# Integral Bifurcation Method together with a Translation-Dilation Transformation for Solving an Integrable 2-Component Camassa-Holm Shallow Water System

Authors: Weiguo Rui; Yao Long
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

An integrable 2-component Camassa-Holm (2-CH) shallow water system is studied by using integral bifurcation method together with a translation-dilation transformation. Many traveling wave solutions of nonsingular type and singular type, such as solitary wave solutions, kink wave solutions, loop soliton solutions, compacton solutions, smooth periodic wave solutions, periodic kink wave solution, singular wave solution, and singular periodic wave solution are obtained. Further more, their dynamic behaviors are investigated. It is found that the waveforms of some traveling wave solutions vary with the changes of parameter, that is to say, the dynamic behavior of these waves partly depends on the relation of the amplitude of wave and the level of water.


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## Body

## 1. Introduction

In this paper, employing the integral bifurcation method together with a translation-dilation transformation, we will study an integrable 2-component Camassa-Holm (2-CH) shallow water system [1] as follows:
(1.1)mt+εmux+12εumx+σε(ρ2)x=0,ρt+ε2(ρu)x=0,
which is a nonlinear dispersive wave equation that models the propagation of unidirectional irrotational shallow water waves over a flat bed [2], as well as water waves moving over an underlying shear flow [3]. Equation (1.1) also arises in the study of a certain non-Newtonian fluids [4] and also models finite length, small amplitude radial deformation waves in cylindrical hyperelastic rods [5], where m=u-(1/3)δ2uxx+κ/2,σ=±1 and ε=a~/ℏ,δ=ℏ/λ are two dimensionless parameters. The interpretation of (u,ρ), respectively, describes the horizontal fluid velocity and the density in the shallow water regime, where the variable u(x,t) describes the horizontal velocity of the fluid in x direction at time t, and the variable ρ(x,t) is related to the free surface elevation from equilibrium position (or scalar density) with the boundary assumptions. The parameter ℏ denotes the level of water, the parameter a~ denotes the typical amplitude of the water wave, and the parameter λ denotes the typical wavelength of the water wave. The constant κ denotes the speed of the water current which is related to the shallow water wave speed. The case σ=±1, respectively, corresponds to the two situations in which the gravity acceleration points downwards and upwards. Especially, when the speed of the water current κ=0 and the parameter σ=1, (1.1) becomes the following form
(1.2)m~t+εm~ux+12εum~x+1ε(ρ2)x=0,ρt+ε2(ρu)x=0,
where m~=u-(1/3)δ2uxx. The system (1.2) appeared in [1], which was first derived by Constantin and Ivanov from the Green-Naghdi equations [6, 7] via the hydrodynamical point of view. Under the scaling u↦(2/ε)u,x↦(δ/3)x,t↦(δ/3)t, (1.1) can be reduced to the following two-component generalization of the well-known 2-component Camassa-Holm (2-CH) system [1, 8–10]:
(1.3)mt+2mux+umx+σρρx=0,ρt+(ρu)x=0,
where m=u-(1/3)δ2uxx+κ/2. The equation (1.3) attracts much interest since it appears. Attention was more paid on the local well-posedness, blow-up phenomenon, global existence, and so forth. When κ=0, (1.3) has been studied by many authors, see [11–26] and references cited therein. Especially, under the parametric conditions δ=1,ρ=0,σ=-1, (1.3) can be reduced to the celebrated Camassa-Holm equation [27, 28],
(1.4)ut+κux-uxxt+3uux=2uxuxx+uuxxx.
In 2006, the integrability of (1.3) for σ=-1 was proved and some peakon and multikink solutions of this system were presented by Chen et al. in [9]. In 2008, the Lax pair of (1.3) for any value of σ was given by Constantin and Ivanov in [1]. In [29], by using the method of dynamical systems, under the traveling wave transformation u(x,t)=ϕ(x-ct),ρ(x,t)=ψ(x-ct), some explicit parametric representations of exact traveling wave solutions of (1.3) were obtained. But the loop solitons were not obtained in [29]. In [30], under σ=-1, one-loop and two-loop soliton solutions and multisoliton solutions of (1.3) are obtained by using the Darboux transformations.Although (1.1) can be reduced to (1.3) by the scaling transformation u↦(2/ε)u,x↦(δ/3)x,t↦(δ/3)t, the dynamic properties of some traveling wave solutions for these two equations are very different. In fact, the dynamic behaviors of some traveling waves of system (1.1) partly depends on the relation (ε=a~/ℏ) of the amplitude of wave and the level (deepness) of water. In the other words, their dynamic behavior vary with the changes of parameter ε, that is, the changes of ratio for the amplitude of wave a~ and the deepness of water ℏ. In addition, compared with the research results of (1.3), the research results for (1.1) are few in the existing literatures. Thus, (1.1) is very necessary to be further studied.It is worthy to mention that the solutions obtained by us in this paper are different from others in existing references, such as [9, 12–17, 29]. On the other hand, under different transformations, by using different methods, different results will be presented. In this paper, by using the integral bifurcation method [31], we will investigate different kinds of new traveling wave solutions of (1.1) and their dynamic properties under the translation-dilation transformation u(x,t)=v[1+ϕ(x-vt)],ρ(x,t)=v[1+ψ(x-vt)]. By the way, the integral bifurcation method possessed some advantages of the bifurcation theory of the planar dynamic system [32] and auxiliary equation method (see [33, 34] and references cited therein), it is easily combined with computer method [35] and useful for many nonlinear partial diffential equations (PDEs) including some PDEs with high power terms, such as K(m,n) equation [36]. So, by using this method, we will obtain some new traveling wave solutions of (1.1). Some interesting phenomena will be presented.The rest of this paper is organized as follows. In Section2, we will derive the two-dimensional planar system of (1.1) and its first integral equations. In Section 3, by using the integral bifurcation method, we will obtain some new traveling wave solutions of nonsingular type and singular type and investigate their dynamic behaviors.


## 2. The Two-Dimensional Planar System of (1.1) and Its First Integral Equations

Obviously, (1.1) can be rewritten as the following form
(2.1)ut+12εκux+32εuux-13δ2uxxt+2σερρx=16εδ2(2uxuxx+uuxxx),ρt+ε2(ρxu+ρux)=0.
In order to change the PDE (2.1) into an ordinary differential equation, we make a transformation
(2.2)u(x,t)=v[1+ϕ(ξ)],ρ(x,t)=v[1+ψ(ξ)],
where ξ=x-vt and v is an arbitrary nonzero constant. In fact, (2.2) is a translation-dilation transformation, it has been used extensively in many literatures. Its idea came from many existing references. For example, in [37], the expression u=λ1/(m-l)U[λ(m-n)/2(m-l)(x+λt)] is a dilation transformation. In [38], the expression U(x,t)=u(x-ωt)+ω is a translation transformation. In [39, 40], the expression z=(u-v)/|v| which was first used by Parkes and Vakhnenko is also a translation-dilation transformation.After substituting (2.2) into (2.1), integrating them once yields
(2.3)(3ε2+εκ2v-1)ϕ+3ε4ϕ2+13δ2ϕ′′+2σεψ+σεψ2=εδ26[12(ϕ′)2+ϕ′′+ϕϕ′′],[1-ε2(ϕ+1)]ψ=ε2ϕ+A,
where the integral constant A≠0 and ϕ′ denotes ϕξ. From the second equation of (2.3), we easily obtain
(2.4)ψ=2A+εϕ(2-ε)-εϕ,
where ε≠0. Substituting (2.4) into the first equation of (2.3), we obtain
(2.5)ϕ′′=[(3ε/2+εκ/2v-1)ϕ+(3ε/4)ϕ2-(εδ2/12)(ϕ′)2]𝔸2-(2σ/ε)(2A+εϕ)𝔸+(σ/ε)(2A+εϕ)2(1/6)δ2𝔸3,where 𝔸 denotes (ε-2+εϕ).Letϕ′=dϕ/dξ=y. Thus (2.5) can be reduced to 2-dimensional planar system as follows:
(2.6)dϕdξ=y,dydξ=[(3ε/2+εκ/2v-1)ϕ+(3ε/4)ϕ2]𝔸2-(2σ/ε)(2A+εϕ)𝔸+(σ/ε)(2A+εϕ)2-(εδ2/12)𝔸2y2(1/6)δ2𝔸3,where 𝔸 denotes (ε-2+εϕ).Making a transformation(2.7)dξ=(ε-2+εϕ)dτ,
(2.6) becomes
(2.8)dϕdτ=(ε-2+εϕ)y,dydτ=6δ2[(3ε2+εκ2v-1)ϕ+3ε4ϕ2]+6σ(2A+εϕ)(2A+4-2ε-εϕ)εδ2(ε-2+εϕ)2-ε2y2,
where τ is a parameter. From the point of view of the geometric theory, the parameter τ is a fast variable, but the parameter ξ is a slow one. The system (2.8) is still a singular system through the system (2.6) becomes (2.8) by the transformation (2.7). This case is not same as that in [29]. Of course, as in [29], we can also change the system (2.6) into a regular system under another transformation dξ=(ε-2+εϕ)3dτ. But we do not want to do that due to the following two reasons: (i) we need to keep the singularity of the original system; (ii) we need not to make any analysis of phase portraits as in [29].Obviously, systems (2.6) and (2.8) have the same first integral as follows:
(2.9)y2=1(ε-2+εϕ)2×[2ε2δ2ϕ4+3ε(4εv-4v+εκ)vδ2ϕ3+3(ε2κ+3ε2v+4v-4σv-8εv-2εκ)vδ2ϕ2+ε2δ2h-12εσ+24σεδ2ϕ+(ε-2)h-48σA2+96σA+12σε2+48σ-48σε-48σεAε2δ2],
where h is an integral constant.


## 3. Traveling Wave Solutions of Nonsingular Type and Singular Type for (1.1) and Their Dynamic Behaviors

In this section, we will investigate different kinds of exact traveling wave solutions for  (1.1) and their dynamic behaviors.It is easy to know that (2.9) can be reduced to four kinds of simple equations when the parametric conditions satisfy the following four cases:Case 1.
A
=
ε
-
2
,
h
=
12
σ
(
ε
-
2
)
/
ε
2
δ
2 and ε≠0,2,v≠εκ/4(1-ε).Case 2.
v
=
ε
κ
/
4
(
1
-
ε
)
,
h
=
12
σ
(
ε
-
2
)
/
ε
2
δ
2 and ε≠0,1,A≠ε-2.Case 3.
A
=
ε
-
2
,
h
=
12
σ
(
ε
-
2
)
/
ε
2
δ
2
,
v
=
ε
κ
/
4
(
1
-
ε
) and ε≠0,1,2.Case 4.
h
=
12
σ
(
2
A
+
2
-
ε
)
2
/
ε
2
δ
2
(
ε
-
2
).We mainly aim to consider the new results of (1.1), so we only discuss the first two typical cases in this section. The other two cases can be similarly discussed, here we omit them.


### 3.1. The Exact Traveling Wave Solutions under Case1

Under the parametric conditionsA=ε-2,h=12σ(ε-2)/ε2δ2,v≠εκ/4(1-ε) and ε≠0or2,σ=±1, (2.9) can be reduced to
(3.1)y=±cϕ4+bϕ3+aϕ2ε-2+εϕ,
where c=3ε2/δ2≥0,b=(3ε/δ2)[4(ε-1)+εκ/v],a=(3/δ2)[4(1-σ)+ε(3ε-8)+εκ(ε-2)/v]. Substituting (3.1) into the first expression in (2.8) yields
(3.2)dϕdτ=±cϕ4+bϕ3+aϕ2.
Write
(3.3)ϵ=±1andΔ=9ε2(4v2ε2+4vεκ+ε2κ2+16σv2)v2δ4.
The Δ=0 if only if κ=2v(ε±2-σ)/ε. By using the exact solutions of (3.2), we can obtain different kinds of exact traveling wave solutions of parametric type of (1.1), see the following discussion.(i)
Ifa>0, then (3.2) has one exact solution as follows:
(3.4)ϕ=-absech2((a/2)τ)b2-ac[1+ϵtanh((a/2)τ)]2,a>0.
Substituting (3.4) into (2.7), and then integrating it yields
(3.5)ξ=(ε-2)τ-2εϵctanh-1[acb(1+ϵtanh(a2τ))].
Substituting (3.4) into (2.2) and (2.4), then combining with (3.5), we obtain a couple of soliton-like solutions of (1.1) as follows:
(3.6)u=v[1-absech2((a/2)τ)b2-ac[1+ϵtanh((a/2)τ)]2],ξ=(ε-2)τ-2εϵctanh-1[acb(1+ϵtanh(a2τ))],(3.7)ρ=v[1+2(ε-2)(b2-ac[1+ϵtanh((a/2)τ)]2)-abεsech2((a/2)τ)(2-ε)(b2-ac[1+ϵtanh((a/2)τ)]2)+abεsech2((a/2)τ)],ξ=(ε-2)τ-2εϵctanh-1[acb(1+ϵtanh(a2τ))].(ii)
Ifa>0,c>0,Δ>0, then (3.2) has one exact solution as follows:
(3.8)ϕ=2asech(aτ)ϵΔ-bsech(aτ).
Similarly, by using (3.8), (2.7), (2.2), and (2.4), we obtain two couples of soliton-like solutions of  (1.1) as follows:
(3.9)u=v[1+2asech(aτ)ϵΔ-bsech(aτ)],ξ=(ε-2)τ-2εctanh-1[b+ϵΔ2actanh(a2τ)],ρ=v[1+2(ε-2)(ϵΔ-bsech(aτ))+2aεsech(aτ)(2-ε)(ϵΔ-bsech(aτ))-2aεsech(aτ)],ξ=(ε-2)τ-2εctanh-1[b+ϵΔ2actanh(a2τ)].(iii)
Ifa>0 and Δ<0, then (1.1) has one couple of soliton-like solutions as follows:
(3.10)u=v[1+2acsch(aτ)ϵ-Δ-bcsch(aτ)],ξ=(ε-2)τ-2εctanh-1[btanh((a/2)τ)+ϵ-Δ2ac],ρ=v[1+2(ε-2)(ϵ-Δ-bcsch(aτ))+2aεcsch(aτ)(2-ε)(ϵ-Δ-bcsch(aτ))-2aεcsch(aτ)],ξ=(ε-2)τ-2εctanh-1[btanh((a/2)τ)+ϵ-Δ2ac].(iv)
Ifa>0 and c≠0 (i.e., ε≠0), then (1.1) has one couple of soliton-like solutions as follows:
(3.11)u=v[1-asech2((a/2)τ)b+2ϵactanh((a/2)τ)],ξ=(ε-2)τ-εϵcln|b+2ϵactanh(a2τ)|,ρ=v[1+2(ε-2)[b+2ϵactanh((a/2)τ)]-aεsech2((a/2)τ)(2-ε)[b+2ϵactanh((a/2)τ)]+aεsech2((a/2)τ)],ξ=(ε-2)τ-εϵcln|b+2ϵactanh(a2τ)|.(v)
Ifa>0 and Δ=0 (i.e. κ=(v/ε)[(4-4v+2vε-4ε)±2v(v-1)(ε-2)2-4σ]), then  (1.1) has one couple of kink and antikink wave solutions as follows:
(3.12)u=v[1-ab(1+ϵtanh(a2τ))],ξ=(ε-2)τ-ε[abτ+2ϵabln|cosh(a2τ)|],ρ=v[1+2b(ε-2)-aε[1+ϵtanh((a/2)τ)]b(2-ε)+aε[1+ϵtanh((a/2)τ)]],ξ=(ε-2)τ-ε[abτ+2ϵabln|cosh(a2τ)|],u=v[1-ab(1+ϵcoth(a2τ))],ξ=(ε-2)τ-ε[abτ+2ϵabln|sinh(a2τ)|],ρ=v[1+2b(ε-2)-aε[1+ϵcoth((a/2)τ)]b(2-ε)+aε[1+ϵcoth((a/2)τ)]],ξ=(ε-2)τ-ε[abτ+2ϵabln|sinh(a2τ)|].(vi)
Ifa>0 and b=0 (i.e. κ=4v(1/ε-1)), then (1.1) has one couple of soliton-like solutions as follows:
(3.13)u=v[1-accsch(ϵaτ)],ξ=(ε-2)τ-εϵcln[tanh(12ϵaτ)],ρ=v[1+2(ε-2)c-εacsch(ϵaτ)(2-ε)c+εacsch(ϵaτ)],ξ=(ε-2)τ-εϵcln[tanh(12ϵaτ)].In order to show the dynamic properties of above soliton-like solutions and kink and antikink wave solutions intuitively, as examples, we plot their graphs of some solutions, see Figures 1, 2, 3, and 4. Figures 1(a)–1(h) show the profiles of multiwaveform to solution the first solution of (3.6) for fixed parameters ϵ=1,σ=-1,κ=4,v=4,δ=5 and different ε-values. Figures 2(a)–2(d) show the profiles of multiwaveform to solution (3.7) for fixed parameters ϵ=1,σ=-1,κ=4,v=3,δ=5 and different ε-values. Figures 3(a)–3(d) show the profiles of multiwaveform to solution (3.9) for fixed parameters ϵ=-1,σ=-1,κ=-2,v=-4,δ=3.5 and different ε-values. Figures 4(a)-4(b) show the profiles of kink wave solution (the first formula of (3.12)) for fixed parameters ϵ=1,σ=-1,v=0.4,δ=3.5 and different (ε,κ)-values.The profiles of multiwaveform of (3.6) for the given parameters and different ε-values: (a) ε=1.2; (b) ε=1.26; (c) ε=1.4; (d) ε=1.99; (e) ε=2; (f) ε=2.2; (g) ε=3; (h) ε=5.
(a)
Antikink wave
(b)
Transmutative wave of antikink waveform
(c)
Dark soliton of thin waveform
(d)
Dark soliton of fat waveform
(e)
Compacton
(f)
Loop soliton of fat waveform
(g)
Loop soliton of thin waveform
(h)
Loop soliton of oblique waveformThe profiles of multiwaveform of (3.7) for the given parameters and different ε-values: (a) ε=1.2; (b) ε=1.20000001; (c) ε=1.22; (d) ε=2.4.
(a)
Kink wave
(b)
Bright soliton of fat waveform
(c)
Bright soliton of thin waveform
(d)
Singular wave of cracked loop solitonThe profiles of multiwaveform of the first solution of (3.9) for the given parameters and different ε-values: (a) ε=1.8; (b) ε=2; (c) ε=2.2; (d) ε=2.8.
(a)
Bright soliton
(b)
Bright compacton
(c)
Loop soliton of fat waveform
(d)
Loop soliton of thin waveformThe profiles of kink wave solution (the first solution of (3.12)) for the given parameters and different (ε,κ)-values: (a) ε=0.2,κ=7.2,t=0,n∈[-30,30]; (b) ε=1.22,κ=0.5333333334,t=0.2,n∈[-110,-50].
(a)
Smooth kink wave
(b)
Smooth kink waveWe observe that some profiles of above soliton-like solutions are very much sensitive to one of parameters, that is, their profiles are transformable (see Figures1–3). But the others are not, their waveforms do not vary no matter how the parameters vary (see Figure 4). Some phenomena are very similar to those in [41, 42]. In [41], the waveforms of soliton-like solution of the generalized KdV equation vary with the changes of parameter and depend on the velocity c extremely. Similarly, in [42], the properties of some traveling wave solutions of the generalized KdV-Burges equation depend on the dissipation coefficient α; if dissipation coefficient α≥λ1, it appears as a monotonically kink profile solitary wave; if 0<α≤λ1, it appears as a damped oscillatory wave.From Figures1(a)–1(h), it is easy to know that the profiles of solution (3.6) vary gradually, its properties depend on the parameter ε. When parametric values of ε increase from 1.2 to 5, the solution (3.6) has eight kinds of waveforms: Figure 1(a) shows a shape of antikink wave when ε=1.2; Figure 1(b) shows a shape of transmutative antikink wave when ε=1.6; Figure 1(c) shows a shape of thin and dark solitary wave when ε=1.4;  Figure 1(d) shows a shape of fat and dark solitary wave when ε=1.99; Figure 1(e) shows a shape of compacton wave when ε=2; Figure 1(f) shows a shape of fat loop soliton when ε=2.2; the Figure 1(g) shows a shape of thin loop soliton when ε=3; Figure 1(h) shows a shape of oblique loop soliton when ε=5.Similarly, from Figures2(a)–2(d), it is also easy to know that the profiles of solution (3.7) are transformable, but their changes are not gradual, it depends on the parameter ε extremely. When parametric values of ε increase from 1.2 to 2.4, the solution (3.7) has four kinds of waveforms: Figure 2(a) shows a shape of smooth kink wave when ε=1.2; Figure 2(b) shows a shape of fat and bright solitary wave when ε=1.20000001; Figure 2(c) shows a shape of thin and bright solitary wave when ε=1.22; Figure 2(d) shows a shape of singular wave of cracked loop soliton when ε=2.4. Especially, the changes of waveforms from Figure 2(a) to Figure 2(b) and from Figure 2(c) to Figure 2(d) happened abruptly.As in Figure1, the profiles of the first solution of (3.9) in Figure 3 vary gradually. When parametric values of ε increase from 1.8 to 2.8, the waveform becomes a bright compacton from a shape of bright solitary wave, then becomes a shape of fat loop soliton and a shape of thin loop soliton at last.Different from the properties of solutions (3.6), (3.7) and (3.9), the property of the first solution of (3.12) is stable. The profile of the first solution of (3.12) is not transformable no matter how the parameters vary. Both Figures 4(a) and 4(b) show a shape of smooth kink wave.


### 3.2.  The Traveling Wave Solutions Under Case2

Under the parametric conditionsv=εκ/4(1-ε),h=12σ(ε-2)/ε2δ2 and ε≠0or1, (2.9) becomes
(3.14)y=±Pϕ4+Qϕ2+Rε-2+εϕ,
where P=3ε2/δ2>0,Q=(-3/δ2)[4(σ+1)+ε(ε-4)],R=48σA(ε-2-A)/ε2δ2. Substituting (3.14) into the first expression in (2.8) yields
(3.15)dϕdτ=±Pϕ4+Qϕ2+R.
We know that the case σ=-1 corresponds to the situation in which the gravity acceleration points upwards. As an example, in this subsection, we only discuss the case σ=-1. The case σ=1 can be similarly discussed, but we omit them here. Especially, when σ=-1, the above values of P,Q,R can be reduced to P=3ε2/δ2>0,Q=-3ε(ε-4)/δ2,R=48A(A+2-ε)/ε2δ2.(i) IfA=(-1+ε/2)±(1-ε)(4+ε2)/2,m=-ε/2,δ=±2-3ε,-4<ε<0, then the P=m2,Q=-(1+m2),R=1. Under these parametric conditions, (3.15) has a Jacobi elliptic function solution as follows:
(3.16)ϕ=sn(τ,m)
or
(3.17)ϕ=cd(τ,m),
where m=-ε/2 is the model of Jacobi elliptic function and 0<m<1.Substituting (3.16) and (3.17) into (2.7), respectively, we obtain
(3.18)ξ=(ε-2)τ-2ε-εcosh-1[2dn(τ,-ε/2)4+ε],(3.19)ξ=(ε-2)τ+2ε-εln[1+(-ε/2)sn(τ,-ε/2)dn(τ,-ε/2)].
Substituting (3.16) into (2.2) and (2.4), using (3.18) and the transformation ξ=x-vt=x-(εκ/4(1-ε))t, we obtain a couple of periodic solutions of (1.1) as follows:
(3.20)u=εκ4(1-ε)[1+sn(τ,-ε/2)],x=(ε-2)τ-2ε-εcosh-1[2dn(τ,-ε/2)4+ε]+εκ4(1-ε)t,ρ=εκ4(1-ε)[1+2A+εsn(τ,-ε/2)(ε-2)-εsn(τ,-ε/2)],x=(ε-2)τ-2ε-εcosh-1[2dn(τ,-ε/2)4+ε]+εκ4(1-ε)t.
Substituting (3.17) into (2.2) and (2.4), using (3.19) and the transformation ξ=x-vt=x-(εκ/4(1-ε))t, we also obtain a couple of periodic solutions of (1.1) as follows:
(3.21)u=εκ4(1-ε)[1+cd(τ,m)],x=(ε-2)τ+εmln[1+msn(τ,m)dn(τ,m)]+εκ4(1-ε)t,ρ=εκ4(1-ε)[1+2A+εcd(τ,m)(ε-2)-εcd(τ,m)],x=(ε-2)τ+εmln[1+msn(τ,m)dn(τ,m)]+εκ4(1-ε)t.(ii)  IfA=-m2-2m±m6+4m4+4m2+16/m3,0<m<1,δ=±43/m,ε=-4/m2, then the P=1,Q=-(1+m2),R=m2. In the case of these parametric conditions, (3.15) has a Jacobi elliptic function solution as follows:
(3.22)ϕ=ns(τ,m)orϕ=dc(τ,m).As in the first case (i) using the same method, we obtain a couple of periodic solutions of (1.1) as follows:
(3.23)u=-κm2+4[1+ns(τ,m)],x=(-4m2-2)τ-4m2ln[sn(τ,m)cn(τ,m)+dn(τ,m)]-κm2+4t,(3.24)ρ=-κm2+4[1+2m2A-4ns(τ,m)-(4+2m2)+4ns(τ,m)],x=(-4m2-2)τ-4m2ln[sn(τ,m)cn(τ,m)+dn(τ,m)]-κm2+4t
or
(3.25)u=-κm2+4[1+dc(τ,m)],x=(-4m2-2)τ-4m2ln[1+sn(τ,m)cn(τ,m)]-κm2+4t,ρ=-κm2+4[1+2m2A-4dc(τ,m)-(4+2m2)+4dc(τ,m)],x=(-4m2-2)τ-4m2ln[1+sn(τ,m)cn(τ,m)]-κm2+4t.
Especially, when m→1 (i.e., A=2or-8,δ=±42,ε=-4,v=-κ/5), ns(τ,m)→coth(τ),sn(τ,m)→tanh(τ),cn(τ,m)→sech(τ),dn(τ,m)→sech(τ). From (3.23) and (3.24), we obtain a couple of kink-like solutions of (1.1) as follows:
(3.26)u=-κ5[1+coth(τ)],x=-6τ-4ln[12sinh(τ)]-κ5t,ρ=-κ5[1+A-2coth(τ)-3+2coth(τ)],x=-6τ-4ln[12sinh(τ)]-κ5t.(iii)  IfA=(ε/2-1)±(1-ε)(ε2+4)/2,m=21/(ε+4),δ=±3ε2+12ε,0<ε≤1, then the P=1-m2,Q=2m2-1,R=-m2. In the case of these parametric conditions, (3.15) has a Jacobi elliptic function solution as follows:
(3.27)ϕ=nc(τ,m).
As in the first case (i), similarly, we obtain a couple of periodic solutions of (1.1) as follows:
(3.28)u=εκ4(1-ε)[1+nc(τ,m)],x=(ε-2)τ+ε1-m2ln[1-m2sn(τ,m)+dn(τ,m)cn(τ,m)]+εκ4(1-ε)t,ρ=εκ4(1-ε)[1+2A+εnc(τ,m)(ε-2)-εnc(τ,m)],x=(ε-2)τ+ε1-m2ln[1-m2sn(τ,m)+dn(τ,m)cn(τ,m)]+εκ4(1-ε)t,
where m=21/(ε+4).(iv)  IfA=(ε/2-1)±2(4-ε2)(ε2-2ε+2)/4,m=(3ε-4)/(2ε-4),δ=±-6ε2+12ε,0<ε<1and1<ε<4/3, then the P=1-m2,Q=2-m2,R=1. In the case of these parametric conditions, (3.15) has a Jacobi elliptic function solution as follows:
(3.29)ϕ=sc(τ,m).
Similarly, we obtain a couple of periodic solutions of (1.1) as follows:
(3.30)u=εκ4(1-ε)[1+sc(τ,m)],x=(ε-2)τ+ε21-m2ln[dn(τ,m)+1-m2dn(τ,m)-1-m2]+εκ4(1-ε)t,ρ=εκ4(1-ε)[1+2A+εsc(τ,m)(ε-2)-εsc(τ,m)],x=(ε-2)τ+ε21-m2ln[dn(τ,m)+1-m2dn(τ,m)-1-m2]+εκ4(1-ε)t,
where m=(3ε-4)/(2ε-4).(v)  IfA=(-m4+4m2-3±m8-8m6+22m4-40m2+25)/(m2-3)2,0<m<1,δ=±43/(m2-3),ε=4/(3-m2), then the P=1,Q=2-m2,R=1-m2. In the case of these parametric conditions,  (3.15) has a Jacobi elliptic function solution as follows:
(3.31)ϕ=cs(τ,m).As in the first case (i) using the same method, we obtain a couple of periodic solutions of (1.1) as follows:
(3.32)u=-κ1+m2[1+cs(τ,m)],x=(43-m2-2)τ+43-m2ln[1-dn(τ,m)sn(τ,m)]-κ1+m2t,ρ=-κ1+m2[1+2(3-m2)A+4cs(τ,m)4-2(3-m2)-4cs(τ,m)],x=(43-m2-2)τ+43-m2ln[1-dn(τ,m)sn(τ,m)]-κ1+m2t.In order to show the dynamic properties of above periodic solutions intuitively, as examples, we plot their graphs of the solutions (3.20), (3.23) and (3.24), see Figures 5 and 6.The profiles of smooth periodic waves of solutions (3.20) for the given parameters:  ϵ=1,σ=-1,κ=-4,v=3,δ=5,t=1 and different ε-values: (a) ε=-0.2; (b) ε=-2.
(a)
Smooth periodic wave
(b)
Smooth periodic waveThe profiles of noncontinuous periodic waves of solutions (3.23) and (3.24) for the given parameters: (a) κ=-4,m=0.6,t=1; (b) κ=-2,m=0.6,t=1.
(a)
Periodic kink wave
(b)
Singular periodic waveFigures5(a) and 5(b) show two shapes of smooth and continuous periodic waves, all of them are nonsingular type. Figure 6(a) shows a shape of periodic kink wave. Figure 6(b) shows a shape of singular periodic wave. Both Figures 6(a) and 6(b) show two discontinuous periodic waves, all of them are singular type.From the above illustrations, we find that the waveforms of some solutions partly depend on wave parameters. Indeed, in 2006, Vakhnenko and Parkes’s work [43] successfully explained similar phenomena. In [43], the graphical interpretation of the solution for gDPE is presented. In this analysis, the 3D-spiral (whether one loop from a spiral or a half loop of a spiral) has the different projections that is the essence of the possible solutions. Of caurse, this approach also can be employed to the 2-component Camassa-Holm shallow water system (1.1). By using the Vakhnenko and Parkes’s theory, the phenomena which appeared in this work are easily understood, so we omit this analysis at here. However, it is necessary to say something about the cracked loop soliton (Figure 2(d)) and the singular periodic wave (Figure 6(b)), what are the 3D-curves with projections associated with two peculiar solutions (3.7) and (3.24) which are shown in Figure 2(d) and Figure 6(b)? In order to answer this question, by using the Vakhnenko and Parkes’s approach, as an example we give the 3D-curves of solution (3.24) and their projection curve in the xoρ-plane, which are shown in Figure 7. For convenience to distinguish them, we colour the 3D-curves red and colour their projection curve green. From Figure 7, we can see that the 3D-curves are not intersected, but their projection curve is intersected in xoρ-plane.Evolvement graphs of 3D-curves along with the time and their projection curve.
(a)
3D-curves of solution (3.24)
(b)
3D-curves and their projection


## 3.1. The Exact Traveling Wave Solutions under Case1

Under the parametric conditionsA=ε-2,h=12σ(ε-2)/ε2δ2,v≠εκ/4(1-ε) and ε≠0or2,σ=±1, (2.9) can be reduced to
(3.1)y=±cϕ4+bϕ3+aϕ2ε-2+εϕ,
where c=3ε2/δ2≥0,b=(3ε/δ2)[4(ε-1)+εκ/v],a=(3/δ2)[4(1-σ)+ε(3ε-8)+εκ(ε-2)/v]. Substituting (3.1) into the first expression in (2.8) yields
(3.2)dϕdτ=±cϕ4+bϕ3+aϕ2.
Write
(3.3)ϵ=±1andΔ=9ε2(4v2ε2+4vεκ+ε2κ2+16σv2)v2δ4.
The Δ=0 if only if κ=2v(ε±2-σ)/ε. By using the exact solutions of (3.2), we can obtain different kinds of exact traveling wave solutions of parametric type of (1.1), see the following discussion.(i)
Ifa>0, then (3.2) has one exact solution as follows:
(3.4)ϕ=-absech2((a/2)τ)b2-ac[1+ϵtanh((a/2)τ)]2,a>0.
Substituting (3.4) into (2.7), and then integrating it yields
(3.5)ξ=(ε-2)τ-2εϵctanh-1[acb(1+ϵtanh(a2τ))].
Substituting (3.4) into (2.2) and (2.4), then combining with (3.5), we obtain a couple of soliton-like solutions of (1.1) as follows:
(3.6)u=v[1-absech2((a/2)τ)b2-ac[1+ϵtanh((a/2)τ)]2],ξ=(ε-2)τ-2εϵctanh-1[acb(1+ϵtanh(a2τ))],(3.7)ρ=v[1+2(ε-2)(b2-ac[1+ϵtanh((a/2)τ)]2)-abεsech2((a/2)τ)(2-ε)(b2-ac[1+ϵtanh((a/2)τ)]2)+abεsech2((a/2)τ)],ξ=(ε-2)τ-2εϵctanh-1[acb(1+ϵtanh(a2τ))].(ii)
Ifa>0,c>0,Δ>0, then (3.2) has one exact solution as follows:
(3.8)ϕ=2asech(aτ)ϵΔ-bsech(aτ).
Similarly, by using (3.8), (2.7), (2.2), and (2.4), we obtain two couples of soliton-like solutions of  (1.1) as follows:
(3.9)u=v[1+2asech(aτ)ϵΔ-bsech(aτ)],ξ=(ε-2)τ-2εctanh-1[b+ϵΔ2actanh(a2τ)],ρ=v[1+2(ε-2)(ϵΔ-bsech(aτ))+2aεsech(aτ)(2-ε)(ϵΔ-bsech(aτ))-2aεsech(aτ)],ξ=(ε-2)τ-2εctanh-1[b+ϵΔ2actanh(a2τ)].(iii)
Ifa>0 and Δ<0, then (1.1) has one couple of soliton-like solutions as follows:
(3.10)u=v[1+2acsch(aτ)ϵ-Δ-bcsch(aτ)],ξ=(ε-2)τ-2εctanh-1[btanh((a/2)τ)+ϵ-Δ2ac],ρ=v[1+2(ε-2)(ϵ-Δ-bcsch(aτ))+2aεcsch(aτ)(2-ε)(ϵ-Δ-bcsch(aτ))-2aεcsch(aτ)],ξ=(ε-2)τ-2εctanh-1[btanh((a/2)τ)+ϵ-Δ2ac].(iv)
Ifa>0 and c≠0 (i.e., ε≠0), then (1.1) has one couple of soliton-like solutions as follows:
(3.11)u=v[1-asech2((a/2)τ)b+2ϵactanh((a/2)τ)],ξ=(ε-2)τ-εϵcln|b+2ϵactanh(a2τ)|,ρ=v[1+2(ε-2)[b+2ϵactanh((a/2)τ)]-aεsech2((a/2)τ)(2-ε)[b+2ϵactanh((a/2)τ)]+aεsech2((a/2)τ)],ξ=(ε-2)τ-εϵcln|b+2ϵactanh(a2τ)|.(v)
Ifa>0 and Δ=0 (i.e. κ=(v/ε)[(4-4v+2vε-4ε)±2v(v-1)(ε-2)2-4σ]), then  (1.1) has one couple of kink and antikink wave solutions as follows:
(3.12)u=v[1-ab(1+ϵtanh(a2τ))],ξ=(ε-2)τ-ε[abτ+2ϵabln|cosh(a2τ)|],ρ=v[1+2b(ε-2)-aε[1+ϵtanh((a/2)τ)]b(2-ε)+aε[1+ϵtanh((a/2)τ)]],ξ=(ε-2)τ-ε[abτ+2ϵabln|cosh(a2τ)|],u=v[1-ab(1+ϵcoth(a2τ))],ξ=(ε-2)τ-ε[abτ+2ϵabln|sinh(a2τ)|],ρ=v[1+2b(ε-2)-aε[1+ϵcoth((a/2)τ)]b(2-ε)+aε[1+ϵcoth((a/2)τ)]],ξ=(ε-2)τ-ε[abτ+2ϵabln|sinh(a2τ)|].(vi)
Ifa>0 and b=0 (i.e. κ=4v(1/ε-1)), then (1.1) has one couple of soliton-like solutions as follows:
(3.13)u=v[1-accsch(ϵaτ)],ξ=(ε-2)τ-εϵcln[tanh(12ϵaτ)],ρ=v[1+2(ε-2)c-εacsch(ϵaτ)(2-ε)c+εacsch(ϵaτ)],ξ=(ε-2)τ-εϵcln[tanh(12ϵaτ)].In order to show the dynamic properties of above soliton-like solutions and kink and antikink wave solutions intuitively, as examples, we plot their graphs of some solutions, see Figures 1, 2, 3, and 4. Figures 1(a)–1(h) show the profiles of multiwaveform to solution the first solution of (3.6) for fixed parameters ϵ=1,σ=-1,κ=4,v=4,δ=5 and different ε-values. Figures 2(a)–2(d) show the profiles of multiwaveform to solution (3.7) for fixed parameters ϵ=1,σ=-1,κ=4,v=3,δ=5 and different ε-values. Figures 3(a)–3(d) show the profiles of multiwaveform to solution (3.9) for fixed parameters ϵ=-1,σ=-1,κ=-2,v=-4,δ=3.5 and different ε-values. Figures 4(a)-4(b) show the profiles of kink wave solution (the first formula of (3.12)) for fixed parameters ϵ=1,σ=-1,v=0.4,δ=3.5 and different (ε,κ)-values.The profiles of multiwaveform of (3.6) for the given parameters and different ε-values: (a) ε=1.2; (b) ε=1.26; (c) ε=1.4; (d) ε=1.99; (e) ε=2; (f) ε=2.2; (g) ε=3; (h) ε=5.
(a)
Antikink wave
(b)
Transmutative wave of antikink waveform
(c)
Dark soliton of thin waveform
(d)
Dark soliton of fat waveform
(e)
Compacton
(f)
Loop soliton of fat waveform
(g)
Loop soliton of thin waveform
(h)
Loop soliton of oblique waveformThe profiles of multiwaveform of (3.7) for the given parameters and different ε-values: (a) ε=1.2; (b) ε=1.20000001; (c) ε=1.22; (d) ε=2.4.
(a)
Kink wave
(b)
Bright soliton of fat waveform
(c)
Bright soliton of thin waveform
(d)
Singular wave of cracked loop solitonThe profiles of multiwaveform of the first solution of (3.9) for the given parameters and different ε-values: (a) ε=1.8; (b) ε=2; (c) ε=2.2; (d) ε=2.8.
(a)
Bright soliton
(b)
Bright compacton
(c)
Loop soliton of fat waveform
(d)
Loop soliton of thin waveformThe profiles of kink wave solution (the first solution of (3.12)) for the given parameters and different (ε,κ)-values: (a) ε=0.2,κ=7.2,t=0,n∈[-30,30]; (b) ε=1.22,κ=0.5333333334,t=0.2,n∈[-110,-50].
(a)
Smooth kink wave
(b)
Smooth kink waveWe observe that some profiles of above soliton-like solutions are very much sensitive to one of parameters, that is, their profiles are transformable (see Figures1–3). But the others are not, their waveforms do not vary no matter how the parameters vary (see Figure 4). Some phenomena are very similar to those in [41, 42]. In [41], the waveforms of soliton-like solution of the generalized KdV equation vary with the changes of parameter and depend on the velocity c extremely. Similarly, in [42], the properties of some traveling wave solutions of the generalized KdV-Burges equation depend on the dissipation coefficient α; if dissipation coefficient α≥λ1, it appears as a monotonically kink profile solitary wave; if 0<α≤λ1, it appears as a damped oscillatory wave.From Figures1(a)–1(h), it is easy to know that the profiles of solution (3.6) vary gradually, its properties depend on the parameter ε. When parametric values of ε increase from 1.2 to 5, the solution (3.6) has eight kinds of waveforms: Figure 1(a) shows a shape of antikink wave when ε=1.2; Figure 1(b) shows a shape of transmutative antikink wave when ε=1.6; Figure 1(c) shows a shape of thin and dark solitary wave when ε=1.4;  Figure 1(d) shows a shape of fat and dark solitary wave when ε=1.99; Figure 1(e) shows a shape of compacton wave when ε=2; Figure 1(f) shows a shape of fat loop soliton when ε=2.2; the Figure 1(g) shows a shape of thin loop soliton when ε=3; Figure 1(h) shows a shape of oblique loop soliton when ε=5.Similarly, from Figures2(a)–2(d), it is also easy to know that the profiles of solution (3.7) are transformable, but their changes are not gradual, it depends on the parameter ε extremely. When parametric values of ε increase from 1.2 to 2.4, the solution (3.7) has four kinds of waveforms: Figure 2(a) shows a shape of smooth kink wave when ε=1.2; Figure 2(b) shows a shape of fat and bright solitary wave when ε=1.20000001; Figure 2(c) shows a shape of thin and bright solitary wave when ε=1.22; Figure 2(d) shows a shape of singular wave of cracked loop soliton when ε=2.4. Especially, the changes of waveforms from Figure 2(a) to Figure 2(b) and from Figure 2(c) to Figure 2(d) happened abruptly.As in Figure1, the profiles of the first solution of (3.9) in Figure 3 vary gradually. When parametric values of ε increase from 1.8 to 2.8, the waveform becomes a bright compacton from a shape of bright solitary wave, then becomes a shape of fat loop soliton and a shape of thin loop soliton at last.Different from the properties of solutions (3.6), (3.7) and (3.9), the property of the first solution of (3.12) is stable. The profile of the first solution of (3.12) is not transformable no matter how the parameters vary. Both Figures 4(a) and 4(b) show a shape of smooth kink wave.


## 3.2.  The Traveling Wave Solutions Under Case2

Under the parametric conditionsv=εκ/4(1-ε),h=12σ(ε-2)/ε2δ2 and ε≠0or1, (2.9) becomes
(3.14)y=±Pϕ4+Qϕ2+Rε-2+εϕ,
where P=3ε2/δ2>0,Q=(-3/δ2)[4(σ+1)+ε(ε-4)],R=48σA(ε-2-A)/ε2δ2. Substituting (3.14) into the first expression in (2.8) yields
(3.15)dϕdτ=±Pϕ4+Qϕ2+R.
We know that the case σ=-1 corresponds to the situation in which the gravity acceleration points upwards. As an example, in this subsection, we only discuss the case σ=-1. The case σ=1 can be similarly discussed, but we omit them here. Especially, when σ=-1, the above values of P,Q,R can be reduced to P=3ε2/δ2>0,Q=-3ε(ε-4)/δ2,R=48A(A+2-ε)/ε2δ2.(i) IfA=(-1+ε/2)±(1-ε)(4+ε2)/2,m=-ε/2,δ=±2-3ε,-4<ε<0, then the P=m2,Q=-(1+m2),R=1. Under these parametric conditions, (3.15) has a Jacobi elliptic function solution as follows:
(3.16)ϕ=sn(τ,m)
or
(3.17)ϕ=cd(τ,m),
where m=-ε/2 is the model of Jacobi elliptic function and 0<m<1.Substituting (3.16) and (3.17) into (2.7), respectively, we obtain
(3.18)ξ=(ε-2)τ-2ε-εcosh-1[2dn(τ,-ε/2)4+ε],(3.19)ξ=(ε-2)τ+2ε-εln[1+(-ε/2)sn(τ,-ε/2)dn(τ,-ε/2)].
Substituting (3.16) into (2.2) and (2.4), using (3.18) and the transformation ξ=x-vt=x-(εκ/4(1-ε))t, we obtain a couple of periodic solutions of (1.1) as follows:
(3.20)u=εκ4(1-ε)[1+sn(τ,-ε/2)],x=(ε-2)τ-2ε-εcosh-1[2dn(τ,-ε/2)4+ε]+εκ4(1-ε)t,ρ=εκ4(1-ε)[1+2A+εsn(τ,-ε/2)(ε-2)-εsn(τ,-ε/2)],x=(ε-2)τ-2ε-εcosh-1[2dn(τ,-ε/2)4+ε]+εκ4(1-ε)t.
Substituting (3.17) into (2.2) and (2.4), using (3.19) and the transformation ξ=x-vt=x-(εκ/4(1-ε))t, we also obtain a couple of periodic solutions of (1.1) as follows:
(3.21)u=εκ4(1-ε)[1+cd(τ,m)],x=(ε-2)τ+εmln[1+msn(τ,m)dn(τ,m)]+εκ4(1-ε)t,ρ=εκ4(1-ε)[1+2A+εcd(τ,m)(ε-2)-εcd(τ,m)],x=(ε-2)τ+εmln[1+msn(τ,m)dn(τ,m)]+εκ4(1-ε)t.(ii)  IfA=-m2-2m±m6+4m4+4m2+16/m3,0<m<1,δ=±43/m,ε=-4/m2, then the P=1,Q=-(1+m2),R=m2. In the case of these parametric conditions, (3.15) has a Jacobi elliptic function solution as follows:
(3.22)ϕ=ns(τ,m)orϕ=dc(τ,m).As in the first case (i) using the same method, we obtain a couple of periodic solutions of (1.1) as follows:
(3.23)u=-κm2+4[1+ns(τ,m)],x=(-4m2-2)τ-4m2ln[sn(τ,m)cn(τ,m)+dn(τ,m)]-κm2+4t,(3.24)ρ=-κm2+4[1+2m2A-4ns(τ,m)-(4+2m2)+4ns(τ,m)],x=(-4m2-2)τ-4m2ln[sn(τ,m)cn(τ,m)+dn(τ,m)]-κm2+4t
or
(3.25)u=-κm2+4[1+dc(τ,m)],x=(-4m2-2)τ-4m2ln[1+sn(τ,m)cn(τ,m)]-κm2+4t,ρ=-κm2+4[1+2m2A-4dc(τ,m)-(4+2m2)+4dc(τ,m)],x=(-4m2-2)τ-4m2ln[1+sn(τ,m)cn(τ,m)]-κm2+4t.
Especially, when m→1 (i.e., A=2or-8,δ=±42,ε=-4,v=-κ/5), ns(τ,m)→coth(τ),sn(τ,m)→tanh(τ),cn(τ,m)→sech(τ),dn(τ,m)→sech(τ). From (3.23) and (3.24), we obtain a couple of kink-like solutions of (1.1) as follows:
(3.26)u=-κ5[1+coth(τ)],x=-6τ-4ln[12sinh(τ)]-κ5t,ρ=-κ5[1+A-2coth(τ)-3+2coth(τ)],x=-6τ-4ln[12sinh(τ)]-κ5t.(iii)  IfA=(ε/2-1)±(1-ε)(ε2+4)/2,m=21/(ε+4),δ=±3ε2+12ε,0<ε≤1, then the P=1-m2,Q=2m2-1,R=-m2. In the case of these parametric conditions, (3.15) has a Jacobi elliptic function solution as follows:
(3.27)ϕ=nc(τ,m).
As in the first case (i), similarly, we obtain a couple of periodic solutions of (1.1) as follows:
(3.28)u=εκ4(1-ε)[1+nc(τ,m)],x=(ε-2)τ+ε1-m2ln[1-m2sn(τ,m)+dn(τ,m)cn(τ,m)]+εκ4(1-ε)t,ρ=εκ4(1-ε)[1+2A+εnc(τ,m)(ε-2)-εnc(τ,m)],x=(ε-2)τ+ε1-m2ln[1-m2sn(τ,m)+dn(τ,m)cn(τ,m)]+εκ4(1-ε)t,
where m=21/(ε+4).(iv)  IfA=(ε/2-1)±2(4-ε2)(ε2-2ε+2)/4,m=(3ε-4)/(2ε-4),δ=±-6ε2+12ε,0<ε<1and1<ε<4/3, then the P=1-m2,Q=2-m2,R=1. In the case of these parametric conditions, (3.15) has a Jacobi elliptic function solution as follows:
(3.29)ϕ=sc(τ,m).
Similarly, we obtain a couple of periodic solutions of (1.1) as follows:
(3.30)u=εκ4(1-ε)[1+sc(τ,m)],x=(ε-2)τ+ε21-m2ln[dn(τ,m)+1-m2dn(τ,m)-1-m2]+εκ4(1-ε)t,ρ=εκ4(1-ε)[1+2A+εsc(τ,m)(ε-2)-εsc(τ,m)],x=(ε-2)τ+ε21-m2ln[dn(τ,m)+1-m2dn(τ,m)-1-m2]+εκ4(1-ε)t,
where m=(3ε-4)/(2ε-4).(v)  IfA=(-m4+4m2-3±m8-8m6+22m4-40m2+25)/(m2-3)2,0<m<1,δ=±43/(m2-3),ε=4/(3-m2), then the P=1,Q=2-m2,R=1-m2. In the case of these parametric conditions,  (3.15) has a Jacobi elliptic function solution as follows:
(3.31)ϕ=cs(τ,m).As in the first case (i) using the same method, we obtain a couple of periodic solutions of (1.1) as follows:
(3.32)u=-κ1+m2[1+cs(τ,m)],x=(43-m2-2)τ+43-m2ln[1-dn(τ,m)sn(τ,m)]-κ1+m2t,ρ=-κ1+m2[1+2(3-m2)A+4cs(τ,m)4-2(3-m2)-4cs(τ,m)],x=(43-m2-2)τ+43-m2ln[1-dn(τ,m)sn(τ,m)]-κ1+m2t.In order to show the dynamic properties of above periodic solutions intuitively, as examples, we plot their graphs of the solutions (3.20), (3.23) and (3.24), see Figures 5 and 6.The profiles of smooth periodic waves of solutions (3.20) for the given parameters:  ϵ=1,σ=-1,κ=-4,v=3,δ=5,t=1 and different ε-values: (a) ε=-0.2; (b) ε=-2.
(a)
Smooth periodic wave
(b)
Smooth periodic waveThe profiles of noncontinuous periodic waves of solutions (3.23) and (3.24) for the given parameters: (a) κ=-4,m=0.6,t=1; (b) κ=-2,m=0.6,t=1.
(a)
Periodic kink wave
(b)
Singular periodic waveFigures5(a) and 5(b) show two shapes of smooth and continuous periodic waves, all of them are nonsingular type. Figure 6(a) shows a shape of periodic kink wave. Figure 6(b) shows a shape of singular periodic wave. Both Figures 6(a) and 6(b) show two discontinuous periodic waves, all of them are singular type.From the above illustrations, we find that the waveforms of some solutions partly depend on wave parameters. Indeed, in 2006, Vakhnenko and Parkes’s work [43] successfully explained similar phenomena. In [43], the graphical interpretation of the solution for gDPE is presented. In this analysis, the 3D-spiral (whether one loop from a spiral or a half loop of a spiral) has the different projections that is the essence of the possible solutions. Of caurse, this approach also can be employed to the 2-component Camassa-Holm shallow water system (1.1). By using the Vakhnenko and Parkes’s theory, the phenomena which appeared in this work are easily understood, so we omit this analysis at here. However, it is necessary to say something about the cracked loop soliton (Figure 2(d)) and the singular periodic wave (Figure 6(b)), what are the 3D-curves with projections associated with two peculiar solutions (3.7) and (3.24) which are shown in Figure 2(d) and Figure 6(b)? In order to answer this question, by using the Vakhnenko and Parkes’s approach, as an example we give the 3D-curves of solution (3.24) and their projection curve in the xoρ-plane, which are shown in Figure 7. For convenience to distinguish them, we colour the 3D-curves red and colour their projection curve green. From Figure 7, we can see that the 3D-curves are not intersected, but their projection curve is intersected in xoρ-plane.Evolvement graphs of 3D-curves along with the time and their projection curve.
(a)
3D-curves of solution (3.24)
(b)
3D-curves and their projection


## 4. Conclusion

In this work, by using the integral bifurcation method together with a translation-dilation transformation, we have obtained some new traveling wave solutions of nonsingular type and singular type of 2-component Camassa-Holm equation. These new solutions include soliton solutions, kink wave solutions, loop soliton solutions, compacton solutions, smooth periodic wave solutions, periodic kink wave solution, singular wave solution, and singular periodic wave solution. By investigating these new exact solutions of parametric type, we found some new traveling wave phenomena, that is, the waveforms of some solutions partly depend on wave parameters. For example, the waveforms of solutions (3.6), (3.7), and (3.9) vary with the changes of parameter. These are three peculiar solutions. The solution (3.6) has five kinds of waveforms, which contain antikink wave, transmutative antikink wave, dark soliton, compacton, and loop soliton according as the parameter ε varies. The solution (3.7) has three kinds of waveforms, which contain kink wave, bright soliton, and singular wave (cracked loop soliton) according as the parameter ε varies. The solution (3.9) also has three kinds of waveforms, which contain bright soliton, compacton, and loop soliton according as the parameter ε varies. These phenomena show that the dynamic behavior of these waves partly depends on the relation of the amplitude of wave and the level of water.



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*Source: 736765-2012-12-25.xml*