# Exact Traveling Wave Solutions for a Nonlinear Evolution Equation of Generalized Tzitzéica-Dodd-Bullough-Mikhailov Type

Authors: Weiguo Rui
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

By using the integral bifurcation method, a generalized Tzitzéica-Dodd-Bullough-Mikhailov (TDBM) equation is studied. Under different parameters, we investigated different kinds of exact traveling wave solutions of this generalized TDBM equation. Many singular traveling wave solutions with blow-up form and broken form, such as periodic blow-up wave solutions, solitary wave solutions of blow-up form, broken solitary wave solutions, broken kink wave solutions, and some unboundary wave solutions, are obtained. In order to visually show dynamical behaviors of these exact solutions, we plot graphs of profiles for some exact solutions and discuss their dynamical properties.


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## Body

## 1. Introduction

In this paper, we consider the following nonlinear evolution equation:(1)uxt=αemu+βenu,
where α, β are two non-zero real numbers and m, n are two integers. We call (1) generalized Tzitzéica-Dodd-Bullough-Mikhailov equation because it contains Tzitzéica equation, Dodd-Bullough-Mikhailov equation, and Tzitzéica-Dodd-Bullough equation. When α=1, β=-1, m=1, n=-2 or α=-1, β=1, m=-2, n=1, especially (1) becomes classical Tzitzéica equation [1–3] as follows:
(2)uxt=eu-e-2u,
which was originally found in the field of geometry in 1907 by G. Tzitzéica and appeared in the fields of mathematics and physics alike. Equation (2) usually called the “Dodd-Bullough equation,’’ which was initiated by Bullough and Dodd [4] and Žiber and Šabat [5]. Indeed, (2) has another form
(3)uxx-utt=eu-e-2u;
see [6, 7] and the references cited therein.Whenα=-1, β=-1 and m=1, n=-2, (1) becomes Dodd-Bullough-Mikhailov equation
(4)uxt+eu+e-2u=0.
When α=1, β=1 and m=1, n=-2, (1) becomes Tzitzéica-Dodd-Bullough equation
(5)uxt-eu-e-2u=0.
The Dodd-Bullough-Mikhailov equation and Tzitzéica-Dodd-Bullough equation appeared in many problems varying from fluid flow to quantum field theory.Moreover, whenm=1, n=-1, α=1/2, β=-1/2 or m=-1, n=1, α=-1/2, β=1/2, (1) becomes the sinh-Gordon equation
(6)uxt=sinhu,
which was shown in [7–10] and the references cited therein. When m=1, n=-1, α=β=1/2 or m=-1, n=1, α=β=1/2, (1) becomes the cosh-Gordon equation
(7)uxt=coshu,
which was shown in [11, 12] and the references cited therein. When m=1, n=0, especially (1) becomes the Liouville equation
(8)uxt=αeu.
All the equations mentioned above play very significant roles in many scientific applications. And some of them were studied by many authors in recent years; see the following brief statements.In [13], by using the tanh method, Wazwaz considered some solitary wave and periodic wave solutions for the Dodd-Bullough-Mikhailov and Dodd-Bullough equations. In [14], Andreev studied the Bäcklund transformation for Bullough-Dodd-Jiber-Shabat equation. In [15], Cherdantzev and Sharipov obtained finite-gap solutions of the Bullough-Dodd-Jiber-Shabat equation. In [16], Cherdantzev and Sharipov investigated solitons on the finite-gap background in the Bullough-Dodd-Jiber-Shabat model. In addition, the Darboux transformation, self-dual Einstein spaces and consistency and general solution of Tzitzéica equation were studied in [17–19].In this paper, by using integral bifurcation method [20], we will study (1). Indeed, the integral bifurcation method has successfully been combined with the computer method [21] and some transformations [22, 23] for investigating exact traveling wave solutions of some nonlinear PDEs. Therefore, using this method, we will obtain some new results which are different from those in the above references.The rest of this paper is organized as follows: in Section2, we will derive two-dimensional planar system which is equivalent to (1) and give its first integral equation. In Sections 3 and 4, by using the integral bifurcation method, we will obtain some new traveling wave solutions of (1) and discuss their dynamic properties.


## 2. Two-Dimensional Planar Dynamical System of (1) and Its First Integral

Applying the transformationsu(x,t)=ln|v(x,t)|, we change (1) into the following form:
(9)vvxt-vxvt=αvm+2+βvn+2.
Let v(x,t)=ϕ(x-ct)=ϕ(ξ), substituting ϕ(x-ct) into (9), respectively, we obtain
(10)c(ϕ′)2-cϕϕ′′=αϕm+2+βϕn+2,
where ‘‘′’’ is the derivative with respect to ξ and c is wave speed.Clearly, (10) is equivalent to the following two-dimensional systems:
(11)dϕdξ=y,dydξ=cy2-αϕm+2-βϕ2+ncϕ.
However, when ϕ=0, (11) is not equivalent to (10) and the dy/dξ cannot be defined, so we call the ϕ=0 singular line and call (11) singular system. In order to obtain an equivalent system of (10), making a scalar transformation
(12)dξ=cϕdτ,
equation (11) can be changed into a regular two-dimensional system as follows:
(13)dϕdτ=cϕy,dydτ=cy2-αϕm+2-βϕn+2.
Systems (11) and (13) are two integrable systems, and they have the same first integral as follows:
(14)y2=hϕ2-2αcmϕm+2-2βcnϕn+2,
where h is integral constant.Systems (11) and (13) are planar dynamical systems defined by the 5-parameter (α,β,c,m,n). Usually their first integral (14) can be rewritten as
(15)H(ϕ,y)≡2αcmϕm+2βcnϕn+y2ϕ2=h.Whenn-m=2k-1, (k∈N), system (13) has two equilibrium points O(0,0) and A((-α/β)1/(n-m),0) in the ϕ-axes. When n-m=2k, (k∈N) and αβ<0, system (13) has three equilibrium point O(0,0) and B1,2(±(-α/β)1/(n-m),0) in the ϕ-axes. When n-m=2k, (k∈N) and αβ>0, system (13) has only one equilibrium points O(0,0). When n=m=k, (k∈N), system (13) has only one equilibrium point O(0,0).Respectively, substituting every equilibrium point (the pointO(0,0) is exception) into (15), we have
(16)hA=H((-αβ)1/(n-m),0)=2α(n-m)cmn(-αβ)m/(n-m),hB1=H(B1)=H((-αβ)1/(n-m),0)=2α(n-m)cmn(-αβ)m/(n-m),hB2=H(B2)=H(-(-αβ)1/(n-m),0)=(-1)mhB1.From the transformationu(x,t)=ln|ϕ(x-ct)|, we know that ln|ϕ(x-ct)| approaches to -∞ if a solution ϕ(ξ) approaches to 0 in (14). In other words, this solution determines an unboundary wave solution or blow-up wave solution of (1). It is easy to see that the second equation of system (11) is not continuous when ϕ=0. In other words, on such straight line ϕ=0 in the phase plane (ϕ,y), the function ϕ′′ is not defined, and this implies that the smooth wave solutions of (1) sometimes become nonsmooth wave solutions.


## 3. Exact Solutions of (1) and Their Properties under the Conditions n=±m, ±2m, m∈Z+, and m=2, n=-1

In this section, we investigate exact solutions of (1) under different kinds of parametric conditions and their properties.


### 3.1. Different Kinds of Solitary Wave Solutions and Unboundary Wave Solutions in the Special Casesn=2m∈Z+ or n=m∈Z+

(i) Whenh>0, βc<0, n=2m and m is even, then (14) can be reduced to
(17)y=±ϕh-2αcmϕm-βcm(ϕm)2.
Taking ξ0=0 and ϕ(0)=ϕ1 as initial values, substituting (17) into the dϕ/dξ=y of (11), and then integrating it yield
(18)v(x,t)=±(γ1Ω1cosh(ω1(x-ct))+δ1)1/m,
where ϕ1=(-α/β+(α2+βcmh)/β2)1/m,Ω1=ϕ1m(α2+βcmh), ω1=-mh,γ1=cmh(cmh-αϕ1m), δ1=α(cmh-αϕ1m). By using program of Maple, it is easy to validate that (18) is the solution equation (9) when m is given. Substituting (18) into the transformation u(x,t)=ln|v(x,t)|, we obtain a solitary wave solutions of (1) as follows:
(19)u(x,t)=1mln|γ1Ω1cosh(ω1(x-ct))+δ1|.(ii) Whenn=m, h>0, c/(α+β)>0 and m is even, (14) can be reduced to
(20)y=±ϕh-2(α+β)cmϕm.
Taking ξ0=0 and ϕ(0)=(mhc/2(α+β))1/m as initial values, substituting (20) into the dϕ/dξ=y of (11), and then integrating it yield
(21)v(x,t)=±(mhcsech2ω2(x-ct)2(α+β))1/m,
where ω2=(1/2)mh. Substituting (21) into the transformation u(x,t)=ln|v(x,t)|, we obtain a solution without boundary of (1) as follows:
(22)u(x,t)=1mln|mhcsech2ω2(x-ct)2(α+β)|.
Equation (21) is smooth solitary wave solution of (9) but (22) is an exact solution without boundary of (1) because u→-∞ as v→0. In order to visually show dynamical behaviors of solutions v and u of (21) and (22), we plot graphs of their profiles which are shown in Figures 1(a) and 1(b), respectively.The phase portraits of solutions (21) and (22) for m=4, α=3, β=-1, c=2, h=9, t=0.1.
(a)
Smooth solitary wave
(b)
Unboundary wave(iii) Whenn=2m, h=0, βc>0, αc<0 and m is even, (14) can be reduced to
(23)y=±ϕ-2αcmϕm-βcm(ϕm)2.
Taking ξ0=0 and ϕ(0)=0 as initial values, substituting (23) into the dϕ/dξ=y of (11), and then integrating it yield
(24)v(x,t)=±(2(-αc)βc+mα2(x-ct)2)1/m.
Substituting (24) into the transformation u(x,t)=ln|v(x,t)|, we obtain an exact solutions of rational function type of (1) as follows:
(25)u(x,t)=1mln|2αcβc+mα2(x-ct)2|.(iv) Whenn=2m, h>0, αβ>0, βc>0 (or αβ<0, βc>0), and m is odd, (14) can be reduced to
(26)y=±βcmϕcmhβ-2αβϕm-(ϕm)2.
Respectively, taking ξ0=0, ϕ(0)=ϕ1, and ξ0=0, ϕ(0)=ϕ2 as initial values, substituting (26) into the dϕ/dξ=y of (11), and then integrating them yield
(27)v(x,t)=(γ1Ω1cosh(ω1(x-ct))+δ1)1/m,(28)v(x,t)=(γ2Ω2cosh(ω2(x-ct))+δ2)1/m,
where ϕ1 is given above and ϕ2=(-α/β-(α2+βcmh)/β2)1/m, Ω2=ϕ2m(α2+βcmh), ω2=mh, γ2=cmh(cmh-αϕ2m), δ2=α(cmh-αϕ2m). Respectively, substituting (27) and (28) into the transformation u(x,t)=ln|v(x,t)|, we obtain two smooth unboundary wave solutions of (1) as follows:
(29)u(x,t)=1mln|γ1Ω1cosh(ω1(x-ct))+δ1|,(30)u(x,t)=1mln|γ2Ω2cosh(ω2(x-ct))+δ2|.(v) Whenn=2m, 0<h<hA, αβ<0, βc<0 (or αβ>0, βc<0) and m is odd, (14) can be reduced to
(31)y=±-βcmϕ-cmhβ+2αβϕm+(ϕm)2.
Similarly, by using (31), we obtain two smooth solitary wave solutions of (1) which are the same as the solutions (29) and (30).(vi) Whenn=m,h>0 and m is odd, (14) can be reduced to (20), so the obtained solution is as the same as the solution (22).(vii) Whenn=2m, h=0, βc>0 and m is odd, (14) can be reduced to
(32)y=±βcmϕ-2αβϕm-(ϕm)2.
Taking ξ0=0 and ϕ(0)=(-2α/β)1/m as initial values, substituting (32) into the dϕ/dξ=y of (11), and then integrating it yield
(33)v(x,t)=(-2αβ[1+(mα2/βc)(x-ct)2])1/m.
Substituting (33) into the transformation u(x,t)=ln|v(x,t)|, we obtain a smooth unboundary wave solution of rational function type of (1) as follows:
(34)u(x,t)=1mln|2αβ[1+(mα2/βc)(x-ct)2]|.


### 3.2. Periodic Blow-Up Wave Solutions, Broken Kink Wave, and Antikink Wave Solutions in the Special Casen=2m∈Z+

(1) Whenn=2m, αβ<0, βc>0, hB1<h<0 (or hB2<h<0) and m is even, (14) can be reduced to
(35)y=±βcmϕcmhβ-2αβϕm-(ϕm)2.
Taking ξ0=0, ϕ(0)=(-α/β+α2/β2+mhc/β)1/m and ξ0=0, ϕ(0)=(-α/β-α2/β2+mhc/β)1/m as the initial values, substituting (35) into the dϕ/dξ=y of (11), and then integrating them yield
(36)v(x,t)=(cmhα-βΩ3sin(arcsinD1-ω3(x-ct)))1/m,(37)v(x,t)=-(cmhα+βΩ3sin(arcsinD2+ω3(x-ct)))1/m,
where Ω3=(α2+cmhβ)/β2, ω3=m-h, D1=(α2+αβΩ3+βcmh)/βΩ3(α+βΩ3), D2=(α2-αβΩ3+βcmh)/βΩ3(-α+βΩ3). Respectively, substituting (36) and (37) into the transformation u(x,t)=ln|v(x,t)|, we obtain two periodic blow-up wave solutions of (1) as follows:
(38)u(x,t)=1mln|cmhα-βΩ3sin(arcsinD1-ω3(x-ct))|,(39)u(x,t)=1mln|cmhα+βΩ3sin(arcsinD2+ω3(x-ct))|.(2) Whenn=2m, αβ<0, βc>0, hB2<h<0 and m is odd, (14) can be reduced to
(40)y=±βcmϕcmhβ-2αβϕm-(ϕm)2,
which is the same as (35), so the obtained solution is the same as solution (39). Similarly, when n=2m, αβ<0, βc>0, hB1<h<0 and m is odd, the obtained solution is also the same as solution (38).(3) whenn=2m, αβ<0, βc<0, h=hB1=hB2=-α2/βcm and m is even, (14) can be reduced to
(41)y=±-1cmβϕ(βϕm+α).
Substituting (41) into the dϕ/dξ=y of (11) and then integrating it and setting the integral constants as zero yield
(42)v(x,t)=(-α2β(1±tanhΩ42(x-ct)))1/m,
where Ω4=α-m/cβ. Substituting (42) into the transformation u(x,t)=ln|v(x,t)|, we obtain an unboundary wave solution of (1) as follows:
(43)u(x,t)=1mln|α2β[1±tanhΩ42(x-ct)]|.
Equation (42) is smooth kink wave solution of (9), but (43) is a broken kink wave solution without boundary of (1) because u→-∞ as v→0. In order to visually show dynamical behaviors of solutions v and u of (42) and (43), we plot graphs of their profiles which are shown in Figures 2(a) and 2(b), respectively.The phase portraits of solutions (42) and (43) for m=4,α=0.3,β=-0.3,c=2,t=0.1.
(a)
Smooth kink wave
(b)
Unboundary wave(4) Whenn=2m, αβ>0, βc<0, h=hA or αβ<0, βc<0, h=hA and m is odd, (14) can be reduced to (41). So the obtained solution is also the same as the solution (43).


### 3.3. Periodic Blow-Up Wave Solutions and Solitary Wave Solutions of Blow-Up Form in the Special Casem=2, n=-1

Whenn=-1, (14) can be reduced to
(44)y=±hϕ2-2αcmϕm+2+2βcϕ.(a) Under the conditionsαβ>0, αc>0, h∈(-∞,+∞) and m=2, (44) becomes
(45)y=±αc(ϑ1-ϕ)(ϕ-0)(ϕ-γ1)(ϕ-δ1),ϕ∈(0,ϑ1],
where ϑ1, γ1, δ1 are three roots of equation 2β/α+(ch/α)ϕ-ϕ3 and ϑ1>0>γ1>δ1. These three roots ϑ1, γ1, δ1 can be obtained by Cardano formula as long as the parameters α, β, c, h are fixed (or given) concretely. For example, taking α=4.0, β=2.0, c=4.0, h=9.0, the ϑ1≐3.054084215, γ1≐-0.1112641576, δ1≐-2.942820058. Taking (0,ϑ1) as the initial values of the variables (ξ,ϕ), substituting (45) into the dϕ/dξ=y of (11), and then integrating it yield
(46)v(x,t)=ϕ(ξ)=ϑ1(-δ1)cn2(Ω1(ξ),k1)(-δ1)+ϑ1sn2(Ω1ξ,k1),ξ≠T1,
where Ω1=(1/2)-αδ1(ϑ1-δ1)/c, k1=ϑ1(γ1-δ1)/-δ1(ϑ1-γ1), T1=(1/Ω1)sn-1(1,k1). By using the transformation u(x,t)=ln|v(x,t)| and (46), we obtain a periodic blow-up wave solution of (1) as follows:
(47)u(x,t)=ln|ϑ1(-δ1)cn2(Ω1(x-ct),k1)(-δ1)+ϑ1sn2(Ω1(x-ct),k1)|,x-ct≠T1.(b) Under the conditionsαβ>0, αc>0, h∈(hA,+∞) and m=2, (44) becomes
(48)y=±αc(ϑ1-ϕ)(0-ϕ)(γ1-ϕ)(ϕ-δ1),ϕ∈[δ1,γ1].
Similarly, taking (0,δ1) as the initial values of the variables (ξ,ϕ), substituting (48) into the dϕ/dξ=y of (11), and then integrating it yield
(49)v(x,t)=ϕ(ξ)=γ1δ1δ1+(γ1-δ1)sn2(Ω1ξ,k1).
By using the transformation u(x,t)=ln|v(x,t)| and (49), we obtain a periodic blow-up wave solution of (1) as follows:
(50)u(x,t)=ln|γ1δ1δ1+(γ1-δ1)sn2(Ω1(x-ct),k1)|.(c) Under the conditionsαβ<0, αc>0, h∈(-∞,+∞) and m=2, (44) becomes
(51)y=±αc(ϑ2-ϕ)(η2-ϕ)(0-ϕ)(ϕ-δ2),ϕ∈[δ2,0),
where ϑ2, η2, 0, δ2 are roots of equation ϕ(2β/α+(ch/α)ϕ-ϕ3)=0 and ϑ2>η2>0>δ2. As in the above cases, these three roots can be obtained by Cardano formula as long as the parameters α, β, c, h are given concretely, so the similar cases will be not commentated anymore in the following discussions. Similarly, taking (0,δ2) as the initial values of the variables (ξ,ϕ), substituting (51) into the dϕ/dξ=y of (11), and then integrating it yield
(52)v(x,t)=ϕ(ξ)=η2δ2sn2(Ω2(x-ct),k2)(η2-δ2)+δ2sn2(Ω2ξ,k2),ξ≠T2,
where Ω2=(1/2)αϑ2(η2-δ2)/c, k2=-δ2(ϑ2-η2)/ϑ2(η2-δ2), T2=(1/Ω2)sn-1(0,k2). By using the transformation u(x,t)=ln|v(x,t)| and (52), we obtain a periodic blow-up wave solution of (1) as follows:
(53)u(x,t)=ln|η2δ2sn2(Ω2(x-ct),k2)(η2-δ2)+δ2sn2(Ω2ξ,k2)|,x-ct≠T2.(d) Under the conditionsαβ<0, αc>0, h∈(hA,+∞) and m=2, (44) becomes
(54)y=±αc(ϑ2-ϕ)(ϕ-η2)(ϕ-0)(ϕ-δ2).
Taking the (0,ϑ2) as the initial values of the variables (ξ,ϕ), substituting (54) into the dϕ/dξ=y of (11), and then integrating it yield
(55)v(x,t)=ϕ(ξ)=ϑ2(η2-δ2)dn2(Ω2(x-ct),k2)(η2-δ2)+(ϑ2-η2)sn2(Ω2ξ,k2).
By using the transformations u(x,t)=ln|v(x,t)| and (55), we obtain a periodic blow-up wave solution of (1) as follows:
(56)u(x,t)=ln|ϑ2(η2-δ2)dn2(Ω2(x-ct),k2)(η2-δ2)+(ϑ2-η2)sn2(Ω2(x-ct),k2)|.(e) Under the conditionsαβ<0, αc<0, h∈(-∞,hA) and m=2, (44) becomes
(57)y=±-αc(ϑ3-ϕ)(η3-ϕ)(ϕ-0)(ϕ-δ3),ϕ∈(0,η3],
where ϑ3, η3, 0, δ3 are roots of equation ϕ(-2β/α-(ch/α)ϕ+ϕ3)=0 and ϑ3>η3>0>δ3. Taking the (0,η3) as the initial values of the variables (ξ,ϕ), substituting (57) into the dϕ/dξ=y of (11), and then integrating it yield
(58)v(x,t)=ϕ(ξ)=ϑ3η3cn2(Ω3ξ,k3)δ3-η3sn2(Ω3ξ,k3),ξ≠T3,
where Ω3=(1/2)-αϑ3(η3-δ3)/c, k3=η3(ϑ3-δ3)/ϑ3(η3-δ3), T3=(1/Ω3)sn-1(1,k3). By using the transformation u(x,t)=ln|v(x,t)| and (58), we obtain a periodic blow-up wave solution of (1) as follows:
(59)u(x,t)=ln|ϑ3η3cn2(Ω3(x-ct),k3)δ3-η3sn2(Ω3(x-ct),k3)|,x-ct≠T3.(f) Under the conditionsαβ<0, αc<0, h∈(-∞,hA) and m=2, (44) becomes
(60)y=±-αc(ϑ4-ϕ)(0-ϕ)(ϕ-γ4)(ϕ-δ4),ϕ∈[γ4,0),
where ϑ4, 0, γ4, δ4 are roots of equation ϕ(-2β/α-(ch/α)ϕ+ϕ3)=0 and ϑ4>0>γ4>δ4. Taking the (0,γ4) as the initial values of the variables (ξ,ϕ), substituting (60) into the dϕ/dξ=y of (11), and then integrating it yield
(61)v(x,t)=ϕ(ξ)=δ4γ4cn2(Ω4ξ,k4)δ4-γ4cn2(Ω4ξ,k4),ξ≠T4,
where Ω4=(1/2)αδ4(ϑ4-γ4)/c, k4=γ4(ϑ4-δ4)/ϑ4(ϑ4-γ4), T4=(1/Ω4)sn-1(1,k4). By using the transformation u(x,t)=ln|v(x,t)| and (61), we obtain a periodic below-up wave solutions of (1) as follows:
(62)u(x,t)=ln|δ4γ4cn2(Ω4(x-ct),k4)δ4-γ4cn2(Ω4(x-ct),k4)|,x-ct≠T4.(g) Under the conditionsm=2, αc<0, h=hA=-α2/2cβ, (44) becomes
(63)y=±-αc|ϕ1-ϕ|ϕ(ϕ+2ϕ1),ϕ≠0,
where ϕ1=(-β/α)1/3. Taking (0,(1/2)ϕ1) as the initial values of the variables (ξ,ϕ), substituting (63) into the dϕ/dξ=y of (11), and then integrating it yield
(64)v(x,t)=ϕ(ξ)=ϕ1[1-3cosh(Ω5ξ)+2],ξ≠T5,
where Ω5=ϕ1-3α/c, T5=(1/Ω5)cosh-1(1). Clearly, we have ϕ(T5)=0, ϕ(±∞)=ϕ1. By using the transformation u(x,t)=ln|v(x,t)| and (64), we obtain a solitary wave solution of blow-up form of (1) as follows:
(65)u(x,t)=ln|ϕ1[1-3cosh(Ω5(x-ct))+2]|,x-ct≠T5.
Equation (64) is smooth solitary wave solution of (9), but (65) is a solitary wave solution of blow-up form for (1) because u→-∞ as v→0. In order to visually show dynamical behaviors of solutions v and u of (64) and (65), we plot graphs of their profiles which are shown in Figures 3(a) and 3(b), respectively.The phase portraits of solutions (64) and (65) for m=2, α=0.3, β=-0.3, c=-2, t=0.1.
(a)
Smooth solitary wave
(b)
Solitary wave of blow-up form


### 3.4. Different Kinds of Periodic Wave, Broken Solitary Wave Solutions in the Special Casem=1, n=-1

(1) Under the conditionsh∈(hA,+∞), αβ>0, βc>0 (or αβ<0, βc>0), αc>0 and m=1, (44) becomes
(66)y=±2αc(ϕM-ϕ)(ϕ-ϕm)(ϕ-0),ϕ∈[ϕm,ϕM],
where ϕM,m=(hc±h2c2+16αβ)/4α and 0 are three roots of equation hϕ2-(2α/c)ϕ3+(2β/c)ϕ=0 and ϕM>ϕm>0. Taking (0,ϕm) as the initial values of the variables (ξ,ϕ), substituting (66) into the dϕ/dξ=y of (11), and then integrating it yield
(67)v(x,t)=ϕ(ξ)=ϕm-(ϕM-ϕm)sn2(Ω6ξ,k6),
where Ω6=αϕM/2c, k6=(ϕM-ϕm)/ϕM. By using the transformation u(x,t)=ln|v(x,t)| and (67), we obtain a double periodic wave solution of (1) as follows:
(68)u(x,t)=ln|ϕm-(ϕM-ϕm)sn2(Ω6(x-ct),k6)|.
Equation (67) is smooth periodic wave solution of (9), but (68) is a double periodic wave solution of blow-up form of (1) because u→-∞ as v→0. In order to visually show dynamical behaviors of solutions v and u of (67) and (68), we plot graphs of their profiles which are shown in Figures 4(a) and 4(b), respectively.The phase portraits of solutions (67) and (68) for h=8, α=2.5, β=-2.5, c=2, t=0.1.
(a)
Smooth periodic wave
(b)
Double periodic wave of blow-up type(2) Under the conditionh∈(-∞,hA),αβ<0, βc>0 (or αβ<0, βc<0),αc>0 and m=1, (44) becomes
(69)y=±2αc(0-ϕ)(ϕ-ϕM)(ϕ-ϕm),ϕ∈[ϕM,0),
where ϕM,ϕm,0 are roots of equation hϕ2-(2α/c)ϕ3+(2β/c)ϕ=0 and ϕm<ϕM<0. Taking (0,ϕM) as the initial values of the variables (ξ,ϕ), substituting (69) into the dϕ/dξ=y of (11), and then integrating it yield
(70)v(x,t)=ϕ(ξ)=ϕMsn2(Ω7ξ,k7),ξ≠T7,
where Ω7=-αϕM/2c, k7=ϕM/ϕm, T7=(1/Ω7)sn-1(0,k7). By using the transformation u(x,t)=ln|v(x,t)| and (70), we obtain a periodic blow-up wave solution of (1) as follows:
(71)u(x,t)=ln|ϕMsn2(Ω7(x-ct),k7)|,x-ct≠T7.(3) Under the conditionsh∈(hA,+∞), αβ<0, βc>0 (or αβ>0, βc>0) and m=1, (44) becomes
(72)y=±-2αc(0-ϕ)(ϕM-ϕ)(ϕ-ϕm),ϕ∈[ϕm,ϕM],
where ϕM, ϕm, 0 are roots of equation hϕ2-(2α/c)ϕ3+(2β/c)ϕ=0 and 0>ϕM>ϕm. Taking (0,ϕM) as the initial values of the variables (ξ,ϕ), substituting (72) into the dϕ/dξ=y of (11), and then integrating it yield
(73)v(x,t)=ϕ(ξ)=ϕMdn2(Ω8ξ,k8),
where Ω8=αϕm/2c, k8=(ϕM-ϕm)/-ϕm. By using the transformation u(x,t)=ln|v(x,t)| and (73), we obtain a periodic blow-up wave solutions of (1) as follows:
(74)u(x,t)=ln|ϕMdn2(Ω8(x-ct),k8)|.(4) Under the conditionsh∈(-∞,hA),αβ<0, βc>0 (or αβ<0, βc<0) and m=1, (44) becomes
(75)y=±-2αc(ϕM-ϕ)(ϕm-ϕ)(ϕ-0),ϕ∈(0,ϕm],
where ϕM, ϕm, 0 are roots of equation hϕ2-(2α/c)ϕ3+(2β/c)ϕ=0 and ϕM>ϕm>0. Taking (0,ϕm) as the initial values of the variables (ξ,ϕ), substituting (75) into the dϕ/dξ=y of (11), and then integrating it yield
(76)v(x,t)=ϕ(ξ)=ϕmcn2(Ω9ξ,k9)dn2(Ω9ξ,k9),ξ≠T9,
where Ω9=-αϕM/2c, k9=ϕm/ϕM, T9=(1/Ω9)sn-1(0,k9). By using the transformation u(x,t)=ln|v(x,t)| and (76), we obtain periodic blow-up wave solutions of (1) as follows:
(77)u(x,t)=ln|ϕmcn2(Ω9(x-ct),k9)dn2(Ω9(x-ct),k9)|,x-ct≠T9.(5) Under the conditionsh=hA=-4αϕ1/c, αβ<0, βc>0 and m=1, (44) becomes
(78)y=±2αc|ϕ1+ϕ|-ϕ,ϕ∈[-ϕ1,0),
where ϕ1=(-β/α)1/2. Taking (0,-ϕ1) as the initial values of the variables (ξ,ϕ), substituting (78) into the dϕ/dξ=y of (11), and then integrating it yield
(79)v(x,t)=ϕ(ξ)=-ϕ1tanh2(0.5Ω10ξ),ξ≠0,
where Ω10=2αϕ1/c. By using the transformation u(x,t)=ln|v(x,t)| and (79), we obtain a solitary wave solution of blow-up form of (1) as follows:
(80)u(x,t)=ln|ϕ1tanh2(0.5Ω10(x-ct))|,x-ct≠0.(6) Under the conditionsh=hA=4αϕ1/c, αβ<0, βc>0 and m=1, (44) becomes
(81)y=±-2αc|ϕ-ϕ1|ϕ,ϕ∈(0,ϕ1],
where ϕ1=(-β/α)1/2. Taking (0,ϕ1) as the initial values of the variables (ξ,ϕ), substituting (81) into the dϕ/dξ=y of (11), and then integrating it yield
(82)v(x,t)=ϕ(ξ)=ϕ1tanh2(0.5Ω11ξ),ξ≠0,
where Ω11=-2αϕ1/c. By using the transformation u(x,t)=ln|v(x,t)| and (82), we obtain a broken solitary wave solution of (1) as follows:
(83)u(x,t)=ln|ϕ1tanh2(0.5Ω11(x-ct))|,x-ct≠0.(7) Under the conditionsh∈(-∞,0], αβ>0, βc<0 and m=1, (44) becomes
(84)y=±-2αc(ϕM-ϕ)(0-ϕ)(ϕ-ϕm),ϕ∈[ϕm,0),
where ϕM, ϕm, 0 are roots of equation ϕ2h-(2α/c)ϕ3+(2β/c)ϕ=0 and ϕM>0>ϕm. Taking (0,ϕM) as the initial values of the variables (ξ,ϕ), substituting (84) into the dϕ/dξ=y of (11), and then integrating it yield
(85)v(x,t)=ϕ(ξ)=-ϕMk122sn2(Ω12ξ,k12)dn2(Ω12ξ,k12),ξ≠T12,
where Ω12=-α(ϕM-ϕm)/2c, k12=-ϕm/(ϕM-ϕm), T12=(1/Ω12)sn-1(0,k12). By using the transformation u(x,t)=ln|v(x,t)| and (85), we obtain a periodic blow-up wave solution of (1) as follows:
(86)u(x,t)=ln|-ϕMk122sn2(Ω12(x-ct),k12)dn2(Ω12(x-ct),k12)|,x-ct≠T12.(8) Under the conditionsh∈(-∞,0], αβ>0, βc>0 and m=1, (44) becomes
(87)y=±2αc(ϕM-ϕ)(ϕ-0)(ϕ-ϕm),ϕ∈(0,ϕM],
where ϕM, ϕm, 0 are roots of equation hϕ2-(2α/c)ϕ3+(2β/c)ϕ=0 and ϕM>0>ϕm. Taking (0,ϕM) as the initial values of the variables (ξ,ϕ), substituting (87) into the dϕ/dξ=y of (11), and then integrating it yield
(88)v(x,t)=ϕMcn2(Ω13ξ,k13),ξ≠T13,
where Ω13=α(ϕM-ϕm)/2c, k13=ϕM/(ϕM-ϕm), T13=(1/Ω13)sn-1(0,k13). By using the transformation u(x,t)=ln|v(x,t)| and (88), we obtain a periodic blow-up wave solution of (1) as follows:
(89)u(x,t)=ln|ϕMcn2(Ω13(x-ct),k13)|,x-ct≠T13.


### 3.5. Periodic Blow-Up Wave, Broken Solitary Wave, Broken Kink, and Antikink Wave Solutions in the Special Casesm=2 or m=1 and n=-2

Whenm=2 is even and n=-2, (14) can be reduced to
(90)y=±hϕ2-2αcmϕm+2+βc.
Because the singular straight line ϕ=0, this implies that (1) has broken kink and antikink wave solutions: see the following discussions.(i) Under the conditionsm=2, n=-2, h=-4αβ/c2, αβ<0, αc<0, (90) becomes
(91)y=±-αc|ϕ2-ϕ12|,(ϕ≠0),
where ϕ1=-β/α. Taking (0,ϕ1) as the initial values of the variables (ξ,ϕ), substituting (91) into the dϕ/dξ=y of (11), and then integrating it yield
(92)v(x,t)=ϕ(ξ)=±ϕ1tanh[Ω14ξ],
where Ω14=ϕ1-α/c. By using the transformations u(x,t)=ln|v(x,t)| and (92), we obtain a broken kink wave solutions of (1) as follows:
(93)u(x,t)=ln|ϕ1tanh[Ω14(x-ct)]|,
where (x-ct)∈(-∞,0) and (x-ct)∈(0,+∞).(ii) Under the conditionsh∈(hB1,+∞), αβ>0, αc>0 and m=2, n=-2, (90) becomes
(94)y=±αc(a2-ϕ2)(ϕ2-b2),
where a2=(1/2α)(ch+c2h2+4αβ), b2=(1/2α)(ch-c2h2+4αβ). Taking (0,a) as the initial values of the variables (ξ,ϕ), substituting (94) into the dϕ/dξ=y of (11), and then integrating it yield
(95)v(x,t)=ϕ(ξ)=±a2-(a2-b2)sn2(Ω15ξ,k15),
where a2=(1/2α)(ch+c2h2+4αβ), b2=(1/2α)(ch-c2h2+4αβ), Ω15=aα/c, k15=(a2-b2)/a2. By using the transformations u(x,t)=ln|v(x,t)| and (95), we obtain a periodic blow-up wave solution of (1) as follows:
(96)u(x,t)=0.5ln|a2-(a2-b2)sn2(Ω15ξ,k15)|.(iii) Under the conditionsm=1, n=-2, h=hB1=3αϕ2/c, αβ>0, αc>0, (90) becomes
(97)y=±αc|-ϕ2+ϕ|-ϕ2-2ϕ,(ϕ≠0),
where ϕ2=(-β/α)1/3. Taking (0,-0.5ϕ2) as the initial values of the variables (ξ,ϕ), substituting (97) into the dϕ/dξ=y of (11), and then integrating it yield
(98)v(x,t)=ϕ(ξ)=-ϕ2[3cosh(Ω16ξ)+1-1],ξ≠T16,
where Ω16=-3αϕ2/c, T16=(1/Ω16)cosh-1(2). By using the transformations u(x,t)=ln|v(x,t)| and (98), when (x-ct)∈(-∞,T16)⋃‍(T16,+∞), we obtain a broken solitary wave solution of (1) as follows:
(99)u(x,t)=ln|3ϕ2cosh(Ω16(x-ct))+1-ϕ2|.(iv) Under the conditionsh=hB1=3αϕ2/c, αβ<0, αc<0 and m=1, n=-2, (90) becomes
(100)y=±-αc|ϕ2-ϕ|ϕ2+2ϕ,(ϕ≠0),
where ϕ2=(-β/α)1/3. Taking (0,-0.5ϕ2) as the initial values of the variables (ξ,ϕ), substituting (100) into the dϕ/dξ=y of (11), and then integrating it yield
(101)v(x,t)=ϕ(ξ)=ϕ2[1-3cosh(Ω17ξ)+1],ξ≠T17,
where Ω17=3αϕ1/c, T17=(1/Ω17)cosh-1(2). By using the transformations u(x,t)=ln|v(x,t)| and (101), when (x-ct)∈(-∞,T17)⋃‍(T17,+∞), we obtain a broken solitary wave solution of (1) as follows:
(102)u(x,t)=ln|ϕ2-3ϕ2cosh(Ω17(x-ct))+1|.(v) Under the conditionsh∈(hB1,+∞), αβ<0, βc<0 and m=1, n=-2, (90) becomes
(103)y=±2αc(ϑ5-ϕ)(ϕ-η5)(ϕ-γ5),ϕ∈[η5,ϑ5],
where ϑ5, η5, γ5 are roots of equation β/2α+(ch/2α)ϕ2-ϕ3 and γ5<0<η5<ϑ5. Taking (0,ϑ5) as the initial values of the variables (ξ,ϕ), substituting (103) into the dϕ/dξ=y of (11), and then integrating it yield
(104)v(x,t)=ϕ(ξ)=ϑ5-(ϑ5-η5)sn2(Ω18ξ,k18),
where Ω18=α(ϑ5-γ5)/2c, k18=(ϑ5-η5)/(ϑ5-γ5). By using the transformations u(x,t)=ln|v(x,t)| and (104), we obtain a periodic blow-up wave solution of (1) as follows:
(105)u(x,t)=ln|ϑ5-(ϑ5-η5)sn2(Ω18(x-ct),k18)|.(vi) Under the conditionsh∈(hB1,+∞), αβ>0, βc<0 and m=1, n=-2, (90) becomes
(106)y=±2αc(ϑ5-ϕ)(η5-ϕ)(ϕ-γ5),ϕ∈[γ5,η5],
where γ5<η5<0<ϑ5. Taking (0,ϑ5) as the initial values of the variables (ξ,ϕ), substituting (106) into the dϕ/dξ=y of (11), and then integrating it yield
(107)v(x,t)=ϕ(ξ)=γ5+(η5-γ5)sn2(Ω19ξ,k19),
where γ5<η5<0<ϑ5, Ω19=-α(ϑ5-γ5)/2c, k19=(ϑ5-η5)/(ϑ5-γ5). By using the transformations u(x,t)=ln|v(x,t)| and (113), we obtain a periodic blow-up wave solution of (1) as follows:
(108)u(x,t)=ln|γ5+(η5-γ5)sn2(Ω19(x-ct),k19)|.


### 3.6. Periodic Wave, Broken Solitary Wave, Broken Kink, and Antikink Wave Solutions in the Special Casesn=-m or n=-2m and m∈Z+

(a) Whenn=-m, m∈Z+ and αβ<0, βc<0, αc>0, h>4-αβ/cm, (14) can be reduced to
(109)y=±(2β/cm)ϕm+h(ϕm)2-(2α/cm)(ϕm)3ϕm-1=±2α/cm(ϕgm-ϕm)(ϕm-ϕlm)(ϕm-0)ϕm-1,
where ϕg,l=[(1/4α)(cmh±c2m2h2+16αβ)]1/m. Taking (0,ϕg) as the initial values of the variables (ξ,ϕ), substituting (109) into the dϕ/dξ=y of (11), and then integrating it yields
(110)v(x,t)=ϕ(ξ)=±[ϕg-(ϕg-ϕl)sn2(Ω19ξ,k19)]1/m,
where Ω19=mαϕg/2cm, k19=(ϕg-ϕl)/ϕg. By using the transformations u(x,t)=ln|v(x,t)| and (110), we obtain a periodic wave solution of (1) as follows:
(111)u(x,t)=1mln|[ϕg-(ϕg-ϕl)sn2(Ω19(x-ct),k19)]|.(b) Whenn=-2m,m∈Z+and αβ<0, βc<0, αc>0, h>hB1, (14) can be reduced to
(112)y=±β/cm+h(ϕm)2-(2α/cm)(ϕm)3ϕm-1=±2α/cm(ϑm-ϕm)(ϕm-ηm)(ϕm-γm)ϕm-1,
where ϑm, ηm, γm are roots of equation β/cm+h(ϕm)2-(2α/cm)(ϕm)3=0 and ϑm>ηm>0>γm. Taking (0,ϑ) as the initial values of the variables (ξ,ϕ), substituting (112) into the dϕ/dξ=y of (11), and then integrating it yield
(113)v(x,t)=ϕ(ξ)=[ϑm-(ϑm-ηm)sn2(Ω20ξ,k20)]1/m,
where Ω20=mα(ϑm-γm)/2cm, k20=(ϑm-ηm)/(ϑm-γm). By using the transformations u(x,t)=ln|v(x,t)| and (113), we obtain periodic wave solutions of (1) as follows:
(114)u(x,t)=1mln|ϑm-(ϑm-ηm)sn2(Ω20(x-ct),k20)|.(c) Whenn=-2m, m∈Z+ and αβ>0, βc<0, αc>0, h>hB2, (14) can be reduced to
(115)y=±β/cm+h(ϕm)2-(2α/cm)(ϕm)3ϕm-1=±-2α/cm(ϑm-ϕm)(ηm-ϕm)(ϕm-γm)ϕm-1,
where ϑm>0>ηm>γm. Taking (0,η) as the initial values of the variables (ξ,ϕ), substituting (115) into the dϕ/dξ=y of (11), and then integrating it yield
(116)v(x,t)=[ηm-ϑmk212sn2(Ω21ξ,k21)dn2(Ω21ξ,k21)]1/m,
where Ω21=m-α(ϑm-γm)/2cm, k21=(ηm-ηm)/(ϑm-γm). By using the transformations u(x,t)=ln|v(x,t)| and (116), we obtain a periodic wave solution of (1) as follows:
(117)u(x,t)=1mln|ηm-ϑmk212sn2(Ω21ξ,k21)dn2(Ω21ξ,k21)|.


## 3.1. Different Kinds of Solitary Wave Solutions and Unboundary Wave Solutions in the Special Casesn=2m∈Z+ or n=m∈Z+

(i) Whenh>0, βc<0, n=2m and m is even, then (14) can be reduced to
(17)y=±ϕh-2αcmϕm-βcm(ϕm)2.
Taking ξ0=0 and ϕ(0)=ϕ1 as initial values, substituting (17) into the dϕ/dξ=y of (11), and then integrating it yield
(18)v(x,t)=±(γ1Ω1cosh(ω1(x-ct))+δ1)1/m,
where ϕ1=(-α/β+(α2+βcmh)/β2)1/m,Ω1=ϕ1m(α2+βcmh), ω1=-mh,γ1=cmh(cmh-αϕ1m), δ1=α(cmh-αϕ1m). By using program of Maple, it is easy to validate that (18) is the solution equation (9) when m is given. Substituting (18) into the transformation u(x,t)=ln|v(x,t)|, we obtain a solitary wave solutions of (1) as follows:
(19)u(x,t)=1mln|γ1Ω1cosh(ω1(x-ct))+δ1|.(ii) Whenn=m, h>0, c/(α+β)>0 and m is even, (14) can be reduced to
(20)y=±ϕh-2(α+β)cmϕm.
Taking ξ0=0 and ϕ(0)=(mhc/2(α+β))1/m as initial values, substituting (20) into the dϕ/dξ=y of (11), and then integrating it yield
(21)v(x,t)=±(mhcsech2ω2(x-ct)2(α+β))1/m,
where ω2=(1/2)mh. Substituting (21) into the transformation u(x,t)=ln|v(x,t)|, we obtain a solution without boundary of (1) as follows:
(22)u(x,t)=1mln|mhcsech2ω2(x-ct)2(α+β)|.
Equation (21) is smooth solitary wave solution of (9) but (22) is an exact solution without boundary of (1) because u→-∞ as v→0. In order to visually show dynamical behaviors of solutions v and u of (21) and (22), we plot graphs of their profiles which are shown in Figures 1(a) and 1(b), respectively.The phase portraits of solutions (21) and (22) for m=4, α=3, β=-1, c=2, h=9, t=0.1.
(a)
Smooth solitary wave
(b)
Unboundary wave(iii) Whenn=2m, h=0, βc>0, αc<0 and m is even, (14) can be reduced to
(23)y=±ϕ-2αcmϕm-βcm(ϕm)2.
Taking ξ0=0 and ϕ(0)=0 as initial values, substituting (23) into the dϕ/dξ=y of (11), and then integrating it yield
(24)v(x,t)=±(2(-αc)βc+mα2(x-ct)2)1/m.
Substituting (24) into the transformation u(x,t)=ln|v(x,t)|, we obtain an exact solutions of rational function type of (1) as follows:
(25)u(x,t)=1mln|2αcβc+mα2(x-ct)2|.(iv) Whenn=2m, h>0, αβ>0, βc>0 (or αβ<0, βc>0), and m is odd, (14) can be reduced to
(26)y=±βcmϕcmhβ-2αβϕm-(ϕm)2.
Respectively, taking ξ0=0, ϕ(0)=ϕ1, and ξ0=0, ϕ(0)=ϕ2 as initial values, substituting (26) into the dϕ/dξ=y of (11), and then integrating them yield
(27)v(x,t)=(γ1Ω1cosh(ω1(x-ct))+δ1)1/m,(28)v(x,t)=(γ2Ω2cosh(ω2(x-ct))+δ2)1/m,
where ϕ1 is given above and ϕ2=(-α/β-(α2+βcmh)/β2)1/m, Ω2=ϕ2m(α2+βcmh), ω2=mh, γ2=cmh(cmh-αϕ2m), δ2=α(cmh-αϕ2m). Respectively, substituting (27) and (28) into the transformation u(x,t)=ln|v(x,t)|, we obtain two smooth unboundary wave solutions of (1) as follows:
(29)u(x,t)=1mln|γ1Ω1cosh(ω1(x-ct))+δ1|,(30)u(x,t)=1mln|γ2Ω2cosh(ω2(x-ct))+δ2|.(v) Whenn=2m, 0<h<hA, αβ<0, βc<0 (or αβ>0, βc<0) and m is odd, (14) can be reduced to
(31)y=±-βcmϕ-cmhβ+2αβϕm+(ϕm)2.
Similarly, by using (31), we obtain two smooth solitary wave solutions of (1) which are the same as the solutions (29) and (30).(vi) Whenn=m,h>0 and m is odd, (14) can be reduced to (20), so the obtained solution is as the same as the solution (22).(vii) Whenn=2m, h=0, βc>0 and m is odd, (14) can be reduced to
(32)y=±βcmϕ-2αβϕm-(ϕm)2.
Taking ξ0=0 and ϕ(0)=(-2α/β)1/m as initial values, substituting (32) into the dϕ/dξ=y of (11), and then integrating it yield
(33)v(x,t)=(-2αβ[1+(mα2/βc)(x-ct)2])1/m.
Substituting (33) into the transformation u(x,t)=ln|v(x,t)|, we obtain a smooth unboundary wave solution of rational function type of (1) as follows:
(34)u(x,t)=1mln|2αβ[1+(mα2/βc)(x-ct)2]|.


## 3.2. Periodic Blow-Up Wave Solutions, Broken Kink Wave, and Antikink Wave Solutions in the Special Casen=2m∈Z+

(1) Whenn=2m, αβ<0, βc>0, hB1<h<0 (or hB2<h<0) and m is even, (14) can be reduced to
(35)y=±βcmϕcmhβ-2αβϕm-(ϕm)2.
Taking ξ0=0, ϕ(0)=(-α/β+α2/β2+mhc/β)1/m and ξ0=0, ϕ(0)=(-α/β-α2/β2+mhc/β)1/m as the initial values, substituting (35) into the dϕ/dξ=y of (11), and then integrating them yield
(36)v(x,t)=(cmhα-βΩ3sin(arcsinD1-ω3(x-ct)))1/m,(37)v(x,t)=-(cmhα+βΩ3sin(arcsinD2+ω3(x-ct)))1/m,
where Ω3=(α2+cmhβ)/β2, ω3=m-h, D1=(α2+αβΩ3+βcmh)/βΩ3(α+βΩ3), D2=(α2-αβΩ3+βcmh)/βΩ3(-α+βΩ3). Respectively, substituting (36) and (37) into the transformation u(x,t)=ln|v(x,t)|, we obtain two periodic blow-up wave solutions of (1) as follows:
(38)u(x,t)=1mln|cmhα-βΩ3sin(arcsinD1-ω3(x-ct))|,(39)u(x,t)=1mln|cmhα+βΩ3sin(arcsinD2+ω3(x-ct))|.(2) Whenn=2m, αβ<0, βc>0, hB2<h<0 and m is odd, (14) can be reduced to
(40)y=±βcmϕcmhβ-2αβϕm-(ϕm)2,
which is the same as (35), so the obtained solution is the same as solution (39). Similarly, when n=2m, αβ<0, βc>0, hB1<h<0 and m is odd, the obtained solution is also the same as solution (38).(3) whenn=2m, αβ<0, βc<0, h=hB1=hB2=-α2/βcm and m is even, (14) can be reduced to
(41)y=±-1cmβϕ(βϕm+α).
Substituting (41) into the dϕ/dξ=y of (11) and then integrating it and setting the integral constants as zero yield
(42)v(x,t)=(-α2β(1±tanhΩ42(x-ct)))1/m,
where Ω4=α-m/cβ. Substituting (42) into the transformation u(x,t)=ln|v(x,t)|, we obtain an unboundary wave solution of (1) as follows:
(43)u(x,t)=1mln|α2β[1±tanhΩ42(x-ct)]|.
Equation (42) is smooth kink wave solution of (9), but (43) is a broken kink wave solution without boundary of (1) because u→-∞ as v→0. In order to visually show dynamical behaviors of solutions v and u of (42) and (43), we plot graphs of their profiles which are shown in Figures 2(a) and 2(b), respectively.The phase portraits of solutions (42) and (43) for m=4,α=0.3,β=-0.3,c=2,t=0.1.
(a)
Smooth kink wave
(b)
Unboundary wave(4) Whenn=2m, αβ>0, βc<0, h=hA or αβ<0, βc<0, h=hA and m is odd, (14) can be reduced to (41). So the obtained solution is also the same as the solution (43).


## 3.3. Periodic Blow-Up Wave Solutions and Solitary Wave Solutions of Blow-Up Form in the Special Casem=2, n=-1

Whenn=-1, (14) can be reduced to
(44)y=±hϕ2-2αcmϕm+2+2βcϕ.(a) Under the conditionsαβ>0, αc>0, h∈(-∞,+∞) and m=2, (44) becomes
(45)y=±αc(ϑ1-ϕ)(ϕ-0)(ϕ-γ1)(ϕ-δ1),ϕ∈(0,ϑ1],
where ϑ1, γ1, δ1 are three roots of equation 2β/α+(ch/α)ϕ-ϕ3 and ϑ1>0>γ1>δ1. These three roots ϑ1, γ1, δ1 can be obtained by Cardano formula as long as the parameters α, β, c, h are fixed (or given) concretely. For example, taking α=4.0, β=2.0, c=4.0, h=9.0, the ϑ1≐3.054084215, γ1≐-0.1112641576, δ1≐-2.942820058. Taking (0,ϑ1) as the initial values of the variables (ξ,ϕ), substituting (45) into the dϕ/dξ=y of (11), and then integrating it yield
(46)v(x,t)=ϕ(ξ)=ϑ1(-δ1)cn2(Ω1(ξ),k1)(-δ1)+ϑ1sn2(Ω1ξ,k1),ξ≠T1,
where Ω1=(1/2)-αδ1(ϑ1-δ1)/c, k1=ϑ1(γ1-δ1)/-δ1(ϑ1-γ1), T1=(1/Ω1)sn-1(1,k1). By using the transformation u(x,t)=ln|v(x,t)| and (46), we obtain a periodic blow-up wave solution of (1) as follows:
(47)u(x,t)=ln|ϑ1(-δ1)cn2(Ω1(x-ct),k1)(-δ1)+ϑ1sn2(Ω1(x-ct),k1)|,x-ct≠T1.(b) Under the conditionsαβ>0, αc>0, h∈(hA,+∞) and m=2, (44) becomes
(48)y=±αc(ϑ1-ϕ)(0-ϕ)(γ1-ϕ)(ϕ-δ1),ϕ∈[δ1,γ1].
Similarly, taking (0,δ1) as the initial values of the variables (ξ,ϕ), substituting (48) into the dϕ/dξ=y of (11), and then integrating it yield
(49)v(x,t)=ϕ(ξ)=γ1δ1δ1+(γ1-δ1)sn2(Ω1ξ,k1).
By using the transformation u(x,t)=ln|v(x,t)| and (49), we obtain a periodic blow-up wave solution of (1) as follows:
(50)u(x,t)=ln|γ1δ1δ1+(γ1-δ1)sn2(Ω1(x-ct),k1)|.(c) Under the conditionsαβ<0, αc>0, h∈(-∞,+∞) and m=2, (44) becomes
(51)y=±αc(ϑ2-ϕ)(η2-ϕ)(0-ϕ)(ϕ-δ2),ϕ∈[δ2,0),
where ϑ2, η2, 0, δ2 are roots of equation ϕ(2β/α+(ch/α)ϕ-ϕ3)=0 and ϑ2>η2>0>δ2. As in the above cases, these three roots can be obtained by Cardano formula as long as the parameters α, β, c, h are given concretely, so the similar cases will be not commentated anymore in the following discussions. Similarly, taking (0,δ2) as the initial values of the variables (ξ,ϕ), substituting (51) into the dϕ/dξ=y of (11), and then integrating it yield
(52)v(x,t)=ϕ(ξ)=η2δ2sn2(Ω2(x-ct),k2)(η2-δ2)+δ2sn2(Ω2ξ,k2),ξ≠T2,
where Ω2=(1/2)αϑ2(η2-δ2)/c, k2=-δ2(ϑ2-η2)/ϑ2(η2-δ2), T2=(1/Ω2)sn-1(0,k2). By using the transformation u(x,t)=ln|v(x,t)| and (52), we obtain a periodic blow-up wave solution of (1) as follows:
(53)u(x,t)=ln|η2δ2sn2(Ω2(x-ct),k2)(η2-δ2)+δ2sn2(Ω2ξ,k2)|,x-ct≠T2.(d) Under the conditionsαβ<0, αc>0, h∈(hA,+∞) and m=2, (44) becomes
(54)y=±αc(ϑ2-ϕ)(ϕ-η2)(ϕ-0)(ϕ-δ2).
Taking the (0,ϑ2) as the initial values of the variables (ξ,ϕ), substituting (54) into the dϕ/dξ=y of (11), and then integrating it yield
(55)v(x,t)=ϕ(ξ)=ϑ2(η2-δ2)dn2(Ω2(x-ct),k2)(η2-δ2)+(ϑ2-η2)sn2(Ω2ξ,k2).
By using the transformations u(x,t)=ln|v(x,t)| and (55), we obtain a periodic blow-up wave solution of (1) as follows:
(56)u(x,t)=ln|ϑ2(η2-δ2)dn2(Ω2(x-ct),k2)(η2-δ2)+(ϑ2-η2)sn2(Ω2(x-ct),k2)|.(e) Under the conditionsαβ<0, αc<0, h∈(-∞,hA) and m=2, (44) becomes
(57)y=±-αc(ϑ3-ϕ)(η3-ϕ)(ϕ-0)(ϕ-δ3),ϕ∈(0,η3],
where ϑ3, η3, 0, δ3 are roots of equation ϕ(-2β/α-(ch/α)ϕ+ϕ3)=0 and ϑ3>η3>0>δ3. Taking the (0,η3) as the initial values of the variables (ξ,ϕ), substituting (57) into the dϕ/dξ=y of (11), and then integrating it yield
(58)v(x,t)=ϕ(ξ)=ϑ3η3cn2(Ω3ξ,k3)δ3-η3sn2(Ω3ξ,k3),ξ≠T3,
where Ω3=(1/2)-αϑ3(η3-δ3)/c, k3=η3(ϑ3-δ3)/ϑ3(η3-δ3), T3=(1/Ω3)sn-1(1,k3). By using the transformation u(x,t)=ln|v(x,t)| and (58), we obtain a periodic blow-up wave solution of (1) as follows:
(59)u(x,t)=ln|ϑ3η3cn2(Ω3(x-ct),k3)δ3-η3sn2(Ω3(x-ct),k3)|,x-ct≠T3.(f) Under the conditionsαβ<0, αc<0, h∈(-∞,hA) and m=2, (44) becomes
(60)y=±-αc(ϑ4-ϕ)(0-ϕ)(ϕ-γ4)(ϕ-δ4),ϕ∈[γ4,0),
where ϑ4, 0, γ4, δ4 are roots of equation ϕ(-2β/α-(ch/α)ϕ+ϕ3)=0 and ϑ4>0>γ4>δ4. Taking the (0,γ4) as the initial values of the variables (ξ,ϕ), substituting (60) into the dϕ/dξ=y of (11), and then integrating it yield
(61)v(x,t)=ϕ(ξ)=δ4γ4cn2(Ω4ξ,k4)δ4-γ4cn2(Ω4ξ,k4),ξ≠T4,
where Ω4=(1/2)αδ4(ϑ4-γ4)/c, k4=γ4(ϑ4-δ4)/ϑ4(ϑ4-γ4), T4=(1/Ω4)sn-1(1,k4). By using the transformation u(x,t)=ln|v(x,t)| and (61), we obtain a periodic below-up wave solutions of (1) as follows:
(62)u(x,t)=ln|δ4γ4cn2(Ω4(x-ct),k4)δ4-γ4cn2(Ω4(x-ct),k4)|,x-ct≠T4.(g) Under the conditionsm=2, αc<0, h=hA=-α2/2cβ, (44) becomes
(63)y=±-αc|ϕ1-ϕ|ϕ(ϕ+2ϕ1),ϕ≠0,
where ϕ1=(-β/α)1/3. Taking (0,(1/2)ϕ1) as the initial values of the variables (ξ,ϕ), substituting (63) into the dϕ/dξ=y of (11), and then integrating it yield
(64)v(x,t)=ϕ(ξ)=ϕ1[1-3cosh(Ω5ξ)+2],ξ≠T5,
where Ω5=ϕ1-3α/c, T5=(1/Ω5)cosh-1(1). Clearly, we have ϕ(T5)=0, ϕ(±∞)=ϕ1. By using the transformation u(x,t)=ln|v(x,t)| and (64), we obtain a solitary wave solution of blow-up form of (1) as follows:
(65)u(x,t)=ln|ϕ1[1-3cosh(Ω5(x-ct))+2]|,x-ct≠T5.
Equation (64) is smooth solitary wave solution of (9), but (65) is a solitary wave solution of blow-up form for (1) because u→-∞ as v→0. In order to visually show dynamical behaviors of solutions v and u of (64) and (65), we plot graphs of their profiles which are shown in Figures 3(a) and 3(b), respectively.The phase portraits of solutions (64) and (65) for m=2, α=0.3, β=-0.3, c=-2, t=0.1.
(a)
Smooth solitary wave
(b)
Solitary wave of blow-up form


## 3.4. Different Kinds of Periodic Wave, Broken Solitary Wave Solutions in the Special Casem=1, n=-1

(1) Under the conditionsh∈(hA,+∞), αβ>0, βc>0 (or αβ<0, βc>0), αc>0 and m=1, (44) becomes
(66)y=±2αc(ϕM-ϕ)(ϕ-ϕm)(ϕ-0),ϕ∈[ϕm,ϕM],
where ϕM,m=(hc±h2c2+16αβ)/4α and 0 are three roots of equation hϕ2-(2α/c)ϕ3+(2β/c)ϕ=0 and ϕM>ϕm>0. Taking (0,ϕm) as the initial values of the variables (ξ,ϕ), substituting (66) into the dϕ/dξ=y of (11), and then integrating it yield
(67)v(x,t)=ϕ(ξ)=ϕm-(ϕM-ϕm)sn2(Ω6ξ,k6),
where Ω6=αϕM/2c, k6=(ϕM-ϕm)/ϕM. By using the transformation u(x,t)=ln|v(x,t)| and (67), we obtain a double periodic wave solution of (1) as follows:
(68)u(x,t)=ln|ϕm-(ϕM-ϕm)sn2(Ω6(x-ct),k6)|.
Equation (67) is smooth periodic wave solution of (9), but (68) is a double periodic wave solution of blow-up form of (1) because u→-∞ as v→0. In order to visually show dynamical behaviors of solutions v and u of (67) and (68), we plot graphs of their profiles which are shown in Figures 4(a) and 4(b), respectively.The phase portraits of solutions (67) and (68) for h=8, α=2.5, β=-2.5, c=2, t=0.1.
(a)
Smooth periodic wave
(b)
Double periodic wave of blow-up type(2) Under the conditionh∈(-∞,hA),αβ<0, βc>0 (or αβ<0, βc<0),αc>0 and m=1, (44) becomes
(69)y=±2αc(0-ϕ)(ϕ-ϕM)(ϕ-ϕm),ϕ∈[ϕM,0),
where ϕM,ϕm,0 are roots of equation hϕ2-(2α/c)ϕ3+(2β/c)ϕ=0 and ϕm<ϕM<0. Taking (0,ϕM) as the initial values of the variables (ξ,ϕ), substituting (69) into the dϕ/dξ=y of (11), and then integrating it yield
(70)v(x,t)=ϕ(ξ)=ϕMsn2(Ω7ξ,k7),ξ≠T7,
where Ω7=-αϕM/2c, k7=ϕM/ϕm, T7=(1/Ω7)sn-1(0,k7). By using the transformation u(x,t)=ln|v(x,t)| and (70), we obtain a periodic blow-up wave solution of (1) as follows:
(71)u(x,t)=ln|ϕMsn2(Ω7(x-ct),k7)|,x-ct≠T7.(3) Under the conditionsh∈(hA,+∞), αβ<0, βc>0 (or αβ>0, βc>0) and m=1, (44) becomes
(72)y=±-2αc(0-ϕ)(ϕM-ϕ)(ϕ-ϕm),ϕ∈[ϕm,ϕM],
where ϕM, ϕm, 0 are roots of equation hϕ2-(2α/c)ϕ3+(2β/c)ϕ=0 and 0>ϕM>ϕm. Taking (0,ϕM) as the initial values of the variables (ξ,ϕ), substituting (72) into the dϕ/dξ=y of (11), and then integrating it yield
(73)v(x,t)=ϕ(ξ)=ϕMdn2(Ω8ξ,k8),
where Ω8=αϕm/2c, k8=(ϕM-ϕm)/-ϕm. By using the transformation u(x,t)=ln|v(x,t)| and (73), we obtain a periodic blow-up wave solutions of (1) as follows:
(74)u(x,t)=ln|ϕMdn2(Ω8(x-ct),k8)|.(4) Under the conditionsh∈(-∞,hA),αβ<0, βc>0 (or αβ<0, βc<0) and m=1, (44) becomes
(75)y=±-2αc(ϕM-ϕ)(ϕm-ϕ)(ϕ-0),ϕ∈(0,ϕm],
where ϕM, ϕm, 0 are roots of equation hϕ2-(2α/c)ϕ3+(2β/c)ϕ=0 and ϕM>ϕm>0. Taking (0,ϕm) as the initial values of the variables (ξ,ϕ), substituting (75) into the dϕ/dξ=y of (11), and then integrating it yield
(76)v(x,t)=ϕ(ξ)=ϕmcn2(Ω9ξ,k9)dn2(Ω9ξ,k9),ξ≠T9,
where Ω9=-αϕM/2c, k9=ϕm/ϕM, T9=(1/Ω9)sn-1(0,k9). By using the transformation u(x,t)=ln|v(x,t)| and (76), we obtain periodic blow-up wave solutions of (1) as follows:
(77)u(x,t)=ln|ϕmcn2(Ω9(x-ct),k9)dn2(Ω9(x-ct),k9)|,x-ct≠T9.(5) Under the conditionsh=hA=-4αϕ1/c, αβ<0, βc>0 and m=1, (44) becomes
(78)y=±2αc|ϕ1+ϕ|-ϕ,ϕ∈[-ϕ1,0),
where ϕ1=(-β/α)1/2. Taking (0,-ϕ1) as the initial values of the variables (ξ,ϕ), substituting (78) into the dϕ/dξ=y of (11), and then integrating it yield
(79)v(x,t)=ϕ(ξ)=-ϕ1tanh2(0.5Ω10ξ),ξ≠0,
where Ω10=2αϕ1/c. By using the transformation u(x,t)=ln|v(x,t)| and (79), we obtain a solitary wave solution of blow-up form of (1) as follows:
(80)u(x,t)=ln|ϕ1tanh2(0.5Ω10(x-ct))|,x-ct≠0.(6) Under the conditionsh=hA=4αϕ1/c, αβ<0, βc>0 and m=1, (44) becomes
(81)y=±-2αc|ϕ-ϕ1|ϕ,ϕ∈(0,ϕ1],
where ϕ1=(-β/α)1/2. Taking (0,ϕ1) as the initial values of the variables (ξ,ϕ), substituting (81) into the dϕ/dξ=y of (11), and then integrating it yield
(82)v(x,t)=ϕ(ξ)=ϕ1tanh2(0.5Ω11ξ),ξ≠0,
where Ω11=-2αϕ1/c. By using the transformation u(x,t)=ln|v(x,t)| and (82), we obtain a broken solitary wave solution of (1) as follows:
(83)u(x,t)=ln|ϕ1tanh2(0.5Ω11(x-ct))|,x-ct≠0.(7) Under the conditionsh∈(-∞,0], αβ>0, βc<0 and m=1, (44) becomes
(84)y=±-2αc(ϕM-ϕ)(0-ϕ)(ϕ-ϕm),ϕ∈[ϕm,0),
where ϕM, ϕm, 0 are roots of equation ϕ2h-(2α/c)ϕ3+(2β/c)ϕ=0 and ϕM>0>ϕm. Taking (0,ϕM) as the initial values of the variables (ξ,ϕ), substituting (84) into the dϕ/dξ=y of (11), and then integrating it yield
(85)v(x,t)=ϕ(ξ)=-ϕMk122sn2(Ω12ξ,k12)dn2(Ω12ξ,k12),ξ≠T12,
where Ω12=-α(ϕM-ϕm)/2c, k12=-ϕm/(ϕM-ϕm), T12=(1/Ω12)sn-1(0,k12). By using the transformation u(x,t)=ln|v(x,t)| and (85), we obtain a periodic blow-up wave solution of (1) as follows:
(86)u(x,t)=ln|-ϕMk122sn2(Ω12(x-ct),k12)dn2(Ω12(x-ct),k12)|,x-ct≠T12.(8) Under the conditionsh∈(-∞,0], αβ>0, βc>0 and m=1, (44) becomes
(87)y=±2αc(ϕM-ϕ)(ϕ-0)(ϕ-ϕm),ϕ∈(0,ϕM],
where ϕM, ϕm, 0 are roots of equation hϕ2-(2α/c)ϕ3+(2β/c)ϕ=0 and ϕM>0>ϕm. Taking (0,ϕM) as the initial values of the variables (ξ,ϕ), substituting (87) into the dϕ/dξ=y of (11), and then integrating it yield
(88)v(x,t)=ϕMcn2(Ω13ξ,k13),ξ≠T13,
where Ω13=α(ϕM-ϕm)/2c, k13=ϕM/(ϕM-ϕm), T13=(1/Ω13)sn-1(0,k13). By using the transformation u(x,t)=ln|v(x,t)| and (88), we obtain a periodic blow-up wave solution of (1) as follows:
(89)u(x,t)=ln|ϕMcn2(Ω13(x-ct),k13)|,x-ct≠T13.


## 3.5. Periodic Blow-Up Wave, Broken Solitary Wave, Broken Kink, and Antikink Wave Solutions in the Special Casesm=2 or m=1 and n=-2

Whenm=2 is even and n=-2, (14) can be reduced to
(90)y=±hϕ2-2αcmϕm+2+βc.
Because the singular straight line ϕ=0, this implies that (1) has broken kink and antikink wave solutions: see the following discussions.(i) Under the conditionsm=2, n=-2, h=-4αβ/c2, αβ<0, αc<0, (90) becomes
(91)y=±-αc|ϕ2-ϕ12|,(ϕ≠0),
where ϕ1=-β/α. Taking (0,ϕ1) as the initial values of the variables (ξ,ϕ), substituting (91) into the dϕ/dξ=y of (11), and then integrating it yield
(92)v(x,t)=ϕ(ξ)=±ϕ1tanh[Ω14ξ],
where Ω14=ϕ1-α/c. By using the transformations u(x,t)=ln|v(x,t)| and (92), we obtain a broken kink wave solutions of (1) as follows:
(93)u(x,t)=ln|ϕ1tanh[Ω14(x-ct)]|,
where (x-ct)∈(-∞,0) and (x-ct)∈(0,+∞).(ii) Under the conditionsh∈(hB1,+∞), αβ>0, αc>0 and m=2, n=-2, (90) becomes
(94)y=±αc(a2-ϕ2)(ϕ2-b2),
where a2=(1/2α)(ch+c2h2+4αβ), b2=(1/2α)(ch-c2h2+4αβ). Taking (0,a) as the initial values of the variables (ξ,ϕ), substituting (94) into the dϕ/dξ=y of (11), and then integrating it yield
(95)v(x,t)=ϕ(ξ)=±a2-(a2-b2)sn2(Ω15ξ,k15),
where a2=(1/2α)(ch+c2h2+4αβ), b2=(1/2α)(ch-c2h2+4αβ), Ω15=aα/c, k15=(a2-b2)/a2. By using the transformations u(x,t)=ln|v(x,t)| and (95), we obtain a periodic blow-up wave solution of (1) as follows:
(96)u(x,t)=0.5ln|a2-(a2-b2)sn2(Ω15ξ,k15)|.(iii) Under the conditionsm=1, n=-2, h=hB1=3αϕ2/c, αβ>0, αc>0, (90) becomes
(97)y=±αc|-ϕ2+ϕ|-ϕ2-2ϕ,(ϕ≠0),
where ϕ2=(-β/α)1/3. Taking (0,-0.5ϕ2) as the initial values of the variables (ξ,ϕ), substituting (97) into the dϕ/dξ=y of (11), and then integrating it yield
(98)v(x,t)=ϕ(ξ)=-ϕ2[3cosh(Ω16ξ)+1-1],ξ≠T16,
where Ω16=-3αϕ2/c, T16=(1/Ω16)cosh-1(2). By using the transformations u(x,t)=ln|v(x,t)| and (98), when (x-ct)∈(-∞,T16)⋃‍(T16,+∞), we obtain a broken solitary wave solution of (1) as follows:
(99)u(x,t)=ln|3ϕ2cosh(Ω16(x-ct))+1-ϕ2|.(iv) Under the conditionsh=hB1=3αϕ2/c, αβ<0, αc<0 and m=1, n=-2, (90) becomes
(100)y=±-αc|ϕ2-ϕ|ϕ2+2ϕ,(ϕ≠0),
where ϕ2=(-β/α)1/3. Taking (0,-0.5ϕ2) as the initial values of the variables (ξ,ϕ), substituting (100) into the dϕ/dξ=y of (11), and then integrating it yield
(101)v(x,t)=ϕ(ξ)=ϕ2[1-3cosh(Ω17ξ)+1],ξ≠T17,
where Ω17=3αϕ1/c, T17=(1/Ω17)cosh-1(2). By using the transformations u(x,t)=ln|v(x,t)| and (101), when (x-ct)∈(-∞,T17)⋃‍(T17,+∞), we obtain a broken solitary wave solution of (1) as follows:
(102)u(x,t)=ln|ϕ2-3ϕ2cosh(Ω17(x-ct))+1|.(v) Under the conditionsh∈(hB1,+∞), αβ<0, βc<0 and m=1, n=-2, (90) becomes
(103)y=±2αc(ϑ5-ϕ)(ϕ-η5)(ϕ-γ5),ϕ∈[η5,ϑ5],
where ϑ5, η5, γ5 are roots of equation β/2α+(ch/2α)ϕ2-ϕ3 and γ5<0<η5<ϑ5. Taking (0,ϑ5) as the initial values of the variables (ξ,ϕ), substituting (103) into the dϕ/dξ=y of (11), and then integrating it yield
(104)v(x,t)=ϕ(ξ)=ϑ5-(ϑ5-η5)sn2(Ω18ξ,k18),
where Ω18=α(ϑ5-γ5)/2c, k18=(ϑ5-η5)/(ϑ5-γ5). By using the transformations u(x,t)=ln|v(x,t)| and (104), we obtain a periodic blow-up wave solution of (1) as follows:
(105)u(x,t)=ln|ϑ5-(ϑ5-η5)sn2(Ω18(x-ct),k18)|.(vi) Under the conditionsh∈(hB1,+∞), αβ>0, βc<0 and m=1, n=-2, (90) becomes
(106)y=±2αc(ϑ5-ϕ)(η5-ϕ)(ϕ-γ5),ϕ∈[γ5,η5],
where γ5<η5<0<ϑ5. Taking (0,ϑ5) as the initial values of the variables (ξ,ϕ), substituting (106) into the dϕ/dξ=y of (11), and then integrating it yield
(107)v(x,t)=ϕ(ξ)=γ5+(η5-γ5)sn2(Ω19ξ,k19),
where γ5<η5<0<ϑ5, Ω19=-α(ϑ5-γ5)/2c, k19=(ϑ5-η5)/(ϑ5-γ5). By using the transformations u(x,t)=ln|v(x,t)| and (113), we obtain a periodic blow-up wave solution of (1) as follows:
(108)u(x,t)=ln|γ5+(η5-γ5)sn2(Ω19(x-ct),k19)|.


## 3.6. Periodic Wave, Broken Solitary Wave, Broken Kink, and Antikink Wave Solutions in the Special Casesn=-m or n=-2m and m∈Z+

(a) Whenn=-m, m∈Z+ and αβ<0, βc<0, αc>0, h>4-αβ/cm, (14) can be reduced to
(109)y=±(2β/cm)ϕm+h(ϕm)2-(2α/cm)(ϕm)3ϕm-1=±2α/cm(ϕgm-ϕm)(ϕm-ϕlm)(ϕm-0)ϕm-1,
where ϕg,l=[(1/4α)(cmh±c2m2h2+16αβ)]1/m. Taking (0,ϕg) as the initial values of the variables (ξ,ϕ), substituting (109) into the dϕ/dξ=y of (11), and then integrating it yields
(110)v(x,t)=ϕ(ξ)=±[ϕg-(ϕg-ϕl)sn2(Ω19ξ,k19)]1/m,
where Ω19=mαϕg/2cm, k19=(ϕg-ϕl)/ϕg. By using the transformations u(x,t)=ln|v(x,t)| and (110), we obtain a periodic wave solution of (1) as follows:
(111)u(x,t)=1mln|[ϕg-(ϕg-ϕl)sn2(Ω19(x-ct),k19)]|.(b) Whenn=-2m,m∈Z+and αβ<0, βc<0, αc>0, h>hB1, (14) can be reduced to
(112)y=±β/cm+h(ϕm)2-(2α/cm)(ϕm)3ϕm-1=±2α/cm(ϑm-ϕm)(ϕm-ηm)(ϕm-γm)ϕm-1,
where ϑm, ηm, γm are roots of equation β/cm+h(ϕm)2-(2α/cm)(ϕm)3=0 and ϑm>ηm>0>γm. Taking (0,ϑ) as the initial values of the variables (ξ,ϕ), substituting (112) into the dϕ/dξ=y of (11), and then integrating it yield
(113)v(x,t)=ϕ(ξ)=[ϑm-(ϑm-ηm)sn2(Ω20ξ,k20)]1/m,
where Ω20=mα(ϑm-γm)/2cm, k20=(ϑm-ηm)/(ϑm-γm). By using the transformations u(x,t)=ln|v(x,t)| and (113), we obtain periodic wave solutions of (1) as follows:
(114)u(x,t)=1mln|ϑm-(ϑm-ηm)sn2(Ω20(x-ct),k20)|.(c) Whenn=-2m, m∈Z+ and αβ>0, βc<0, αc>0, h>hB2, (14) can be reduced to
(115)y=±β/cm+h(ϕm)2-(2α/cm)(ϕm)3ϕm-1=±-2α/cm(ϑm-ϕm)(ηm-ϕm)(ϕm-γm)ϕm-1,
where ϑm>0>ηm>γm. Taking (0,η) as the initial values of the variables (ξ,ϕ), substituting (115) into the dϕ/dξ=y of (11), and then integrating it yield
(116)v(x,t)=[ηm-ϑmk212sn2(Ω21ξ,k21)dn2(Ω21ξ,k21)]1/m,
where Ω21=m-α(ϑm-γm)/2cm, k21=(ηm-ηm)/(ϑm-γm). By using the transformations u(x,t)=ln|v(x,t)| and (116), we obtain a periodic wave solution of (1) as follows:
(117)u(x,t)=1mln|ηm-ϑmk212sn2(Ω21ξ,k21)dn2(Ω21ξ,k21)|.


## 4. Exact Traveling Wave Solutions of (1) and Their Properties under the Conditions m∈Z-, n∈Z-

In this sections, we consider the casem∈Z-, n∈Z- of (1); that is, we will investigate different kinds of exact traveling wave solutions of (1) under the conditions m∈Z-, n∈Z-. Letting m=-q, n=-p, p∈Z+, q∈Z+, (11) becomes
(118)dϕdξ=y,dydξ=cy2-αϕ2-q-βϕ2-pcϕ.First, we consider a special casep=q=1. When p=q=1, under the transformation dξ=cϕdτ, system (118) can be reduced to the following regular system:
(119)dϕdτ=cϕy,dydτ=cy2-(α+β)ϕ,
which is an integrable system and it has the following first integral:
(120)y2=ϕ2h~+2(α+β)cϕ,
where h~ is integral constant. When ϕ≠0, we define
(121)H(ϕ,y)=y2ϕ2-2(α+β)cϕ=h~.
Obviously, when α≠-β, system (119) has only one equilibrium point O(0,0) in the ϕ-axis.Second, we consider the special casep=2, q=1, and another case p=1, q=2 is very similar to the front case, so we only discuss the front case here. When p=2, q=1, under the transformation dξ=cϕdτ, system (118) can be reduced to the following regular system:
(122)dϕdτ=cϕy,dydτ=cy2-αϕ-β,
which is an integrable system and it has the following first integral:
(123)y2=h~ϕ2+2αcϕ+βc,
also we define
(124)H(ϕ,y)=y2ϕ2-2αcϕ-βcϕ2=h~.
Indeed, system (122) has only one equilibrium point A~(ϕ0,0) in the ϕ-axis, where ϕ0=-β/α. When cβ>0, system (122) has two equilibrium points S±(0,Y±) in the singular line ϕ=0, where Y±=±β/c.Finally, we consider the general casep>2. When p>2, under the transformation dξ=cϕp-1dτ, system (118) can be reduced to the following regular system:
(125)dϕdτ=cϕp-1y,dydτ=cϕp-2y2-αϕp-q-β
which is an integrable system and it has the following first integral:
(126)y2=ϕ2h~+2αcqϕ2-q+2βcpϕ2-p.
When ϕ≠0, we define
(127)H(ϕ,y)=y2ϕ2-2αcqϕ-q-2βcpϕ-p=h~.
Obviously, when p=q, system (125) has not any equilibrium point in the ϕ-axis. When p-q is even number and αβ<0, system (125) has two equilibrium points B1,2(±ϕ+,0) in the ϕ-axis, where ϕ+=(-β/α)1/(p-q). When p-q is odd number, system (125) has only one equilibrium point at B1(ϕ+,0) in the ϕ-axis.Respectively substituting the above equilibrium points into (121), (124), and (127), we have
(128)h~A~=H(ϕ0,0)=-α2cβ,(129)h~1=H(ϕ+,0)=-2αcqϕ+-q-2βcpϕ+-p,(130)h~2=H(-ϕ+,0)=(-1)q+12αcqϕ+-q+(-1)p+12βcpϕ+-p.


### 4.1. Nonsmooth Peakon Solutions and Periodic Blow-Up Wave Solutions under Two Special Casesp=q=1 and p=2, q=1

(1) Whenp=q=1 and αc>0, h~<0, (120) can be reduced to
(131)y=±h~ϕ2-2(α+β)cϕ.
Substituting (131) into the first equation of (118), taking (0,ch~/(2(α+β))) as initial value conditions, and integrating it, we obtain
(132)v(x,t)=ϕ(ξ)=α+βch(cos-h~ξ-1).
From the transformation u(x,t)=ln|v(x,t)| and (132), we obtain a periodic blow-up wave solution of (1) as follows:
(133)u(x,t)=ln|α+βch[cos(-h~(x-ct))-1]|,
where (x-ct)≠2Kπ/-h~,K∈Z.(2) Whenp=2, q=1 and αβ>0, βc>0 (or αβ<0, βc>0), h~=h~A~=α2/βc, (123) can be reduced to
(134)y=±αβc|ϕ+βα|.
Similarly, substituting (134) into the first equation of (118) and then integrating it, and setting the integral constant as zero, we have
(135)v(x,t)=ϕ(ξ)=-βα[1-exp(-|αξ|βc)].
From the transformation u(x,t)=ln|v(x,t)| and (135), we obtain a broken solitary cusp wave solution of (1) as follows:
(136)u(x,t)=ln|-βα[1-exp(-|α(x-ct)|βc)]|,(x-ct)≠0.(3) Whenp=2,q=1 and αβ>0, βc>0 (or αβ<0,βc>0),h~<0, (123) can be reduced to
(137)y=±-h~-βch~-2αch~ϕ-ϕ2=±-h~(ϕM-ϕ)(ϕ-ϕm),
where ϕM=(-α+α2-ch~β)/ch~, ϕm=(-α-α2-ch~β)/ch~ are roots of equation -β/ch~-(2α/ch~)ϕ-ϕ2=0. Substituting (137) into the first equation of (118), taking (0,ϕM) as initial value conditions, and then integrating it, we have
(138)v(x,t)=ϕ(ξ)=12[(-hξ)(ϕM+ϕm)-(ϕM-ϕm)cos(-hξ)],ξ∈(-T1,T1),
where T1=(1/-h~)arccos((ϕM+ϕm)/(ϕM-ϕm)). From the transformation u(x,t)=ln|v(x,t)| and (138), we obtain a periodic blow-up wave solution of (1) as follows:
(139)u(x,t)=ln|12[(ϕM+ϕm)-(ϕM-ϕm)cos(-h~(x-ct))]|,
where x-ct≠T1+Kπ,K∈Z.(4) Whenp=2,q=1 and αβ>0, βc<0 (or αβ<0,βc<0),h~>-α2/cβ, (123) can be reduced to
(140)y=±h~ϕ2+2αcϕ+αc.
Similarly, we have
(141)v(x,t)=ϕ(ξ)=-αch~-α2-βch~ch~cos(-h~ξ).
From the transformation u(x,t)=ln|v(x,t)| and (141), we obtain a periodic blow-up wave solution of (1) as follows:
(142)u(x,t)=ln|αch~+α2-βch~ch~cos-h~(x-ct)|.


### 4.2. Periodic Blow-Up Wave Solutions in the Special Casep=2q>1

Whenp=2q>2 and αβ<0, αc>0 (or αβ>0, αc<0), h~∈(h~1,0), (118) becomes
(143)y=±ϕ2(h~+2αcqϕ-q+βcqϕ-2q)=±-h~-β/cqh~-(2α/cqh~)ϕq-(ϕq)2ϕq-1.
Substituting (143) into the first equation of (118) and then integrating it, we have
(144)v(x,t)=ϕ(ξ)=[-αcqh~+α2-cqh~βc2q2h~2cos(q-h~ξ)]1/q.
From the transformation u(x,t)=ln|v(x,t)| and (144), we obtain periodic blow-up wave solutions of (1) as follows:
(145)u(x,t)=1qln|-αcqh~+α2-cqh~βc2q2h~2cos(q-h~(x-ct))|.


## 4.1. Nonsmooth Peakon Solutions and Periodic Blow-Up Wave Solutions under Two Special Casesp=q=1 and p=2, q=1

(1) Whenp=q=1 and αc>0, h~<0, (120) can be reduced to
(131)y=±h~ϕ2-2(α+β)cϕ.
Substituting (131) into the first equation of (118), taking (0,ch~/(2(α+β))) as initial value conditions, and integrating it, we obtain
(132)v(x,t)=ϕ(ξ)=α+βch(cos-h~ξ-1).
From the transformation u(x,t)=ln|v(x,t)| and (132), we obtain a periodic blow-up wave solution of (1) as follows:
(133)u(x,t)=ln|α+βch[cos(-h~(x-ct))-1]|,
where (x-ct)≠2Kπ/-h~,K∈Z.(2) Whenp=2, q=1 and αβ>0, βc>0 (or αβ<0, βc>0), h~=h~A~=α2/βc, (123) can be reduced to
(134)y=±αβc|ϕ+βα|.
Similarly, substituting (134) into the first equation of (118) and then integrating it, and setting the integral constant as zero, we have
(135)v(x,t)=ϕ(ξ)=-βα[1-exp(-|αξ|βc)].
From the transformation u(x,t)=ln|v(x,t)| and (135), we obtain a broken solitary cusp wave solution of (1) as follows:
(136)u(x,t)=ln|-βα[1-exp(-|α(x-ct)|βc)]|,(x-ct)≠0.(3) Whenp=2,q=1 and αβ>0, βc>0 (or αβ<0,βc>0),h~<0, (123) can be reduced to
(137)y=±-h~-βch~-2αch~ϕ-ϕ2=±-h~(ϕM-ϕ)(ϕ-ϕm),
where ϕM=(-α+α2-ch~β)/ch~, ϕm=(-α-α2-ch~β)/ch~ are roots of equation -β/ch~-(2α/ch~)ϕ-ϕ2=0. Substituting (137) into the first equation of (118), taking (0,ϕM) as initial value conditions, and then integrating it, we have
(138)v(x,t)=ϕ(ξ)=12[(-hξ)(ϕM+ϕm)-(ϕM-ϕm)cos(-hξ)],ξ∈(-T1,T1),
where T1=(1/-h~)arccos((ϕM+ϕm)/(ϕM-ϕm)). From the transformation u(x,t)=ln|v(x,t)| and (138), we obtain a periodic blow-up wave solution of (1) as follows:
(139)u(x,t)=ln|12[(ϕM+ϕm)-(ϕM-ϕm)cos(-h~(x-ct))]|,
where x-ct≠T1+Kπ,K∈Z.(4) Whenp=2,q=1 and αβ>0, βc<0 (or αβ<0,βc<0),h~>-α2/cβ, (123) can be reduced to
(140)y=±h~ϕ2+2αcϕ+αc.
Similarly, we have
(141)v(x,t)=ϕ(ξ)=-αch~-α2-βch~ch~cos(-h~ξ).
From the transformation u(x,t)=ln|v(x,t)| and (141), we obtain a periodic blow-up wave solution of (1) as follows:
(142)u(x,t)=ln|αch~+α2-βch~ch~cos-h~(x-ct)|.


## 4.2. Periodic Blow-Up Wave Solutions in the Special Casep=2q>1

Whenp=2q>2 and αβ<0, αc>0 (or αβ>0, αc<0), h~∈(h~1,0), (118) becomes
(143)y=±ϕ2(h~+2αcqϕ-q+βcqϕ-2q)=±-h~-β/cqh~-(2α/cqh~)ϕq-(ϕq)2ϕq-1.
Substituting (143) into the first equation of (118) and then integrating it, we have
(144)v(x,t)=ϕ(ξ)=[-αcqh~+α2-cqh~βc2q2h~2cos(q-h~ξ)]1/q.
From the transformation u(x,t)=ln|v(x,t)| and (144), we obtain periodic blow-up wave solutions of (1) as follows:
(145)u(x,t)=1qln|-αcqh~+α2-cqh~βc2q2h~2cos(q-h~(x-ct))|.


## 5. Conclusion

In this work, by using the integral bifurcation method, a generalized Tzitzéica-Dodd-Bullough-Mikhailov (TDBM) equation is studied. In some special cases, we obtained many exact traveling wave solutions of TDBM equation. These exact solutions include smooth solitary wave solutions, singular periodic wave solutions of blow-up form, singular solitary wave solutions of blow-up form, broken solitary wave solutions, broken kink wave solutions, and some unboundary wave solutions. Comparing with the results in many references (such as [24–27]), these types of solutions in this paper are very different than those types of solutions which are obtained by Exp-function method. Though the types of solutions derived from two methods are different, they are all useful in nonlinear science. The smooth solitary wave solutions and singular solitary wave solutions obtained in this paper especially can be explained phenomenon of soliton surface that the total curvature at each point is proportional to the fourth power of the distance from a fixed point to the tangent plane.



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*Source: 395628-2013-06-10.xml*