Applied Mathematics and Computation 217 (2010) 1697–1703

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Applied Mathematics and Computation

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

New exact explicit peakon and smooth periodic wave solutionsof the Kð3;2Þ equation

Bin He a,*, Qing Meng b

a College of Mathematics, Honghe University, Mengzi, Yunnan 661100, PR Chinab Department of Physics, Honghe University, Mengzi, Yunnan 661100, PR China

a r t i c l e i n f o a b s t r a c t

Keywords:Kð3;2Þ equationIndependent variable transformationPeakonSmooth periodic waveExact explicit solution

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.10.007

* Corresponding author.E-mail addresses: hebinmtc@163.com, hbhhu@y

Based on an independent variable transformation, the Kð3;2Þ equation is investigated bythe bifurcation method of planar systems and qualitative theory of polynomial differentialsystem. In different regions of the parametric space, some new exact explicit peakon andsmooth periodic wave solutions are obtained. The results presented in this paper improvethe previous results.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

To study the role of nonlinear dispersion in the formation of patterns in the liquid drop, Rosenau and Hyman [1] showedthat in a particular generalization of the KdV equation:

ut þ ðumÞx þ ðunÞxxx ¼ 0; m > 0; 1 < n 6 3; ð1Þ

which is called Kðm;nÞ equation. They obtained solitary wave solutions with compact support in it, which they calledcompactons. For the case m ¼ n (m is an integer), these compactons had the property that the width was independentof the amplitude. In Ref. [1], Rosenau and Hyman studied Kð2;2Þ and Kð3;3Þ equations further and they stated thatKð3;2Þ equation had an elliptic function solution. In a later work, Rosenau [2] obtained elliptic function compactonsfor the cases of Kð4;2Þ and Kð5;3Þ. Phase compactons have also been investigated [3]. Rosenau [4] also studied theKðm;nÞ equation:

ut þ aðumÞx þ ðunÞxxx ¼ 0; ð2Þ

where a is a constant. He investigated nonlinear dispersion and compact structures [5], nonanalytic solitary waves [6], anda class of nonlinear dispersive–dissipative interactions [7]. Lately, Ismail and Taha [8] implemented a finite differencemethod and a finite element method to study the two types of equations Kð2;2Þ and Kð3;3Þ. A single compacton as wellas the interaction of compactons has been numerically studied. Then, Ismail [9] made an extension to the work in [8] andapplied a finite difference method on Kð2;3Þ equation and obtained numerical solutions of Kð2;3Þ equation [10]. Frutosand Lopez-Marcos [11] presented a finite difference method for the numerical integration of Kð2;2Þ equation. Zhou andTian [12] studied soliton solution of Kð2;2Þ equation. Xu and Tian [13] investigated the peaked wave solutions ofKð2;2Þ equation. Zhou et al. [14] obtained kink-like wave solutions and antikink-like wave solutions of Kð2;2Þ equation.In [15,16], general solutions to the Kðn;nÞ equation were studied. In [17], the nonlinear equation Kðm;nÞ was studied forall possible values of m and n. In [18], Tian and Yin investigated shock-peakon and shock-compacton solutions for Kðm;nÞ

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ahoo.com.cn (B. He).

1698 B. He, Q. Meng / Applied Mathematics and Computation 217 (2010) 1697–1703

equation by variational iteration method. In 2008, Biswas [19] considered the following Kðm;nÞ equation with generalizedevolution term:

ðulÞt þ aumux þ bðunÞxxx ¼ 0; ð3Þ

he presented a solitary wave ansatz and obtained a 1-soliton solution.In this paper, we will introduce an independent variable transformation to study the Kð3;2Þ equation:

ut þ ðu3Þx þ ðu2Þxxx ¼ 0; ð4Þ

and obtained its some new exact explicit peakon and smooth periodic wave solutions using the bifurcation method of planarsystems and qualitative theory of polynomial differential system [20,21].

Using the following independent variable transformation:

uðx; tÞ ¼ lþ wðnÞ ¼ lþ wðx� ctÞ; ð5Þ

where c ðc – 0Þ is the wave speed, l is a constant, and substituting (5) into (4), we obtain

�cw0 þ ððlþ wÞ3Þ0 þ ððlþ wÞ2Þ00 ¼ 0; ð6Þ

where ‘‘0” is the derivative with respect to n.Integrating (6) once with respect to n and setting the constant of integration to �l3, we have the following travelling

wave equation of (6):

�2ðwþ lÞw00 ¼ w3 þ 3lw2 þ ð3l2 � cÞwþ 2ðw0Þ2: ð7Þ

Letting y ¼ dwdn, we get the following planar system

dwdn¼ y;

dydn¼ �w3 þ 3lw2 þ ð3l2 � cÞwþ 2y2

2ðwþ lÞ : ð8Þ

System (8) is a two-parameter planar dynamical system depending on the parameter set ðl; cÞ. Since the phase orbits de-fined by the vector field of system (8) determine all travelling wave solutions of Eq. (4), we should investigate the bifurca-tions of phase portraits of system (8) in ðw; yÞ-phase plane as the parameters l; c are changed.

Clearly, on such straight line w ¼ �l in the phase plane ðw; yÞ, system (8) is discontinuous. Such system is called a singulartravelling wave system by one of authors [22].

The rest of this paper is organized as follows: In Section 2, we discuss the bifurcations of phase portraits of system (8),where explicit parametric conditions will be derived. In Section 3, we give some exact parametric representations of peakonand smooth periodic wave solutions of Eq. (4) in explicit form. A short conclusion will be given in Section 4.

2. Bifurcation sets and phase portraits of system (8)

Using the transformation dn ¼ �2ðwþ lÞds, it carries (8) into the Hamiltonian system

dwds¼ �2ðwþ lÞy; dy

ds¼ w3 þ 3lw2 þ ð3l2 � cÞwþ 2y2: ð9Þ

Since both system (8) and (9) have the same first integral

ðwþ lÞ2y2 þ 15

w3 þ lw2 þ 13ð6l2 � cÞwþ 1

2lð3l2 � cÞ

� �w2 ¼ h; ð10Þ

then the two systems above have the same topological phase portraits except the line w ¼ �l. Obviously, w ¼ �l is aninvariant straight-line solution of system (9).

Write D1 ¼ 4c � 3l2; D2 ¼ 12 lðl2 � cÞ; ws ¼ �l. Clearly, when D1 > 0, system (9) has three equilibrium point at

Oð0; 0Þ;A1;2ðw1;2;0Þ in w-axis, where w1;2 ¼�3l�

ffiffiffiffiD1

p2 . When D1 ¼ 0, system (9) has two equilibrium points at

Oð0; 0Þ; A12ðw12;0Þ in w-axis, where w12 ¼ � 3l2 . When D1 < 0, system (9) has only one equilibrium point at Oð0;0Þ in w-axis.

When D2 > 0, there exist two equilibrium points of system (9) in line w ¼ ws at S�ðws;Y�Þ; Y� ¼ �ffiffiffiffiffiffiD2p

.Let Mðwe; yeÞ be the coefficient matrix of the linearized system of the system (9) at an equilibrium point ðwe; yeÞ. Then we

have

Mðwe; yeÞ ¼�2ye �2ðwe þ lÞ

3w2e þ 6lwe þ ð3l2 � cÞ 4ye

� �;

and at this equilibrium point, we have

TraceðMð/i; yeÞÞ ¼ 2ye;

Jðwe; yeÞ ¼ detMðwe; yeÞ ¼ �8y2e þ 6w3

e þ 18lw2e þ 2ð9l2 � cÞwe þ 2lð3l2 � cÞ:

B. He, Q. Meng / Applied Mathematics and Computation 217 (2010) 1697–1703 1699

By the theory of planar dynamical systems, we know that for an equilibrium point ðwe; yeÞ of a planar integrable system, ifJðwe; yeÞ < 0 then the equilibrium point is a saddle point. If Jðwe; yeÞ > 0 and TraceðMð/i; yeÞÞ ¼ 0 then it is a center point. IfJðwe; yeÞ ¼ 0 and the Poincaré index of the equilibrium point is zero then it is a cusp.

Using the property of equilibrium points and bifurcation theory, we can obtain seven bifurcation curves as follows:

L1 : c ¼ 34l2; L2 : c ¼ 9

32ð13�

ffiffiffiffiffiffiffiffiffi105p

Þl2 ðl > 0Þ; L3 : c ¼ l2;

L4 : c ¼ 95l2 ðl < 0Þ; L5 : c ¼ 3l2; L6 : c ¼ 9

32ð13þ

ffiffiffiffiffiffiffiffiffi105p

Þl2 ðl < 0Þ; L7 : l ¼ 0;

which divide the ðl; cÞ-parameter plane into 22 subregions:

A1 ¼ ðl; cÞjl < 0; c <34l2

� �; A2 ¼ ðl; cÞjl < 0; c ¼ 3

4l2

� �; A3 ¼ ðl; cÞjl < 0;

34l2 < c < l2

� �;

A4 ¼ ðl; cÞjl < 0; c ¼ l2� �; A5 ¼ ðl; cÞjl < 0;l2 < c <

95l2

� �; A6 ¼ ðl; cÞjl < 0; c ¼ 9

5l2

� �;

A7 ¼ ðl; cÞjl < 0;95l2 < c < 3l2

� �; A8 ¼ ðl; cÞjl < 0; c ¼ 3l2� �

;

A9 ¼ ðl; cÞjl < 0;3l2 < c <9

32ð13þ

ffiffiffiffiffiffiffiffiffi105p

Þl2� �

; A10 ¼ ðl; cÞjl < 0; c ¼ 932ð13þ

ffiffiffiffiffiffiffiffiffi105p

Þl2� �

;

A11 ¼ ðl; cÞjl < 0; c >9

32ð13þ

ffiffiffiffiffiffiffiffiffi105p

Þl2� �

; A12 ¼ ðl; cÞjl ¼ 0; c > 0f g; A13 ¼ ðl; cÞjl ¼ 0; c < 0f g;

A14 ¼ ðl; cÞjl > 0; c <34l2

� �; A15 ¼ ðl; cÞjl > 0; c ¼ 3

4l2

� �;

A16 ¼ ðl; cÞjl > 0;34l2 < c <

932ð13�

ffiffiffiffiffiffiffiffiffi105p

Þl2� �

; A17 ¼ ðl; cÞjl > 0; c ¼ 932ð13�

ffiffiffiffiffiffiffiffiffi105p

Þl2� �

;

A18 ¼ ðl; cÞjl > 0;9

32ð13�

ffiffiffiffiffiffiffiffiffi105p

Þl2 < c < l2� �

; A19 ¼ ðl; cÞjl > 0; c ¼ l2� �

;

A20 ¼ ðl; cÞjl > 0;l2 < c < 3l2� �; A21 ¼ ðl; cÞjl > 0; c ¼ 3l2� �

; A22 ¼ ðl; cÞjl > 0; c > 3l2� �:

The bifurcation sets and phase portraits of system (8) shown in Fig. 1.

3. Exact explicit peakon and smooth periodic wave solutions

3.1. Peakon solutions

(i) From Fig. 1(1-6), it is seen that there are two heteroclinic orbits connecting with saddle points ðws;Y�Þ and ð0;0Þ. Theirexpressions are

y ¼ �ffiffiffi5p

5w

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwM � w

p; 0 < w 6 ws; ð11Þ

where ws ¼ �l; wM ¼ �3l; Y� ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffi� 2

5 l3q

; l < 0.

Substituting (11) into the dwdn ¼ y and integrating it along the heteroclinic orbits, yield equation

Z ws

w

dssffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwM � s

p ¼ffiffiffi5p

5jnj: ð12Þ

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

wðnÞ ¼ wMws

ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwM � ws

psinh jXnj þ

ffiffiffiffiffiffiffiwM

pcoshðXnÞÞ2

; ð13Þ

where X ¼ 110

ffiffiffiffiffiffiffiffiffiffi5wM

p.

Noting that uðx; tÞ ¼ lþ wðnÞ; n ¼ x� ct; c ¼ 95 l

2, we get the peakon solution uðx; tÞ as follows:

uðx; tÞ ¼ lþ wMwsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwM � ws

psinh jX x� 9

5 l2t

j þffiffiffiffiffiffiffiwM

pcosh X x� 9

5 l2t 2 : ð14Þ

The profile of (14) is shown in Fig. 2(2-1).

Fig. 1. The bifurcation sets and phase portraits of system (8). ð1-1Þ ðl; cÞ 2 A1: ð1-2Þ ðl; cÞ 2 A2: ð1-3Þ ðl; cÞ 2 A3: ð1-4Þ ðl; cÞ 2 A4: ð1-5Þ ðl; cÞ2 A5: ð1-6Þ ðl; cÞ 2 A6: ð1-7Þ ðl; cÞ 2 A7: ð1-8Þ ðl; cÞ 2 A8: ð1-9Þ ðl; cÞ 2 A9: ð1-10Þ ðl; cÞ 2 A10: ð1-11Þ ðl; cÞ 2 A11: ð1-12Þ ðl; cÞ 2 A12 : ð1-13Þ ðl; cÞ 2 A13:

ð1-14Þ ðl; cÞ 2 A14 : ð1-15Þ ðl; cÞ 2 A15: ð1-16Þ ðl; cÞ 2 A16: ð1-17Þ ðl; cÞ 2 A17: ð1-18Þ ðl; cÞ 2 A18: ð1-19Þ ðl; cÞ 2 A19: ð1-20Þ ðl; cÞ 2 A20: ð1-21Þ ðl; cÞ 2 A21 :

ð1-22Þ ðl; cÞ 2 A22.

1700 B. He, Q. Meng / Applied Mathematics and Computation 217 (2010) 1697–1703

—2

0t

—40

—20

0t

—5

200t

Fig. 2. Peakons of the Kð3;2Þ equation. ð2-1Þ l ¼ �2; c ¼ 365 ; x ¼ �1: ð2-2Þ l ¼ �2; c ¼ 1

8 ð117þ 9ffiffiffiffiffiffiffiffiffi105p

Þ; x ¼ 1: ð2-3Þ l ¼ 1; c ¼ 132 ð117� 9

ffiffiffiffiffiffiffiffiffi105p

Þ; x ¼ 1.

B. He, Q. Meng / Applied Mathematics and Computation 217 (2010) 1697–1703 1701

(ii) From Fig. 1(1-10), it is seen that there are two heteroclinic orbits connecting with saddle points ðws;Y�Þ and ðw2; 0Þ.Their expressions are

y ¼ �ffiffiffi5p

5ðw� w2Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwM � w

p; w2 < w 6 ws; ð15Þ

where ws ¼ �l; w2 ¼ � 12 l 3� 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6ð31þ 3

ffiffiffiffiffiffiffiffiffi105p

Þq� �

; wM ¼ � 14 l

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6ð31þ 3

ffiffiffiffiffiffiffiffiffi105p

Þq

; Y� ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� 1

64 l3ð85þ 9ffiffiffiffiffiffiffiffiffi105p

Þq

; l < 0.

Substituting (15) into the dwdn ¼ y, integrating it along the heteroclinic orbits and completing the integral, we have

wðnÞ ¼ wM � ðwM � w2Þtanh2 jXnj þ tanh�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwM � ws

wM � w2

s ! !; ð16Þ

where X ¼ 110

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5ðwM � w2Þ

p.

Noting that uðx; tÞ ¼ lþ wðnÞ; n ¼ x� ct; c ¼ 932 ð13þ

ffiffiffiffiffiffiffiffiffi105p

Þl2, we get the peakon solution uðx; tÞ as follows:

uðx; tÞ ¼ lþ wM � ðwM � w2Þtanh2 X x� 932ð13þ

ffiffiffiffiffiffiffiffiffi105p

Þl2t� �����

����þ tanh�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwM � ws

wM � w2

s ! !: ð17Þ

The profile of (17) is shown in Fig. 2(2-2).

(iii) From Fig. 1(1-17), it is seen that there are two heteroclinic orbits connecting with saddle points ðws;Y�Þ and ðw2;0Þ.Their expressions are

y ¼ �ffiffiffi5p

5ðw� w2Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwM � w

p; w2 < w 6 ws; ð18Þ

where ws ¼ �l; w2 ¼ 12 l �3� 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6ð31� 3

ffiffiffiffiffiffiffiffiffi105p

Þq� �

; wM ¼ 14 l

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6ð31� 3

ffiffiffiffiffiffiffiffiffi105p

Þq

; Y� ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 l3ð1� 9

32 ð13�ffiffiffiffiffiffiffiffiffi105p

ÞÞq

; l > 0.

Substituting (18) into the dwdn ¼ y, integrating it along the heteroclinic orbits and completing the integral, we have

wðnÞ ¼ wM � ðwM � w2Þtanh2 jXnj þ tanh�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwM � ws

wM � w2

s ! !; ð19Þ

where X ¼ 110

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5ðwM � w2Þ

p.

Noting that uðx; tÞ ¼ lþ wðnÞ; n ¼ x� ct; c ¼ 932 ð13�

ffiffiffiffiffiffiffiffiffi105p

Þl2, we get the peakon solution uðx; tÞ as follows:

uðx; tÞ ¼ lþ wM � ðwM � w2Þtanh2 X x� 932ð13�

ffiffiffiffiffiffiffiffiffi105p

Þl2t� �����

����þ tanh�1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwM � ws

wM � w2

s ! !: ð20Þ

The profile of (20) is shown in Fig. 2(2-3).

3.2. Smooth periodic wave solutions

(i) From Fig. 1(1-3), we see that there is one periodic orbit enclosing the center point ðw1;0Þ and passing pointsðc2;0Þ; ðc3;0Þ. Its expression is

0

1

t0

2

t0

2

t

Fig. 3. Smooth periodic waves of the Kð3;2Þ equation. ð3-1Þ l ¼ �1; c ¼ 1; x ¼ �1: ð3-2Þ l ¼ 0; c ¼ 3; x ¼ 1: ð3-3Þ l ¼ 2; c ¼ 4; x ¼ 1.

1702 B. He, Q. Meng / Applied Mathematics and Computation 217 (2010) 1697–1703

y ¼ �ffiffiffi5p

5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðc3 � wÞðw� c2Þðw� c1Þ

q; c2 6 w 6 c3; ð21Þ

where w1 ¼�3lþ

ffiffiffiffiffiffiffiffiffiffiffiffi4c�3l2p

2 ; l < 0; 34 l

2 < c < l2 and c1; c2; c3 ðc1 < c2 < c3Þ are three real roots of x3 þ 3lx2þ3l2 � 5

3 c

xþ 56 lc � 3

2 l3 ¼ 0 and can be obtained by the Cardan formula.Substituting (21) into the dw

dn ¼ y, integrating it alongthe periodic orbit and completing the integral, we have

wðnÞ ¼ c3cn2ðXn; kÞ þ c2sn2ðXn; kÞ; ð22Þ

where X ¼ffiffi5p

10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic3 � c1p

; k ¼ffiffiffiffiffiffiffiffiffiffic3�c2c3�c1

q; snð�; �Þ and cnð�; �Þ are two Jacobian elliptic functions (see [23]).Noting that uðx; tÞ

¼ lþ wðnÞ; n ¼ x� ct, we get the smooth periodic wave solution uðx; tÞ as follows:

uðx; tÞ ¼ lþ c3cn2ðXðx� ctÞ; kÞ þ c2sn2ðXðx� ctÞ; kÞ: ð23Þ

(ii) From Fig. 1(1-4), we see that there is one periodic orbit enclosing the center point ðw1;0Þ and passing pointsðws;0Þ; ðwM ;0Þ. Its expression is

y ¼ �ffiffiffi5p

5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðwM � wÞðw� wsÞðw� wmÞ

p; ws 6 w 6 wM; ð24Þ

where w1 ¼ �2l; ws ¼ �l; wM;m ¼ �lð1� 13

ffiffiffiffiffiffi15pÞ; l < 0.Substituting (24) into the dw

dn ¼ y, integrating it along the periodicorbit and completing the integral, we have

wðnÞ ¼ wMcn2ðXn; kÞ þ wssn2ðXn; kÞ; ð25Þ

where X ¼ffiffi5p

10

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwM � wm

p; k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiwM�wswM�wm

q.Noting that uðx; tÞ ¼ lþ wðnÞ; n ¼ x� ct; c ¼ l2, we get the smooth periodic wave

solution uðx; tÞ as follows:

uðx; tÞ ¼ lþ wMcn2ðXðx� l2tÞ; kÞ þ wssn2ðXðx� l2tÞ; kÞ: ð26Þ

The profile of (26) is shown in Fig. 3(3-1).(iii) From Fig. 1(1-12), we see that there is one periodic orbit enclosing the center point ðw1;0Þ and passing points

ð0;0Þ; ðwM ;0Þ. Its expression is

y ¼ �ffiffiffi5p

5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðwM � wÞðw� 0Þðw� wmÞ

p; 0 6 w 6 wM; ð27Þ

where w1 ¼ffiffifficp; wM;m ¼ � 1

3

ffiffiffiffiffiffiffiffi15cp

; c > 0.Substituting (27) into the dwdn ¼ y, integrating it along the periodic orbit and complet-

ing the integral, we have

wðnÞ ¼ wMcn2ðXn; kÞ; ð28Þ

where X ¼ffiffi5p

10

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiwM � wm

p; k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiwM

wM�wm

q.Noting that uðx; tÞ ¼ lþ wðnÞ; n ¼ x� ct; l ¼ 0, we get the smooth periodic wave

solution uðx; tÞ as follows:

uðx; tÞ ¼ wMcn2ðXðx� ctÞ; kÞ: ð29Þ

The profile of (29) is shown in Fig. 3(3-2).(iv) From Fig. 1(1-19), we see that there is one periodic orbit enclosing the center point ð0;0Þ and passing points

ðws;0Þ; ðwM ;0Þ. Its expression is

y ¼ �ffiffiffi5p

5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðwM � wÞðw� wsÞðw� wmÞ

p; ws 6 w 6 wM; ð30Þ

B. He, Q. Meng / Applied Mathematics and Computation 217 (2010) 1697–1703 1703

where ws ¼ �l; wM;m ¼ lð�1� 13

ffiffiffiffiffiffi15pÞ; l > 0.Substituting (30) into the dw

dn ¼ y, integrating it along the periodic orbit andcompleting the integral, we can get the smooth periodic wave solution uðx; tÞ as (26). Its profile is shown in Fig. 3(3-3).

(v) From Fig. 1(1-20), we see that there is one periodic orbit enclosing the center point ð0;0Þ and passing pointsðc2;0Þ; ðc3;0Þ. Its expression is

y ¼ �ffiffiffi5p

5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðc3 � wÞðw� c2Þðw� c1Þ

q; c2 6 w 6 c3; ð31Þ

where l > 0; l2 < c < 3l2 and c1; c2; c3ðc1 < c2 < c3Þ are three real roots of x3 þ 3lx2 þ ð3l2 � 53 cÞxþ 5

6 lc � 32 l

3 ¼ 0 andcan be obtained by the Cardan formula.Substituting (31) into the dw

dn ¼ y, integrating it along the periodic orbit and complet-ing the integral, we can get the smooth periodic wave solution uðx; tÞ as (23).

4. Conclusion

In this paper, based on the transformation (5) and using the bifurcation theory and the method of phase portrait analysis,we investigated the bifurcations of Eq. (4) and obtained some new exact explicit peakon and smooth periodic wave solutions.Using the method given in this paper, we also can obtain some new exact travelling wave solutions for the Kð3;3Þ equation.We would like to study the Kðm;nÞ equation further.

Acknowledgements

The authors thank the referees for some perceptive comments and for some valuable suggestions. This work is supportedby the Natural Science Foundation of YunNan Province, China (2007A080M).

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