a Translation-Dilation Transformation for Solving an Integrable 2-Component Camassa-Holm

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Volume 2012, Article ID 736765,21pages doi:10.1155/2012/736765

Research Article

Integral Bifurcation Method together with

a Translation-Dilation Transformation for Solving an Integrable 2-Component Camassa-Holm

Shallow Water System

Weiguo Rui and Yao Long

Center for Nonlinear Science Research, College of Mathematics, Honghe University, Yunnan, Mengzi 661100, China

Correspondence should be addressed to Weiguo Rui,weiguorhhu@yahoo.com.cn Received 12 September 2012; Accepted 15 November 2012

Academic Editor: Michael Meylan

Copyrightq2012 W. Rui and Y. Long. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 system1as follows:

mtεmux 1

2εumxσ ε

ρ²

x0, ρtε 2

ρu

x0, 1.1

which is a nonlinear dispersive wave equation that models the propagation of unidirectional irrotational shallow water waves over a flat bed2, as well as water waves moving over

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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−1/3δ²uxx κ/2, σ ±1 and εa/, δ /λare two dimensionless parameters. The interpretation ofu, ρ, respectively, describes the horizontal fluid velocity and the density in the shallow water regime, where the variableux, tdescribes the horizontal velocity of the fluid inxdirection at timet, and the variable ρx, t is related to the free surface elevation from equilibrium positionor scalar densitywith the boundary assumptions. The parameter denotes the level of water, the parameteradenotes the typical amplitude of the water wave, and the parameterλdenotes the typical wavelength of the water wave. The constant κdenotes 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 currentκ0 and the parameterσ1,1.1 becomes the following form

mtεmu x1

2εumx 1 ε

ρ²

x0, ρt ε 2

ρu

x 0, 1.2

where m u−1/3δ²uxx. The system1.2appeared in 1, which was first derived by Constantin and Ivanov from the Green-Naghdi equations6,7via the hydrodynamical point of view. Under the scalingu→2/εu, x→δ/√

3x, t→δ/√

3t,1.1can be reduced to the following two-component generalization of the well-known 2-component Camassa-Holm 2-CHsystem1,8–10:

mt2muxumxσρρx0, ρt ρu

x0, 1.3

where m u − 1/3δ²uxx 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. Whenκ0,1.3has been studied by many authors, see11–26and references cited therein. Especially, under the parametric conditionsδ 1, ρ 0, σ −1, 1.3can be reduced to the celebrated Camassa-Holm equation27,28,

utκux−uxxt3uux2uxuxxuuxxx. 1.4 In 2006, the integrability of1.3for σ −1 was proved and some peakon and multikink solutions of this system were presented by Chen et al. in9. In 2008, the Lax pair of1.3 for any value ofσwas given by Constantin and Ivanov in1. In29, by using the method of dynamical systems, under the traveling wave transformationux, t φx−ct, ρx, t ψx−ct, some explicit parametric representations of exact traveling wave solutions of1.3 were obtained. But the loop solitons were not obtained in29. In30, underσ −1, one- loop and two-loop soliton solutions and multisoliton solutions of1.3are obtained by using the Darboux transformations.

Although1.1can be reduced to1.3by the scaling transformationu→2/εu, x→ δ/√

3x, t → δ/√

3t, 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 system1.1partly depends on the relationεa/ of the amplitude of wave and the level deepnessof water. In the other words, their dynamic behavior vary with the changes of

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parameterε, that is, the changes of ratio for the amplitude of wave aand the deepness of water. In addition, compared with the research results of1.3, the research results for1.1 are few in the existing literatures. Thus,1.1is 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 as9,12–17,29. On the other hand, under different transformations, by using different methods, different results will be presented. In this paper, by using the integral bifurcation method31, we will investigate different kinds of new traveling wave solutions of1.1and their dynamic properties under the translation- dilation transformation ux, t v1φx−vt, ρx, t v1ψx−vt. By the way, the integral bifurcation method possessed some advantages of the bifurcation theory of the planar dynamic system32and auxiliary equation methodsee33,34and references cited therein, it is easily combined with computer method35and useful for many nonlinear partial diffential equations PDEsincluding some PDEs with high power terms, such as Km, nequation36. So, by using this method, we will obtain some new traveling wave solutions of1.1. 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,1.1can be rewritten as the following form

ut1

2εκux 3

2εuux−1

3δ²uxxt2σ

ε ρρx 1

6εδ²2uxuxxuuxxx, ρt ε 2

ρxuρux

0.

2.1 In order to change the PDE 2.1 into an ordinary diﬀerential equation, we make a transformation

ux, t v

1φξ

, ρx, t v

1ψξ

, 2.2

whereξx−vtandvis an arbitrary nonzero constant. In fact,2.2is a translation-dilation transformation, it has been used extensively in many literatures. Its idea came from many existing references. For example, in37, the expressionu λ^1/m−lUλ^m−n/2m−lxλt is a dilation transformation. In38, the expressionUx, t ux−ωt ω is a translation transformation. In39,40, the expressionz u−v/|v|which was first used by Parkes and Vakhnenko is also a translation-dilation transformation.

After substituting2.2into2.1, integrating them once yields 3ε

2 εκ 2v−1

φ3ε

4 φ² 1

3δ²φ 2σ ε ψσ

εψ² εδ² 6

1 2

φ2φφφ

,

1− ε 2

φ1 ψ ε

2φA,

2.3

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where the integral constantA /0 andφdenotesφξ. From the second equation of2.3, we easily obtain

ψ 2Aεφ

2−ε−εφ, 2.4

whereε /0. Substituting2.4into the first equation of2.3, we obtain

φ

3ε/2εκ/2v−1φ3ε/4φ²−

εδ²/12 φ₂

A²−2σ/ε

2Aεφ

Aσ/ε

2Aεφ₂ 1/6δ²A³ ,

2.5 whereAdenotesε−2εφ.

Letφ dφ/dξ y. Thus 2.5 can be reduced to 2-dimensional planar system as follows:

dφ dξ y, dy

dξ

3ε/2εκ/2v−1φ3ε/4φ²

A²−2σ/ε

2Aεφ

Aσ/ε

2Aεφ₂

−

εδ²/12 A²y² 1/6δ²A³ ,

2.6 whereAdenotesε−2εφ.

Making a transformation

dξ

ε−2εφ

dτ, 2.7

2.6becomes

dφ dτ

ε−2εφ y, dy

dτ 6 δ²

3ε 2 εκ

2v−1

φ3ε 4 φ²

6σ

2Aεφ

2A4−2ε−εφ εδ²

ε−2εφ₂ − ε 2y²,

2.8

whereτ is a parameter. From the point of view of the geometric theory, the parameterτ is a fast variable, but the parameterξ is a slow one. The system2.8is still a singular system through the system2.6becomes2.8by the transformation2.7. This case is not same as that in29. Of course, as in29, we can also change the system2.6into a regular system under another transformationdξ ε−2εφ³dτ. But we do not want to do that due to the following two reasons:iwe need to keep the singularity of the original system;iiwe need not to make any analysis of phase portraits as in29.

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Obviously, systems2.6and2.8have the same first integral as follows:

y² 1

ε−2εφ₂

× 2ε²

δ²φ⁴3ε4εv−4vεκ vδ² φ³ 3

ε²κ3ε²v4v−4σv−8εv−2εκ

vδ² φ²

ε²δ²h−12εσ24σ

εδ² φ ε−2h− 48σA²96σA12σε²48σ−48σε−48σεA ε²δ²

, 2.9

wherehis 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 for1.1 and their dynamic behaviors.

It is easy to know that2.9can be reduced to four kinds of simple equations when the parametric conditions satisfy the following four cases:

Case 1. Aε−2, h12σε−2/ε²δ²andε /0,2, v /εκ/41−ε.

Case 2. vεκ/41−ε, h12σε−2/ε²δ²andε /0,1, A /ε−2.

Case 3. Aε−2, h12σε−2/ε²δ², vεκ/41−εandε /0,1,2.

Case 4. h12σ2A2−ε²/ε²δ²ε−2.

We mainly aim to consider the new results of1.1, 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 conditions A ε− 2, h 12σε− 2/ε²δ², v /εκ/41 − ε and ε /0 or 2, σ±1,2.9can be reduced to

y±

cφ⁴bφ³aφ²

ε−2εφ , 3.1

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wherec3ε²/δ²≥0, b 3ε/δ²4ε−1εκ/v, a 3/δ²41−σε3ε−8εκε−2/v.

Substituting3.1into the first expression in2.8yields dφ

dτ ±

cφ⁴bφ³aφ². 3.2

Write

±1 andΔ 9ε²

4v²ε²4vεκε²κ²16σv²

v²δ⁴ . 3.3

TheΔ 0 if only ifκ2vε±2√

−σ/ε. By using the exact solutions of3.2, we can obtain diﬀerent kinds of exact traveling wave solutions of parametric type of1.1, see the following discussion.

iIfa >0, then3.2has one exact solution as follows:

φ −ab sech²√ a/2

τ b²−ac

1tanh√ a/2

τ₂, a >0. 3.4

Substituting3.4into2.7, and then integrating it yields

ξ ε−2τ− 2ε √

ctanh⁻¹

√ac

b 1tanh

√a 2 τ

. 3.5

Substituting3.4into2.2and2.4, then combining with3.5, we obtain a couple of soliton-like solutions of1.1as follows:

uv

1− absech²√ a/2

τ b²−ac

1tanh√ a/2

τ2

,

ξ ε−2τ− 2ε √

ctanh⁻¹

√ac

b 1 tanh

√a 2 τ

,

3.6

ρv

⎡

⎢⎣12ε−2

b²−ac

1tanh√ a/2

τ2

−abε sech²√ a/2

τ 2−ε

b²−ac

1tanh√ a/2

τ₂

abεsech²√ a/2

τ

⎤

⎥⎦,

ξ ε−2τ− 2ε √

ctanh⁻¹ √

ac

b 1tanh

√a 2 τ

.

3.7

iiIfa >0, c >0, Δ>0, then3.2has one exact solution as follows:

φ 2asech√ aτ √

Δ−bsech√

aτ. 3.8

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Similarly, by using3.8,2.7,2.2, and2.4, we obtain two couples of soliton-like solutions of1.1as follows:

uv

1 2asech√ aτ √

Δ−bsech√ aτ

,

ξ ε−2τ− 2ε

√ctanh⁻¹

b√ Δ 2√

ac tanh

√a 2 τ

,

ρv

⎡

⎢⎣12ε−2 √

Δ−bsech√ aτ

2aεsech√ aτ 2−ε

√

Δ−bsech√ aτ

−2aεsech√ aτ

⎤

⎥⎦,

ξ ε−2τ−√2ε ctanh⁻¹

b√ Δ 2√

ac tanh

√a 2 τ

.

3.9

iiiIfa >0 andΔ<0, then1.1has one couple of soliton-like solutions as follows:

uv

1 2acsch√ aτ √

−Δ−bcsch√ aτ

,

ξ ε−2τ−√2ε ctanh⁻¹

btanh√ a/2

τ √

−Δ 2√

ac

,

ρv

⎡

⎢⎣12ε−2 √

2aεcsch√ aτ 2−ε

√

−2aεcsch√ aτ

⎤

⎥⎦,

ξ ε−2τ− 2ε

√ctanh⁻¹

btanh√ a/2

τ √

−Δ 2√

ac

.

3.10

ivIfa > 0 andc /0i.e.,ε /0, then1.1has one couple of soliton-like solutions as follows:

uv

1− asech²√ a/2

τ b2√

actanh√ a/2

τ

,

ξ ε−2τ− ε √

cln b2√

actanh

√a 2 τ

, ρv

1 2ε−2

b2√

actanh√ a/2

τ

−aεsech²√ a/2

τ 2−ε

b2√

actanh√ a/2

τ

aεsech²√ a/2

τ

,

ξ ε−2τ− ε √

cln b2√

actanh

√a 2 τ

.

3.11

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vIfa > 0 andΔ 0i.e.κ v/ε4−4v2vε−4ε±2

vv−1ε−2²−4σ, then1.1has one couple of kink and antikink wave solutions as follows:

uv

1−a

b 1tanh

√a 2 τ

,

ξ ε−2τ−ε a

bτ2√ a b ln

cosh

√a 2 τ

,

ρv

1 2bε−2−aε

1tanh√ a/2

τ b2−ε aε

1tanh√ a/2

τ

,

ξ ε−2τ−ε a

bτ2√ a b ln

cosh

√a 2 τ

,

uv

1−a

b 1coth

√a 2 τ

,

ξ ε−2τ−ε a

bτ2√ a b ln

sinh

√a 2 τ

,

ρv

12bε−2−aε

1coth√ a/2

τ b2−ε aε

1coth√ a/2

τ

,

ξ ε−2τ−ε a

bτ2√ a b ln

sinh

√a 2 τ

.

3.12

viIfa > 0 andb 0i.e.κ 4v1/ε−1, then1.1has one couple of soliton-like solutions as follows:

uv

1− a

ccsch √

aτ

, ξ ε−2τ− ε √

cln

tanh 1 2√

aτ

,

ρv

12ε−2√ c−ε√

acsch √

aτ 2−ε√

cε√ acsch

√ aτ

, ξ ε−2τ− ε √

cln

tanh 1 2√

aτ

. 3.13

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 Figures1, 2,3, and4. Figures1a–1hshow the profiles of multiwaveform to solution the first solution of 3.6 for fixed parameters 1, σ −1, κ 4, v 4, δ 5 and diﬀerentε-values.

Figures2a–2dshow the profiles of multiwaveform to solution3.7for fixed parameters 1, σ −1, κ 4, v 3, δ 5 and diﬀerentε-values. Figures 3a–3d show the profiles of multiwaveform to solution3.9for fixed parameters−1, σ−1, κ−2, v

−4, δ3.5 and diﬀerentε-values. Figures4a-4bshow the profiles of kink wave solution the first formula of3.12for fixed parameters1, σ−1, v0.4, δ3.5 and diﬀerent ε, κ-values.

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−200

−150

−100

−50 0

−20 −10 0

ξ u

20 10

a Antikink wave

−50

−40

−30

−20

−10 0

−20 −10 0 10

ξ u

bTransmutative wave of antikink waveform

−6

−4

−2 0 2 4

−30 −20 −10 ξ u

20 10 0 c Dark soliton of thin waveform

3.2 3.4 3.6 3.8 4

−5 −4 −3 −2 −1 0

ξ u

dDark soliton of fat waveform

3.2 3.4 3.6 3.8 4

0 ξ

u

−4 −3 −2 −1

e Compacton

3.5 3.6 3.7 3.8 3.9 4

−4 −3.5 −3 −2.5 −2 −1.5 −1 ξ

u

fLoop soliton of fat waveform Figure 1: Continued.

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3.86 3.88 3.9 3.92 3.94 3.96 3.98 4

−4 −3.5 −3 −2.5 −2 −1.5 −1 ξ

u

gLoop soliton of thin waveform

3.982 3.984 3.986 3.988

−2.35 −2.3 −2.25 −2.2 −2.15 ξ

u

h Loop soliton of oblique waveform

Figure 1: The profiles of multiwaveform of3.6for the given parameters and diﬀerentε-values:aε1.2;

bε1.26;cε1.4;dε1.99;eε2;fε2.2;gε3;hε5.

We observe that some profiles of above soliton-like solutions are very much sensitive to one of parameters, that is, their profiles are transformablesee 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 in41,42. In41, the waveforms of soliton-like solution of the generalized KdV equation vary with the changes of parameter and depend on the velocitycextremely. Similarly, in42, the properties of some traveling wave solutions of the generalized KdV-Burges equation depend on the dissipation coeﬃcientα; if dissipation coeﬃcientα ≥λ1, it appears as a monotonically kink profile solitary wave; if 0< α ≤ λ1, it appears as a damped oscillatory wave.

From Figures 1a–1h, it is easy to know that the profiles of solution 3.6 vary gradually, its properties depend on the parameterε. When parametric values ofεincrease from 1.2 to 5, the solution3.6has eight kinds of waveforms: Figure1ashows a shape of antikink wave whenε1.2; Figure1bshows a shape of transmutative antikink wave when ε 1.6; Figure1cshows a shape of thin and dark solitary wave whenε1.4; Figure1d shows a shape of fat and dark solitary wave when ε 1.99; Figure1eshows a shape of compacton wave whenε 2; Figure 1fshows a shape of fat loop soliton whenε 2.2;

the Figure1gshows a shape of thin loop soliton whenε3; Figure1hshows a shape of oblique loop soliton whenε5.

Similarly, from Figures2a–2d, 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: Figure2a shows a shape of smooth kink wave whenε 1.2;

Figure2bshows a shape of fat and bright solitary wave whenε 1.20000001; Figure2c shows a shape of thin and bright solitary wave whenε 1.22; Figure 2dshows a shape of singular wave of cracked loop soliton whenε2.4. Especially, the changes of waveforms from Figure2ato Figure2band from Figure2cto Figure2dhappened abruptly.

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−3

−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−80 −60 −40 −20 0 ξ u

20 aKink wave

−3

−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−1.2

−100−80 −60 −40 −20 ξ u

0 20 40 60 b Bright soliton of fat waveform

−3

−2.8

−2.6

−2.4

−2.2

−2

−1.8

−1.6

−1.4

−40 −20 0 20 40

ξ u

c Bright soliton of thin waveform

−10

−8

−6

−4

−2 0 2 4 6 8 10

−4 −3 −2 −1 0

ξ u

dSingular wave of cracked loop soliton

Figure 2: The profiles of multiwaveform of3.7for the given parameters and diﬀerentε-values:aε1.2;

bε1.20000001;cε1.22;dε2.4.

As in Figure1, the profiles of the first solution of3.9in Figure3vary gradually. When parametric values ofεincrease 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 solutions3.6,3.7and3.9, the property of the first solution of3.12is stable. The profile of the first solution of3.12is not transformable no matter how the parameters vary. Both Figures4aand 4bshow a shape of smooth kink wave.

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−4

−3.5

−3

−2.5

−2

−6 −4 −2 0 2 4 6

ξ u

aBright soliton

−4

−3.5

−3

−2.5

−2

−2 −1 0 1 2

ξ u

bBright compacton

−4

−3.5

−3

−2.5

−1.5 −1 −0.5 0 0.5 1 1.5 ξ

u

c Loop soliton of fat waveform

−4

−3.5

−3

−2.5

−2

−2 −1 0 1 2

ξ u

d Loop soliton of thin waveform

Figure 3: The profiles of multiwaveform of the first solution of3.9for the given parameters and diﬀerent ε-values:aε1.8;bε2;cε2.2;dε2.8.

3.2. The Traveling Wave Solutions Under Case2

Under the parametric conditionsv εκ/41−ε, h 12σε−2/ε²δ²andε /0 or 1,2.9 becomes

y±

P φ⁴Qφ²R

ε−2εφ , 3.14

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0.4 0.5 0.6 0.7 0.8

−150 −100 −50 50 0 100 150 ξ

u

aSmooth kink wave

0.4 0.5 0.6 0.7 0.8

−40 −30 −20 −10 0 10 20 ξ

u

b Smooth kink wave

Figure 4: The profiles of kink wave solutionthe first solution of3.12for the given parameters and diﬀerentε, κ-values:aε 0.2,κ 7.2,t 0,n ∈−30,30;bε 1.22,κ 0.5333333334,t 0.2, n∈−110,−50.

whereP 3ε²/δ²>0, Q −3/δ²4σ1εε−4, R48σAε−2−A/ε²δ². Substituting 3.14into the first expression in2.8yields

dφ dτ ±

P φ⁴Qφ²R. 3.15

We know that the caseσ −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 caseσ 1 can be similarly discussed, but we omit them here. Especially, when σ −1, the above values of P, Q, Rcan be reduced to P 3ε²/δ² > 0, Q −3εε−4/δ², R 48AA2−ε/ε²δ².

iIfA −1ε/2±

1−ε4ε²/2, m√

−ε/2, δ±2√

−3ε, −4< ε <0, then theP m², Q −1m², R 1. Under these parametric conditions,3.15has a Jacobi elliptic function solution as follows:

φsnτ, m 3.16

or

φcdτ, m, 3.17

wherem√

−ε/2 is the model of Jacobi elliptic function and 0< m <1.

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Substituting3.16and3.17into2.7, respectively, we obtain

ξ ε−2τ−√2ε

−εcosh⁻¹ 2dn

τ,√

−ε/2

√4ε

, 3.18

ξ ε−2τ√2ε

−εln

1√

−ε/2 sn

τ,√

−ε/2 dn

τ,√

−ε/2

. 3.19

Substituting 3.16 into2.2 and 2.4, using 3.18 and the transformationξ x−vt x−εκ/41−εt, we obtain a couple of periodic solutions of1.1as follows:

u εκ

41−ε

1sn τ,√

−ε/2 ,

x ε−2τ−√2ε

−εcosh⁻¹ 2dn

τ,√

−ε/2

√4ε

εκ 41−εt,

ρ εκ 41−ε

1 2Aεsn τ,√

−ε/2 ε−2−εsn

τ,√

−ε/2

,

x ε−2τ− 2ε

√−εcosh⁻¹ 2dn

τ,√

−ε/2

√4ε

εκ 41−εt.

3.20

Substituting 3.17 into2.2 and 2.4, using 3.19 and the transformationξ x−vt x−εκ/41−εt, we also obtain a couple of periodic solutions of1.1as follows:

u εκ

41−ε1cdτ, m, x ε−2τ ε

mln

1msnτ, m dnτ, m

εκ

41−εt, ρ εκ

41−ε

1 2Aεcdτ, m ε−2−εcdτ, m

,

x ε−2τ ε mln

1msnτ, m dnτ, m

εκ

41−εt.

3.21

iiIfA−m²−2m±√

m⁶4m⁴4m²16/m³, 0< m <1, δ±4√

3/m, ε−4/m², then theP 1, Q−1m², Rm². In the case of these parametric conditions,3.15has a Jacobi elliptic function solution as follows:

φnsτ, m or φdcτ, m. 3.22

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As in the first caseiusing the same method, we obtain a couple of periodic solutions of1.1as follows:

u− κ

m²41nsτ, m, x − 4

m² −2

τ− 4 m²ln

snτ, m

cnτ, m dnτ, m

− κ m²4t,

3.23

ρ− κ m²4

1 2m²A−4nsτ, m

−42m² 4nsτ, m

,

x − 4 m² −2

τ− 4

m²ln

snτ, m

cnτ, m dnτ, m

− κ m²4t

3.24

or

u− κ

m²41dcτ, m, x − 4

m² −2

τ− 4 m²ln

1snτ, m cnτ, m

− κ m²4t, ρ− κ

m²4

1 2m²A−4dcτ, m

−42m² 4dcτ, m

,

x − 4 m² −2

τ− 4

m²ln

1snτ, m cnτ, m

− κ m²4t.

3.25

Especially, whenm → 1 i.e.,A 2 or −8, δ ±4√

2, ε −4, v −κ/5, nsτ, m → cothτ, snτ, m → tanhτ, cnτ, m → sechτ, dnτ, m → sechτ. From3.23and 3.24, we obtain a couple of kink-like solutions of1.1as follows:

u−κ

51cothτ, x−6τ−4 ln 1

2sinhτ

−κ 5t, ρ−κ

5

1 A−2 cothτ

−32 cothτ

, x−6τ−4 ln 1

2sinhτ

−κ 5t.

3.26

iiiIfA ε/2−1±

1−εε²4/2, m2

1/ε4, δ±√

3ε²12ε, 0< ε≤1, then theP 1−m², Q2m²−1, R−m². In the case of these parametric conditions,3.15 has a Jacobi elliptic function solution as follows:

φncτ, m. 3.27

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As in the first casei, similarly, we obtain a couple of periodic solutions of1.1as follows:

u εκ

41−ε1ncτ, m, x ε−2τ ε

√1−m²ln √

1−m²snτ, m dnτ, m cnτ, m

εκ

41−εt, ρ εκ

41−ε

1 2Aεncτ, m ε−2−εncτ, m

,

x ε−2τ ε

√1−m²ln √

1−m²snτ, m dnτ, m cnτ, m

εκ

41−εt,

3.28

wherem2

1/ε4.

ivIf A ε/2 − 1 ±

24−ε²ε²−2ε2/4, m

3ε−4/2ε−4, δ

±√

−6ε²12ε, 0< ε <1 and 1< ε <4/3, then theP 1−m², Q2−m², R1. In the case of these parametric conditions,3.15has a Jacobi elliptic function solution as follows:

φscτ, m. 3.29

Similarly, we obtain a couple of periodic solutions of1.1as follows:

u εκ

41−ε1scτ, m,

x ε−2τ ε 2√

1−m²ln

dnτ, m √ 1−m² dnτ, m−√

1−m²

εκ 41−εt,

ρ εκ

41−ε

1 2Aεscτ, m ε−2−εscτ, m

,

x ε−2τ ε 2√

1−m²ln

dnτ, m √ 1−m² dnτ, m−√

1−m²

εκ 41−εt,

3.30

wherem

3ε−4/2ε−4.

vIfA −m⁴4m²−3±√

m⁸−8m⁶22m⁴−40m²25/m²−3², 0< m <1, δ

±4√

3/m²−3, ε4/3−m², then theP 1, Q2−m², R1−m². In the case of these parametric conditions,3.15has a Jacobi elliptic function solution as follows:

φcsτ, m. 3.31

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As in the first caseiusing the same method, we obtain a couple of periodic solutions of1.1as follows:

u− κ

1m²1csτ, m,

x 4

3−m² −2

τ 4 3−m²ln

1−dnτ, m snτ, m

− κ 1m²t, ρ− κ

1m²

1 2 3−m²

A4csτ, m 4−23−m²−4csτ, m

,

x 4

3−m² −2

τ 4 3−m²ln

1−dnτ, m snτ, m

− κ 1m²t.

3.32

In order to show the dynamic properties of above periodic solutions intuitively, as examples, we plot their graphs of the solutions3.20,3.23and3.24, see Figures5and6.

Figures5aand5bshow two shapes of smooth and continuous periodic waves, all of them are nonsingular type. Figure6ashows a shape of periodic kink wave. Figure6b shows a shape of singular periodic wave. Both Figures6aand6bshow 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 work43successfully explained similar phenomena. In43, the graphical interpretation of the solution for gDPE is presented. In this analysis, the 3D-spiralwhether one loop from a spiral or a half loop of a spiralhas 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 solitonFigure2dand the singular periodic wave Figure6b, what are the 3D-curves with projections associated with two peculiar solutions 3.7and 3.24 which are shown in Figure 2d and Figure6b? In order to answer this question, by using the Vakhnenko and Parkes’s approach, as an example we give the 3D- curves of solution3.24 and their projection curve in the xoρ-plane, which are shown in Figure7. For convenience to distinguish them, we colour the 3D-curves red and colour their projection curve green. From Figure7, we can see that the 3D-curves are not intersected, but their projection curve is intersected inxoρ-plane.

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 solutions3.6,3.7, and3.9

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x u

0 0.05 0.1 0.15 0.2 0.25 0.3

−30 −20 −10 0 10 20 30 a Smooth periodic wave

x ρ

1.5 2 2.5 3 3.5 4

−40 −20 0 20 40 60

b Smooth periodic wave

Figure 5: The profiles of smooth periodic waves of solutions3.20for the given parameters:1,σ−1, κ−4,v3,δ5,t1 and diﬀerentε-values:aε−0.2;bε−2.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

−300 −200 −100 0 100 200 300 x

u

aPeriodic kink wave

−20

−10 0 10 20 30 40

−200 −100 0 100 200 300 x

ρ

b Singular periodic wave

Figure 6: The profiles of noncontinuous periodic waves of solutions 3.23 and 3.24 for the given parameters:aκ−4,m0.6,t1;bκ−2,m0.6,t1.

vary with the changes of parameter. These are three peculiar solutions. The solution3.6has five kinds of waveforms, which contain antikink wave, transmutative antikink wave, dark soliton, compacton, and loop soliton according as the parameterεvaries. The solution3.7 has three kinds of waveforms, which contain kink wave, bright soliton, and singular wave cracked loop solitonaccording as the parameterεvaries. The solution3.9also has three

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t

x ρ

50 0

−50

−100

0 100

−200

−100 0

100 200

a 3D-curves of solution3.24

50 0

−50

−40−20 0 20

−200

−100 0

100 200

t ρ x

b 3D-curves and their projection Figure 7: Evolvement graphs of 3D-curves along with the time and their projection curve.

kinds of waveforms, which contain bright soliton, compacton, and loop soliton according as the parameterεvaries. 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.

Acknowledgments

The authors thank reviewers very much for their useful comments and helpful suggestions.

This work was financially supported by the Natural Science Foundation of ChinaGrant no.

11161020and was also supported by the Natural Science Foundation of Yunnan Province no. 2011FZ193.

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a Translation-Dilation Transformation for Solving an Integrable 2-Component Camassa-Holm

Research Article

Integral Bifurcation Method together with

a Translation-Dilation Transformation for Solving an Integrable 2-Component Camassa-Holm

Shallow Water System

Weiguo Rui and Yao Long

1. Introduction

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

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

4. Conclusion

Acknowledgments

References

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