∞, and ξ40→∞, completing the integrals in (21), we get the kink wave solution and the antikink wave solution as (29).
Hereto, we have completed the proof for the Proposition9.


## 4. Bifurcation of Smooth Periodic Wave

Proposition 10.
Forab≠0 and δ<0, (1) has a nonlinear wave solution
(55)uh=(-2αβ2-4αsin(τ2ξ+η)+β)2,
where
(56)τ2=12-cδ,
and η is an arbitrary real number. The solution possesses the following properties.(1)
if(a,b)belongs to any one of the regionsA1,A6, thenuhrepresents periodic blow-up wave solution,(2)
if(a,b)belongs toA2, thenuhrepresents periodic wave solution.
In particular, whenb→0-0, the periodic wave becomes a periodic blow-up wave. For the varying process, see Figure 18. When b→-16a2/75c+0, the periodic wave tends to a trivial wave u=225c2/16a2. For the varying process, see Figure 19.(The periodic blow-up wave is bifurcated from the periodic wave). The varying process for the graph ofuh when c>0, δ<0, a>0, η>0, and b→0-0, where c=1, δ=-1, a=1, η=1.57, and (a) b=0-0.2. (b) b=0-0.1. (c) b=0-0.05. (d) b=0-0.01.
(a)
(b)
(c)
(d)(The periodic wave become the trivial wave). The varying process for the graph ofuh when c>0, δ<0, a>0, η<0, and b→-16a2/75c+0, where c=1, δ=-1, a=1, η=-4.7, and (a) b=-16a2/75c+10-2. (b) b=-16a2/75c+10-3. (c) b=-16a2/75c+10-4. (d) b=-16a2/75c+10-7.
(a)
(b)
(c)
(d)Proof.
Completing the integral in (38), we get uh as (55). η=η(ν) is an arbitrary real number.
When(a,b)∈A2, in (38) letting ν=q (see (28)), we have
(57)η=arcsin-βq-2αqβ2-4α.
Letting b→0-0, then
(58)limb→0-0β2-4αβ2=limb→0-075bc16a2+1=1,limb→0-0η=limb→0-0arcsin-βq-2αqβ2-4α=limb→0-0arcsin(1(β2-4α)/β200000000000000+2q(β2-4α)/α2)=arcsin1=π2.
We have
(59)limb→0-0uh=limb→0-0(-2αβ2-4αsin(τ2ξ+η)+β)2=limb→0-0(2α/β(β2-4α)/β2sin(τ2ξ+η)-1)2=225c216a2[cos(τ2ξ)-1]2.
Obviously,uh will blow up when ξ=2kπ/τ2(k∈z).
Hereto, we have completed the proofs for all propositions.


## 5. Conclusion

In this paper, we have studied the bifurcation behavior of S-KdV equation. Two new types of nonlinear waves called kink-like waves and compacton-like waves have been displayed in Propositions1–3. Furthermore, two kinds of new bifurcation phenomena have been revealed. The first phenomenon is that the kink waves can be bifurcated from five types of nonlinear waves which have been stated in Propositions 5–9. The second phenomenon is that the periodic blow-up wave can be bifurcated from the periodic wave which has been explained in Proposition 10. At the same time, we have got three new explicit expressions for traveling waves which were given in (25) and (55). Two previous results are our some special cases (see (30) and (32)).



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*Source: 483492-2013-08-20.xml*