A generalized auxiliary equation method and its application to nonlinear Klein–Gordon and generalized nonlinear Camassa–Holm equations

Physics Letters A 372 (2008) 1048–1060 www.elsevier.com/locate/pla

A generalized auxiliary equation method and its application to nonlinear Klein–Gordon and generalized nonlinear Camassa–Holm equations Emmanuel Yomba a,b a School of Mathematics, University of Minnesota, 127 Vincent Hall, Minneapolis, MN 55455, USA b Department of Physics, Faculty of Sciences, University of Ngaoundéré, PO Box 454, Ngaoundéré, Cameroon

Received 2 July 2007; accepted 5 September 2007 Available online 8 September 2007 Communicated by R. Wu

Abstract With the aid of symbolic computation, a generalized auxiliary equation method is proposed to construct more general exact solutions to two types of NLPDEs. First, we present new family of solutions to a nonlinear Klein–Gordon equation, by using this auxiliary equation method including a new first-order nonlinear ODE with six-degree nonlinear term proposed by Sirendaoreji. Then, we apply an indirect F-function method very close to the F-expansion method to solve the generalized Camassa–Holm equation with fully nonlinear dispersion and fully nonlinear convection C(l, n, p). Taking advantage of the new first-order nonlinear ODE with six degree nonlinear term, this indirect F-function method is used to map the solutions of C(l, n, p) equations to those of that nonlinear ODE. As a result, we can successfully obtain in a unified way, many exact solutions. © 2007 Published by Elsevier B.V.

1. Introduction In the recent years, seeking exact solutions of nonlinear partial differential equations (NLPDEs) is of great significance, since the nonlinear complex physical phenomena related to the NLPDEs are involved in many fields from physics, mechanics, biology, chemistry and engineering. As mathematical models of the phenomena, the investigation of exact solutions of NLPDEs will help one to understand the mechanism that governs these physical models or to better provide knowledge of the physical problem and possible applications. To this aim, a vast variety of powerful and direct methods for finding the exact significant solutions of NLPDEs though it is rather difficult have been derived. Some of the most important methods are Bäcklund transformation [1–3], Hirota’s method [4], tanh-function method [5–8], extended tanh-function method [9–12], homogeneous balance method [13,14], variational iteration methods [15,16], collocation method [17–19], Adomian Padé approximation [20], Jacobi elliptic function expansion method [21], F-expansion method [22], auxiliary equation method [23–25], Fan sub-equation method [26–30], extended Fan sub-equation method [31,32], modified extended Fan sub-equation method [33–35] and so on. Recently, Sirendaoreji [36,37] proposed a new auxiliary equation method by introducing a new first order nonlinear ordinary differential equation (NLODE) with six-degree nonlinear terms and its solutions to construct exact travelling wave solutions of NLPDEs in a unified way. Later, Zhang and Xia [38] improved this method and obtained new formal solutions of some NLPDEs. The aim of this Letter, is to use the generalized auxiliary equation method to solve two different types of NLPDEs such as nonlinear Klein–Gordon equation [39], the generalized Camassa–Holm equations with fully nonlinear dispersion and fully nonlinear convection term (called C(l, n, p) equation) [40]. E-mail address: [email protected]. 0375-9601/$ – see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.physleta.2007.09.003

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The rest of this Letter is organized as follows: in Section 2, we give a description of this new auxiliary equation method; in Section 3, we apply this method to the nonlinear Klein–Gordon equation, the generalized Camassa–Holm equations C(l, n, p); in Section 4, some conclusions are given. 2. A generalized auxiliary equation method To illustrate the basic concepts of a generalized auxiliary equation method, we consider a given PDE in two independent variables (x, t) and dependent variable u: Step 1. We use the wave transformation u(x, t) = u(ξ ), ξ = x − ct or ξ = x + ct, and reduce the given NLPDE H (u, ux , ut , uxx , . . .) = 0

(1)

to the following ODE G(u, uξ , uξ ξ , . . .) = 0.

(2)

Step 2. We seek for the solution of Eq. (2) in the following generalized ansätze [37] u(ξ ) =

2M 

ai F i (ξ ),

(3)

i=0

with F (ξ ) satisfying the following new auxiliary equation F  2 (ξ ) = h0 + h1 F (ξ ) + h2 F 2 (ξ ) + h3 F 3 (ξ ) + h4 F 4 (ξ ) + h5 F 5 (ξ ) + h6 F 6 (ξ ),

(4)

F

= dF where dξ , the positive integer M can be terms [37], ai (where i = 0, . . . , 2M) and c are

determined by balancing the highest-order derivative term with the nonlinear constants to be determined and F (ξ ) satisfies the variable separated ODE (4), where hj (j = 0, 1, . . . , 6) are real constants. We have found that the auxiliary equation (4) possesses several types of solutions depending of the value of the hj (j = 0, 1, . . . , 6). Step 3. By choosing the different values of hj (j = 0, 1, . . . , 6), Eq. (4) has many kinds of special solutions. The solutions which are obtained under the conditions: Case 1. Suppose that h1 = h3 = h5 = 0, h0 =

8h22 27h4

and h6 =

h24 4h2

the solutions are listed in Table 1 [41].

Case 2. Suppose that h0 = h1 = h3 = h5 = 0, the solutions are listed in Table 2. Case 3. Suppose that h5 = h6 = 0, and for α = h4 , β =

h3 4 ,

γ=

h2 6 ,

δ=

h1 4 ,

 = h0 , Eq. (4) can be written as

F  2 (ξ ) =  + 4δF (ξ ) + 6γ F 2 (ξ ) + 4βF 3 (ξ ) + αF 4 (ξ ) = R(F ).

(5)

As is well known the general solution of Eq. (5) reads [42–44] √ ;g2 ,g3 ) 1  1 + 12 [℘ (ξ ; g2 , g3 ) − 24 R (f0 )] + 24 R(f0 )R  (f0 ) R(f0 ) d℘ (ξdξ F (ξ ) = f0 + , 1  1 2[℘ (ξ ; g2 , g3 ) − 24 R (f0 )]2 − 48 R(f0 )R  (f0 )

(6)

where the primes denote differentiation with respect to F and f0 is any constant, not necessarily a real root of R(F ). If there exists a simple root f0 of R(F ), Eq. (6) can be simplified to [42,45] F (ξ ) = f0 +

R  (f0 ) 4[℘ (ξ ; g2 , g3 ) −

Table 1

8h2

1  24 R (f0 )]

(7)

.

h2

Solutions of Case 1 with h0 = 27h2 , h6 = 4h4 ,  = ±1 4 2 No.

FjI (ξ ) 

1

2

No. 

h

1

2 8h2 tanh2 ( − 32 ξ )  − , h2 < 0, h4 > 0 h2 2 3h4 [3+tanh ( − 3 ξ )]  1  h 2 8h2 tan2 ( 32 ξ )  , h2 > 0, h4 < 0 h2 2 3h4 [3−tan ( − 3 ξ )]

FjI (ξ )

 1 h 2 8h2 coth2 ( − 32 ξ )  − , h2 < 0, h4 > 0 h2 2 3h4 [3+coth ( 3 ξ )]  1  h 2 8h2 cot2 ( 32 ξ )  , h2 > 0, h4 < 0 h2 2 3h4 [3−cot ( 3 ξ )]

 3

4

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Table 2 Solutions of Eq. (3) with Δ1 = h24 − 4h2 h6 ,  = ±1 FjII (ξ )

No.



1

No. √

1 −h2 h4 sech2 ( h2 ξ ) 2 , h2 > 0 √ h24 −h2 h6 (1+ tanh( h2 ξ ))2 √

1 h2 h4 csh2 ( h2 ξ ) 2 , h2 > 0 √ h24 −h2 h6 (1+ coth( h2 ξ ))2 1  2h2 2 , h2 > 0, Δ1 > 0 √ √  Δ1 cosh(2 h2 ξ )−h4  1 2h2 2 , h2 < 0, Δ1 > 0 √ √  Δ1 cos(2 h2 ξ )−h4  1 2h2 2 , h2 > 0, Δ1 < 0 √ √  −Δ1 sinh(2 h2 ξ )−h4 1  2h2 2 , h2 < 0, Δ1 > 0 √ √  Δ1 sin(2 −h2 ξ )−h4 √  1 −h2 sech2 ( h2 ξ ) 2 , h2 > 0, h6 > 0 √ √ h4 +2 h2 h6 tanh( h2 ξ ) 

2 3 4 5 6 7

8 9 10 11 12 13 14

FjII (ξ )

√



1 −h2 sec2 ( −h2 ξ ) 2 , h2 < 0, h6 > 0 √ √ h4 +2 −h2 h6 tan( −h2 ξ )



1 h2 csch2 ( −h2 ξ ) 2 , h2 > 0, h6 > 0 √ √ h4 +2 h2 h6 coth( h2 ξ )



1 −h2 csc2 ( −h2 ξ ) 2 , h2 < 0, h6 > 0 √ √ h4 +2 −h2 h6 cot( −h2 ξ )

√

√

√  h2 1 h − h (1 +  tanh( 2 2 ξ )) 2 , 4 √  h2 1 h − h (1 +  coth( 2 2 ξ )) 2 , 4

 4

2

√

h2 ξ

h2 > 0, Δ1 = 0 h2 > 0, Δ1 = 0

1

2 , h2 > 0 √ h2 e (e2 h2 ξ −4h4 )2 −64h2 h6 √  ±h2 e2 h2 ξ  1 2 , h2 > 0, h4 = 0 √ 4 1−64h2 h6 e4 h2 ξ

Table 3 Eq. (6) reads [47,48] where f0 is the simple root of R(F ) No.

FjIII (ξ )

1

f0 +

2 3 4 5

R  (f0 ) 1√ 3 1 R  (f )+3e csch2 (√3e ξ )] , Δ = 0, g3 < 0, e1 = 2 −g3 4[e1 − 24 0 1 1  √ R (f0 )  f0 + , Δ = 0, g3 > 0, e1 = 12 3 g3 1 R  (f )+3e csc2 ( 3 e ξ )] 4[−e1 − 24 0 1 1 2

6R  (f0 )ξ 2 , Δ = 0, g2 = g3 = 0, R  (f0 ) < 0 24−R  (f0 )ξ 2 √ R  (f0 )−cn(2 H2 ξ,m)R  (f0 ) 3e g √ , Δ < 0, m = 12 − 4H2 , H22 = 3e22 − 42 f0 + 2 [4H2 + 16 R  (f0 )−4e2 ] cn(2 H2 ξ,m)+4H2 +4e2 − 16 R  (f0 ) √ √ R  (f0 ) sn2 ( e1 −e3 ξ,m) R  (f0 )(1−cn2 ( e1 −e3 ξ,m)) f0 + = f0 + √ √ 1 1  2  2 [4e3 − 6 R (f0 )] sn ( e1 −e3 ξ,m)+4(e1 −e3 ) [4e3 − 6 R (f0 )](1−cn ( e1 −e3 ξ,m))+4(e1 −e3 ) √ 1−dn2 ( e1 −e3 ξ,m) R  (f0 )( ) e −e = f0 + , Δ > 0, m = e2 −e3 , e1  e2  e3 √m 1−dn2 ( e1 −e3 ξ,m) 1 3 1  [4e3 − 6 R (f0 )]( )+4(e1 −e3 ) m

f0 +

The invariants g2 , g3 of Weierstrass elliptic function ℘ (ξ ; g2 , g3 ) are related to the coefficients of R(F ) by [46] g2 = α − 4βδ + 3γ 2 ,

(8)

g3 = αγ  + 2βγ δ − αδ − γ − β . 2

3

2

(9)

The discriminant (of ℘ and R [46]) Δ = g23 − 27g32 ,

(10)

is suitable to classify the behavior of F (ξ ). The solutions are listed in Table 3, where e1 , e2 , e3 are the roots of the equation 4s 3 − g2 s − g3 = 0.

(11)

Case 4. If now, h4 = h5 = h6 = 0, the solutions are obtained from those in Table 3 by replacing the g2 and g3 values by the following: g2 = −4βδ + 3γ 2 ,

(12)

g3 = 2βγ δ − γ 3 − β 2 .

(13)

Case 5. If now, h0 = h1 = h5 = h6 = 0, the solutions are obtained from those in Table 4. Case 6. Suppose that h1 = h3 = h5 = h6 = 0, the solutions are given in Table 5, where the modulus m of the Jacobi elliptic function satisfies (O  m  1).

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Table 4 Solutions of Case 5 with Δ2 = h23 − 4h2 h4 No.

FjV (ξ )

No.

FjV (ξ )

1

√

2h2 sech( h2 ξ ) √ , h2 > 0, Δ2 > 0 Δ2 −h4 sech( h2 ξ )

2

−2h2 sech( h2 ξ ) √ √ , h2 > 0, Δ2 > 0 Δ2 +h4 sech( h2 ξ )

3

√

2h2 csch( h2 ξ ) √ , h2 > 0, Δ2 < 0 −Δ2 −h4 csch( h2 ξ )

4

√

√

√

√

√ h h − h2 [1 ± tanh( 2 2 ξ )], h2 > 0, Δ2 = 0 4

5

√

6

−2h2 csch( h2 ξ ) √ , h2 > 0, Δ2 < 0 −Δ2 +h4 csch( h2 ξ )

√ h h − h2 [1 ± coth( 2 2 ξ )], h2 > 0, Δ2 = 0 4

Table 5 Solutions of Case 6 h4

h2

h0

FjVI

m2

−(1 − m2 )

1

cn ξ F1VI = sn ξ, F2VI = cd ξ = dn ξ

−m2

2m2 − 1

1 − m2

F3VI = cn ξ

−1

2 − m2

m2 − 1

F4VI = dn ξ

1

−(1 + m2 )

m2

ξ F5VI = ns ξ = (sn ξ )−1 , F6VI = dc ξ = dn cn ξ

1 − m2

2m2 − 1

−m2

F7VI = nc ξ = (cn ξ )−1

m2 − 1

2 − m2

−1

F8VI = nd ξ = (dn ξ )−1

1 − m2

2 − m2

1

sn ξ F9VI = sc ξ = cn ξ

−m2 (1 − m2 )

2m2 − 1

1

VI = sd ξ = sn ξ F10 dn ξ

1

2 − m2

1 − m2

VI = cs ξ = cn ξ F11 sn ξ

1

2m2 − 1

−m2 (1 − m2 )

VI = ds ξ = dn ξ F12 sn ξ

1 4 1−m2 4 1 4 m2 4

1−2m2 2 1+m2 2 m2 −2 2 m2 −2 2

1 4 1−m2 4 m2 4 m2 4

VI = ns ξ ± cs ξ F13 VI = nc ξ ± sc ξ F14 VI = ns ξ ± ds ξ F15 VI = sn ξ ± ics ξ F16

Step 4. Substitute ansatz (3) along with Eq. (4) into (2) and equate the coefficients of all powers of F (ξ ) to zero yields a set of algebraic equations for unknowns hj (j = 0, 1, . . . , 6), ai (i = 0, 1, . . . , 2M) and c. Step 5. Solve the set of algebraic equations by use of MATHEMATICA can permit obtention of explicit expressions of hj (j = 0, 1, . . . , 6), ai (i = 0, 1, . . . , 2M), and c. Step 6. Obtain exact solutions. By using the results obtained in the above steps, we can derive a series of travelling wave solutions of Eq. (1) depending on the solution F (ξ ) of Eq. (4). Selecting appropriate Fj (ξ ) from Tables 1–5 and substituting it into the travelling wave solutions Eq. (3), we can obtain exact solutions of Eq. (1). 3. Applications of the method 3.1. Nonlinear Klein–Gordon equation Let us first consider the nonlinear Klein–Gordon equation [39] utt − μ2 uxx + σ u − κun + τ u2n−1 = 0,

n > 1,

(14)

where μ, σ , κ and τ are nonzero constants. To look for travelling wave solution of Eq. (14), we make transformation u(x, t) = u(k(x − λt)) and change Eq. (14) into the form   k 2 λ2 − μ2 u + σ u − κun + τ u2n−1 = 0, n > 1, (15) balancing u with u2n−1 gives M = 1/(n − 1) [37]. Thus we make the transformation u = v 1/(n−1) ,

(16)

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E. Yomba / Physics Letters A 372 (2008) 1048–1060

and transform Eq. (15) to the following ODE [37]    k 2 λ2 − μ2 (n − 1)vv  + (2 − n)v  2 + (n − 1)2 σ v 2 − κv 3 + τ v 4 = 0.

(17)

Now balancing vv  and v 4 , we obtain M = 1. This suggests that v(ξ ) = a0 + a1 F (ξ ) + a2 F 2 (ξ ),

(18)

where a0 , a1 and a2 are constants to be determined. Substitution of (18) with (4) into (17) shows that the set of algebraic equation possesses the solutions: Case 1. h1 = h3 = h5 = 0, h0 = (i):

nκ + h4 | nκ h4 | , 2 2τ (1 + n)

and h6 =

h24 4h2 ,





h6

nκ



, a2 = 2

τ (n + 1)

h

h2 (n + 1)2 τ σ h2 = 4 , 4nh6 κ 2 a0 =

(ii):

8h22 27h4

4

a1 = 0,

h0 =

(−1 + n)2 nh6 κ 2 > 0, τ h24 (n + 1)2 (μ2 − λ2 )

2 2 n(nκ + h4 | nκ h4 |)(−nκ + τ σ (1 + n) ) 3 8h26 | nκ h4 |





k nh6 (μ2 − λ2 )

a2 = 2

, −1 + n

τ

a0 = a1 = 0,

k2 =

;

σ > 0, k = h2 (μ2 − λ2 ) 2

(19) √ 3κh6 h4 = −  . 2 τ σh2h6

(20)

Case 2. h0 = h1 = h3 = h5 = 0, (i):

(ii):

(−1 + n)2 τ 2 2nκh6 + (1 + n)τ h4 a2 a > 0, a0 = , 4(1 + n)τ h6 4nh6 (μ2 − λ2 ) 2  



h6 n(−3nκ 2 + 2τ σ (1 + n)2 ) n(−3nκ 2 + 2τ σ (1 + n)2 )



, nκ + h = 0; a2 = 2

4 (n + 1)τ

8h2 h6 − 3h24 8h2 h6 − 3h24

a1 = 0,

k2 =

a0 = a1 = 0,

k2 =

(−1 + n)2 τ 2 a > 0, 4nh6 (μ2 − λ2 ) 2

a2 = −

2nh6 κ , h4 (1 + n)τ

h2 =

(21)

h24 (1 + n)2 τ σ . 4nh6 κ 2

(22)

For Cases 3 and 4, no solution was found in that class of solutions. Case 5. h0 = h1 = h5 = h6 = 0, and τ = 0, (i):

(ii):

(1 − n)σ > 0, h2 (μ2 − λ2 )

a0 =

σ , κ

h23 =

12h2 h4 (−1 + n)2 , n2 + 4n − 9

a0 = 0,

k2 =

k2 =

a2 =

2(1 + n)h4 σ , (−1 + n)h2 κ

a1 =

(1 + n)σ h3 , (−1 + n)κh2

(−5 + n)(−3 + n)(−2 + n) = 0;

(1 − n)σ > 0, h2 (μ2 − λ2 )

a2 =

−2(1 + n)h4 σ , h2 κ

a2 =

−(1 + n)h4 σ , 2h2 κ

(23)

a1 =

−(1 + n)σ h3 , κh2

h23 − 4h2 h4 = 0.

(24)

Case 6. h0 = h1 = h5 = h6 = 0, and τ = 0, a0 = a1 = 0,

k2 =

(1 − n)2 σ > 0, 4h2 (μ2 − λ2 )

n = 3.

(25)

The solution (22) was already found by Sirendaoreji [37], but the solutions (19)–(21), (23)–(26) are new ones for Eq. (14). Thus, we will focus only on the new solutions. From Eq. (18), Cases 1, 2 and 5, we can obtain many kinds of solutions of Eq. (14) depending on the special choices for hj (j = 1, 2, . . . , 6). If Case 1(i), Eq. (19) is used, then F (ξ ) is one of the four FLI (L = 1, 2, 3, 4). For example, if we select L = 1, then we may write down explicitly the following soliton-like solution of Eq. (14)   1 u = v 1/(n−1) = a0 + a1 F (ξ ) + a2 F 2 (ξ ) n−1 ,

(26)

E. Yomba / Physics Letters A 372 (2008) 1048–1060

which gives u=

1053

 1



 8h2 tanh2 ( − h32 ξ )  n−1

h6

nκ



−  + 2 ,

τ (1 + n)

h

2τ 2 (1 + n) 4 3h4 (3 + tanh2 ( − h32 ξ ))

nκ + h4 | nκ h4 |

h2 (n + 1)2 τ σ h2 = 4 , 4nh6 κ 2

h0 =

2 2 n(nκ + h4 | nκ h4 |)(−nκ + σ τ (1 + n) ) 3 8h26 | nκ h4 |

,





(−1 + n)κ

nh6

(x − λt). ξ =

h4 (1 + n) τ (μ2 − λ2 )

(27)

If Case 2(i), Eq. (21) is used, then F (ξ ) is one of the Fourteen FLII (L = 1, 2, . . . , 14). For example, if we select L = 7, we find another soliton-like solution of Eq. (14) √



 1  n−1

h0 n(−3nκ 2 + 2(1 + n)2 σ τ ) 2nκh0 + (1 + n)τ h4 −h2 sech2 ( h2 ξ )



+ , u = 2

√ √

2 (1 + n)τ 4(1 + n)τ h6 h4 + 2 h2 h6 tanh( h2 ξ ) 8h2 h6 − 3h4  



2 + 2(1 + n)2 σ τ )

h6

(−1 + n)τ n(−3nκ n(−3nκ 2 + 2(1 + n)2 σ τ )

(x − λt), nκ + h = 0. (28) ξ =

4

(1 + n)τ 4nh6 (μ2 − λ2 ) 8h2 h6 − 3h24 8h2 h6 − 3h24 Now, if Case 5(i), Eq. (23) is exploited, then F (ξ ) is one of the six FLV (L = 1, 2, . . . , 6). For example, if we select L = 5, we have another soliton-like solution of Eq. (14)   2 1   √  √ n−1 h2 h2 (1 + n)σ h3 2(1 + n)h4 σ h2 h2 σ , + ξ − − ξ − 1 ± tanh 1 ± tanh u= κ (−1 + n)κh2 h4 2 (−1 + n)h2 κ h4 2  (−1 + n)σ 12h2 h4 (−1 + n)2 2 , ξ= (x − λt). h3 = (29) 2 n + 4n − 9 h2 (μ2 − λ2 ) 3.2. Fully nonlinear generalized Camassa–Holm equations The generalized Camassa–Holm equations with fully nonlinear dispersion and fully nonlinear convection term C(l, n, p) [40] is considered       ut + kux + β1 uxxt + β2 ul x + β3 ux un xx + β4 u up xxx = 0, (30) where k, β1 , β2 , β3 and β4 are arbitrary real constants. Eq. (30) is a class of physically important equations. In fact, if one takes β1 = −1, β2 = 3/2, β3 = −2, β4 = −1, l = 2, n = p = 1, (30) becomes the Camassa–Holm (CH) equation ut + kux − uxxt + 3uux = 2ux uxx + uuxxx .

(31)

Much has been written in recent years on the CH equation and the literature concerning (31) continues to expand. In reality, the importance of Eq. (31) is known since it resurfaced as model for shallow water waves in Refs. [49,50] and later on Ref. [51]. There is now a substantial body of work that attests to the integrability status of (31) and an extensive bibliography can be found in Ref. [52]. Recently, explicit analytic multisoliton solutions of CH equation have been found in parametric form by a number of authors using various methods [53–57]. Parker and Matsumo [58] proposed a method for obtaining peakon limits of multisoliton solutions of the CH equation and used it to recover the peakon and two-peakon limits of the solitary wave and two-soliton solution. They also obtained a criterion that discriminates between the dynamical behavior of the two soliton solutions and determines the break down point of the interaction. When β1 = −1, β2 = a/(L + 1), β3 = −2, β4 = −1, l = L + 1, n = p = 1, (30) becomes another form of generalized Camassa– Holm equation ut + kux − uxxt + auL ux = 2ux uxx + uuxxx ,

(32)

studied by Tian and Song [59]. They derived some exact peaked solitary wave solutions. When k = 0, β1 = −1, β2 = 3/2, β3 = −2γ , β4 = −γ , l = 2, n = p = 1, (30) becomes ut − uxxt + 3uux = γ (2ux uxx + uuxxx ),

(33)

which was derived by Day and Huo [60] when they studied disturbances in an initially stretched or compressed rod which is composed of compressible Mooney–Rivlin material. They showed that Eq. (33) includes solitary shock, solitary wave and periodic shock wave solutions. Liu and Chen [61] showed that Eq. (33) also generated compacton structures by using the bifurcation method of planar dynamical systems.

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E. Yomba / Physics Letters A 372 (2008) 1048–1060

When k = 0, β1 = −1, β2 = (b + 1)/2, β3 = −b, β4 = −1, l = 2, n = p = 1, (30) becomes ut − uxxt + (b + 1)uux = bux uxx + uuxxx ,

(34)

which has been investigated in the literature [62–72]. Degasperis and Procesi [64] showed that the family of Eq. (34) cannot be integrable unless b = 2 or b = 3 by using the method of asymptotic integrability. It was found by researchers that all equations in the family (34) possess not only the peakon solutions but also the multipeakon solutions [64]. More recently, by using four direct ansätze, Tian and Yin [40] obtained abundant solutions, compactons, solitary patterns, solitary patterns, singular periodic wave solutions to the generalized CH equation C(l, n, p) Eq. (30). Later on in a recent paper [73], we use a form of indirect F-function method which takes advantage of elliptic equation to successfully present in an unified way various single and combined nondegenerative Jacobi elliptic function solutions. In the following, we will use the generalized auxiliary NLODE with six-degree nonlinear terms method to obtain new solution of C(l, n, p) Eq. (30) different to those found in [74]. We first make the following formal travelling wave transformation u(x, t) = u(ξ ),

ξ = λ1 x + λ2 t,

(35)

where λ1 and λ2 are undetermined constants. Substituting (35) into Eq. (30), we obtained the ODE for u(ξ ),       λ2 u + ku + β1 λ2 λ21 u + β2 λ1 ul + β3 λ31 u un + β4 λ31 u up = 0.

(36)

Let us assume that Eq. (36) has the solution in the form u(ξ ) = ηF μ (ξ ),

(37)

where η and μ are parameters to be determined later. We also propose that the function F (ξ ) should be mapped to the solutions of the Eq. (4). It is important to observe that, we use here a form of indirect F-function method (37), not the F-expansion method as Eq. (3) because the complexity of Eq. (30) cannot allow use of this ansatz (Eq. (3)). But our indirect F method will be very close to the F-expansion method in the sense that, the indirect F-function method will also take advantage of nonlinear ODE (4). To be more concise, only cases with βi (i = 1, 2, 3) different to 0 will be considered in this work. Now, substituting the ansatz (37) along with Eq. (4) into Eq. (36), collecting coefficients of power F (ξ ) with the aid of MATHEMATICA, we can obtain the following results (the method we followed in this resolution, is the one explained in [73]): Case 1. h1 = h3 = h5 = 0, h0 = (i):

l = 1 + n, η=−

(ii):

(iii):

p = n,

and h6 =

h24 4h2 ,

2 μ= , n

β3 = −3β4 ,

n = 1,

4kh6 β1 , h4 β4 + 4(h2 h4 − 6h0 h6 )β1 β4 λ21

λ21 =

2 μ= , n = 2, n 5kh6 β1 > 0, η2 = − 4(h4 β4 + (h2 h4 − 15h0 h6 )β1 β4 λ21 )

l = 1 + n,

l = 1 + n, λ2 =

p = n,

β3 = −6β4 ,

2 n = 1, μ=− , n kλ1 (β2 − 2h2 β4 λ21 )

λ21 =

μ = −2,

n = 2,

4kh4 β4 λ1 , 4(12h0 h6 β1 λ21 − h4 )β4 − 3h4 β1 β2

η2 =

λ2 =

(38) 4η2 h4 β4 λ1 , 5h6 β1 (39)

β3 = −2β4 , ,

−β2 (1 + 4h2 β1 λ21 ) + 2β4 λ21 (h2 + 4(h22 − 3h0 h4 )β1 λ21 ) p = n,

ηh4 β4 λ1 , 4h6 β1

15h6 β2 ; 8β4 (10h2 h6 − 3h24 )

6kh0 β1 λ21

l = 1 + n, λ2 =

p = n,

λ2 =

2h6 β2 > 0; β4 (8h2 h6 − 3h24 )

−β2 (1 + 4h2 β1 λ21 ) + 2β4 λ21 (h2 + 4(h22 − 3h0 h4 )β1 λ21 )

η=−

(iv):

8h22 27h4

;

(40)

β3 = −3β4 ,

λ21 =

3β2 , 16h2 β4

2kh0 β1 ; − h4 )β4 − 3h4 β1 β2

4(12h0 h6 β1 λ21

(41)

E. Yomba / Physics Letters A 372 (2008) 1048–1060

2 n = 1, μ=− , n 3h0 β1 (k + β2 ) ; η=− β4 (h2 + 4(h22 − 3h0 h4 )β1 λ21 )

(v): l = 1,

p = n,

β3 = −2β4 ,

β3 = −3β4 ,

2 μ= , n = 2, n 5h6 β1 (k + β2 ) , η2 = − 4β4 (h4 + (h2 h4 − 15h0 h6 )β1 λ21 )

β3 = −6β4 ,

p = n,

(vii): l = 1,

ηh2 β4 λ1 , 3h0 β1

λ2 =

(42)

2 μ= , n = 1, n 4h6 β1 η=− ; β4 (h4 + 4(h2 h4 − 6h0 h6 )β1 λ21 )

(vi): l = 1,

1055

3h24 − 8h2 h6 = 0, (43)

p = n,

4η2 h4 β4 βλ1 , 5h6 β1

λ2 =

3h24 − 10h2 h6 = 0.

(44)

Case 2. h0 = h1 = h3 = h5 = 0, (i): l = 1 + n, ηn = −

β3 = −3nβ4 ,

2kh6 (1 + 3n + 2n2 )β1 , 3nh4 β4 (n2 + 4h2 β1 λ21 )

(ii): l = 1 + n,

(iii):

2 μ= , n

p = n,

(iv): l = 1 + n,

2 μ=− , n

p = n,

l = 1 + n,

p = n,

p = n,

2 μ=− , n

(vi): l = 1,

p = n,

2 μ= , n

kn2 λ1 , n2 + 4h2 β1 λ21

h6 (1 + 3n + 2n2 )β2 > 0; n(−3(2 + n)h24 + 8(1 + 2n)h2 h6 )β4 n = 1,

λ2 = −

λ1 (k + 2ηh4 β3 λ21 ) 1−

2β1 β2 β3 +β4

(45)

λ21 = −

,

β2 > 0; 2h2 (β3 + β4 ) (46)

− 8η2 h6 β4 λ21 )

4λ1 β4 (k (47) ; 3β1 β2 − 4β4 1 3β4 λ1 β4 k 3β2 n= , β3 = , λ2 = , λ21 = − > 0; 2 2 3β1 β2 − β4 4h2 β4 (48) λ1 (−k − β2 + 2ηh4 β4 λ21 ) (49) n = 1, β3 = −nβ4 , λ2 = ; 1 + 4h2 β1 λ21

4 μ=− , n 1 μ=− , n

p = n,

(v): l = 1,

ηn = −

λ21 =

λ2 = −

n = 2,

β3 = −3nβ4 ,

(2 + 3n + n2 )h4 β1 (k + β2 ) , 4nh2 β4 (n2 + 4h2 β1 λ21 )

β3 = −

λ2 = −

nβ4 , 2

λ2 =

n2 λ1 (k + β2 ) , n2 + 4h2 β1 λ21

3(2 + n)h24 − 8(1 + 2n)h2 h6 = 0.

(50)

Case 3. h5 = h6 = 0; (i): l = 1 + n, λ21 =

1 μ= , n

8h4 β2 > 0, (8h2 h4 − 3h23 )β4

(ii): l = 1 + n, λ21 =

p = n,

p = n,

η=

1 μ=− , n

8h0 β2 > 0, (8h0 h2 − 3h21 )β4

η=

n = 1,

β3 = −3β4 ,

λ2 =

ηh3 β4 λ1 , 4h4 β1

4kh4 (8h2 h4 − 3h23 )β1 8h4 (6h1 h4 − h2 h3 )β1 β2 + h3 (3h23 − 8h2 h4 )β4 n = 1,

β3 = −3β4 ,

λ2 =

;

(51)

ηh1 β4 λ1 , 4h0 β1

4kh4 (8h0 h2 − 3h21 )β1 8h0 (6h0 h3 − h1 h2 )β1 β2 + h1 (3h21 − 8h0 h2 )β4

;

(52)

1056

E. Yomba / Physics Letters A 372 (2008) 1048–1060

(iii):

(iv):

1 μ=− , n 4h0 β1 (k + β2 ) + ηh1 β4 λ21 = > 0, ηβ1 β4 (6h0 h3 − h1 h2 )

l = 1,

n = 1,

p = n,

β3 = −3β4 ,

ηh1 β4 λ1 , 4h0 β1

8h0 h2 − 3h21 = 0;

1 n = 1, β3 = −3β4 , μ= , n 4h4 β1 (k + β2 ) , 8h2 h4 − 3h23 = 0. η=− β4 (h3 + (h2 h3 − 6h1 h4 )β1 λ21 ) l = 1,

λ2 =

p = n,

(53)

λ2 =

ηh3 β4 λ1 , 4h4 β1 (54)

Case 4. h4 = h5 = h6 = 0, (i):

l = 1 + n,

3β2 > 0, 4h2 β4

λ21 =

(ii):

l = 1 + n, η=

(iii):

p = n,

p = n,

p = n,

3h21 , 10h2

l = 1,

η2 =

p = n,

1 μ=− , n

η=−

(55)

λ2 =

λ1 (ηh1 β4 λ21 − k) 1 + h2 β1 λ21

,

;

(56)

β3 = −3β4 ,

λ2 =

ηh1 β4 λ1 , 4h0 β1

8h0 (h1 h2 − 6h0 h3 )β1 β2 − h1 (3h21 − 8h0 h2 )β4

;

4η2 h1 β4 λ1 , 5h0 β1 10kh0 (−3h21 + 10h0 h2 )β1

β3 = −6β4 ,

n = 2,

η2 =

β3 = −6β4 ,

(57)

λ2 =

15h0 (−h1 h2 + 15h0 h3 )β1 β2 + 8h1 (3h21 − 10h0 h2 )β4

n = 2,

λ2 =

> 0;

μ = −1,

n = 1,

(59)

λ2 =

ηh1 β4 λ1 , 4h0 β1

3h21 − 8h0 h2 = 0;

,

1 μ= , n

β3 = −3β4 ,

n = 1,

β3 = −2β4 ,

3h3 β1 (k + β2 ) . β4 (h2 + (h22 − 3h1 h3 )β1 λ21 )

(58)

8η2 h2 β4 λ1 , 3h1 β1

3h1 (k + β2 ) > 0; β4 (−8h2 + (−2h22 + 9h1 h3 )β1 λ21 )

2β1 (−2k − 2β2 + 3ηh3 β4 λ21 ) p = n,

2η2 h1 β4 λ1 , h3 β1

4h0 (3h21 − 8h0 h2 )kβ1

1 μ=− , n

ηh1 β4 (1 + h2 β1 λ21 )

l = 1,

n = 1,

η=

15h0 β2 > 0, 2(10h0 h2 − 3h21 )β4

l = 1,

λ2 =

β3 = −2β4 ,

n = 1,

1 μ=− , n

8h0 β2 > 0, (8h0 h2 − 3h21 )β4

l = 1 + n,

h0 =

(vii):

β3 = −3β4 ,

2kh2 h3 β1 > 0; 3(3h0 h3 − h1 h2 )β1 β2 − 4h1 h2 β4 1 μ= , n

p = n,

l = 1 + n,

h0 =

(vi):

η2 =

n = 2,

2β2 (1 + h2 β1 λ21 ) − β4 λ21 (h2 + (h22 − 3h1 h3 )β1 λ21 )

λ21 =

(v):

μ = 1,

3kh3 β1 λ21

λ21 =

(iv):

p = n,

(60)

λ2 = −

h2 (k + β2 )λ1 , h2 + (h22 − 3h1 h3 )β1 λ21 (61)

Case 5. h0 = h1 = h5 = h6 = 0, (i):

l = 1 + n,

p = n,

μ=−

1 , 2n

1 n= , 2

3 β3 = β4 , 2

λ2 = −

kλ1 , 1 + h2 β1 λ21

λ21 = −

3β2 > 0; h2 β4 (62)

E. Yomba / Physics Letters A 372 (2008) 1048–1060

(ii): l = 1 + n,

1 μ=− , n

p = n,

λ2 = −

n = 1,

1057

λ1 (2k + ηh3 β3 λ21 ) 2(1 + h2 β1 λ21 )

,

λ21 = −

2β2 (β3 + β4 ) > 0; h2 (63)

(iii): l = 1 + n, ηn = −

(iv):

2(n2 + h2 β1 λ21 )((1 + n)β2

− 2nh2 β4 λ21 )

2 μ=− , n

p = n,

,

λ2 = − λ21 =

kn2 λ1 , n2 + h2 β1 λ21

4h4 β2 (1 + 3n + 2n2 ) > 0; nβ4 (8h2 h4 (1 + 2n) − 3h23 (2 + n))

β3 = −β4 ,

n = 2,

λ2 = −

λ1 (k − 2η2 h4 β4 λ21 ) 1 + h2 β1 λ21

(64)

,

3β2 > 0; 4h2 β4

(65)

(v): l = 1,

p = n,

1 μ=− , n

(vi): l = 1,

p = n,

1 μ= , n

ηn = −

β3 = −3nβ4 ,

kh3 β1 λ21 (2 + 3n + n2 )

l = 1 + n, λ21 = −

1 μ= , n

p = n,

β3 = −nβ4 ,

n = 1,

β3 = −3nβ4 ,

(2 + 3n + n2 )(k + β2 )β1 h3 , 4nh2 β4 (n2 + h2 β1 λ21 )

λ2 = −

λ2 = −

λ1 (2(k + β2 ) − ηh3 β4 λ21 ) 2(1 + h2 β1 λ21 )

;

(66)

n2 (k + β2 ) , n2 + h2 β1 λ21 )

3(2 + n)h23 − 8(1 + 2n)h2 h4 = 0.

(67)

Case 6. h1 = h3 = h5 = h6 = 0. As the readers can find the solutions of Case 6 in [73], we will not focus on it. Substituting Case 1(iii), Eq. (40) along with F (ξ ) in Table 1 into (37), we obtain exact travelling wave solutions of Eq. (30). So F (ξ ) is one of fourteen FL (ξ ) (L = 1, 2, . . . , 14). For example, if we select L = 1, then we may write down explicitly the following solution   −8h tanh2 ( −h2 ξ )  μ 2 2 3  u(x, t) = η (68) , 2 3h4 [3 + tanh2 ( −h ξ )] 3 μ = 2, η=−

l = 2,

p = n = 1,

ηh4 β4 λ1 , 4h6 β1 2h6 β2 > 0; λ21 = β4 (8h2 h6 − 3h24 )

β3 = −3β4 ,

4kh6 β1 , (h4 β4 + 4(h2 h4 − 6h0 h6 ))β1 β4 λ21

ξ = λ1 x + λ2 t,

h0 =

8h22 , 27h4

h6 =

λ2 =

h24 , 4h2

(69)

where λ1 is arbitrary function. If we choose Case 3(i), Eq. (51) along with F (ξ ) in Table 3 into (37), such that F (ξ ) is one five FLIII (L = 1, 2, . . . , 5). For example, if we select L = 5, the solution is √   R  (f0 ) sn2 ( e1 − e3 ξ, m) , u(x, t) = η f0 + (70) √ [4e3 − 16 R  (f0 )] sn2 ( e1 − e3 ξ, m) + 4(e1 − e3 ) l = 1 + n,

p = n,

1 μ= , n

8h4 β2 > 0, (8h2 h4 − 3h23 )β4 e 2 − e3 m= , e1 > e2 > e3 . e 1 − e3

λ21 =

η=

n = 1,

β3 = −3β4 ,

λ2 =

ηh3 β4 λ1 , 4h4 β1

4kh4 (8h2 h4 − 3h23 )β1 8h4 (6h1 h4 − h2 h3 )β1 β2 + h3 (3h23 − 8h2 h4 )β4

,

ξ = λ1 x + λ2 t, (71)

e1 , e2 , e3 are roots of Eq. (11), with g2 , g3 , and Δ by Eqs. (8), (9) and (10), respectively, where h0 = , h1 = 4δ, h2 = 6γ , h3 = 4β, h4 = α.

1058

E. Yomba / Physics Letters A 372 (2008) 1048–1060

If we choose Case 5(iii), Eq. (64), then F (ξ ) is one of the six FLV (L = 1, 2, . . . , 6). For example, if we select L = 2, we obtain soliton-like solution of Eq. (30) √ μ  −2h2 sech( h2 ξ ) , u(x, t) = η  (72) √ Δ2 + h4 sech( h2 ξ ) l = 1 + n, ηn = −

p = n,

1 μ= , n

β3 = −3nβ4 ,

kh3 β1 λ21 (2 + 3n + n2 ) 2(n2 + h2 β1 λ21 )((1 + n)β2 − 2nh2 β4 λ21 )

ξ = λ1 x + λ2 t,

,

λ2 = − λ21 =

n2

kn2 λ1 , + h2 β1 λ21

h4 β2 (1 + 3n + n2 ) > 0, nβ4 (8h2 h4 (1 + 2n) − 3h23 (2 + n))

Δ2 = h23 − 4h2 h4 .

(73)

4. Conclusion In short, by using the modified ansatz (3) and a new auxiliary equation method including a new first-order nonlinear ODE with six-degree nonlinear term (4) proposed by Sirendaoreji, we have picked up the family of solution of a Klein–Gordon equation, which was not considered by Sirendaoreji. Then, we have applied an indirect F-function method very close to the F-expansion method to solve the generalized Camassa–Holm equation with fully nonlinear dispersion and fully nonlinear convection C(l, n, p). Taking advantage of the new first-order nonlinear ODE with six-degree nonlinear term (4), this F-function has been used to map the solutions of the C(l, n, p) equations to those of (4). The indirect F-function method is imposed itself by the complexity of these two equations. By using this F-function method, we have been able to obtain in a unified way simultaneously many solitary wave solutions and periodic solutions. The following Camassa–Holm family of equations has been solved by the indirect F-function method:    

ut + kux + β1 uxxt + β2 u2 x = β4 3ux (u)xx + u(u)xxx 1,3,4 ,       

 ut + kux + β1 uxxt + β2 u3/2 x = −β4 3/2ux u1/2 xx + u u1/2 xxx 2,5 ,      

ut + kux + β1 uxxt + β2 ux = β4 3nux un xx − u un xxx 2,5 ,        

ut + kux + β1 uxxt + β2 u3 x = β4 6ux u2 xx + u u2 xxx 1,4 ,        

ut + kux + β1 uxxt + β2 un+1 x = β4 3nux un xx − u un xxx 2,5 .    

ut + kux + β1 uxxt + β2 u2 x = β4 2ux (u)xx + u(u)xxx 1,4,6 ,        

ut + kux + β1 uxxt + β2 u3 x = β4 3ux u2 xx − u u2 xxx 1,4,6 ,

  ut + kux + β1 uxxt + β2 u2 x = −β3 ux (u)xx − β4 u(u)xxx 2,5 ,        

ut + kux + β1 uxxt + β2 u3 x = β4 ux u2 xx − u u2 xxx 2,5 ,  

ut + kux + β1 uxxt + β2 (u)x = −β4 ux (u)xx + u(u)xxx 2 ,  

ut + kux + β1 uxxt + β2 (u)x = β4 3ux (u)xx − u(u)xxx 3,4 ,      

ut + kux + β1 uxxt + β2 (u)x = β4 6ux u2 xx − u u2 xxx 4 ,      

ut + kux + β1 uxxt + β2 (u)x = β4 2ux u2 xx − u u2 xxx 4,6 ,  

ut + kux + β1 uxxt + β2 (u)x = β4 ux (u)xx − u(u)xxx 5 ,        

ut + kux + β1 uxxt + β2 u3 x = β4 4ux u2 xx − u u2 xxx 6 ,        

ut + kux + β1 uxxt + β2 u5 x = β4 6ux u4 xx − u u4 xxx 6 ,      

ut + kux + β1 uxxt + β2 (u)x = β4 4ux u2 xx − u u2 xxx 6 . The subscript |i (i = 1, 2, . . . , 6) appearing at the right-hand side of the above mentioned equations states that the equations in which it occurs, are solved by the cases i (i = 1, 2, . . . , 6). Acknowledgement The author is indebted to Professor George Sell of the School of Mathematics for his helpful discussions.

E. Yomba / Physics Letters A 372 (2008) 1048–1060

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66]

M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, Cambridge, 1991. M.R. Miurs, Bäcklund Transformation, Springer-Verlag, Berlin, 1978. A. Coely, et al. (Eds.), Bäcklund and Darboux Transformations, Amer. Math. Soc., Providence, RI, 2001. R. Hirota, Phys. Rev. Lett. 27 (1971). C.L. Bai, H. Zhao, Chaos Solitons Fractals 27 (2006) 1026. M.F. El-Sabbagh, A.T. Ali, Int. J. Nonlinear Sci. Numer. Simul. 6 (2005) 151. Y.G. Zhu, Z.S. Lu, Chaos Solitons Fractals 27 (2006) 836. H.A. Abdusalam, Int. J. Nonlinear Sci. Numer. Simul. 6 (2005) 99. E.G. Fan, Phys. Lett. A 277 (2000) 212. E.G. Fan, Z. Naturforsch. A: Phys. Sci. 56 (2001) 312. Z.Y. Yan, Phys. Lett. A 292 (2001) 100. B. Li, H.Q. Zhang, Chaos Solitons Fractals 15 (2003) 647. M.L. Wang, Phys. Lett. A 213 (1996) 279. Z.Y. Yan, H.Q. Zhang, Phys. Lett. A 285 (2001) 355. M.A. Abdou, A.A. Soliman, Physica D 211 (2005) 1. M.A. Abdou, A.A. Soliman, J. Comput. Appl. Math. 181 (2005) 245. A. A Soliman, K.R. Raslan, Int. J. Comput. Math. 78 (2001) 399. A.A. Soliman, Int. J. Comput. Math. 81 (2004) 325. A.A. Soliman, M.H. Hussein, Appl. Math. Comput. 161 (2005) 623. T.A. Abassy, M.A. El-Tawil, H.K. Saleh, Int. J. Nonlinear Sci. Numer. Simul. 5 (2004) 327. S.K. Liu, Z.T. Fu, S.D. Liu, Q. Zhao, Phys. Lett. A 289 (2001) 69. Y.B. Zhou, M.L. Wang, Y.M. Wang, Phys. Lett. A 308 (2003) 31. Sirendaoreji, J. Sun, Phys. Lett. A 309 (2003) 387. E. Yomba, Chaos Solitons Fractals 22 (2004) 321. E. Yomba, Chaos Solitons Fractals 21 (2004) 75. E.G. Fan, Phys. Lett. A 300 (2003) 243. E.G. Fan, Y. Hon, Chaos Solitons Fractals 15 (2003) 559. J.Q. Hu, Chaos Solitons Fractals 23 (2005) 391. R. Sabry, M.A. Zahran, E.G. Fan, Phys. Lett. A 326 (2004) 93. Y. Chen, Q. Wang, Chaos Solitons Fractals 23 (2005) 801. E. Yomba, Chin. J. Phys. 43 (2005) 789. E. Yomba, Phys. Lett. A 336 (2005) 463. S. Zhang, T.C. Xia, Phys. Lett. A 356 (2006) 119. E. Yomba, Chaos Solitons Fractals 27 (2006) 187. E. Yomba, Chaos Solitons Fractals 26 (2005) 785. Sirendaoreji, Phys. Lett. A 356 (2006) 124. Sirendaoreji, Phys. Lett. A 363 (2007) 440. S. Zhang, T.C. Xia, Phys. Lett. A 363 (2007) 356. A.M. Wazwaz, Chaos Solitons Fractals 27 (2006) 1005. L. Tian, J. Yin, Chaos Solitons Fractals 20 (2004) 289. S. Zhang, T.C. Xia, J. Phys. A: Math. Theor. 40 (2007) 227. H.W. Schürmann, Phys. Rev. E 54 (1996) 4312. K. Weierstrass, Mathematische Werke V, Johnson, New York, 1915, pp. 4–16. E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge, UK, 1927, pp. 452–454. H.W. Schürmann, V.S. Serov, J. Nickel, Int. J. Theor. Phys. 45 (2006) 1057. K. Chandrasekharan, Elliptic Functions, Springer-Verlag, Berlin, 1985, p. 44. J. Nickel, Phys. Lett. A 364 (2007) 221. J. Nickel, H.W. Schürmann, Phys. Rev. E 75 (2007) 038601. R. Camassa, D.D. Holm, Phys. Rev. Lett. 71 (1993) 1661. R. Camassa, D.D. Holm, J.M. Hyman, Adv. Appl. Mech. 31 (1994) 1. R.S. Johnson, J. Fluid Mech. 455 (2002) 63. A. Parker, Proc. R. Soc. London, Ser. A 460 (2005) 2929. Y. Li, J.E. Zhang, Proc. R. Soc. London, Ser. A 460 (2004) 2617. Y. Li, J. Nonlinear Math. Phys. 12 (2005) 466. Y. Matsumo, J. Phys. Soc. Jpn. 74 (2005) 1983. A. Parker, Proc. R. Soc. London, Ser. A 461 (2005) 3611. A. Parker, Proc. R. Soc. London, Ser. A 461 (2005) 3893. A. Parker, Y. Matsumo, J. Phys. Soc. Jpn. 75 (2007) 124001. L. Tian, X. Song, Chaos Solitons Fractals 9 (2004) 627. H.H. Day, Y. Huo, Proc. R. Soc. London, Ser. A 456 (2000) 331. Z. Liu, C. Chen, Chaos Solitons Fractals 22 (2004) 627. O.G. Mustafa, J. Math. Phys. 12 (2005) 10. H. Lundmark, J. Szmigielski, Inverse Problems 19 (2003) 1241. A. Degasperis, M. Procesi, Asymptotic integrability, in: Symmetry and Perturbation Theory, World Scientific, 2002, 23p. J. Shen, W. Xu, W. Li, Chaos Solitons Fractals 27 (2006) 413. C. Chen, M. Tang, Chaos Solitons Fractals 27 (2006) 698.

1059

1060

[67] [68] [69] [70] [71] [72] [73] [74]

E. Yomba / Physics Letters A 372 (2008) 1048–1060

R. Camassa, Discrete Contin. Dyn. Syst. Ser. B 3 (2003) 115. Z. Liu, R. Wang, Z. Jing, Chaos Solitons Fractals 19 (2004) 77. Z. Liu, T. Qian, Appl. Math. Model. 26 (2002) 473. T. Qian, M. Tang, Chaos Solitons Fractals 12 (2001) 1347. L. Tian, S. Song, Chaos Solitons Fractals 19 (2004) 621. A.M. Wazwaz, Phys. Lett. A 352 (2007) 500. E. Yomba, J. Math. Phys. 46 (2005) 123504. E. Yomba, Phys. Lett. A 372 (2008) 215.