Nonlinear Analysis 71 (2009) 6011–6018

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

New peakon, solitary wave and periodic wave solutions for the modifiedCamassa–Holm equationBin He ∗Center for Nonlinear Science Studies, College of Mathematics of Honghe University, Mengzi, Yunnan, 661100, PR China

a r t i c l e i n f o

Article history:Received 28 March 2009Accepted 11 May 2009

MSC:34C3734G2034K1874J30

PACS:02.30.Ik02.30.Jr05.10.-a05.45.Yv

Keywords:Modified Camassa–Holm equationPeakonSolitary wavePeriodic waveExact solution

a b s t r a c t

In this paper, an independent variable transformation is introduced to solve the modifiedCamassa–Holm equation using the bifurcation theory and the method of phase portraitanalysis. Some peakons, solitary waves and periodic waves are found and their exactparametric representations in explicit form and in implicit form are obtained.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Camassa and Holm [1] derived a shallow water wave equation

ut − 2kux − uxxt + 3uux = 2uxuxx + uuxxx, (1)

which is called Camassa–Holm (CH) equation. The alternative derivation of (1) as a shallow water equation provided in theRefs. [2–4]. For k = 0, Camassa and Holm [1] showed that Eq. (1) has peakons of the form u(x, t) = ce−|x−ct|. The peakonsreplicate a feature that is characteristic for the waves of great height-waves of largest amplitude that are exact solutions ofthe governing equations for water waves [5,6].The CH equation is a truly remarkable equation. Its solitary waves are solitons [7–10]. It is a completely integrable

Hamiltonian system [11,12]. Moreover, the CH equation models breaking waves [13,14]. It is a re-expression of geodesicflow and satisfies the Least Action Principle (see the discussions in [15–17]), and some authors have even argued recently(see [18]). The CH equation might be relevant to the modeling of tsunamis (see also the discussion in [19]).

∗ Tel.: +86 873 3699153; fax: +86 873 3698806.E-mail addresses: hebinmtc@163.com, hbhhu@yahoo.com.cn.

0362-546X/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2009.05.057

6012 B. He / Nonlinear Analysis 71 (2009) 6011–6018

In 2006, Wazwaz [20] suggested studying the following modified Camassa–Holm equation:

ut − uxxt + 3u2ux = 2uxuxx + uuxxx. (2)

Wazwaz [21] showed that Eq. (2) has the following solitary wave solution:

u(x, t) = −2 sech212(x− 2t). (3)

In Ref. [22], using the bifurcation method of planar systems and numerical simulation of differential equations, Liu andOuyang showed the following fact:For the wave speed c = 2, the solitary wave and the peakon coexist in Eq. (2). The solitary wave is given by (3) and the

peakon is expressed by

u(x, t) =2(

cosh( x2 − t)+√2 sinh | x2 − t|

)2 . (4)

Through some special phase orbits, Wang and Tang [23] showed that Eq. (2) has the following solitary wave and peakonsolutions:

u(x, t) =13

[1− 4 sech2

1√6

(x−

t3

)], (5)

and

u(x, t) =8

(√2+ |x− 3t|)2

− 1. (6)

Using the Homotopy perturbation method, Zhang et al. [24] considered Eq. (2). Yusufoglu [25] investigated Eq. (2) usingthe Exp-function method and obtained some solitary wave solutions. Rui et al. [26] obtained some exact travelling wavesolutions of Eq. (2) using the integral bifurcation method.In this paper, we introduce an independent variable transformation to study Eq. (2) using the bifurcation method of

planar systems [27,28]. Some new peakon, solitary wave and periodic wave solutions will be obtained.

2. Peakon, solitary wave and periodic wave solutions of Eq. (2)

Using the following independent variable transformation:

u(x, t) = λ+ φ(ξ), ξ = x− ct, (7)

where c (c > 0) is the wave speed, λ is a constant, and substituting (7) into (2), we have

− cφ′ + cφ′′′ + 3(φ + λ)2φ′ = 2φ′φ′′ + (φ + λ)φ′′′, (8)

where ‘‘′’’ is the derivative with respect to ξ .Integrating (8) once with respect to ξ and setting the constant of integration to −λ3, we have the following travelling

wave equation of (8)

φ3 + 3λφ2 + (3λ2 − c)φ = (φ + λ− c)φ′′ +12(φ′)2. (9)

Letting y = dφdξ , we get the following planar system

dφdξ= y,

dydξ=φ3 + 3λφ2 + (3λ2 − c)φ − 1

2y2

φ + λ− c. (10)

Using the transformation dξ = (φ + λ− c)dτ , it carries (10) into the Hamiltonian system

dφdτ= (φ + λ− c)y,

dydτ= φ3 + 3λφ2 + (3λ2 − c)φ −

12y2. (11)

Since both system (10) and (11) have the same first integral

(φ + λ− c)y2 −(12φ2 + 2λφ + (3λ2 − c)

)φ2 = h, (12)

then the two systems above have the same topological phase portraits except the line φ = c − λ.

B. He / Nonlinear Analysis 71 (2009) 6011–6018 6013

Fig. 1. The bifurcation phase portraits of system (11) when λ = − 13 c +13

√2c(3− c), 0 < c ≤ 3. (1-1) 0 < c < 1

3 . (1-2) c =13 . (1-3)

13 < c < 3.

(1-4) c = 3.

Fig. 2. The bifurcation phase portraits of system (11) when λ = −√c, c > 0. (2-1) 0 < c < 1. (2-2) c = 1. (2-3) 1 < c < 2. (2-4) c = 2. (2-5) c > 2.

Write∆1 = 4c − 3λ2, ∆2 = 2(c − λ)(λ2 + cλ+ c(c − 1)), φs = c − λ. Clearly, when∆1 > 0, system (11) has threeequilibrium point at O(0, 0), A1,2(φ1,2, 0) in φ-axis, where φ1,2 =

−3λ±√∆1

2 . When∆1 = 0, system (11) has two equilibriumpoints at O(0, 0), A12(φ12, 0) in φ-axis, where φ12 = − 3λ2 . When∆2 > 0, there exist two equilibrium points of system (11)in line φ = φs at S±(φs, Y±), Y± = ±

√∆2.

For case λ = − 13 (c −√2c(3− c)), 0 < c ≤ 3, one easily sees that system (11) has five equilibrium points ( 12 (c −

√2c(3− c))± 16

√3c(6+ c + 2

√2c(3− c)), 0), (0, 0), ( 13 (4c−

√2c(3− c)),± 19

√6c(2c(11c − 9)+ (3− c)

√2c(3− c)))

when 0 < c < 13 or

13 < c < 3, and equilibrium points (

12 (c −

√2c(3− c)) ± 1

6

√3c(6+ c + 2

√2c(3− c)), 0) are both

center points, while the latter three are saddle points. When c = 13 , system (11) has two equilibrium points (−1, 0), (0, 0),

and the (−1, 0) is a center point, the (0, 0) is a saddle point. When c = 3, system (11) has four equilibrium points(0, 0), (3, 0), (4,±4

√2) and the (0, 0) is a degenerate saddle point, the (3, 0) is a center point, while the latter two are

saddle points. We get the bifurcation phase portraits of system (11) when λ = − 13 (c−√2c(3− c)) and 0 < c ≤ 3 as Fig. 1.

For case λ = −√c, c > 0, one easily sees that system (11) has three equilibrium points (0, 0), (2

√c, 0), (

√c, 0)when

0 < c ≤ 1, and equilibrium point (0, 0) is a center point, while the latter two are saddle points. When c > 1, system (11)has five equilibrium points (0, 0), (2

√c, 0), (

√c, 0), (c +

√c,±

√2c2(c − 1)), and equilibrium points (0, 0), (2

√c, 0)

are both center points, while the latter three are saddle points. We get the bifurcation phase portraits of system (11) whenλ = −

√c and c > 0 as Fig. 2.

For case λ = − 12 (c −√c(4− 3c)), 0 < c ≤ 4

3 , one easily sees that system (11) has four equilibrium points (0, 0),( 34 (c −

√c(4− 3c)) ± 1

4

√2c(2+ 3c + 3

√c(4− 3c)), 0), ( 12 (3c −

√c(4− 3c)), 0) when 1 < c < 1

3 or13 < c <

43 , and

equilibrium point (0, 0) is a saddle point, equilibrium point ( 34 (c −√c(4− 3c)) − 1

4

√2c(2+ 3c + 3

√c(4− 3c)), 0) is a

center point, the ( 12 (3c −√c(4− 3c)), 0) is a complex equilibrium point. When c = 1

3 , system (11) has two equilibriumpoints (0, 0), (−1, 0), and equilibrium point (0, 0) is a saddle point, the (−1, 0) is a center point. When c = 4

3 , system (11)has two equilibrium points (0, 0), (2, 0), and equilibrium point (0, 0) is a degenerate saddle point. We get the bifurcationphase portraits of system (11) when λ = − 12 (c −

√c(4− 3c)) and 0 < c ≤ 4

3 as Fig. 3.

2.1. Peakon solutions

From Fig. 1(1-1), it is seen that there are two heteroclinic orbits connecting with equilibrium points (φs, Y±) and (0, 0)when 0 < c < 1

3 . Their expressions are

y = ±1√2φ√φ − φm, φm < φs ≤ φ < 0, (13)

where φs = 13 (4c −

√2c(3− c)), Y± = ± 19

√6c(2c(11c − 9)+ (3− c)

√2c(3− c)), φm = −

√2c(3− c).

6014 B. He / Nonlinear Analysis 71 (2009) 6011–6018

Fig. 3. The bifurcation phase portraits of system (11) when λ = − 12 c +12

√c(4− 3c), 0 < c ≤ 4

3 . (3-1) 0 < c <13 . (3-2) c =

13 . (3-3)

13 < c <

43 .

(3-4) c = 43 .

Substituting (13) into the dφdξ = y and integrating it along the heteroclinic orbits, yield equation∫ φ

φs

dss√s− φm

= −1√2|ξ |. (14)

Completing the integral and solving the equation for φ, it follows that

φ(ξ) = −φmφs

(√−φm cosh(Ωξ)+

√φs − φm sinh |Ωξ |)2

, (15)

whereΩ = 14

√−2φm.

Noting that u(x, t) = λ+ φ(ξ), ξ = x− ct , we get the peakon solution u(x, t) as follows:

u(x, t) = λ−φmφs

(√−φm cosh(Ω(x− ct))+

√φs − φm sinh |Ω(x− ct)|)2

, (16)

where λ = − 13 (c −√2c(3− c)), 0 < c < 1

3 .On the other hand, from Fig. 1(1-3), we see that there are two heteroclinic orbits connecting with equilibrium points

(φs, Y±), (0, 0) and at the same time, there is a homoclinic orbit connecting with equilibrium point (0, 0) and passing point(φm, 0)when 13 < c < 3. Their expressions respectively are

y = ±1√2φ√φ − φm, 0 < φ ≤ φs, (17)

y = ±1√2φ√φ − φm, φm ≤ φ < 0, (18)

where φs = 13 (4c −

√2c(3− c)), Y± = ± 19

√6c(2c(11c − 9)+ (3− c)

√2c(3− c)), φm = −

√2c(3− c).

Substituting (17) into the dφdξ = y, integrating it along the heteroclinic orbits and completing the integral, we can get thepeakon solution u(x, t) as follows:

u(x, t) = λ−φmφs

(√−φm cosh(Ω(x− ct))+

√φs − φm sinh |Ω(x− ct)|)2

, (19)

whereΩ = 14

√−2φm, λ = − 13 (c −

√2c(3− c)), 13 < c < 3.

2.2. Solitary wave solutions

Substituting (18) into the dφdξ = y and integrating it along the homoclinic orbit, we have∫ φ

φm

dss√s− φm

= −1√2|ξ |. (20)

Completing the integral and solving the equation for φ, it follows that

φ(ξ) = φm sech2(Ωξ), (21)

where φm = −√2c(3− c), Ω = 1

4

√−2φm.

B. He / Nonlinear Analysis 71 (2009) 6011–6018 6015

Using transformation (7), we get the following solitary wave solution:

u(x, t) = λ+ φm sech2(Ω(x− ct)), (22)

where λ = − 13 (c −√2c(3− c)), 13 < c < 3.

From Fig. 2(2-1), (2-2) and (2-3), they are seen that there is a homoclinic orbit connecting with equilibrium point (φ2, 0)and passing point (φm, 0)when 0 < c < 2. Its expression is

y = ±1√2

φ2 − φ

φs − φ

√(φM − φ)(φs − φ)(φ − φm), φm ≤ φ < φ2, (23)

where φ2 =√c, φs = c +

√c, φM,m = (1±

√2)√c.

Substituting (23) into the dφdξ = y and integrating it along the homoclinic orbit, yield equation∫ φ

φm

(φs − s)ds(φ2 − s)

√(φM − s)(φs − s)(s− φm)

=1√2|ξ |. (24)

Completing the integral and solving the equation for φ, it follows that

(α2 − 1)Π

(sin−1

(√φ − φm

φs − φm

), α2, k

)+ sn−1

(√φ − φm

φs − φm, k

)= Ω|ξ |, (25)

where α2 = φs−φmφ2−φm

, k =√

φs−φmφM−φm

, Ω =√24

√φM − φm, Π(·, ·, ·) is Legendre’s incomplete elliptic integral of the third

kind and sn−1(·, ·) is the inverse function of the Jacobian elliptic function sn(·, ·) (see [29]).Using transformation (7), we get the following solitary wave solution:

u(x, t) = λ+ φ,

(α2 − 1)Π

(sin−1

(√φ − φm

φs − φm

), α2, k

)+ sn−1

(√φ − φm

φs − φm, k

)= Ω|x− ct|,

(26)

where λ = −√c, 0 < c < 2.

From Fig. 2(2-5), it is seen that there are two homoclinic orbits connecting with equilibrium point (φ2, 0) and passingpoints (φm, 0), (φM , 0) respectively when c > 2. Their expressions respectively are

y = ±1√2

φ2 − φ

φs − φ

√(φs − φ)(φM − φ)(φ − φm), φm ≤ φ < φ2, (27)

y = ±1√2

φ2 − φ

φs − φ

√(φs − φ)(φM − φ)(φ − φm), φ2 < φ ≤ φM , (28)

where φ2 =√c, φs = c +

√c, φM,m = (1±

√2)√c.

Substituting (27) into the dφdξ = y and integrating it along the homoclinic orbit, yield equation∫ φ

φm

(φs − s)ds(φ2 − s)

√(φs − s)(φM − s)(s− φm)

=1√2|ξ |. (29)

Completing the integral and solving the equation for φ, it follows that

(α2 − α21)Π

(sin−1

(√φ − φm

φM − φm

), α2, k

)+ α21sn

−1

(√φ − φm

φM − φm, k

)= Ω|ξ |, (30)

where α2 = φM−φmφ2−φm

, α21 =φM−φmφs−φm

, k =√φM−φmφs−φm

, Ω =√2(φM−φm)4√φs−φm

.

Noting that u(x, t) = λ+ φ(ξ), ξ = x− ct , we get the solitary wave solution u(x, t) as follows:u(x, t) = λ+ φ,

(α2 − α21)Π

(sin−1

(√φ − φm

φM − φm

), α2, k

)+ α21sn

−1

(√φ − φm

φM − φm, k

)= Ω|x− ct|,

(31)

where λ = −√c, c > 2.

6016 B. He / Nonlinear Analysis 71 (2009) 6011–6018

Substituting (28) into the dφdξ = y and integrating it along the homoclinic orbit, yield equation∫ φM

φ

(φs − s)ds(φ2 − s)

√(φs − s)(φM − s)(s− φm)

= −1√2|ξ |. (32)

Completing the integral and solving the equation for φ, it follows that

α2Π

(sin−1

(√(φs − φm)(φM − φ)

(φM − φm)(φs − φ)

), α2, k

)= Ω|ξ |, (33)

where α2 = (φM−φm)(φs−φ2)(φs−φm)(φM−φ2)

, k =√φM−φmφs−φm

, Ω =√2(φM−φm)(φs−φ2)4(φs−φM )

√φs−φm

.

Noting that u(x, t) = λ+ φ(ξ), ξ = x− ct , we get the solitary wave solution u(x, t) as follows:u(x, t) = λ+ φ,

α2Π

(sin−1

(√(φs − φm)(φM − φ)

(φM − φm)(φs − φ)

), α2, k

)= Ω|x− ct|,

(34)

where λ = −√c, c > 2.

2.3. Periodic wave solutions

From Fig. 2(2-3), it is seen that there is one periodic orbit enclosing the center point (0, 0) and passing points (γ3, 0),(γ2, 0)when 1 < c < 2. Its expression is

y = ±1√2

√(γ1 − φ)(γ2 − φ)(φ − γ3), γ3 ≤ φ ≤ γ2, (35)

where γ1, γ2, γ3 (γ3 < 0 < γ2 < γ1) are three real roots of x3+ (c− 3√c)x2+ (c−

√c)2x+ (c+

√c)(c−

√c)2 = 0. For

examples:γ1 = 23 (√3+√2), γ2 = 2

3 (√3−√2), γ3 = 2

3 (√3−2)when c = 4

3 . γ1 =12 (√6+√3), γ2 = 1

2 (√6−√3), γ3 =

12 (√6− 3)when c = 3

2 .

Substituting (35) into the dφdξ = y and integrating it along the periodic orbit, yield equation∫ φ

γ3

ds√(γ1 − s)(γ2 − s)(s− γ3)

=1√2|ξ |. (36)

Completing the integral and solving the equation for φ, it follows that

φ(ξ) = γ3 + (γ2 − γ3)sn2(Ωξ, k), (37)

whereΩ =√24

√γ1 − γ3, k =

√γ2−γ3γ1−γ3

.

Using transformation (7), we get the following periodic wave solution:

u(x, t) = λ+ γ3 + (γ2 − γ3)sn2(Ω(x− ct), k), (38)

where λ = −√c, 1 < c < 2.

From Fig. 3(3-1), it is seen that there is one periodic orbit enclosing the center point (φ2, 0) and passing points (β1, 0),(β2, 0) and at the same time, there is one periodic orbit enclosing the center point (φ2, 0) and passing points (γ2, 0),(φs, 0)when 0 < c < 1

3 . Their expressions respectively are

y = ±φ√(φs − φ)(β1 − φ)(φ − β2)√2(φ − φs)

, β1 ≤ φ ≤ β1, (39)

y = ±1√2

√(γ1 − φ)(φs − φ)(φ − γ2), γ2 ≤ φ ≤ φs, (40)

whereβ1,2 = c−√c(4− 3c)±

√c(c +

√c(4− 3c)), γ1,2 = − 12 (c+

√c(4− 3c))±

√2c(1− c), φs = 1

2 (3c−√c(4− 3c)).

Substituting (39) into the dφdξ = y and integrating it along the periodic orbit, yield equation∫ β1

φ

(s− φs)dss√(φs − s)(β1 − s)(s− β2)

=1√2|ξ |. (41)

B. He / Nonlinear Analysis 71 (2009) 6011–6018 6017

Completing the integral and solving the equation for φ, it follows that

α2Π

(sin−1

(√(φs − β2)(β1 − φ)

(β1 − β2)(φs − φ)

), α2, k

)= Ω|ξ |, (42)

where α2 = (β1−β2)φs(φs−β2)β1

, k =√β1−β2φs−β2

, Ω =√2(β2−β1)φs

4(φs−β1)√φs−β2

.

Using transformation (7), we get the following periodic wave solution:u(x, t) = λ+ φ,

α2Π

(sin−1

(√(φs − β2)(β1 − φ)

(β1 − β2)(φs − φ)

), α2, k

)= Ω|x− ct|,

(43)

where λ = − 12 (c −√c(4− 3c)), 0 < c < 1

3 .

Substituting (40) into the dφdξ = y and integrating it along the periodic orbit, yield equation∫ φ

γ2

ds√(γ1 − s)(φs − s)(s− γ2)

=1√2|ξ |. (44)

Completing the integral and solving the equation for φ, it follows that

φ(ξ) = γ2 + (φs − γ2)sn2(Ωξ, k), (45)

where k =√φs−γ2γ1−γ2

, Ω =√24

√γ1 − γ2.

Using transformation (7), we get the following periodic wave solution:

u(x, t) = λ+ γ2 + (φs − γ2)sn2(Ω(x− ct), k), (46)

where λ = − 12 (c −√c(4− 3c)), 0 < c < 1

3 .

3. Conclusion

In this paper, using transformation (7) and the bifurcation theory and the method of phase portrait analysis, weinvestigated Eq. (2) and obtained some new peakon, solitary wave and periodic wave solutions. We show that when thewave speed c satisfies 13 < c < 3, the peakon and the solitary wave coexist in Eq. (2) (see (19), (22)). It is easy to see thatour method is a useful tool to explore the exact travelling wave solution for nonlinear partial differential equations.

Acknowledgements

The author would like to thank the referees for some perceptive comments and for some valuable suggestions. This workis supported by the Natural Science Foundation of YunNan Province, China (2007A080M).

References

[1] R. Camassa, D.D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993) 1661–1664.[2] R.S. Johnson, Camassa–Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech. 455 (2002) 63–82.[3] D. Ionescu-Kruse, Variational derivation of the Camassa–Holm shallow water equation, J. Nonlinear Math. Phys. 14 (2007) 303–312.[4] A. Constantin, D. Lannes, The hydrodynamical relevance of the Camassa–Holm andDegasperis–Procesi equations, Arch. Ration.Mech. Anal. 192 (2009)165–186.

[5] A. Constantin, The trajectories of particles in stokes waves, Invent. Math. 166 (2006) 523–535.[6] A. Constantin, J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc. 44 (2007) 423–431.[7] A. Parker, On the Camassa–Holm equation and a direct method of solution. II. Soliton solutions, Proc. R. Soc. London A 461 (2005) 3611–3632.[8] R.S. Johnson, On solutions of the Camassa–Holm equation, Proc. R. Soc. London A 459 (2003) 1687–1708.[9] R. Beals, D. Sattinger, J. Szmigielski, Multi-peakons and a theorem of Stieltjes, Inverse Problems 15 (1999) L1–L4.[10] A. Constantin, W. Strauss, Stability of the Camassa–Holm solitons, J. Nonlinear Sci. 12 (2002) 415–422.[11] A. Constantin, On the inverse spectral problem for the Camassa–Holm equation, J. Funct. Anal. 155 (1998) 352–363.[12] A. Constantin, V. Gerdjikov, R. Ivanov, Inverse scattering transform for the Camassa–Holm equation, Inverse Problems 22 (2006) 2197–2207.[13] A. Constantin, J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998) 229–243.[14] A. Constantin, Existence of permanent and breaking waves for a shallowwater equation: A geometric approach, Ann. Inst. Fourier 50 (2000) 321–362.[15] A. Constantin, B. Kolev, Geodesic flow on the diffeomorphism group of the circle, Comment. Math. Helv. 78 (2003) 787–804.[16] A. Constantin, T. Kappeler, B. Kolev, P. Topalov, On geodesic exponential maps of the Virasoro group, Ann. Global Anal. Geom. 31 (2007) 155–180.[17] B. Kolev, Poisson brackets in hydrodynamics, Discrete Contin. Dyn. Syst. 19 (2007) 555–574.[18] M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics, in: Tsunami and Nonlinear Waves, Springer,

Berlin, 2007, pp. 31–49.[19] A. Constantin, R.S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis, Fluid Dynam.

Res. 40 (2008) 175–211.[20] A.M. Wazwaz, Solitary wave solutions for modified forms of Degasperis–Procesi and Camassa–Holm equations, Phys. Lett. A 352 (2006) 500–504.

6018 B. He / Nonlinear Analysis 71 (2009) 6011–6018

[21] A.M. Wazwaz, New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa–Holm equations, Appl. Math. Comput. 186(2007) 130–141.

[22] Z.R. Liu, Z.Y. Ouyang, A note on solitary waves for modified forms of Camassa–Holm and Degasperis–Procesi equations, Phys. Lett. A 366 (2007)377–381.

[23] Q.D. Wang, M.Y. Tang, New exact solutions for two nonlinear equations, Phys. Lett. A 372 (2008) 2995–3000.[24] B.G. Zhang, S.Y. Li, Z.R. Liu, Homotopy perturbation method for modified Camassa–Holm and Degasperis–Procesi equations, Phys. Lett. A 372 (2008)

1867–1872.[25] E. Yusufoğlu, New solitonary solutions for modified forms of DP and CH equations using Exp-function method, Chaos Solitons Fractals (2007) doi:10.

1016/j.chaos.2007.07.009.[26] W.G. Rui, B. He, S.L. Xie, Y. Long, Application of the integral bifurcationmethod for solvingmodified Camassa–Holm andDegasperis–Procesi equations,

Nonlinear Anal. (2009) doi:10.1016/j.na.2009.02.026.[27] B. He, J.B. Li, Y. Long, W.G. Rui, Bifurcations of travelling wave solutions for a variant of Camassa–Holm equation, Nonlinear Anal. RWA 9 (2008)

222–232.[28] B. He, Y. Long, W.G. Rui, New exact bounded travelling wave solutions for the Zhiber–Shabat equation, Nonlinear Anal. (2009) doi:10.1016/j.na.2009.

01.029.[29] P.F. Byrd, M.D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag, 1971.