The extended Jacobi elliptic function method to solve a generalized Hirota–Satsuma coupled KdV equations

Chaos, Solitons and Fractals 26 (2005) 1415–1421 www.elsevier.com/locate/chaos

The extended Jacobi elliptic function method to solve a generalized Hirota–Satsuma coupled KdV equations Yaxuan Yu b

a,*

, Qi Wang

a,b

, Hongqing Zhang

a,b

a Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China Key Laboratory of Mathematics Mechanization, Chinese Academy of Sciences, Beijing 100080, China

Accepted 31 March 2005

Abstract In this paper, we extend the Jacobi elliptic function rational expansion method by using a new generalized ansa¨tz. With the help of symbolic computation, we construct more new explicit exact solutions of nonlinear evolution equations (NLEEs). We apply this method to a generalized Hirota–Satsuma coupled KdV equations and gain more general solutions. The general solutions not only contain the solutions by the existing Jacobi elliptic function expansion methods but also contain many new solutions. When the modulus of the Jacobi elliptic functions m ! 1 or 0, the corresponding solitary wave solutions and triangular functional (singly periodic) solutions are also obtained. Ó 2005 Elsevier Ltd. All rights reserved.

1. Introduction Nonlinear evolution equations are related to nonlinear science such as physics, mechanics, biology, etc. To further explain some physical phenomena, searching for exact solutions of NLEEs is very important. Up to now, there have been many powerful methods, such as Ba¨cklund transformation, Darboux transformation, variable separation approach, various tanh methods, Painleve´ method, generalized hyperbolic-function method, homogeneous balance method, similarity reduction method and so on [1–10]. In recent years, some researchers have paid attention to the relations between the elliptic functions (e.g., Jacobi elliptic functions and Weierstrass elliptic functions) and some NLEEs. They study how to express more exact solutions of NLEEs using the elliptic functions. Liu et al. [11] proposed Jacobi elliptic function expansion method. Fan and coworker [12] and Yan [13] extended this method. Chen et al. presented a new Jacobi elliptic function rational expansion method in [14,15]. Based on the above ideas, we extend ChenÕs method by means of a new general ansa¨tz. It is more powerful than the above existing Jacobi elliptic function method [11–15] to construct more new exact doubly-periodic rational formal solutions. For illustration, we apply it to a generalized Hirota–Satsuma coupled KdV equations, which is

*

Corresponding author. E-mail address: [email protected] (Y. Yu).

0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.011

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Y. Yu et al. / Chaos, Solitons and Fractals 26 (2005) 1415–1421

1 ut ¼ uxxx
3uux þ 3ðvwÞx ; 2

ð1:1:1Þ

vt ¼
vxxx þ 3uvx ;

ð1:1:2Þ

wt ¼
wxxx þ 3uwx .

ð1:1:3Þ

The generalized Hirota–Satsuma coupled KdV equations was first derived by Wu et al. [16]. When w = v* and w = v Eqs. (1.1) reduce respectively to a new complex coupled KdV equation in [16] and the Hirota–Satsuma equation in [17,18]. Hu [19] derived five kinds of solutions for Eqs. (1.1) by ansa¨tz method. Using an extended tanh-function, Fan obtained four kinds of solitary solutions in [20]. Chen et al. [21], using the well-known Riccati equation, gained more new exact solutions of this system. Recently, Fan and coworkers [12] construct itÕs exact solitary solutions by Jacobi elliptic function. In this paper we study it by the extended Jacobi elliptic function rational expansion method. This paper is organized as follows. In Section 2, the extended Jacobi elliptic function rational expansion method is presented. In Section 3, as an example, we apply this method to a generalized Hirota–Satsuma coupled KdV equations. Finally, we will give conclusion.

2. Summary of the new method The key steps of this method are presented as follows: Step 1. For a given NLEEs with some physical fields ui(x, y, t) (i = 1, 2,   ) in three variables x, y, t, F i ðui ; uit ; uix ; uiy ; uitt ; uixt ; uiyt ; uixx ; uiyy ; uixy ; . . .Þ ¼ 0;

ð2:1Þ

we consider itÕs travelling wave solutions in the following form: ui ðx; y; tÞ ¼ ui ðnÞ;

n ¼ ax þ ky þ bt;

ð2:2Þ

where a, b, k are constants to be determined later. Then Eq. (2.1) is reduced to a nonlinear ordinary differential equation (ODE): Gi ðui ; u0i ; u00i ; . . .Þ ¼ 0.

ð2:3Þ

Step 2. We search for itÕs solutions in the form: ui ðnÞ ¼ ai;0 þ

mi X j¼1

ai;2j
1 snj n ai;2j snj
1 n cn n ; j þ ðl1 sn n þ l2 cn n þ 1Þ ðl1 sn n þ l2 cn n þ 1Þj

ð2:4Þ

where sn n, cn n, dn n, ns n, cs n, and ds n etc. are Jacobi elliptic functions. They have the following properties: cn2 n þ sn2 n ¼ dn2 n þ m2 sn2 n ¼ 1; ns2 n ¼ 1 þ cs2 n;

ns2 n ¼ m2 þ ds2 n;

sn0 n ¼ cn n dn n;

cn0 n ¼
sn n dn n;

ns0 n ¼
ds n cs n;

ds0 n ¼
cs n ns n;

ð2:5:1Þ m2
1 þ ds2 n ¼ cs2 n;

ð2:5:2Þ

dn0 n ¼
m2 sn n cn n;

ð2:5:3Þ

cs0 n ¼
ns n ds n;

ð2:5:4Þ

and m (0 < m < 1) is the modulus. The Jacobi–Glaisher functions for elliptic function can be found in Refs. [22– 24]. Step 3. By balancing the highest order derivative term and the nonlinear terms in Eq. (2.3) we can gain values of mi. Step 4. Substitute Eq. (2.4) into Eq. (2.3) with condition of Eq. (2.5). Then set all coefficients of sni n cnj n (i = 1,2, . . .; j = 0, 1) to be zero. We get an over-determined nonlinear algebraic system with respect to ai,0, ai,j (j = 1,2 . . ., 2mi), l1, l2, a, b, k. Step 5. Solving the over-determined nonlinear algebraic system with the help of Maple, we can gain the explicit expressions for ai,0, ai,j (i = 1, 2, . . ., j = 1, 2 . . ., 2mi), l1, l2, a, b, k.

Y. Yu et al. / Chaos, Solitons and Fractals 26 (2005) 1415–1421

1417

In this way, we would get double periodic solutions with Jacobi elliptic function. Since lim sn n ¼ tanh n;

lim cn n ¼ sech n;

m!1

m!1

lim ns n ¼ coth n;

lim cs n ¼ csch n;

m!1

lim sn n ¼ sin n;

m!1

lim cn n ¼ cos n;

m!0

m!0

lim ns n ¼ csc n;

lim cs n ¼ cot n;

m!0

m!0

lim dn n ¼ sech n;

ð2:6:1Þ

lim ds n ¼ csch n;

ð2:6:2Þ

m!1

m!1

lim dn n ¼ 1;

ð2:6:3Þ

lim ds n ¼ csc n;

ð2:6:4Þ

m!0

m!0

ui degenerate respectively as the following form: 1. solitary wave solutions: ui ðnÞ ¼ ai;0 þ

k X j¼1

ai;2j
1 tanhj n ai;2j tanhj
1 n sech n ; j þ ðl1 tanh n þ l2 sech n þ 1Þ ðl1 tanh n þ l2 sech n þ 1Þj

ð2:7Þ

2. triangular function formal solutions: ui ðnÞ ¼ ai;0 þ

k X j¼1

ai;2j
1 sinj n ai;2j sinj
1 n cos n . j þ ðl1 sin n þ l2 cos n þ 1Þ ðl1 sin n þ l2 cos n þ 1Þj

ð2:8Þ

Remark 1. When l2 = 0 we gain the solutions that obtained by the method in [14]. Remark 2. If we replace the Jacobi elliptic functions sn n, cn n in the ansa¨tz (2.4) with other pairs of Jacobi elliptic functions such as sn n and dn n; ns n and cs n; ns n and ds n; sc n and nc n; dc n and nc n; sd n and nd n; cd n and nd n. It is necessary to point out that the above combinations only require solving the recurrent coefficient relation or derivative relation for the terms of polynomial for computation closed. Therefore other new double periodic wave solutions, solitary wave solutions and triangular functional solutions can be obtained for some system. For simplicity, we omit them here. Thus, the extended Jacobi elliptic function rational expansion method is more powerful than the method by Liu et al. [11], the method by Fan and coworkers [12] and the method extended by Yan [13] and Chen and coworkers [14,15]. The solutions which contain solitary wave solutions, singular solitary solutions and triangular function formal solutions can be obtained by the extended method.

3. The Jacobi elliptic function solutions for a generalized Hirota–Satsuma coupled KdV equations Let us consider the generalized Hirota–Satsuma coupled KdV equations, i.e. 1 ut ¼ uxxx
3uux þ 3ðvwÞx ; 2

ð3:1:1Þ

vt ¼
vxxx þ 3uvx ;

ð3:1:2Þ

wt ¼
wxxx þ 3uwx .

ð3:1:3Þ

According to the above steps, to seek travelling wave solutions of Eqs. (3.1), we first make the transformation uðx; tÞ ¼ uðnÞ;

vðx; tÞ ¼ vðnÞ;

wðx; tÞ ¼ wðnÞ;

n ¼ ax þ bt;

ð3:2Þ

where a and b are constants to be determined. We substitute Eqs. (3.2) into Eqs. (3.1) and Eqs. (3.1) becomes bu0
12a3 u000 þ 3uu0
3v0 w
3vw0 ¼ 0;

ð3:3:1Þ

bv0 þ a3 v000
3uv0 ¼ 0;

ð3:3:2Þ

bw0 þ a3 w000
3uw0 ¼ 0.

ð3:3:3Þ

By balancing u000 (n) and (v(n)w(n)) 0 , v000 (n) and uv 0 (n), w000 (n) and uw 0 (n) in Eqs. (3.3), we suppose that Eqs. (3.3) has the following formal solutions

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Y. Yu et al. / Chaos, Solitons and Fractals 26 (2005) 1415–1421

uðnÞ ¼ a0 þ

a1 sn n a2 cn n a3 sn2 n a4 sn n cn n þ þ þ ; l1 sn n þ l2 cn n þ 1 l1 sn n þ l2 cn n þ 1 ðl1 sn n þ l2 cn n þ 1Þ2 ðl1 sn n þ l2 cn n þ 1Þ2

vðnÞ ¼ b0 þ

b1 sn n b2 cn n b3 sn2 n b4 sn n cn n þ ; þ þ l1 sn n þ l2 cn n þ 1 l1 sn n þ l2 cn n þ 1 ðl1 sn n þ l2 cn n þ 1Þ2 ðl1 sn n þ l2 cn n þ 1Þ2

wðnÞ ¼ c0 þ

c1 sn n c2 cn n c3 sn2 n c4 sn n cn n þ þ þ ; l1 sn n þ l2 cn n þ 1 l1 sn n þ l2 cn n þ 1 ðl1 sn n þ l2 cn n þ 1Þ2 ðl1 sn n þ l2 cn n þ 1Þ2

ð3:4Þ

and a0, a1, a2, a3, a4, b0, b1, b2, b3, b4, c0, c1, c2, c3, c4 are constants to be determined later. With the aid of Maple, substituting Eqs. (3.4) into Eqs. (3.3) with condition of Eqs. (2.5), we yield a set of algebraic equations for sni n cnj n, (i, l = 1,2, . . .; j = 0, 1). Setting the coefficients of these terms to zero, we gain a set of over-determined algebraic equations with respect to a0, a1, a2, a3, a4, b0, b1, b2, b3, b4, c0, c1,c2, c3, c4, a and b. By using of Maple, we get many families of solutions. We only choose some of them as follows: Case 1 1 3 a0 ¼ ðb
4a
4a3 m2 Þ; 3 4a3 m2 ð2bb3
3a3 m2 b0
2a3 b3
2a3 b3 m2 Þ c0 ¼ ; a3 ¼ 4a3 m2 ; 3b23

l1 ¼ l2 ¼ a1 ¼ a2 ¼ a4 ¼ b1 ¼ b2 ¼ b4 ¼ c1 ¼ c2 ¼ c4 ¼ 0; c3 ¼

4a6 m4 ; b3

ð3:5Þ

where b0, b3, a and b are arbitrary constants. Case 2 ia6 m4 ; a4 ¼ 2ia3 m2 ; c4 a3 m2 ð3c0 a3 m2  ðia3 c4 m2
4ibc4 þ 4ia3 c4 ÞÞ a6 m4 b0 ¼ ; b4 ¼
; 2 3c4 c4

l1 ¼ l2 ¼ a1 ¼ a2 ¼ b1 ¼ b2 ¼ c1 ¼ c2 ¼ 0; c3 ¼ ic4 ;

1 a0 ¼ ðb
4a3
a3 m2 Þ; 3

a3 ¼ 2a3 m2 ;

b3 ¼

ð3:6Þ

where c0, c4, a and b are arbitrary constants. Case 3 l1 ¼ a1 ¼ a4 ¼ b1 ¼ b4 ¼ c1 ¼ c4 ¼ 0; b0 ¼

l2 ¼ 1;

8ða6 m2
ba3 Þðc2
2c3 Þ
a6 ð7c2
8c3 þ 6c0 Þ 6ðc2
2c3 Þ2

b3 ¼

a6
c2 b2 þ 2c3 b2 ; 2ð2c3
c2 Þ


3b2 ;

a2 ¼ 2a3
2a3 ;

1 a0 ¼ ðb
4a3 m2 þ 5a3 Þ
a3 ; 3 ð3:7Þ

where a3, b2, c0, c2, c3, a and b are arbitrary constants. Case 4 l1 ¼ a1 ¼ a4 ¼ b4 ¼ c4 ¼ 0; c2 ¼ 2c3 ;

b0 ¼

l2 ¼ 1;

a2 ¼ ð2a3
a3 Þ;


a3 ðc0 a3 þ c3 Þð4b þ 2a3 m2
a3 Þ 1
b2 ; 12c21 2

1 a0 ¼ ð2a3 þ b
a3 m2 Þ
a3 ; 3 a3 ð4b þ 2a3 m2
a3 Þ b1 ¼ ; 12c1

1 b3 ¼  b2 ; 2 ð3:8Þ

where a3, b2, c0, c1, c3, a and b are arbitrary constants. Case 5 c1 b4  4a6 ðm2
1Þ2 1 1 ; a0 ¼ ðb þ 5a3
a3 m2 Þ
a4 ; 3 2 c1 2 6 2 6 3 6 4 3 2 6 2 3c1 ð8a m
c1 b4 Þ
20a c1
16bc1 a
4a m c1 þ 16ba c1 m
24a c0 ðm
1Þ ; c3 ¼
c1 ; b0 ¼ 6c21 c2 ¼ 0;

l1 ¼ 1;

a3 ¼ 4a3 ð1
m2 Þ;

l2 ¼ 1;

b3 ¼

b2 ¼
b4 ;


4a6 ð1
2m2 Þ2 ; c1

b1 ¼

a1 ¼ 4a3 ðm2
1Þ þ a4 ;

where a4, b4, c0, c1, c4,a and b are arbitrary constants.

a2 ¼
a4 ;

ð3:9Þ

Y. Yu et al. / Chaos, Solitons and Fractals 26 (2005) 1415–1421

1419

Case 6 a2 ¼ a4 ¼ 0; c0 ¼

l1 ¼
1;

l2 ¼ 1;

c2 ¼ c4 ;

b3 ¼

4a6 ð1
m2 Þ2 ; c3

4a6 c3 m4
16a3 c3 m2 b þ 20a6 c3
6c23 b0 þ 16bc3 a3
3c23 b4 24a6 ð1


a1 ¼ 4a3 ð1
m2 Þ;

m 2 Þ2

1 a0 ¼ ðb þ 5a3
a3 m2 Þ; 3

1
c4 ; 2

c1 ¼
c4 þ c3 ; b1 ¼

b2 ¼ b4 ;

4a6 ð1
m2 Þ2
c3 b4 ; c3

a3 ¼ 4a3 ð1
m2 Þ;

ð3:10Þ

where b0, b4, c3, c4, a and b are arbitrary constants. Therefore we obtain six families of explicit exact travelling wave solutions, as follows: According to Case 1 1 u1 ¼ ðb
4a3
4a3 m2 Þ þ 4a3 m2 sn2 n; 3 v1 ¼ b0 þ b3 sn2 n;

ð3:11Þ

4a3 m2 ð2bb3
3a3 m2 b0
2a3 b3
2a3 b3 m2 Þ 4a6 m4 sn2 n w1 ¼ þ ; b3 3b23 where n = ax + bt and b0, b3, a and b are arbitrary constants. According to Case 2 1 u2 ¼ ðb
4a3
a3 m2 Þ þ 2a3 m2 sn2 n  2ia3 m2 sn n cn n; 3 v2 ¼

a3 m2 ð3c0 a3 m2  ðia3 c4 m2
4ibc4 þ 4ia3 c4 ÞÞ ia6 m4 sn2 n a6 m4 sn n cn n 
; c4 c4 3c24

ð3:12Þ

w2 ¼ c0  ic4 sn2 n þ c4 sn n cn n; where n = ax + bt and c0, c4, a, b are arbitrary constants. According to Case 3 u3 ¼

v3 ¼

b
4a3 m2 þ 5a3
3a3 ð2a3
2a3 Þ cn n a3 sn2 n þ þ ; 1  cn n 3 ð1  cn nÞ2 8ða6 m2
ba3 Þðc2
2c3 Þ
a6 ð7c2
8c3 þ 6c0 Þ 6ðc2
2c3 Þ

w 3 ¼ c0 þ

2


3b2 þ

b2 cn n ða6
c2 b2 þ 2c3 b2 Þ sn2 n
; 1  cn n 2ðc2
2c3 Þð1  cn nÞ2

ð3:13Þ

c2 cn n c3 sn2 n þ ; 1  cn n ð1  cn nÞ2

where n = ax + bt and a3, b2, c0, c2, c3, a, b are arbitrary constants. According to Case 4 u4 ¼

2a3 þ b
a3
a3 m2 ð2a3
a3 Þ cn n a3 sn2 n þ  ; 1  cn n 3 ð1  cn nÞ2

v4 ¼


a3 ðc0 a3 þ c3 Þð4b þ 2a3 m2
a3 Þ 1 a3 ð4b þ 2a3 m2
a3 Þ sn n b2 cn n b2 sn2 n  þ
b2 þ ; 2 12c1 2 12c1 ð1  cn nÞ 1  cn n 2ð1  cn nÞ2

w 4 ¼ c0 þ

c1 sn n 2c3 cn n c3 sn2 n  þ ; 1  cn n 1  cn n ð1  cn nÞ2

where n = ax + bt and a3, b2, c0, c1, c3, a, b are arbitrary constants.

ð3:14Þ

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Y. Yu et al. / Chaos, Solitons and Fractals 26 (2005) 1415–1421

According to Case 5 1 1 ða4 þ 4a3 m2
4a3 Þ sn n a4 cn n 4a3 ðm2
1Þ sn2 n a4 sn n cn n u5 ¼ ðb þ 5a3
a3 m2 Þ
a4 þ þ ;

3 2 sn n  cn n þ 1 sn n  cn n þ 1 ðsn n  cn n þ 1Þ2 ðsn n  cn n þ 1Þ2 v5 ¼


a3 ðc0 a3 þ c3 Þð4b þ 2a3 m2
a3 Þ 1 ðc1 b4  ð4a6 m4 þ 4a6 ðm2
1Þ2 ÞÞ sn n b4 cn n

þ b 2 12c21 2 c1 ðsn n  cn n þ 1Þ sn n  cn n þ 1 4a6 ð1
m2 Þ2 sn2 n

b4 sn n cn n þ ; c1 ðsn n  cn n þ 1Þ2 ðsn n  cn n þ 1Þ2 c1 sn n c1 sn2 n
; w5 ¼ c0 þ sn n  cn n þ 1 ðsn n  cn n þ 1Þ2


ð3:15Þ where n = ax + bt and a4, c0, c1, a, b are arbitrary constants. According to Case 6 1 4a3 ðm2
1Þ sn n 4a3 ðm2
1Þ sn2 n
; u6 ¼ ðb þ 5a3
a3 m2 Þ
3
sn n  cn n þ 1 ð
sn n  cn n þ 1Þ2 v6 ¼ b0 þ w6 ¼

ð4a6 ð1
m2 Þ2
c3 b4 Þ sn n b4 cn n 4a6 ð1
m2 Þ2 sn2 n b4 sn ncn n þ þ þ ; c3 ð
sn n  cn n þ 1Þ
sn n  cn n þ 1 c3 ð
sn n  cn n þ 1Þ2 ð
sn n  cn n þ 1Þ2

4a6 c3 m4
16a3 c3 m2 b þ 20a6 c3
6c23 b0 þ 16bc3 a3
3c23 b4 24a6 ð1
m2 Þ2 þ

ð3:16Þ

1 ðc3
c4 Þ sn n
c4 þ 2
sn n  cn n þ 1

c4 cn n c3 sn2 n c4 sn n cn n þ þ ; 2
sn n  cn n þ 1 ð
sn n  cn n þ 1Þ ð
sn n  cn n þ 1Þ2

where n = ax + bt and b0, b4, c3, c4, a, b are arbitrary constants. Remark 3. It is easily seen that the above obtained solutions including the solutions which can be obtained by Liu et al. [11], Fan and coworkers [12], Yan [13] and Chen and coworkers [14,15]. But to the best of our knowledge, the rest of solutions of Eqs. (3.1) have not been found before. Remark 4. We draw the figures of solution v5 in the following to show the relations among Jacobi elliptic function solutions, solitary wave solutions and triangular functional solutions (Fig. 1).

4. Summary and conclusions In this paper, we extend the Jacobi elliptic function rational expansion method. This method is more powerful than the method proposed recently by Liu et al. [11], Fan and coworkers [12], Yan and coworkers [13] and Chen and coworkers [14,15]. A generalized Hirota–Satsuma coupled KdV equations is chosen to illustrate the method and six families of new Jacobi elliptic function solutions are obtained. When the modulus m ! 1, some of these obtained solutions degen-

Fig. 1. The solution v13 where l2 =
1, a = 1, b = 10, b4 = 4, c1 = 1, c0 = 0 and A, B and C are corresponding to m = 0, m = 0.5, m = 1, respectively.

Y. Yu et al. / Chaos, Solitons and Fractals 26 (2005) 1415–1421

1421

erate as solitary wave solutions. The method can be also applied to many other NLEEs in mathematical physics. Further work about various extensions and improvement of Jacobi function method need us to find more general ansa¨tzs or subequation.

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