# Bifurcations and Parametric Representations of Peakon, Solitary Wave and Smooth Periodic Wave Solutions for a Two-Component Degasperis-Procesi Equation

Authors: Bin He; Qing Meng; Jinhua Zhang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

By using the bifurcation method of dynamical systems and the method of phase portraits analysis, we consider a two-component Degasperis-Procesi equation:mt=-3mux-mxu+kρρx, ρt=-ρxu+2ρux, the existence of the peakon, solitary wave and smooth periodic wave is proved, and exact parametric representations of above travelling wave solutions are obtained in different parameter regions.


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## Body

## 1. Introduction

Based on the deformation of bi-Hamiltonian structure of the hydrodynamic type, Chen et al. [1] obtained a two-component Camassa-Holm equation:
(1)mt=-2mux-mxu+eρρx,ρt=-ρxu-ρux,
where m=u-uxx and e=±1. This system is integrable in the sense that it has Lax pair and it is a generalization form of the Camassa-Holm equation [2]. A good fact is that it has the peakon and multikink solutions [3].A two-component generalization of the Degasperis-Procesi equation [4, 5] is read as follows:
(2)mt=-3mux-mxu+kρρx,ρt=-ρxu+2ρux,
where m=u-uxx, k is a real parameter. When ρ=0, (2) is reduced to the Degasperis-Procesi equation [6]. Jin and Guo [4] analyze some aspects of blowup mechanism, traveling wave solutions and the persistence properties for (2). The self-similar solutions of (2) have been obtained by yuen [5].In the present paper, we will investigate the bifurcation set of (2) using the bifurcation theory and the method of phase portraits analysis [7–9] and obtain some exact peakon, solitary wave and smooth periodic wave solutions.Letξ=x-ct, where c≠0 is the wave speed. By using the travelling wave transformation
(3)u(x,t)=ϕ(ξ)+43c,ρ(x,t)=ρ(ξ),
we reduce (2) to the following ordinary differential equations:
(4)(ϕ+13c)ϕ′′′+3ϕ′ϕ′′-4ϕϕ′-133cϕ′+kρρ′=0,2ρϕ′-(ϕ+13c)ρ′=0,
where "′" is the derivative with respect to ξ.Suppose that(5)ρ=(ϕ+43c)2+a1(ϕ+43c)+a0,
where a1, a0 are constants to be determined later.Substituting (5) into the second equation of (4), we have
(6)(2c+a1)ϕϕ′+(83c2+73a1c+2a0)ϕ′=0.Integrating (6) once with respect to ξ and setting the integral constant to zero yield
(7)12(2c+a1)ϕ2+(83c2+73a1c+2a0)ϕ=0.From (7), we obtain
(8)2c+a1=0,83c2+73a1c+2a0=0.Solving (8), we find the following set of solutions:
(9)a1=-2c,a0=c2.Substituting (9) into (5), we have
(10)ρ=(ϕ+13c)2.Substituting (10) into the first equation of (4), integrating once with respect to ξ, and setting the integral constant to (1/162)kc4-(32/9)c2 yield
(11)(ϕ+13c)ϕ′′-(ϕ+43c)×(-12kϕ3+(2-13kc2)ϕ+1027kc3+53c)+(ϕ′)2=0.Lettingy=dϕ/dξ, we get the following planar system:
(12)dϕdξ=y,dydξ=((ϕ+43c)×(-12kϕ3+(2-13kc2)ϕ+1027kc3+53c)-y2)×(ϕ+13c)-1.System (12) is a two-parameter planar dynamical system depending on the parameter set (c,k). Since the phase orbits defined by the vector field of system (12) determine all travelling wave solutions of (11), we should investigate the bifurcations of phase portraits of system (12) in (ϕ,y)-phase plane as the parameters c,k are changed.Clearly, on such straight lineϕ=-(1/3)c in the phase plane (ϕ,y), system (12) is discontinuous. Such system is called a singular travelling wave system by one of authors in [10].Definition 1 (see [7, 10]).
Suppose thatϕ(ξ) is a solution of system (12) for ξ∈(-∞,∞). (i) ϕ(ξ) is called peakon solution if ϕ(ξ) is smooth locally on either side of ξ0 and limξ→ξ0-0ϕ(ξ)=limξ→ξ0+0ϕ(ξ)=a and a≠0, a≠±∞. (ii) ϕ(ξ) is called a solitary wave solution if limξ→+∞ϕ(ξ)=limξ→-∞ϕ(ξ). Usually, a peakon solution of (2) corresponds to two heteroclinic orbits of system (12) and a solitary wave solution of (2) corresponds to a homoclinic orbit of system (12). Similarly, a periodic orbit of system (12) corresponds to a smooth periodic wave solution of (2).Thus, to investigate peakons, solitary waves and smooth periodic waves of (2), we should find all heteroclinic, homoclinic, and periodic orbits of system (12) depending on the parameter space of this system.The rest of this paper is organized as follows. In Section2, we discuss the bifurcations of phase portraits of system (12), where explicit parametric conditions will be derived. In Section 3, we give some exact parametric representations of peakon, solitary wave and smooth periodic wave solutions of (2). A short conclusion will be given in Section 4.


## 2. Bifurcation Sets and Phase Portraits of System (12)

Using the transformationdξ=(ϕ+(1/3)c)dτ, it carries (12) into the Hamiltonian system
(13)dϕdτ=(ϕ+13c)y,dydτ=(ϕ+43c)×(-12kϕ3+(2-13kc2)ϕ+1027kc3+53c)-y2.Since both systems (12) and (13) have the same first integral
(14)(3ϕ+c)2y2+154ϕ(80c3(2kc2+9)81kϕ5+162kcϕ4+9(15kc2-52)ϕ3+60c(kc2-27)ϕ2-2c2(114kc2+891)ϕ-80c3(2kc2+9))=h,
then two systems above have the same topological phase portraits except for the line ϕ=-(1/3)c. Therefore, we can obtain the bifurcation phase portraits of system (12) from that of system (13). We consider the equilibrium points and their properties for system (13) as follows.Whenk=0, it is easy to see that system (13) has two equilibrium points (-(4/3)c,0),(-(5/6)c,0) on ϕ-axis and two equilibrium points (-(1/3)c,-c),(-(1/3)c,c) on the line ϕ=-(1/3)c.Whenk≠0, let
(15)f(ϕ)=ϕ3-2k(2-13kc2)ϕ-103k(29kc2+1).
Obviously, (-(4/3)c,0) is an equilibrium point of system (13) and (ϕ,0) is a equilibrium point of system (13) if and only if f(ϕ)=0 when ϕ≠-(4/3)c. Denoting that Ω=(5725+157514)1/3, k*=(Ω2-14Ω-125)/6Ωc2, we draw the graph of f=f(ϕ) for given c≠0 as Figure 1.The graphics of the functionf(ϕ) when c≠0. (a) c<0,  k<0. (b) c<0, 0<k<k*. (c) c<0, k=k*. (d): c<0, k>k*. (e) c>0, k<0. (f) c>0, 0<k<k*. (g) c>0, k=k*. (h) c>0, k>k*.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)From Figure1 and paying attention to that (-(4/3)c,0) is always an equilibrium point of system (13), we have the following.(i)
Ifk<0, then system (13) has two equilibrium points on ϕ-axis.(ii)
If0<k<k*, then system (13) has four equilibrium points on ϕ-axis.(iii)
Ifk=k*, then system (13) has three equilibrium points on ϕ-axis.(iv)
Ifk>k*, then system (13) has two equilibrium points on ϕ-axis.On the other hand, it is also easy to see that ifk>-2/c2, then system (13) has two equilibrium points (-(1/3)c,±(2c/2)kc2+2) on the line ϕ=-(1/3)c and if   k=-2/c2, then system (13) has one equilibrium point (-(1/3)c,0) on the line ϕ=-(1/3)c.LetM(ϕe,ye) be the coefficient matrix of the linearized system of the system (13) at an equilibrium point (ϕe,ye). Then, we have
(16)M(ϕe,ye)=(yeϕe+13c-2kϕe3-2kcϕe2+2(2-13kc2)ϕe+13c(13-29kc2)-2ye),
and at this equilibrium point, we have
(17)Trace(M(ϕe,ye))=-ye,J(ϕe,ye)=detM(ϕe,ye)=-2ye2+2kϕe4+kcϕe3+4(13kc2-1)ϕe2+13c(89kc2-17)ϕe+19c2(29kc2-13).By the theory of planar dynamical systems, we know that for an equilibrium point(ϕe,ye) of a planar integrable system if J(ϕe,ye)<0, then the equilibrium point is a saddle point. If J(ϕe,ye)>0 and Trace(M(ϕe,ye))=0, then it is a center point. If J(ϕe,ye)=0 and the Poincaré index of the equilibrium point is zero, then it is a cusp.For a fixedh, the level curve H(ϕ,y)=h defined by (14) determines a set of invariant curves of system (13) which contains different branches of curves. As h is varied, it defines different families of orbits of (13) with different dynamical behaviors.Using the property of the equilibrium points and bifurcation theory, we obtain the following results.(i)
Whenk=0, there is one bifurcation line c=0 which divides the region (-∞,∞) into 2 subregions: A1={c∣c<0}, A2={c∣c>0}.(ii)
Whenk≠0, there are five bifurcation curves as follows:
(18)L1:c=0,L2:k=0,L3:k=μc2,L4:k=12c2,L5:k=-2c2,which divide the (c,k)-parameter plane into 16 subregions as follows:
(19)B1={(c,k)∣c<0,k<-2/c2},B2={(c,k)∣c<0,k=-2/c2},B3={(c,k)∣c<0,-2/c2<k<0},B4={(c,k)∣c<0,0<k<1/2c2},B5={(c,k)∣c<0,k=1/2c2},B6={(c,k)∣c<0,1/2c2<k<μ/c2},B7={(c,k)∣c<0,k=μ/c2},B8={(c,k)∣c<0,k>μ/c2},B9={(c,k)∣c>0,k<-2/c2},B10={(c,k)∣c>0,k=-2/c2},B11={(c,k)∣c>0,-2/c2<k<0},B12={(c,k)∣c>0,0<k<1/2c2},B13={(c,k)∣c>0,k=1/2c2},B14={(c,k)∣c>0,1/2c2<k<μ/c2},B15={(c,k)∣c>0,k=μ/c2},B16={(c,k)∣c>0,k>μ/c2},
where μ=(Ω2-14Ω-125)/6Ω, Ω=(5725+157514)1/3.The bifurcation sets and phase portraits of system (13) are shown in Figures 2 and 3.The bifurcation sets and phase portraits of system (13) when k=0. (a) c∈A1. (b) c∈A2.
(a)
(b)The bifurcation sets and phase portraits of system (13) when k≠0. (a) (c,k)∈B1. (b) (c,k)∈B2. (c) (c,k)∈B3. (d) (c,k)∈B4. (e) (c,k)∈B5 (f) (c,k)∈B6. (g) (c,k)∈B7. (h) (c,k)∈B8. (i) (c,k)∈B9. (j) (c,k)∈B10. (k) (c,k)∈B11. (l) (c,k)∈B12. (m) (c,k)∈B13. (n) (c,k)∈B14. (o) (c,k)∈B15. (p) (c,k)∈B16.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)


## 3. Exact Parametric Representations of Peakon, Solitary Wave and Smooth Periodic Wave Solutions of (2)

In this section, we present all possible exact parametric representations of peakon, solitary wave and smooth periodic wave solutions through some special phase orbits. Next, we always suppose thath1=((16/9)-(608/243)kc2)c4, h2=(c4/23328000)((895537-23801414)Ω2 + (970075+38272514)Ω-45865625), ϕs=-(1/3)c, ϕ1=-(4/3)c, ϕ2=((1/1500)c)((614-23)Ω2-5(314+1)Ω-625), and Ω=(5725+157514)1/3.


### 3.1. Exact Parametric Representations of Peakon Solutions

(i)   For the givenh=(16/9)c4 in Figure 2(a), the level curve is shown in Figure 4(a). From Figure 4(a), we see that there are two heteroclinic orbits connecting with saddle points (ϕs,±c) and (ϕ1,0) of system (13) when (c,k)∈A1. Their expressions are
(20)y=±(ϕ-ϕ1),ϕs≤ϕ<ϕ1.The level curves of system (13) defined by h=h*.
(a)
c
<
0, h*=(16/9)c4
(b)
c
>
0, h*=(16/9)c4
(c)
c
<
0, h*=h1
(d)
c
<
0, h*=(128/243)c4
(e)
c
<
0, h*=h2
(f)
c
>
0, h*=h1
(g)
c
>
0, h*=(128/243)c4
(h)
c
>
0, h*=h2Substituting (20) into the dϕ/dξ=y and integrating it along the heteroclinic orbits yield
(21)∫ϕsϕ‍dss-ϕ1=-|ξ|.Completing the above integral and solving the equation forϕ, it follows that
(22)ϕ(ξ)=ce-|ξ|-43c.Noting thatu(x,t)=ϕ(x-ct)+(4/3)c,ρ(x,t)=(ϕ(ξ)+(1/3)c)2,andξ=x-ct, we get the parametric representation of peakon solution as follows:
(23)u(x,t)=ce-|x-ct|,ρ(x,t)=c2(e-|x-ct|-1)2.(ii)   For the givenh=(16/9)c4 in Figure 2(b), the level curve is shown in Figure 4(b). From Figure 4(b), we see that there are two heteroclinic orbits connecting with saddle points (ϕs,±c) and (ϕ1,0) of system (13) when (c,k)∈A2. Their expressions are
(24)y=±(ϕ-ϕ1),ϕ1<ϕ≤ϕs.Substituting (24) into the dϕ/dξ=y and integrating it along the heteroclinic orbits yield
(25)∫ϕϕs‍dss-ϕ1=|ξ|.Completing the above integral, we can get the parametric representation of peakon solution which is the same as (23).


### 3.2. Exact Parametric Representations of Solitary Wave Solutions

(i) For the givenh=h1 in Figure 3(d), the level curve is shown in Figure 4(c). From Figure 4(c), we see that there is a homoclinic orbit connecting with a saddle point (ϕ1,0) of system (13) and passing point (β2,0), and at the same time, there is a homoclinic orbit connecting with the saddle point (ϕ1,0) and passing point (β1,0) when (c,k)∈B4, and their expressions are
(26)y=±k6(ϕ-ϕ1)(β1-ϕ)(ϕ-β2)(ϕ-β3)(ϕ-β4)ϕ-ϕs,β2≤ϕ<ϕ1,(27)y=±k6(ϕ-ϕ1)(β1-ϕ)(ϕ-β2)(ϕ-β3)(ϕ-β4)ϕ-ϕs,ϕ1<ϕ≤β1,
respectively, where β1, β2, β3, and β4(β4<β3<β2<β1) are four real roots of
(28)ψ4-23cψ3+13k(5kc2-18)ψ2-4c27k(17kc2+27)ψ+2c281k(38kc2-27)=0,
which are obtained by the Cardan formula. For example, β1≈3.392753049, β2≈0.8389789971, β3≈0.09041647394, β4≈-4.988815187 when c=-1, k=0.3; β1≈1.942359082, β2≈0.3828621123, β3≈0.05356030588, β4≈-2.712114834 when c=-0.5, k=1; and β1≈2.712114834, β2≈-0.05356030588, β3≈-0.3828621123, β4≈-1.942359082 when c=0.5, k=1.Substituting (26) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(29)∫β2ϕ(s-ϕs)ds(s-ϕ1)(β1-s)(s-β2)(s-β3)(s-β4)=-k6|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(30)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(31)ϕ(χ)=β2(β3-β1)+β3(β1-β2)sn2(ωχ,m)(β3-β1)+(β1-β2)sn2(ωχ,m),ξ(χ)=Ω~[(α12-α2(β2-ϕ1)ϕs-ϕ1)(α2-α12)Π(arcsin(sn(ωχ,m)),α2,m)+ω(α12-α2(β2-ϕ1)ϕs-ϕ1)χ],
where Ω~=((ϕs-ϕ1)/ωα2(β2-ϕ1))(6/k), α2=(ϕ1-β3)(β1-β2)/(ϕ1-β2)(β1-β3), α12=(β1-β2)/(β1-β3), ω=(1/2)(β1-β3)(β2-β4), m=(β1-β2)(β3-β4)/(β1-β3)(β2-β4), Π(·,·,·) is Legendre's incomplete elliptic integral of the third kind, sn(·,·) is the Jacobian elliptic function (see [11]), and χ is a new variable.Substituting (27) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(32)∫ϕβ1(s-ϕs)ds(s-ϕ1)(β1-s)(s-β2)(s-β3)(s-β4)=k6|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(33)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(34)ϕ(χ)=β1(β2-β4)+β4(β1-β2)sn2(ωχ,m)(β2-β4)+(β1-β2)sn2(ωχ,m),ξ(χ)=Ω~[(α12-α2(β1-ϕ1)ϕs-ϕ1)(α2-α12)Π(arcsin(sn(ωχ,m)),α2,m)+ω(α12-α2(β1-ϕ1)ϕs-ϕ1)χ],
where Ω~=((ϕs-ϕ1)/ωα2(ϕ1-β1))(6/k), α2=(ϕ1-β4)(β1-β2)/(β1-ϕ1)(β2-β4), α12=(β2-β1)/(β2-β4), ω=(1/2)(β1-β3)(β2-β4), and m=(β1-β2)(β3-β4)/(β1-β3)(β2-β4).(ii) For the givenh=(128/243)c4 in Figure 3(e), the level curve is shown in Figure 4(d). From Figure 4(d), we see that there is a homoclinic orbit connecting with a cusp (ϕ1,0) of system (13) and passing point (β1,0) when (c,k)∈B5, and its expression is
(35)y=±123c×(ϕ-ϕ1)(β1-ϕ)(ϕ-ϕ1)(ϕ-β2)(ϕ-β3)ϕ-ϕs,ϕ1<ϕ≤β1,
where β1, β2, and β3(β3<β2<β1) are three real roots of
(36)ψ3-2cψ2-233c2ψ-827c3=0,
which are obtained by the Cardan formula. For example, β1≈1.917558777, β2≈0.03905297305, β3≈-3.956611750 when c=-1; β1≈3.956611750, β2-0.03905297305, β3≈-1.917558777 when c=1; and β1≈0.9587793884, β2≈0.01952648653, β3≈-1.978305875 when c=-0.5.Substituting (35) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(37)∫ϕβ1(s-ϕs)ds(s-ϕ1)(β1-s)(s-ϕ1)(s-β2)(s-β3)=-123c|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(38)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(39)ϕ(χ)=β1(ϕ1-β3)+β3(β1-ϕ1)sn2(ωχ,m)(ϕ1-β3)+(β1-ϕ1)sn2(ωχ,m),ξ(χ)=Ω~[(α12-β1-β3ϕs-ϕ1)(1-α12)Π(arcsin(sn(ωχ,m)),1,m)+ω(α12-β1-β3ϕs-ϕ1)χ],
where Ω~=23(ϕs-ϕ1)c/ω(β1-β3), α12=(ϕ1-β1)/(ϕ1-β3), ω=(1/2)(β1-β2)(ϕ1-β3), and m=(β1-ϕ1)(β2-β3)/(β1-β2)(ϕ1-β3).(iii) For the givenh=h2 in Figure 3(g), the level curve is shown in Figure 4(e). From Figure 4(e), we see that there is a homoclinic orbit connecting with a cusp (ϕ2,0) of system (13) and passing point (β1,0) when (c,k)∈B7, and its expression is
(40)y=±675Ω2-9450Ω-8437524300c2Ω×(ϕ2-ϕ)(ϕ2-ϕ)(ϕ-β1)(ϕ-β2)(ϕ-β3)ϕ-ϕs,β1≤ϕ<ϕ2,
where β1, β2, and β3(β3<β2<β1) are three real roots of
(41)b0+b1ψ+b2ψ2-ψ3=0,
which are obtained by the Cardan formula, and b0=(c3((48614-1663)Ω2-22375Ω+6075014+207875))/(675(Ω2-14Ω-125)), b1=(c2((145814-6489)Ω2-31950Ω+18225014+811125))/(675(Ω2-14Ω-125)), and b2=(c((72914-4482)Ω2+5400Ω+9112514+560250))/(675(Ω2-14Ω-125)). For example, β1≈1.264234987, β2≈0.03471643922, β3≈-3.882082755 when c=-1; β1≈6.321174982, β2≈0.1735822117, β3≈-19.41041364 when c=-5; and β1≈19.41041364, β2≈-0.1735822117, β3≈-6.321174982 when c=5.Substituting (40) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(42)∫β1ϕ(s-ϕs)ds(s-ϕ2)(ϕ2-s)(s-β1)(s-β2)(s-β3)=-675Ω2-9450Ω-8437524300c2Ω|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(43)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(44)ϕ(χ)=β1(ϕ2-β2)+β2(β1-ϕ2)sn2(ωχ,m)(ϕ2-β2)+(β1-ϕ2)sn2(ωχ,m),ξ(χ)=Ω~[ω(α12-ϕ2-β1ϕ2-ϕs)χ(1-α12)Π(arcsin(sn(ωχ,m)),1,m)+ω(α12-ϕ2-β1ϕ2-ϕs)χ],
where Ω~=((ϕ2-ϕs)/ω(ϕ2-β1))24300c2Ω/(675Ω2-9450Ω-84375), α12=(ϕ2-β1)/(ϕ2-β2), ω=(1/2)(ϕ2-β2)(β1-β3), and m=(ϕ2-β1)(β2-β3)/(ϕ2-β2)(β1-β3).(iv) For the givenh=h1 in Figure 3(l), the level curve is shown in Figure 4(f). From Figure 4(f), we see that there is a homoclinic orbit connecting with a saddle point (ϕ1,0) of system (13) and passing point (β4,0), and at the same time, there is a homoclinic orbit connecting with the saddle point (ϕ1,0) and passing point (β3,0) when (c,k)∈B12, and their expressions are
(45)y=±k6×(ϕ1-ϕ)(β1-ϕ)(β2-ϕ)(β3-ϕ)(ϕ-β4)ϕs-ϕ,β4≤ϕ<ϕ1,(46)y=±k6×(ϕ-ϕ1)(β1-ϕ)(β2-ϕ)(β3-ϕ)(ϕ-β4)ϕ-ϕs,ϕ1<ϕ≤β3,
respectively, where β1, β2, β3, and β4(β4<β3<β2<β1) are four real roots of (28).Substituting (45) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(47)∫β4ϕ‍(ϕs-s)ds(ϕ1-s)(β1-s)(β2-s)(β3-s)(s-β4)=k6|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(48)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(49)ϕ(χ)=β4(β1-β3)+β1(β3-β4)sn2(ωχ,m)(β1-β3)+(β3-β4)sn2(ωχ,m),ξ(χ)=Ω~[(α12-α2(β4-ϕ1)ϕs-ϕ1)(α2-α12)Π(arcsin(sn(ωχ,m)),α2,m)+ω(α12-α2(β4-ϕ1)ϕs-ϕ1)χ],
where Ω~=((ϕs-ϕ1)/ωα2(ϕ1-β4))6/k, α2=(ϕ1-β1)(β4-β3)/(ϕ1-β4)(β1-β3), α12=(β4-β3)/(β1-β3), ω=(1/2)(β1-β3)(β2-β4), and m=(β1-β2)(β3-β4)/(β1-β3)(β2-β4).Substituting (46) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(50)∫ϕβ3‍(s-ϕs)ds(s-ϕ1)(β1-s)(β2-s)(β3-s)(s-β4)=-k6|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(51)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(52)ϕ(χ)=β3(β4-β2)+β2(β3-β4)sn2(ωχ,m)(β4-β2)+(β3-β4)sn2(ωχ,m),ξ(χ)=Ω~[(α12-α2(β3-ϕ1)ϕs-ϕ1)(α2-α12)Π(arcsin(sn(ωχ,m)),α2,m)+ω(α12-α2(β3-ϕ1)ϕs-ϕ1)χ],
where Ω~=((ϕ1-ϕs)/ωα2(ϕ1-β3))6/k,α2=(ϕ1-β2)(β3-β4)/(ϕ1-β3)(β2-β4), α12=(β3-β4)/(β2-β4), ω=(1/2)(β1-β3)(β2-β4), and m=(β1-β2)(β3-β4)/(β1-β3)(β2-β4).(v) For the givenh=(128/243)c4 in Figure 3(m), the level curve is shown in Figure 4(g). From Figure 4(g), we see that there is a homoclinic orbit connecting with a cusp (ϕ1,0) of system (13) and passing point (β3,0) when (c,k)∈B13, and its expression is
(53)y=±123c×(ϕ1-ϕ)(β1-ϕ)(β2-ϕ)(ϕ1-ϕ)(ϕ-β3)ϕs-ϕ,β3≤ϕ<ϕ1,
where β1, β2, and β3(β3<β2<β1) are three real roots of (36).Substituting (53) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(54)∫β3ϕ‍(ϕs-s)ds(ϕ1-s)(β1-s)(β2-s)(ϕ1-s)(s-β3)=123c|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(55)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(56)ϕ(χ)=β3(ϕ1-β1)+β1(β3-ϕ1)sn2(ωχ,m)(ϕ1-β1)+(β3-ϕ1)sn2(ωχ,m),ξ(χ)=Ω~[(α12-β3-ϕ1ϕs-ϕ1)(1-α12)Π(arcsin(sn(ωχ,m)),1,m)+ω(α12-β3-ϕ1ϕs-ϕ1)χ],
where Ω~=23(ϕs-ϕ1)c/ω(ϕ1-β3), α12=(ϕ1-β3)/(ϕ1-β1), ω=(1/2)(β2-β3)(β1-ϕ1), and m=(β3-ϕ1)(β2-β1)/(β3-β2)(ϕ1-β1).(vi) For the givenh=h2 in Figure 3(o), the level curve is shown in Figure 4(h). From Figure 4(h), we see that there is a homoclinic orbit connecting with a cusp (ϕ2,0) of system (13) and passing point (β3,0) when (c,k)∈B15, and its expression is
(57)y=±675Ω2-9450Ω-8437524300c2Ω×(ϕ-ϕ2)(ϕ-ϕ2)(β1-ϕ)(ϕ-β2)(ϕ-β3)ϕ-ϕs,ϕ2<ϕ≤β3,
where β1, β2, and β3(β3<β2<β1) are three real roots of (41).Substituting (57) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(58)∫ϕβ3‍(s-ϕs)ds(s-ϕ2)(β1-s)(β2-s)(β3-s)(s-ϕ2)=-675Ω2-9450Ω-8437524300c2Ω|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(59)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(60)ϕ(χ)=β3(ϕ2-β2)+β2(β3-ϕ2)sn2(ωχ,m)(ϕ2-β2)+(β3-ϕ2)sn2(ωχ,m),ξ(χ)=Ω~[(α12-ϕ2-β3ϕ2-ϕs)(1-α12)Π(arcsin(sn(ωχ,m)),1,m)+ω(α12-ϕ2-β3ϕ2-ϕs)χ],
where Ω~=((ϕ2-ϕs)/ω(ϕ2-β3))24300c2Ω/(675Ω2-9450Ω-84375), α12=(ϕ2-β3)/(ϕ2-β2), ω=(1/2)(ϕ2-β2)(β3-β1), and m=(ϕ2-β3)(β2-β1)/(ϕ2-β2)(β3-β1).


### 3.3. Exact Parametric Representations of Smooth Periodic Wave Solutions

(i) For the givenh=h1 in Figure 3(d), the level curve is shown in Figure 4(c). From Figure 4(c), we see that there is one periodic orbit passing points (β3,0) and (β4,0) when (c,k)∈B4, and its expression is
(61)y=±k6×(ϕ-ϕ1)(β1-ϕ)(β2-ϕ)(β3-ϕ)(ϕ-β4)ϕ-ϕs,β4≤ϕ≤β3,
where β1, β2, β3, and β4(β4<β3<β2<β1) are four real roots of (28).Substituting (61) into the dϕ/dξ=y and integrating it along the periodic orbit yield
(62)∫ϕβ3‍(ϕs-s)ds(ϕ1-s)(β1-s)(β2-s)(β3-s)(s-β4)=k6|ξ|.Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:(63)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(64)ϕ(χ)=β3(β4-β2)+β2(β3-β4)sn2(ωχ,m)(β4-β2)+(β3-β4)sn2(ωχ,m),ξ(χ)=Ω~[(α12-α2(β3-ϕ1)ϕs-ϕ1)(α2-α12)Π(arcsin(sn(ωχ,m)),α2,m)+ω(α12-α2(β3-ϕ1)ϕs-ϕ1)χ],
where Ω~=((ϕs-ϕ1)/ωα2(ϕ1-β3))6/k, α2=(ϕ1-β2)(β3-β4)/(ϕ1-β3)(β2-β4), α12=(β3-β4)/(β2-β4), ω=(1/2)(β1-β3)(β2-β4), and m=(β1-β2)(β3-β4)/(β1-β3)(β2-β4).(ii) For the givenh=(128/243)c4 in Figure 3(e), the level curve is shown in Figure 4(d). From Figure 4(d), we see that there is one periodic orbit passing points (β2,0) and (β3,0) when (c,k)∈B5, and its expression is
(65)y=±123c×(ϕ1-ϕ)(β1-ϕ)(ϕ1-ϕ)(β2-ϕ)(ϕ-β3)ϕs-ϕ,β3≤ϕ≤β2,
where β1, β2, and β3(β3<β2<β1) are three real roots of (36).Substituting (65) into the dϕ/dξ=y and integrating it along the periodic orbit yield
(66)∫ϕβ2‍(ϕs-s)ds(ϕ1-s)(β1-s)(ϕ1-s)(β2-s)(s-β3)=-123c|ξ|.Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:(67)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(68)ϕ(χ)=β2(β3-ϕ1)+ϕ1(β2-β3)sn2(ωχ,m)(β3-ϕ1)+(β2-β3)sn2(ωχ,m),ξ(χ)=Ω~[ωm2(ϕ1-β2)ϕ1-ϕs(m2-α2)F(arcsin(sn(ωχ,m)),m)+α2E(arcsin(sn(ωχ,m)),m)-ωm2(ϕ1-β2)ϕ1-ϕsχ],
where Ω~=(23(ϕ1-ϕs)c)/ωm2(ϕ1-β2), α2=(β2-β3)/(ϕ1-β3), ω=(1/2)(β1-β2)(ϕ1-β3), m=(β1-ϕ1)(β2-β3)/(β1-β2)(ϕ1-β3), andF(·,·) and E(·,·) are Legendre's incomplete elliptic integrals of the first and second kinds, respectively (see [11]).(iii) For the givenh=h2 in Figure 3(g), the level curve is shown in Figure 4(e). From Figure 4(e), we see that there is one periodic orbit passing points (β2,0) and (β3,0) when (c,k)∈B7, and its expression is
(69)y=±675Ω2-9450Ω-8437524300c2Ω×(ϕ2-ϕ)(ϕ2-ϕ)(β1-ϕ)(β2-ϕ)(ϕ-β3)ϕs-ϕ,β3≤ϕ≤β2,
where β1, β2, and β3(β3<β2<β1) are three real roots of (41).Substituting (69) into the dϕ/dξ=y and integrating it along the periodic orbit yield
(70)∫ϕβ2‍(ϕs-s)ds(ϕ2-s)(ϕ2-s)(β1-s)(β2-s)(s-β3)=675Ω2-9450Ω-8437524300c2Ω|ξ|.Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:(71)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(72)ϕ(χ)=β2(β3-β1)+β1(β2-β3)sn2(ωχ,m)(β3-β1)+(β2-β3)sn2(ωχ,m),ξ(χ)=Ω~[(α12-m2(ϕ2-β2)ϕ2-ϕs)(m2-α12)Π(arcsin(sn(ωχ,m)),m2,m)+ω(α12-m2(ϕ2-β2)ϕ2-ϕs)χ],
where Ω~=((ϕs-ϕ2)/ωm2(ϕ2-β2))24300c2Ω/(675Ω2-9450Ω-84375), α12=(β2-β3)/(β1-β3), ω=(1/2)(ϕ2-β2)(β1-β3), and m=(ϕ2-β1)(β2-β3)/(ϕ2-β2)(β1-β3).(iv) For the givenh=h1 in Figure 3(l), the level curve is shown in Figure 4(f). From Figure 4(f), we see that there is one periodic orbit passing points (β1,0) and (β2,0) when (c,k)∈B12, and its expression is
(73)y=±k6(ϕ-ϕ1)(β1-ϕ)(ϕ-β2)(ϕ-β3)(ϕ-β4)ϕ-ϕs,β2≤ϕ≤β1,
where β1, β2, β3, and β4(β4<β3<β2<β1) are four real roots of (28).Substituting (73) into the dϕ/dξ=y and integrating it along the periodic orbit yield
(74)∫ϕβ1‍(s-ϕs)ds(s-ϕ1)(β1-s)(s-β2)(s-β3)(s-β4)=k6|ξ|.Completing the above integral, we can get the parametric representation of smooth periodic wave solution which is the same as (33).(v) For the givenh=(128/243)c4 in Figure 3(m), the level curve is shown in Figure 4(g). From Figure 4(g), we see that there is one periodic orbit passing points (β1,0) and (β2,0) when (c,k)∈B13, and its expression is
(75)y=±123c×(ϕ-ϕ1)(β1-ϕ)(ϕ-β2)(ϕ-ϕ1)(ϕ-β3)ϕ-ϕs,β2≤ϕ≤β1,
where β1, β2, and β3(β3<β2<β1) are three real roots of (36).Substituting (75) into the dϕ/dξ=y and integrating it along the periodic orbit yield
(76)∫ϕβ1‍(s-ϕs)ds(s-ϕ1)(β1-s)(s-β2)(s-ϕ1)(s-β3)=123c|ξ|.Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:(77)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(78)ϕ(χ)=β1(β2-β3)+β3(β1-β3)sn2(ωχ,m)(β2-β3)+(β1-β3)sn2(ωχ,m),ξ(χ)=Ω~[(α12-m2(ϕ1-β1)ϕ1-ϕs)(m2-α12)Π(arcsin(sn(ωχ,m)),m2,m)+ω(α12-m2(ϕ1-β1)ϕ1-ϕs)χ],
where Ω~=23(ϕ1-ϕs)c/ωm2(β1-ϕ1), α12=(β2-β1)/(β2-β3), ω=(1/2)(β1-ϕ1)(β2-β3), and m=(β1-β2)(ϕ1-β3)/(β1-ϕ1)(β2-β3).(vi) For the givenh=h2 in Figure 3(o), the level curve is shown in Figure 4(h). From Figure 4(h), we see that there is one periodic orbit passing points (β1,0) and (β2,0) when (c,k)∈B15, and its expression is
(79)y=±675Ω2-9450Ω-8437524300c2Ω×(ϕ-ϕ2)(β1-ϕ)(ϕ-β2)(ϕ-β3)(ϕ-ϕ2)ϕ-ϕs,β2≤ϕ≤β1,
where β1, β2, and β3(β3<β2<β1) are three real roots of (41).Substituting (79) into the dϕ/dξ=y and integrating it along the periodic orbit yield
(80)∫ϕβ1‍(s-ϕs)ds(s-ϕ2)(β1-s)(s-β2)(s-β3)(s-ϕ2)=675Ω2-9450Ω-8437524300c2Ω|ξ|.Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:(81)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(82)ϕ(χ)=β1(β2-ϕ2)+ϕ2(β1-β2)sn2(ωχ,m)(β2-ϕ2)+(β1-β2)sn2(ωχ,m),ξ(χ)=Ω~[ωm2(β2-β1)ϕ2-ϕs(m2-α2)F(arcsin(sn(ωχ,m)),m)+α2E(arcsin(sn(ωχ,m)),m)-ωm2(β2-β1)ϕ2-ϕsχ],
where Ω~=((ϕ2-ϕs)/ωm2(β1-β2))24300c2Ω/(675Ω2-9450Ω-84375), α2=(β2-β1)/(β2-ϕ2), ω=(1/2)(β1-β3)(β2-ϕ2), and m=(β1-β2)(β3-ϕ2)/(β1-β3)(β2-ϕ2).Remark 2.
From Figures3(f) and 3(n), we see that (2) has two solitary wave solutions when (c,k)∈B6 and two solitary wave solutions when (c,k)∈B14, but the exact parametric representations are very complex, so here we omit them.


## 3.1. Exact Parametric Representations of Peakon Solutions

(i)   For the givenh=(16/9)c4 in Figure 2(a), the level curve is shown in Figure 4(a). From Figure 4(a), we see that there are two heteroclinic orbits connecting with saddle points (ϕs,±c) and (ϕ1,0) of system (13) when (c,k)∈A1. Their expressions are
(20)y=±(ϕ-ϕ1),ϕs≤ϕ<ϕ1.The level curves of system (13) defined by h=h*.
(a)
c
<
0, h*=(16/9)c4
(b)
c
>
0, h*=(16/9)c4
(c)
c
<
0, h*=h1
(d)
c
<
0, h*=(128/243)c4
(e)
c
<
0, h*=h2
(f)
c
>
0, h*=h1
(g)
c
>
0, h*=(128/243)c4
(h)
c
>
0, h*=h2Substituting (20) into the dϕ/dξ=y and integrating it along the heteroclinic orbits yield
(21)∫ϕsϕ‍dss-ϕ1=-|ξ|.Completing the above integral and solving the equation forϕ, it follows that
(22)ϕ(ξ)=ce-|ξ|-43c.Noting thatu(x,t)=ϕ(x-ct)+(4/3)c,ρ(x,t)=(ϕ(ξ)+(1/3)c)2,andξ=x-ct, we get the parametric representation of peakon solution as follows:
(23)u(x,t)=ce-|x-ct|,ρ(x,t)=c2(e-|x-ct|-1)2.(ii)   For the givenh=(16/9)c4 in Figure 2(b), the level curve is shown in Figure 4(b). From Figure 4(b), we see that there are two heteroclinic orbits connecting with saddle points (ϕs,±c) and (ϕ1,0) of system (13) when (c,k)∈A2. Their expressions are
(24)y=±(ϕ-ϕ1),ϕ1<ϕ≤ϕs.Substituting (24) into the dϕ/dξ=y and integrating it along the heteroclinic orbits yield
(25)∫ϕϕs‍dss-ϕ1=|ξ|.Completing the above integral, we can get the parametric representation of peakon solution which is the same as (23).


## 3.2. Exact Parametric Representations of Solitary Wave Solutions

(i) For the givenh=h1 in Figure 3(d), the level curve is shown in Figure 4(c). From Figure 4(c), we see that there is a homoclinic orbit connecting with a saddle point (ϕ1,0) of system (13) and passing point (β2,0), and at the same time, there is a homoclinic orbit connecting with the saddle point (ϕ1,0) and passing point (β1,0) when (c,k)∈B4, and their expressions are
(26)y=±k6(ϕ-ϕ1)(β1-ϕ)(ϕ-β2)(ϕ-β3)(ϕ-β4)ϕ-ϕs,β2≤ϕ<ϕ1,(27)y=±k6(ϕ-ϕ1)(β1-ϕ)(ϕ-β2)(ϕ-β3)(ϕ-β4)ϕ-ϕs,ϕ1<ϕ≤β1,
respectively, where β1, β2, β3, and β4(β4<β3<β2<β1) are four real roots of
(28)ψ4-23cψ3+13k(5kc2-18)ψ2-4c27k(17kc2+27)ψ+2c281k(38kc2-27)=0,
which are obtained by the Cardan formula. For example, β1≈3.392753049, β2≈0.8389789971, β3≈0.09041647394, β4≈-4.988815187 when c=-1, k=0.3; β1≈1.942359082, β2≈0.3828621123, β3≈0.05356030588, β4≈-2.712114834 when c=-0.5, k=1; and β1≈2.712114834, β2≈-0.05356030588, β3≈-0.3828621123, β4≈-1.942359082 when c=0.5, k=1.Substituting (26) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(29)∫β2ϕ(s-ϕs)ds(s-ϕ1)(β1-s)(s-β2)(s-β3)(s-β4)=-k6|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(30)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(31)ϕ(χ)=β2(β3-β1)+β3(β1-β2)sn2(ωχ,m)(β3-β1)+(β1-β2)sn2(ωχ,m),ξ(χ)=Ω~[(α12-α2(β2-ϕ1)ϕs-ϕ1)(α2-α12)Π(arcsin(sn(ωχ,m)),α2,m)+ω(α12-α2(β2-ϕ1)ϕs-ϕ1)χ],
where Ω~=((ϕs-ϕ1)/ωα2(β2-ϕ1))(6/k), α2=(ϕ1-β3)(β1-β2)/(ϕ1-β2)(β1-β3), α12=(β1-β2)/(β1-β3), ω=(1/2)(β1-β3)(β2-β4), m=(β1-β2)(β3-β4)/(β1-β3)(β2-β4), Π(·,·,·) is Legendre's incomplete elliptic integral of the third kind, sn(·,·) is the Jacobian elliptic function (see [11]), and χ is a new variable.Substituting (27) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(32)∫ϕβ1(s-ϕs)ds(s-ϕ1)(β1-s)(s-β2)(s-β3)(s-β4)=k6|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(33)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(34)ϕ(χ)=β1(β2-β4)+β4(β1-β2)sn2(ωχ,m)(β2-β4)+(β1-β2)sn2(ωχ,m),ξ(χ)=Ω~[(α12-α2(β1-ϕ1)ϕs-ϕ1)(α2-α12)Π(arcsin(sn(ωχ,m)),α2,m)+ω(α12-α2(β1-ϕ1)ϕs-ϕ1)χ],
where Ω~=((ϕs-ϕ1)/ωα2(ϕ1-β1))(6/k), α2=(ϕ1-β4)(β1-β2)/(β1-ϕ1)(β2-β4), α12=(β2-β1)/(β2-β4), ω=(1/2)(β1-β3)(β2-β4), and m=(β1-β2)(β3-β4)/(β1-β3)(β2-β4).(ii) For the givenh=(128/243)c4 in Figure 3(e), the level curve is shown in Figure 4(d). From Figure 4(d), we see that there is a homoclinic orbit connecting with a cusp (ϕ1,0) of system (13) and passing point (β1,0) when (c,k)∈B5, and its expression is
(35)y=±123c×(ϕ-ϕ1)(β1-ϕ)(ϕ-ϕ1)(ϕ-β2)(ϕ-β3)ϕ-ϕs,ϕ1<ϕ≤β1,
where β1, β2, and β3(β3<β2<β1) are three real roots of
(36)ψ3-2cψ2-233c2ψ-827c3=0,
which are obtained by the Cardan formula. For example, β1≈1.917558777, β2≈0.03905297305, β3≈-3.956611750 when c=-1; β1≈3.956611750, β2-0.03905297305, β3≈-1.917558777 when c=1; and β1≈0.9587793884, β2≈0.01952648653, β3≈-1.978305875 when c=-0.5.Substituting (35) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(37)∫ϕβ1(s-ϕs)ds(s-ϕ1)(β1-s)(s-ϕ1)(s-β2)(s-β3)=-123c|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(38)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(39)ϕ(χ)=β1(ϕ1-β3)+β3(β1-ϕ1)sn2(ωχ,m)(ϕ1-β3)+(β1-ϕ1)sn2(ωχ,m),ξ(χ)=Ω~[(α12-β1-β3ϕs-ϕ1)(1-α12)Π(arcsin(sn(ωχ,m)),1,m)+ω(α12-β1-β3ϕs-ϕ1)χ],
where Ω~=23(ϕs-ϕ1)c/ω(β1-β3), α12=(ϕ1-β1)/(ϕ1-β3), ω=(1/2)(β1-β2)(ϕ1-β3), and m=(β1-ϕ1)(β2-β3)/(β1-β2)(ϕ1-β3).(iii) For the givenh=h2 in Figure 3(g), the level curve is shown in Figure 4(e). From Figure 4(e), we see that there is a homoclinic orbit connecting with a cusp (ϕ2,0) of system (13) and passing point (β1,0) when (c,k)∈B7, and its expression is
(40)y=±675Ω2-9450Ω-8437524300c2Ω×(ϕ2-ϕ)(ϕ2-ϕ)(ϕ-β1)(ϕ-β2)(ϕ-β3)ϕ-ϕs,β1≤ϕ<ϕ2,
where β1, β2, and β3(β3<β2<β1) are three real roots of
(41)b0+b1ψ+b2ψ2-ψ3=0,
which are obtained by the Cardan formula, and b0=(c3((48614-1663)Ω2-22375Ω+6075014+207875))/(675(Ω2-14Ω-125)), b1=(c2((145814-6489)Ω2-31950Ω+18225014+811125))/(675(Ω2-14Ω-125)), and b2=(c((72914-4482)Ω2+5400Ω+9112514+560250))/(675(Ω2-14Ω-125)). For example, β1≈1.264234987, β2≈0.03471643922, β3≈-3.882082755 when c=-1; β1≈6.321174982, β2≈0.1735822117, β3≈-19.41041364 when c=-5; and β1≈19.41041364, β2≈-0.1735822117, β3≈-6.321174982 when c=5.Substituting (40) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(42)∫β1ϕ(s-ϕs)ds(s-ϕ2)(ϕ2-s)(s-β1)(s-β2)(s-β3)=-675Ω2-9450Ω-8437524300c2Ω|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(43)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(44)ϕ(χ)=β1(ϕ2-β2)+β2(β1-ϕ2)sn2(ωχ,m)(ϕ2-β2)+(β1-ϕ2)sn2(ωχ,m),ξ(χ)=Ω~[ω(α12-ϕ2-β1ϕ2-ϕs)χ(1-α12)Π(arcsin(sn(ωχ,m)),1,m)+ω(α12-ϕ2-β1ϕ2-ϕs)χ],
where Ω~=((ϕ2-ϕs)/ω(ϕ2-β1))24300c2Ω/(675Ω2-9450Ω-84375), α12=(ϕ2-β1)/(ϕ2-β2), ω=(1/2)(ϕ2-β2)(β1-β3), and m=(ϕ2-β1)(β2-β3)/(ϕ2-β2)(β1-β3).(iv) For the givenh=h1 in Figure 3(l), the level curve is shown in Figure 4(f). From Figure 4(f), we see that there is a homoclinic orbit connecting with a saddle point (ϕ1,0) of system (13) and passing point (β4,0), and at the same time, there is a homoclinic orbit connecting with the saddle point (ϕ1,0) and passing point (β3,0) when (c,k)∈B12, and their expressions are
(45)y=±k6×(ϕ1-ϕ)(β1-ϕ)(β2-ϕ)(β3-ϕ)(ϕ-β4)ϕs-ϕ,β4≤ϕ<ϕ1,(46)y=±k6×(ϕ-ϕ1)(β1-ϕ)(β2-ϕ)(β3-ϕ)(ϕ-β4)ϕ-ϕs,ϕ1<ϕ≤β3,
respectively, where β1, β2, β3, and β4(β4<β3<β2<β1) are four real roots of (28).Substituting (45) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(47)∫β4ϕ‍(ϕs-s)ds(ϕ1-s)(β1-s)(β2-s)(β3-s)(s-β4)=k6|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(48)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(49)ϕ(χ)=β4(β1-β3)+β1(β3-β4)sn2(ωχ,m)(β1-β3)+(β3-β4)sn2(ωχ,m),ξ(χ)=Ω~[(α12-α2(β4-ϕ1)ϕs-ϕ1)(α2-α12)Π(arcsin(sn(ωχ,m)),α2,m)+ω(α12-α2(β4-ϕ1)ϕs-ϕ1)χ],
where Ω~=((ϕs-ϕ1)/ωα2(ϕ1-β4))6/k, α2=(ϕ1-β1)(β4-β3)/(ϕ1-β4)(β1-β3), α12=(β4-β3)/(β1-β3), ω=(1/2)(β1-β3)(β2-β4), and m=(β1-β2)(β3-β4)/(β1-β3)(β2-β4).Substituting (46) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(50)∫ϕβ3‍(s-ϕs)ds(s-ϕ1)(β1-s)(β2-s)(β3-s)(s-β4)=-k6|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(51)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(52)ϕ(χ)=β3(β4-β2)+β2(β3-β4)sn2(ωχ,m)(β4-β2)+(β3-β4)sn2(ωχ,m),ξ(χ)=Ω~[(α12-α2(β3-ϕ1)ϕs-ϕ1)(α2-α12)Π(arcsin(sn(ωχ,m)),α2,m)+ω(α12-α2(β3-ϕ1)ϕs-ϕ1)χ],
where Ω~=((ϕ1-ϕs)/ωα2(ϕ1-β3))6/k,α2=(ϕ1-β2)(β3-β4)/(ϕ1-β3)(β2-β4), α12=(β3-β4)/(β2-β4), ω=(1/2)(β1-β3)(β2-β4), and m=(β1-β2)(β3-β4)/(β1-β3)(β2-β4).(v) For the givenh=(128/243)c4 in Figure 3(m), the level curve is shown in Figure 4(g). From Figure 4(g), we see that there is a homoclinic orbit connecting with a cusp (ϕ1,0) of system (13) and passing point (β3,0) when (c,k)∈B13, and its expression is
(53)y=±123c×(ϕ1-ϕ)(β1-ϕ)(β2-ϕ)(ϕ1-ϕ)(ϕ-β3)ϕs-ϕ,β3≤ϕ<ϕ1,
where β1, β2, and β3(β3<β2<β1) are three real roots of (36).Substituting (53) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(54)∫β3ϕ‍(ϕs-s)ds(ϕ1-s)(β1-s)(β2-s)(ϕ1-s)(s-β3)=123c|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(55)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(56)ϕ(χ)=β3(ϕ1-β1)+β1(β3-ϕ1)sn2(ωχ,m)(ϕ1-β1)+(β3-ϕ1)sn2(ωχ,m),ξ(χ)=Ω~[(α12-β3-ϕ1ϕs-ϕ1)(1-α12)Π(arcsin(sn(ωχ,m)),1,m)+ω(α12-β3-ϕ1ϕs-ϕ1)χ],
where Ω~=23(ϕs-ϕ1)c/ω(ϕ1-β3), α12=(ϕ1-β3)/(ϕ1-β1), ω=(1/2)(β2-β3)(β1-ϕ1), and m=(β3-ϕ1)(β2-β1)/(β3-β2)(ϕ1-β1).(vi) For the givenh=h2 in Figure 3(o), the level curve is shown in Figure 4(h). From Figure 4(h), we see that there is a homoclinic orbit connecting with a cusp (ϕ2,0) of system (13) and passing point (β3,0) when (c,k)∈B15, and its expression is
(57)y=±675Ω2-9450Ω-8437524300c2Ω×(ϕ-ϕ2)(ϕ-ϕ2)(β1-ϕ)(ϕ-β2)(ϕ-β3)ϕ-ϕs,ϕ2<ϕ≤β3,
where β1, β2, and β3(β3<β2<β1) are three real roots of (41).Substituting (57) into the dϕ/dξ=y and integrating it along the homoclinic orbit yield
(58)∫ϕβ3‍(s-ϕs)ds(s-ϕ2)(β1-s)(β2-s)(β3-s)(s-ϕ2)=-675Ω2-9450Ω-8437524300c2Ω|ξ|.Completing the above integral, we can get the parametric representation of solitary wave solution as follows:(59)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(60)ϕ(χ)=β3(ϕ2-β2)+β2(β3-ϕ2)sn2(ωχ,m)(ϕ2-β2)+(β3-ϕ2)sn2(ωχ,m),ξ(χ)=Ω~[(α12-ϕ2-β3ϕ2-ϕs)(1-α12)Π(arcsin(sn(ωχ,m)),1,m)+ω(α12-ϕ2-β3ϕ2-ϕs)χ],
where Ω~=((ϕ2-ϕs)/ω(ϕ2-β3))24300c2Ω/(675Ω2-9450Ω-84375), α12=(ϕ2-β3)/(ϕ2-β2), ω=(1/2)(ϕ2-β2)(β3-β1), and m=(ϕ2-β3)(β2-β1)/(ϕ2-β2)(β3-β1).


## 3.3. Exact Parametric Representations of Smooth Periodic Wave Solutions

(i) For the givenh=h1 in Figure 3(d), the level curve is shown in Figure 4(c). From Figure 4(c), we see that there is one periodic orbit passing points (β3,0) and (β4,0) when (c,k)∈B4, and its expression is
(61)y=±k6×(ϕ-ϕ1)(β1-ϕ)(β2-ϕ)(β3-ϕ)(ϕ-β4)ϕ-ϕs,β4≤ϕ≤β3,
where β1, β2, β3, and β4(β4<β3<β2<β1) are four real roots of (28).Substituting (61) into the dϕ/dξ=y and integrating it along the periodic orbit yield
(62)∫ϕβ3‍(ϕs-s)ds(ϕ1-s)(β1-s)(β2-s)(β3-s)(s-β4)=k6|ξ|.Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:(63)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(64)ϕ(χ)=β3(β4-β2)+β2(β3-β4)sn2(ωχ,m)(β4-β2)+(β3-β4)sn2(ωχ,m),ξ(χ)=Ω~[(α12-α2(β3-ϕ1)ϕs-ϕ1)(α2-α12)Π(arcsin(sn(ωχ,m)),α2,m)+ω(α12-α2(β3-ϕ1)ϕs-ϕ1)χ],
where Ω~=((ϕs-ϕ1)/ωα2(ϕ1-β3))6/k, α2=(ϕ1-β2)(β3-β4)/(ϕ1-β3)(β2-β4), α12=(β3-β4)/(β2-β4), ω=(1/2)(β1-β3)(β2-β4), and m=(β1-β2)(β3-β4)/(β1-β3)(β2-β4).(ii) For the givenh=(128/243)c4 in Figure 3(e), the level curve is shown in Figure 4(d). From Figure 4(d), we see that there is one periodic orbit passing points (β2,0) and (β3,0) when (c,k)∈B5, and its expression is
(65)y=±123c×(ϕ1-ϕ)(β1-ϕ)(ϕ1-ϕ)(β2-ϕ)(ϕ-β3)ϕs-ϕ,β3≤ϕ≤β2,
where β1, β2, and β3(β3<β2<β1) are three real roots of (36).Substituting (65) into the dϕ/dξ=y and integrating it along the periodic orbit yield
(66)∫ϕβ2‍(ϕs-s)ds(ϕ1-s)(β1-s)(ϕ1-s)(β2-s)(s-β3)=-123c|ξ|.Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:(67)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(68)ϕ(χ)=β2(β3-ϕ1)+ϕ1(β2-β3)sn2(ωχ,m)(β3-ϕ1)+(β2-β3)sn2(ωχ,m),ξ(χ)=Ω~[ωm2(ϕ1-β2)ϕ1-ϕs(m2-α2)F(arcsin(sn(ωχ,m)),m)+α2E(arcsin(sn(ωχ,m)),m)-ωm2(ϕ1-β2)ϕ1-ϕsχ],
where Ω~=(23(ϕ1-ϕs)c)/ωm2(ϕ1-β2), α2=(β2-β3)/(ϕ1-β3), ω=(1/2)(β1-β2)(ϕ1-β3), m=(β1-ϕ1)(β2-β3)/(β1-β2)(ϕ1-β3), andF(·,·) and E(·,·) are Legendre's incomplete elliptic integrals of the first and second kinds, respectively (see [11]).(iii) For the givenh=h2 in Figure 3(g), the level curve is shown in Figure 4(e). From Figure 4(e), we see that there is one periodic orbit passing points (β2,0) and (β3,0) when (c,k)∈B7, and its expression is
(69)y=±675Ω2-9450Ω-8437524300c2Ω×(ϕ2-ϕ)(ϕ2-ϕ)(β1-ϕ)(β2-ϕ)(ϕ-β3)ϕs-ϕ,β3≤ϕ≤β2,
where β1, β2, and β3(β3<β2<β1) are three real roots of (41).Substituting (69) into the dϕ/dξ=y and integrating it along the periodic orbit yield
(70)∫ϕβ2‍(ϕs-s)ds(ϕ2-s)(ϕ2-s)(β1-s)(β2-s)(s-β3)=675Ω2-9450Ω-8437524300c2Ω|ξ|.Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:(71)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(72)ϕ(χ)=β2(β3-β1)+β1(β2-β3)sn2(ωχ,m)(β3-β1)+(β2-β3)sn2(ωχ,m),ξ(χ)=Ω~[(α12-m2(ϕ2-β2)ϕ2-ϕs)(m2-α12)Π(arcsin(sn(ωχ,m)),m2,m)+ω(α12-m2(ϕ2-β2)ϕ2-ϕs)χ],
where Ω~=((ϕs-ϕ2)/ωm2(ϕ2-β2))24300c2Ω/(675Ω2-9450Ω-84375), α12=(β2-β3)/(β1-β3), ω=(1/2)(ϕ2-β2)(β1-β3), and m=(ϕ2-β1)(β2-β3)/(ϕ2-β2)(β1-β3).(iv) For the givenh=h1 in Figure 3(l), the level curve is shown in Figure 4(f). From Figure 4(f), we see that there is one periodic orbit passing points (β1,0) and (β2,0) when (c,k)∈B12, and its expression is
(73)y=±k6(ϕ-ϕ1)(β1-ϕ)(ϕ-β2)(ϕ-β3)(ϕ-β4)ϕ-ϕs,β2≤ϕ≤β1,
where β1, β2, β3, and β4(β4<β3<β2<β1) are four real roots of (28).Substituting (73) into the dϕ/dξ=y and integrating it along the periodic orbit yield
(74)∫ϕβ1‍(s-ϕs)ds(s-ϕ1)(β1-s)(s-β2)(s-β3)(s-β4)=k6|ξ|.Completing the above integral, we can get the parametric representation of smooth periodic wave solution which is the same as (33).(v) For the givenh=(128/243)c4 in Figure 3(m), the level curve is shown in Figure 4(g). From Figure 4(g), we see that there is one periodic orbit passing points (β1,0) and (β2,0) when (c,k)∈B13, and its expression is
(75)y=±123c×(ϕ-ϕ1)(β1-ϕ)(ϕ-β2)(ϕ-ϕ1)(ϕ-β3)ϕ-ϕs,β2≤ϕ≤β1,
where β1, β2, and β3(β3<β2<β1) are three real roots of (36).Substituting (75) into the dϕ/dξ=y and integrating it along the periodic orbit yield
(76)∫ϕβ1‍(s-ϕs)ds(s-ϕ1)(β1-s)(s-β2)(s-ϕ1)(s-β3)=123c|ξ|.Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:(77)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(78)ϕ(χ)=β1(β2-β3)+β3(β1-β3)sn2(ωχ,m)(β2-β3)+(β1-β3)sn2(ωχ,m),ξ(χ)=Ω~[(α12-m2(ϕ1-β1)ϕ1-ϕs)(m2-α12)Π(arcsin(sn(ωχ,m)),m2,m)+ω(α12-m2(ϕ1-β1)ϕ1-ϕs)χ],
where Ω~=23(ϕ1-ϕs)c/ωm2(β1-ϕ1), α12=(β2-β1)/(β2-β3), ω=(1/2)(β1-ϕ1)(β2-β3), and m=(β1-β2)(ϕ1-β3)/(β1-ϕ1)(β2-β3).(vi) For the givenh=h2 in Figure 3(o), the level curve is shown in Figure 4(h). From Figure 4(h), we see that there is one periodic orbit passing points (β1,0) and (β2,0) when (c,k)∈B15, and its expression is
(79)y=±675Ω2-9450Ω-8437524300c2Ω×(ϕ-ϕ2)(β1-ϕ)(ϕ-β2)(ϕ-β3)(ϕ-ϕ2)ϕ-ϕs,β2≤ϕ≤β1,
where β1, β2, and β3(β3<β2<β1) are three real roots of (41).Substituting (79) into the dϕ/dξ=y and integrating it along the periodic orbit yield
(80)∫ϕβ1‍(s-ϕs)ds(s-ϕ2)(β1-s)(s-β2)(s-β3)(s-ϕ2)=675Ω2-9450Ω-8437524300c2Ω|ξ|.Completing the above integral, we can get the parametric representation of smooth periodic wave solution as follows:(81)u(x,t)=ϕ(x-ct)+43c,ρ(x,t)=(ϕ(x-ct)+13c)2.ϕ is confirmed by
(82)ϕ(χ)=β1(β2-ϕ2)+ϕ2(β1-β2)sn2(ωχ,m)(β2-ϕ2)+(β1-β2)sn2(ωχ,m),ξ(χ)=Ω~[ωm2(β2-β1)ϕ2-ϕs(m2-α2)F(arcsin(sn(ωχ,m)),m)+α2E(arcsin(sn(ωχ,m)),m)-ωm2(β2-β1)ϕ2-ϕsχ],
where Ω~=((ϕ2-ϕs)/ωm2(β1-β2))24300c2Ω/(675Ω2-9450Ω-84375), α2=(β2-β1)/(β2-ϕ2), ω=(1/2)(β1-β3)(β2-ϕ2), and m=(β1-β2)(β3-ϕ2)/(β1-β3)(β2-ϕ2).Remark 2.
From Figures3(f) and 3(n), we see that (2) has two solitary wave solutions when (c,k)∈B6 and two solitary wave solutions when (c,k)∈B14, but the exact parametric representations are very complex, so here we omit them.


## 4. Conclusion

In this paper, using the bifurcation theory and the method of phase portraits analysis, we investigated the bifurcations of (2) (which is called a two-component Degasperis-Procesi equation) and obtained some exact parametric representations of peakon, solitary wave and smooth periodic wave solutions (see (23), (30), and (63), etc.). We can say that we obtained some new results of (2) in this paper.



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