Applied Mathematics and Computation 206 (2008) 141–149

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Applied Mathematics and Computation

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

Exact travelling wave solutions of a generalized Camassa–Holmequation using the integral bifurcation method

He Bin *, Rui Weiguo, Chen Can, Li ShaolinDepartment of Mathematics, Honghe University, Mengzi, Yunnan 661100, PR China

a r t i c l e i n f o

Keywords:Generalized Camassa–Holm equationPeaked compactonKink-like waveCompactonPeaked solitary wave

0096-3003/$ - see front matter � 2008 Elsevier Incdoi:10.1016/j.amc.2008.08.043

* Corresponding author.E-mail address: hbhhu@yahoo.com.cn (B. He).

a b s t r a c t

In this paper, a generalized Camassa–Holm equation is studied by using the integral bifur-cation method. Many travelling waves such as peaked compacton, compacton, peaked sol-itary wave, solitary wave and kink-like wave are found. In some parameter conditions,exact parametric representations of these travelling waves in explicit form and implicitform are obtained.

� 2008 Elsevier Inc. All rights reserved.

1. Introduction

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

ut þ 2kux � uxxt þ 3uux ¼ 2uxuxx þ uuxxx; ð1Þ

which is called Camassa–Holm equation (CH equation in short). For k = 0, Camassa and Holm showed that Eq. (1) has solitarywaves of the form c e�jx�ctj, which were called peakons (or called peaked solitary wave) due to the discontinuity of the firstderivative at the wave peak. Boyd [2] derived a perturbation series which converges even at the peakon limit, and gave threeanalytical representation for the spatially periodic generalization of the peakon, called ‘‘coshoidal wave”. Cooper and Shep-ard [3] derived approximate solitary wave solution by using some variational functions. Constantin [4] gave a mathematicaldescription of the existence of interacting solitary waves. Recently, CH equation and some its generalized forms have beenstudied by many authors (see Refs. [5–17] and the references cited therein).

In Ref. [14], Liu and Qian discussed the peakons and their bifurcations when the integral constants taken as zero of thefollowing generalized Camassa–Holm equation (GCH equation in short):

ut þ 2kux � uxxt þ aumux ¼ 2uxuxx þ uuxxx; ð2Þ

where a > 0; k 2 R; m 2 N, and they introduced Eq. (2) from the mathematical point of view. For m ¼ 1;2;3, they obtainedthe explicit expressions for the peakons. In Ref. [15], Tian and Song derived some new exact peaked solitary wave solutions.In Ref. [16], Shen and Xu analyzed the dynamical behavior of travelling wave solutions of Eq. (2) by using the bifurcationtheory and the method of phase portraits analysis.

In this paper, we aim to extend the previous works in Refs. [14–16] to obtain more new exact travelling wave solutions ofEq. (2) using the integral bifurcation method [13]. In some parameter conditions, many new exact parametric representa-tions of the travelling waves such as peaked compacton, compacton, peaked solitary wave, solitary wave and kink-like wavein explicit form and implicit form are obtained.

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142 B. He et al. / Applied Mathematics and Computation 206 (2008) 141–149

We look for travelling wave solutions to Eq. (2) in the form

uðx; tÞ ¼ /ðx� ctÞ ¼ /ðnÞ; ð3Þ

where c ð–0Þ is the wave speed and n ¼ x� ct. Substituting (3) into Eq. (2), we obtain

ð2k� cÞ/0 þ c/000 þ a/m/0 � 2/0/00 � //000 ¼ 0; ð4Þ

where ‘‘0” is the derivative with respect to n.Integrating Eq. (4) once with respect to n and the integral constant taken as zero, we have

ðc � /Þ/00 þ amþ 1

/mþ1 þ ð2k� cÞ/� 12ð/0Þ2 ¼ 0: ð5Þ

Letting /0 ¼ y, we obtain a planar integrable system

d/dn¼ y;

dydn¼

amþ1 /mþ1 þ ð2k� cÞ/� 1

2 y2

/� c: ð6Þ

Transformed by dn ¼ ð/� cÞdf, system (6) becomes a Hamiltonian system

d/df¼ ð/� cÞy; dy

df¼ a

mþ 1/mþ1 þ ð2k� cÞ/� 1

2y2 ð7Þ

with first integral

Hð/; yÞ ¼ ð/� cÞy2 � 2aðmþ 1Þðmþ 2Þ/

mþ2 þ ðc � 2kÞ/2 ¼ h; ð8Þ

i.e.

y2 ¼2a

ðmþ1Þðmþ2Þ/mþ2 � ðc � 2kÞ/2 þ h

/� c: ð9Þ

Denote D1 ¼ ðmþ1Þðc�2kÞa ; /1;2 ¼ �ðD1Þ

1m; D2 ¼ 2cð acm

mþ1þ 2k� cÞ; Ys ¼ffiffiffiffiffiffiD2p

. Obviously, systems (7) have only one equilib-rium point Oð0;0Þ on the /-axis when c ¼ 2k; systems (7) have three equilibrium points Oð0;0Þ;A1;2ð/1;2;0Þ on the /-axiswhen c > 2k and m is an even; systems (7) have two equilibrium points Oð0;0Þ;A1ð/1;0Þ on the /-axis when c–2k and mis an odd; systems (7) have two equilibrium points S�ðc;�YsÞ on the line / ¼ c when D2 > 0. From (8), we write

h0 ¼ Hð0; 0Þ ¼ 0; ð10Þ

h1 ¼ Hð/1;2; 0Þ ¼mðc � 2kÞ

mþ 2ðmþ 1Þðc � 2kÞ

a

� �2m

; ð11Þ

hs ¼ Hðc;�YsÞ ¼ �2a

ðmþ 1Þðmþ 2Þ cmþ2 þ ðc � 2kÞc2: ð12Þ

The rest of this paper is organized as follows. In Section 2, we give some new exact travelling wave solutions of Eq. (2) inexplicit form and implicit form when m ¼ 1;2;3. A short conclusion will be given in Section 3.

2. Exact travelling wave solutions of the GCH equation

2.1. Exact travelling wave solutions of the GCH equation when m ¼ 1

When m ¼ 1, we have

/1 ¼2ðc � 2kÞ

a; ð13Þ

y2 ¼13 a/3 � ðc � 2kÞ/2 þ h

/� c; ð14Þ

h0 ¼ 0; h1 ¼4ðc � 2kÞ3

3a2 ; hs ¼ �13

ac3 þ ðc � 2kÞc2: ð15Þ

Case 2.1.1: Taking h ¼ h0, from (14) we get

y2 ¼13 að/� 3

a ðc � 2kÞÞ/2

/� c: ð16Þ

2.1.1.1 When k ¼ 12 c, (16) can be reduced to

y ¼ �

ffiffiffiffiffiffi13 a

q/

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/ð/� cÞ

p/� c

: ð17Þ

B. He et al. / Applied Mathematics and Computation 206 (2008) 141–149 143

Substituting (17) into the first expression of (6) and integrating it, we get

Z /

/ð0Þ

ðr� cÞdrffiffiffiffiffiffi13 a

qr

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirðr� cÞ

p ¼ �Z n

0dr; ð18Þ

where /ð0Þ and 0 are initial constants.Choosing the /ð0Þ ¼ c and completing (18), we obtain one implicit function expression for the compacton solution of Eq.

(2)

2tanh�1

ffiffiffiffiffiffiffiffiffiffiffiffi/� c

/

s¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffi/� cp ffiffiffiffi

/p þ

ffiffiffiffiffiffiffi13

a

rn: ð19Þ

Let s ¼ 2ffiffiffiffiffiffi/�cp ffiffiffi

/p þ

ffiffiffiffiffiffi13 a

qn, (19) can be rewritten as(

/ðsÞ ¼ ccosh2 12 s� �

;

nðsÞ ¼ 1x1

s� 2 tanh 12 s� �� �

;ð20Þ

where s is a parameter and x1 ¼ffiffiffiffiffiffi13 a

q.

2.1.1.2 When k ¼ 12 c � 1

6 ac, (16) can be reduced to

y ¼ �ffiffiffiffiffiffiffi13

a

r/: ð21Þ

Substituting (21) into the first expression of (6) and integrating it, we get

Z /

/ð0Þ

drffiffiffiffiffiffi13 a

qr¼ �

Z n

0dr: ð22Þ

Choosing the /ð0Þ ¼ c and completing (22), we obtain one explicit function expression for the peaked solitary wave solu-tion of Eq. (2)

/ðx� ctÞ ¼ c e�x2 jx�ctj; ð23Þ

where x2 ¼ffiffiffiffiffiffi13 a

q.

Remark 1. (23) has been obtained in Refs. [14,15].

Case 2.1.2: Taking h ¼ h1, from (14) we get

y2 ¼13 að/� /1Þ

2ð/� 2k�ca Þ

/� c: ð24Þ

2.1.2.1 When k ¼ 12 c � 1

4 ac, (24) can be reduced to

y ¼ �ffiffiffiffiffiffiffi13

a

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/� cÞð/þ 1

2cÞ

r: ð25Þ

Substituting (25) into the first expression of (6) and integrating it, we get

Z /

/ð0Þ

drffiffiffiffiffiffi13 a

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr� cÞ rþ 1

2 c� �q ¼ �

Z n

0dr: ð26Þ

Choosing the /ð0Þ ¼ c and completing (26), we obtain one explicit function expression for the compacton solution of Eq.(2)

/ðx� ctÞ ¼ c12

sinh2ðx3ðx� ctÞÞ þ cosh2ðx3ðx� ctÞÞ� �

; ð27Þ

where x3 ¼ 12

ffiffiffiffiffiffi13 a

q. Choosing the /ð0Þ ¼ � 1

2 c and completing (26), we obtain one explicit function expression for the com-pacton solution of Eq. (2)

/ðx� ctÞ ¼ �cðsinh2ðx4ðx� ctÞÞ þ 12

cosh2ðx4ðx� ctÞÞÞ; ð28Þ

where x4 ¼ 12

ffiffiffiffiffiffi13 a

q.

2.1.2.2 When k ¼ 12 c þ 1

2 ac, (24) can be reduced to

y ¼ �ffiffiffiffiffiffiffi13

a

rð/þ 2cÞ: ð29Þ

144 B. He et al. / Applied Mathematics and Computation 206 (2008) 141–149

Substituting (29) into the first expression of (6) and integrating it, we get

Z /

/ð0Þ

drffiffiffiffiffiffi13 a

qðrþ 2cÞ

¼ �Z n

0dr: ð30Þ

Choosing the /ð0Þ ¼ c and completing (30), we obtain one explicit function expression for the peaked solitary wave solu-tion of Eq. (2)

/ðx� ctÞ ¼ cð3e�x5 jx�ctj � 2Þ; ð31Þ

where x5 ¼ffiffiffiffiffiffi13 a

q.

Remark 2. (31) has been obtained in Ref. [14]

2.2. Exact travelling wave solutions of the GCH equation when m ¼ 2

When m ¼ 2, we have

/1;2 ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ðc � 2kÞ

a

r; ð32Þ

y2 ¼16 a/4 � ðc � 2kÞ/2 þ h

/� c; ð33Þ

h0 ¼ 0; h1 ¼3ðc � 2kÞ2

2a; hs ¼ �

16

ac4 þ ðc � 2kÞc2: ð34Þ

Case 2.2.1: Taking h ¼ h0, from (33) we get

y2 ¼16 að/2 � 6

a ðc � 2kÞÞ/2

/� c: ð35Þ

2.2.1.1 When k ¼ 12 c, (35) can be reduced to

y ¼ �

ffiffiffiffiffiffi16 a

q/2 ffiffiffiffiffiffiffiffiffiffiffiffi

/� cp

/� c: ð36Þ

Substituting (36) into the first expression of (6) and integrating it, we get

Z /

/ð0Þ

ðr� cÞdrffiffiffiffiffiffi16 a

qr2

ffiffiffiffiffiffiffiffiffiffiffiffir� cp ¼ �

Z n

0dr: ð37Þ

Choosing the /ð0Þ ¼ c and completing (37), we obtain one implicit function expression for the peaked solitary wave solu-tion of Eq. (2)

1ffiffiffiffiffiffi�cp tanh�1

ffiffiffiffiffiffiffiffiffiffiffiffi/� c�c

r¼ �

ffiffiffiffiffiffiffiffiffiffiffiffi/� cp

/þ

ffiffiffiffiffiffiffi16

a

rn

!; c < 0 ð38Þ

and one implicit function expression for the peaked compacton solution of Eq. (2)

1ffiffifficp tan�1

ffiffiffiffiffiffiffiffiffiffiffiffi/� c

c

r¼

ffiffiffiffiffiffiffiffiffiffiffiffi/� cp

/þ

ffiffiffiffiffiffiffi16

a

rn; c > 0: ð39Þ� �

Let s ¼ �ffiffiffiffiffiffi/�cp

/ þffiffiffiffiffiffi16 a

qn , l ¼

ffiffiffiffiffiffi/�cp

/ þffiffiffiffiffiffi16 a

qn (38) and (39) can be rewritten respectively asffiffiffiffiffiffip8

/ðsÞ ¼ c sech2ð �csÞ;nðsÞ ¼ 1

x6

12ffiffiffiffi�cp sinhð2

ffiffiffiffiffiffi�cp

sÞ � s� �

; c < 0

<: ð40Þ

and

/ðlÞ ¼ c sec2ðffiffifficp

lÞ;nðlÞ ¼ 1

x6� 1

2ffifficp sinð2

ffiffifficp

lÞ þ l� �

; c > 0;

8<: ð41Þ

where s and l are two parameters and x6 ¼ffiffiffiffiffiffi16 a

q. The profile of (41) is shown in Fig. 1(1-1).

2.2.1.2 When k ¼ 12 c � 1

12 ac2, (35) can be reduced to

y ¼ �ffiffiffiffiffiffiffi16

a

r/

ffiffiffiffiffiffiffiffiffiffiffiffi/þ c

p: ð42Þ

Fig. 1. The graphs of several kinds of travelling wave solutions of Eq. (2). The peaked compacton (corresponding to (41)): c ¼ 1; a ¼ 6; �5 < l < 5. (1-2)The compacton (corresponding to (55)): c ¼ 0:1; a ¼ 1. (1-3) The solitary wave (corresponding to (62)): k ¼ 1

4 ; c ¼ 1; a ¼ 14 ;�3 < X2 < 3. (1-4) The peaked

solitary wave (corresponding to (61)): k ¼ �2; c ¼ �3; a ¼ 14 ; �3 < X1 < 3. (1-5, 1-6) The kink-like wave (corresponding to (73)): c ¼ 1; a ¼ 1.

B. He et al. / Applied Mathematics and Computation 206 (2008) 141–149 145

Substituting (42) into the first expression of (6) and integrating it, we get

Z /

/ð0Þ

drffiffiffiffiffiffi16 a

qrffiffiffiffiffiffiffiffiffiffiffiffirþ cp ¼ �

Z n

0dr: ð43Þ

Choosing the /ð0Þ ¼ �c and completing (43), we obtain one explicit function expression for the compacton solution of Eq.(2)

/ðx� ctÞ ¼ �c sec2ðx7ðx� ctÞÞ; c < 0 ð44Þ

and one explicit function expression for the kink-like wave solution of Eq. (2)

/ðx� ctÞ ¼ 2ccoshðx8ðx� ctÞÞ � 1

; c > 0; ð45Þ

where x7 ¼ 12

ffiffiffiffiffiffiffiffiffiffiffiffi� 1

6 acq

, x8 ¼ffiffiffiffiffiffiffiffi16 ac

q.

Remark 3. The kink-like wave is defined in Ref. [18].

Choosing the /ð0Þ ¼ c and completing (43), we obtain two explicit function expressions for the kink-like wave solutionsof Eq. (2)

/ðx� ctÞ ¼ 4c

ð3� 2ffiffiffi2pÞe�x9ðx�ctÞ þ ð3þ 2

ffiffiffi2pÞe�x9ðx�ctÞ � 2

; c > 0; ð46Þ

where x9 ¼ffiffiffiffiffiffiffiffi16 ac

q.

146 B. He et al. / Applied Mathematics and Computation 206 (2008) 141–149

Case 2.2.2: Taking h ¼ hs, from (33) we get

y2 ¼ 16

a/3 þ 16

ac/2 þ 16

ac2 þ 2k� c� �

/þ 16

ac2 þ 2k� c� �

c: ð47Þ

2.2.2.1 When k ¼ 12 c, (47) can be reduced to

y ¼ �ffiffiffiffiffiffiffi16

a

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/þ cÞð/2 þ c2Þ

q: ð48Þ

Substituting (48) into the first expression of (6) and integrating it, we get

Z /

/ð0Þ

drffiffiffiffiffiffi16 a

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrþ cÞðr2 þ c2Þ

p ¼ �Z n

0dr: ð49Þ

Choosing the /ð0Þ ¼ �c and completing (49), we obtain one explicit function expression of Jacobin elliptic function typefor the compacton solution of Eq. (2)

/ðx� ctÞ ¼ cððffiffiffi2p� 1Þcnðx10ðx� ctÞ;m1Þ � ð

ffiffiffi2pþ 1ÞÞ

1þ cnðx10ðx� ctÞ;m1Þ; c < 0 ð50Þ

and one explicit function expression of Jacobin elliptic function type for the kink-like wave solution of Eq. (2)

/ðx� ctÞ ¼ cððffiffiffi2p� 1Þ � ð

ffiffiffi2pþ 1Þcnðx11ðx� ctÞ;m2ÞÞ

1þ cnðx11ðx� ctÞ;m2Þ; c > 0; ð51Þ

where x10 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffi2p

6 acq

; m1 ¼ffiffiffiffiffiffiffiffiffiffiffi

2p�1

2ffiffi2p

q; x11 ¼

ffiffiffiffiffiffiffiffiffiffiffiffi2p

6 acq

; m2 ¼ffiffiffiffiffiffiffiffiffiffiffi

2pþ1

2ffiffi2p

q.

2.2.2.2 When k ¼ 12 c � 1

6 ac2, (47) can be reduced to

y ¼ �ffiffiffiffiffiffiffi16

a

rð/þ cÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið/� cÞ

p: ð52Þ

Substituting (52) into the first expression of (6) and integrating it, we get

Z /

/ð0Þ

drffiffiffiffiffiffi16 a

qðrþ cÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr� cÞ

p ¼ �Z n

0dr: ð53Þ

Choosing the /ð0Þ ¼ c and completing (53), we obtain one explicit function expression for the kink-like wave solution ofEq. (2)

/ðx� ctÞ ¼ �c � 4ccoshðx12ðx� ctÞÞ � 1

; c < 0 ð54Þ

and one explicit function expression for the compacton solution of Eq. (2)

/ðx� ctÞ ¼ cð1þ 2 tan2ðx13ðx� ctÞÞÞ; c > 0; ð55Þ

where x12 ¼ffiffiffiffiffiffiffiffiffiffiffiffi� 1

3 acq

, x13 ¼ffiffiffiffiffiffiffiffiffiffi1

12 acq

. The profile of (55) is shown in Fig. 1(1-2).

Case 2.2.3: When c > 2k, taking h ¼ h1, from (33) we get

y ¼ �

ffiffiffiffiffiffi16 a

qð/� /1Þð/� /2Þffiffiffiffiffiffiffiffiffiffiffiffi

/� cp ; ð56Þ

where /1; /2 are defined by (32).Substituting (56) into the first expression of (6) and integrating it, we get

Z /

/ð0Þ

ðr� cÞdrffiffiffiffiffiffi16 a

qðr� /1Þðr� /2Þ

ffiffiffiffiffiffiffiffiffiffiffiffir� cp ¼ �

Z n

0dr: ð57Þ

Choosing the /ð0Þ ¼ c and completing (57), we obtain one implicit function expressions for the peaked solitary wavesolution of Eq. (2)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

/2 � cp

/1tanh�1

ffiffiffiffiffiffiffiffiffiffiffiffi/� cpffiffiffiffiffiffiffiffiffiffiffiffiffiffi

/2 � cp !

¼ffiffiffiffiffiffiffi16

a

rnþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi/1 � c

p/1

tanh�1ffiffiffiffiffiffiffiffiffiffiffiffi/� cpffiffiffiffiffiffiffiffiffiffiffiffiffiffi

/1 � cp !

; c < /2 ð58Þ

and one implicit function expressions for the solitary wave solution of Eq. (2)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi/1 � c

ptanh�1

ffiffiffiffiffiffiffiffiffiffiffiffi/� cpffiffiffiffiffiffiffiffiffiffiffiffiffiffi

/1 � cp

!¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffic � /2

ptan�1

ffiffiffiffiffiffiffiffiffiffiffiffi/� cpffiffiffiffiffiffiffiffiffiffiffiffiffiffi

c � /2

p !

�ffiffiffiffiffiffiffi16

a

r/1n; /2 < c < /1 ð59Þ

B. He et al. / Applied Mathematics and Computation 206 (2008) 141–149 147

and one implicit function expressions for the compacton solution of Eq. (2)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffic � /2

p/1

tan�1

ffiffiffiffiffiffiffiffiffiffiffiffi/� cpffiffiffiffiffiffiffiffiffiffiffiffiffiffi

c � /2

p !

¼ffiffiffiffiffiffiffi16

a

rnþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffic � /1

p/1

tan�1

ffiffiffiffiffiffiffiffiffiffiffiffi/� cpffiffiffiffiffiffiffiffiffiffiffiffiffiffi

c � /1

p !

; /1 < c: ð60Þ� � � � � �

Let X1 ¼ffiffiffiffiffiffi16 a

qnþ

ffiffiffiffiffiffiffiffi/1�cp

/1tanh�1

ffiffiffiffiffiffi/�cpffiffiffiffiffiffiffiffi

/1�cp , X2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffic � /2

ptan�1

ffiffiffiffiffiffi/�cpffiffiffiffiffiffiffiffi

c�/2

p �ffiffiffiffiffiffi16 a

q/1n, X3 ¼

ffiffiffiffiffiffi16 a

qnþ

ffiffiffiffiffiffiffiffic�/1

p/1

tan�1ffiffiffiffiffiffi/�cpffiffiffiffiffiffiffiffi

c�/1

p

(58)–(60) can be rewritten respectively as

/ðX1Þ ¼ c þ ð/2 � cÞtanh2 /1ffiffiffiffiffiffiffiffi/2�cp X1

� �;

nðX1Þ ¼ 1x14

X1 �ffiffiffiffiffiffiffiffi/1�cp

/1tanh�1

ffiffiffiffiffiffiffiffi/2�c/1�c

qtanh /1ffiffiffiffiffiffiffiffi

/2�cp X1

� �� �� �; c < /2

8>>><>>>:

ð61Þ

and

/ðX2Þ ¼ c þ ð/1 � cÞtanh2 X2ffiffiffiffiffiffiffiffi/1�cp� �

;

nðX2Þ ¼ 1x14/1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffic � /2

ptan�1

ffiffiffiffiffiffiffiffi/1�cc�/2

qtanh X2ffiffiffiffiffiffiffiffi

/1�cp� �� �

� X2

� �; /2 < c < /1

8>>><>>>:

ð62Þ

and

/ðX3Þ ¼ c þ ðc � /2Þ tan2 /1ffiffiffiffiffiffiffiffic�/2

p X3

� �;

nðX3Þ ¼ 1x14

X3 �ffiffiffiffiffiffiffiffic�/1

p/1

tan�1ffiffiffiffiffiffiffiffic�/2c�/1

qtan /1ffiffiffiffiffiffiffiffi

c�/2

p X3

� �� �� �; /1 < c;

8>>><>>>:

ð63Þ

where X1, X2, and X3 are three parameters and x14 ¼ffiffiffiffiffiffi16 a

q. The profiles of (62) and (61) are shown in Fig. 1(1-3) and (1-4).

2.3. Exact travelling wave solutions of the GCH equation when m ¼ 3

When m ¼ 3, we have

/1 ¼4ðc � 2kÞ

a

� �13

; ð64Þ

y2 ¼1

10 a/5 � ðc � 2kÞ/2 þ h/� c

; ð65Þ

h0 ¼ 0; h1 ¼65ðc � 2kÞ 2

c � 2ka

� �2 !1

3

; hs ¼ �1

10ac5 þ ðc � 2kÞc2: ð66Þ

Case 2.3.1: Taking h ¼ h0 from (65) we get

y2 ¼1

10 a/2ð/3 � 10a ðc � 2kÞÞ

/� c: ð67Þ

2.3.1.1 When k ¼ 12 c, (67) can be reduced to

y ¼ �

ffiffiffiffiffiffiffi1

10 aq

/2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi/ð/� cÞ

p/� c

: ð68Þ

Substituting (68) into the first expression of (6) and integrating it, we get

Z /

/ð0Þ

ðr� cÞdrffiffiffiffiffiffiffi1

10 aq

r2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirðr� cÞ

p ¼ �Z n

0dr: ð69Þ

Choosing the /ð0Þ ¼ c and completing (69), we obtain one explicit function expression for the travelling solutions of Eq.(2)

1823cðð18ðcx15ðx� ctÞÞ2Þ

23 þ 9ðcx15ðx� ctÞÞ2 þ ð218ðcx15ðx� ctÞÞ2Þ

13Þ

9ð9ðcx15ðx� ctÞÞ2 � 4Þððcx15ðx� ctÞÞ2Þ13

; ð70Þ

where x15 ¼ffiffiffiffia

10

p.

148 B. He et al. / Applied Mathematics and Computation 206 (2008) 141–149

2.3.1.2 When k ¼ 12 c � 1

20 ac3, (67) can be reduced to

y ¼ �ffiffiffiffiffiffiffiffiffi1

10a

r/

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 þ c/þ /2

q: ð71Þ

Substituting (71) into the first expression of (6) and integrating it, we get

Z /

/ð0Þ

drffiffiffiffiffiffiffi1

10 aq

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic2 þ crþ r2p ¼ �

Z n

0dr: ð72Þ

Choosing the /ð0Þ ¼ c and completing (72), we obtain two explicit function expressions for the kink-like wave solutionsof Eq. (2)

/ðx� ctÞ ¼ 4cð3þ 2ffiffiffi3pÞ

3ð7þ 4ffiffiffi3pÞe�x16ðx�ctÞ � 3e�x16ðx�ctÞ � 2ð3þ 2

ffiffiffi3pÞ; c > 0; ð73Þ

where x16 ¼ cffiffiffiffia

10

p. The profiles of (73) are shown in Fig. 1(1-5) and (1-6).

Choosing the /ð0Þ ¼ �c and completing (72), we obtain two explicit function expressions for the kink-like wave solutionsof Eq. (2)

/ðx� ctÞ ¼ �4c3e�x17ðx�ctÞ � e�x17ðx�ctÞ þ 2

; c < 0; ð74Þ

where x17 ¼ cffiffiffiffia

10

p.

Case 2.3.2: Taking h ¼ hs, from (65) we get

y ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

10a/4 þ 1

10ac/3 þ 1

10ac2/2 þ 1

10ac3 þ 2k� c

� �/þ 1

10ac3 þ 2k� c

� �c

s: ð75Þ

When k ¼ 12 c � 1

20 ac3, (75) can be reduced to

y ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

10a/4 þ 1

10ac/3 þ 1

10ac2/2

r: ð76Þ

Using the results of Ref. [17], we obtain the travelling wave solutions of Eq. (2) as follows:

/ðx� ctÞ ¼ c sech2ðx18ðx� ctÞÞð1� tanhðx18ðx� ctÞÞÞ2 � 1

; ð77Þ

/ðx� ctÞ ¼ c csch2ðx18ðx� ctÞÞ1� ð1� cothðx18ðx� ctÞÞÞ2

; ð78Þ

/ðx� ctÞ ¼ � c sech2ðx18ðx� ctÞÞ1� 2 tanhðx18ðx� ctÞÞ ; ð79Þ

/ðx� ctÞ ¼ c csch2ðx18ðx� ctÞÞ1� 2 cothðx18ðx� ctÞÞ ; ð80Þ

/ðx� ctÞ ¼ 2c cschðx19ðx� ctÞÞ�

ffiffiffi3p� cschðx19ðx� ctÞÞ

ð81Þ

and

/ðx� ctÞ ¼25 ac2 e�x19ðx�ctÞ

e�x19ðx�ctÞ � 110 ac

� �2 � 125 a2c2

; ð82Þ

where x18 ¼ 12 c

ffiffiffiffiffiffiffi1

10 aq

, x19 ¼ cffiffiffiffiffiffiffi1

10 aq

.

3. Conclusion

In this paper, some exact solutions of GCH equation are obtained by using the integral bifurcation method under someparameter conditions. Compare with Refs. [14–16], many new travelling waves such as peaked compacton, kink-like waveand compacton are found. It is shown that the integral bifurcation method is useful to many nonlinear partial equations.

Acknowledgements

We thank the referees’ valuable suggestions. This work is supported by Natural Science Foundation of YunNan Province(2007A080M) and the National Natural Science Foundation of China (10571062).

B. He et al. / Applied Mathematics and Computation 206 (2008) 141–149 149

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