Chaos, Solitons and Fractals 26 (2005) 393–398

www.elsevier.com/locate/chaos

New Weierstrass elliptic function solutions of theN-coupled nonlinear Klein–Gordon equations

Yong Chen a,*, Zhenya Yan b

a Department of Mathematics, Ningbo University, Ningbo 315211, Chinab Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences,

Beijing 100080, China

Accepted 5 January 2005

Communicated by M. Wadati

Abstract

With the aid of symbolic computation, three families of new doubly periodic solutions are obtained for the N-cou-

pled nonlinear Klein–Gordon equations in terms of the Weierstrass elliptic function. Moreover Jacobi elliptic function

solutions and solitary wave solutions are also given as simple limits of doubly periodic solutions.

� 2005 Elsevier Ltd. All rights reserved.

1. Introduction

With the development of soliton theory, it is interesting and difficult to investigate explicitly exact solutions including

solitary wave solutions of nonlinear wave equations arising from nonlinear science. There have existed some methods to

seek some types of solutions of nonlinear wave equations. Since Jacobi elliptic function solutions include not only sol-

itary wave solutions, for example [1]

0960-0

doi:10.

* Co

E-m

snðn;mÞjm!1 ¼ tanh n; cnðn;mÞjm!1 ¼ sechn; dnðn;mÞjm!1 ¼ sechn; ð1Þ

periodic solutions, for instance [2]

snðn;mÞjm!0 ¼ sin n; cnðn;mÞjm!0 ¼ cos n; ð2Þ

but more types of solutions depending on other different modulus. In addition, Jacobi elliptic functions sn(n;m),cn(n;m), dn(n;m) can be expressed by the unified Weierstrass elliptic function }(n;g2,g3) satisfying nonlinear ordinarydifferential equation

}02ðn; g2; g3Þ ¼ 4}3ðn; g2; g3Þ � g2}ðn; g2; g3Þ � g3; ð3aÞ

779/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

1016/j.chaos.2005.01.003

rresponding author.

ail address: chenyong@nbu.edu.cn (Y. Chen).

394 Y. Chen, Z. Yan / Chaos, Solitons and Fractals 26 (2005) 393–398

where g2, g3 are real parameters and called invariants [1], which has another equivalent form

}00ðn; g2; g3Þ ¼ 6}2ðn; g2; g3Þ �1

2g2; ð3bÞ

Therefore it is of important significance to investigate Weierstrass elliptic function solutions of nonlinear wave equa-

tions. There exist some transformations to study Weierstrass elliptic function solutions of nonlinear wave equations

[3–6].

Recently Alagesan et al. [7] considered the N-coupled nonlinear Klein–Gordon equations

o2ws

ox2� o

2ws

ot2� ws þ 2

XNj¼1

w2j þ q

!ws ¼ 0; ð4Þ

oqox

� oqot

� 2 oot

XNj¼1

w2j ¼ 0; ð5Þ

where s = 1,2,. . .,N. They used the Hirota bilinear method to investigate one-soliton solutions of (4) and (5). In addi-tion, it was shown that when k = 1,2,3, (4) and (5) were Painleve integrability [7–9].

To our knowledge, the doubly periodic solutions of (4) and (5) were not studied before. In this paper we will ex-

tended the transformations [6] to (4) and (5) to derive their doubly periodic solutions in terms of Weierestrass elliptic

function.

2. Weierestrass elliptic function solutions of (4) and (5)

We mainly seek the travelling wave solution of (4) and (5) in the form

wsðx; tÞ ¼ WsðnÞ; s ¼ 1; 2; . . . ;N ; qðx; tÞ ¼ QðnÞ; n ¼ kðxþ ktÞ; ð6Þ

where k,k are constants. Therefore (4) and (5) reduce to the set of N + 1 nonlinear ordinary differential equations

k2ð1� k2Þ d2WsðnÞdn2

� WsðnÞ þ 2XNj¼1

W2j ðnÞ þ QðnÞ

" #WsðnÞ ¼ 0; s ¼ 1; 2; . . . ;N ; ð7Þ

ð1� kÞdQðnÞdn

� 2k ddn

XNj¼1

W2j ðnÞ ¼ 0: ð8Þ

It may be difficult to solve directly the set of N + 1 nonlinear differential equations (7) and (8). We firstly make the

transformations Ws(n) = lsW(n) to reduce (7) and (8) to

k2ð1� k2Þ d2WðnÞdn2

� WðnÞ þ 2XNj¼1

ljW2ðnÞ þ QðnÞ

" #WðnÞ ¼ 0; ð9Þ

ð1� kÞdQðnÞdn

� 2k ddn

XNj¼1

ljW2ðnÞ ¼ 0; ð10Þ

where ls�s are constants and W(n) the new variable, s = 1,2,. . .,N.We have the relationship from (10)

QðnÞ ¼2kPN

j¼1 lj

1� kW2ðnÞ þ C; k 6¼ 1; ð11Þ

when k = 1, we only obtain trivial solutions for variables W or Q.

The substitution of (11) into (9) yields

k2ð1� k2Þ d2WðnÞdn2

þ ð2C � 1ÞWðnÞ þ 2XNj¼1

lj þ4kPN

j¼1 lj

1� k

" #W3ðnÞ ¼ 0: ð12Þ

In the following we obtain solutions of (4) and (5) by researching (12). According to our method [6] (see Appendix

A), we assume that (12) has the solution in the form

Y. Chen, Z. Yan / Chaos, Solitons and Fractals 26 (2005) 393–398 395

WðnÞ ¼ W½}ðn; g2; g3Þ� ¼ a0 þ a1½A}ðn; g2; g3Þ þ B�1=2 þ b1½A}ðn; g2; g3Þ þ B��1=2; ð13Þ

where }(n; g2, g3) satisfies (3a,b), and a0, a1, b1, A, B are constants to be determined.

Therefore we have from (13) and (3a,b)

d2WðnÞdn2

¼ 1

4ðA}ðn; g2; g3Þ þ BÞ5=2�8a1A

3}ðn; g2; g3Þ4 þ 20a1A2B}ðn; g2; g3Þ

3 þ ð12Aa1B2 � 12Ab1BÞ}ðn; g2; g3Þ2

þð�a1A2g2Bþ a1A

3g3 � 2b1A2g2Þ}ðn; g2; g3Þ þ a1A2g3B� Aa1B2g2 � 3b1A2g3 þ Ab1Bg2

�: ð14Þ

With the aid of symbolic computation (Maple), we substitute (13) and (14) into (12) and equate the coefficients of

these terms }jðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA}þ B

pÞiði ¼ 0; 1; j ¼ 0; 1; 2; 3; 4; 5Þ, we get the set of nonlinear algebraic equations with respect to un-

knowns k,k,a0, a1,b1,A,B (Here we omit them, since it is complicated). By solving the set of nonlinear algebraic equa-tions, we can determine these unknowns as follows:

Case 1

a0 ¼ b1 ¼ 0; g3 ¼ð2C � 1Þ½�4ð2C � 1Þ2 þ 9k4ð1� k2Þ2g2�

27k6ð1� k2Þ3; A ¼ � 2k2ð1� k2Þ

a21 2P

j ¼ 1Nlj þ4kPN

j¼1lj

1�k

� ;

B ¼ � 2ð2C � 1Þ

3a21 2PN

j¼1lj þ4kPN

j¼1lj

1�k

� : ð15Þ

Case 2

a0 ¼ a1 ¼ 0; g2 ¼2ð2C � 1Þ 2ð2C � 1ÞBþ b21 2

PNj¼1lj þ

4kPN

j¼1lj

1�k

�� �3Bk4ð1� k2Þ2

g3 ¼2ð2C � 1Þ2 3b21 2

PNj¼1lj þ

4kPN

j¼1lj

1�k

�þ 4Bð2C � 1Þ

� �27Bk6ð2C � 1Þ3

; A ¼ 3k2ð1� k2ÞB2C � 1 : ð16Þ

Case 3

A ¼ � 2k2ð1� k2Þ

a21 2PN

j¼1lj þ4kPN

j¼1lj

1�k

� ; b1 ¼ �2ð2C � 1Þ þ 3a21B 2

PNj¼1lj þ

4kPN

j¼1lj

1�k

�

a1 2PN

j¼1lj þ4kPN

j¼1lj

1�k

� ;

g2 ¼1

36k4ð1� k2Þ299a41B

2 2XNj¼1

lj þ4kPN

j¼1lj

1� k

!2� 4ð2C � 1Þ2

24

35� 12a21Bð2C � 1Þ 2

XNj¼1

lj þ4kPN

j¼1lj

1� k

!;

g3 ¼ � a21B

72k6ð1� k2Þ32XNj¼1

lj þ4kPN

j¼1lj

1� k

!63a41B

2 2XNj¼1

lj þ4kPN

j¼1lj

1� k

!224

�12a21Bð2C � 1Þ 2XNj¼1

lj þ4kPN

j¼1lj

1� k

!� 4ð2C � 1Þ2

35: ð17Þ

Therefore according to (11), (13) and (15)–(17), we get three families of Weierstrass elliptic function solutions of (4)

and (5).

Family 1

wsðx; tÞ ¼ ls

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik � 1

ðk þ 1ÞPN

j¼1lj

k2ð1� k2Þ}ðn; g2; g3Þ þ1

3ð2C � 1Þ

� �s; s ¼ 1; 2; . . . ;N ; ð18Þ

396 Y. Chen, Z. Yan / Chaos, Solitons and Fractals 26 (2005) 393–398

qðx; tÞ ¼ � 2kk þ 1 k2ð1� k2Þ}ðn; g2; g3Þ þ

1

3ð2C � 1Þ

� �þ C; ð19Þ

where n = k(x + kt), g2,g3 are defined by (15).Family 2

wsðx; tÞ ¼b1lsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

3k2ð1� k2ÞBð2C � 1Þ�1}ðn; g2; g3Þ þ Bq ; s ¼ 1; 2; . . . ;N ; ð20Þ

qðx; tÞ ¼2kb1

PNj¼1lj

ðk � 1Þ½3k2ð1� k2ÞBð2C � 1Þ�1}ðn; g2; g3Þ þ B�þ C; ð21Þ

where n = k(x + kt), g2, g3 are determined by (16).Family 3

ws ¼ls

ðk þ 1ÞPN

j¼1lj

ð1þ kÞ½k2ðk2 � 1Þ}ðn; g2; g3Þ þ 1� 2C� � 2a21Bð1þ kÞPN

j¼1ljffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� kÞ2k2

PNj¼1lj

� ��1}ðn; g2; g3Þ þ Ba21

r ; s ¼ 1; 2; . . . ;N ; ð22Þ

q ¼ 2k

ð1� kÞðk þ 1Þ2ð1þ kÞ½k2ðk2 � 1Þ}ðn; g2; g3Þ þ 1� 2C� � 2a21Bð1þ kÞ

PNj¼1lj

n o2ð1� kÞ2k2}ðn; g2; g3Þ þ Ba21

PNj¼1lj

; ð23Þ

where n = k(x + kt), g2, g3 are determined by (17).

Remark. We analysis solutions (18) and (19) of (4) and (5). We know that the Weierstrass elliptic function }(n;g2,g3)can be write as

}ðn; g2; g3Þ ¼ e2 � ðe2 � e3Þcn2ffiffiffiffiffiffiffiffiffiffiffiffiffiffie1 � e3

pn;mð Þ; ð24Þ

in terms of the Jacobi elliptic cosine function, where m2 = (e2 � e3)/ (e1 � e3) is the modulus of the Jacobi elliptic func-

tion, ei(i = 1,2,3;e1Pe2Pe3) are three roots of the cubic equation

4y3 � g2y �ð2C � 1Þ½�4ð2C � 1Þ2 þ 9k4ð1� k2Þ2g2�

27k6ð1� k2Þ3¼ 0: ð25Þ

Therefore solutions (18) and (19) are rewritten as

ws ¼ lk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik � 1

ðk þ 1ÞPN

j¼1lj

k2ð1� k2Þ e2 � ðe2 � e3Þcn2ffiffiffiffiffiffiffiffiffiffiffiffiffiffie1 � e3

pn;mð Þ½ � þ 1

3ð2C � 1Þ

� �s: ð26Þ

q ¼ � 2kk þ 1 k2ð1� k2Þ e2 � ðe2 � e3Þcn2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffie1 � e3

pn;mð Þ

� �þ 13ð2C � 1Þ

� �þ C: ð27Þ

In particular, when m ! 1, i.e., e2! e1, cn(n;m)! sech(n), thus the solitary wave solutions of (4) and (5) can bewritten as

ws ¼ lk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik � 1

ðk þ 1ÞPN

j¼1lj

k2ð1� k2Þ e2 � ðe2 � e3Þsech2ffiffiffiffiffiffiffiffiffiffiffiffiffiffie1 � e3

pnð Þ

� �þ 13ð2C � 1Þ

� �s: ð28Þ

q ¼ � 2kk þ 1 k2ð1� k2Þ e2 � ðe2 � e3Þsech2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffie1 � e3

pnð Þ

� �þ 13ð2C � 1Þ

� �þ C: ð29Þ

Similarly, we also write the solutions (13) and (14) as other forms in terms of Jacobi elliptic function or the hyper-

bolic function.

Y. Chen, Z. Yan / Chaos, Solitons and Fractals 26 (2005) 393–398 397

3. Conclusion

In summary, we firstly transformed the N-coupled Klein–Gordon equations (4) and (5) into a nonlinear ordinary

differential equations (12) using a series of ansatze. And then with the aid of Maple, we used a transformation in terms

of the Weierstrass elliptic function to obtain three families of doubly periodic solutions of (4) and (5). In particular,

new solitary wave solutions are also derived. These solutions are useful to explain the corresponding physical

phenomena.

Acknowledgement

This work is supported by Zhejiang Provincial Natural Science Foundation of China (No. Y604056), Postdoctoral

Science Foundation of China, NNSF of China (No. 10401039), the NKBRP of China (No. 2004CB318000) and the

SRF for ROCS, SEM of China.

Appendix A

The Weierstrass elliptic function expansion method is summarized as follows:

Step 1: For a given nonlinear evolution equation with a physical field u and two independent variables x, t

F ðu; ut; ux; uxx; uxt; utt; . . .Þ ¼ 0: ðA:1Þ

The travelling wave transformation u(x,t) = u(n), n = k(x + kt) reduces (A.1) to a nonlinear ordinary differentialequation

Gðu; u0; u00; u000; . . .Þ ¼ 0; ðA:2Þ

where the prime denotes d/dn.Step 2: We assume that (A.2) has the power series solution in terms of the Weierstrass elliptic function

uðnÞ ¼ uð}ðn; g2; g3ÞÞ ¼ a0 þXni¼1

ai½A}ðn; g2; g3Þ þ B�i=2 þ bi½A}ðn; g2; g3Þ þ B��i=2; ðA:3Þ

where n, A 5 0, B, a0, ai, bi are parameters to be determined later, and }(n;g2,g3) the Weierstrass elliptic functionsatisfying

}02ðn; g2; g3Þ ¼ 4}3ðn; g2; g3Þ � g2}ðn; g2; g3Þ � g3; ðA:4Þ

where g2, g3 are real parameters and called invariants.

According to Eq. (A.4), we define a polynomial degree function as D(u(})) = n, thus we have

D upð}Þ dsuð}Þdns

�q �¼ np þ qðnþ sÞ: ðA:5Þ

Therefore we can determine n in (A.3) by balancing the highest degree linear term and nonlinear terms.

Step 3: The substitution of (A.3) into (A.2) along with (A.4) leads to a polynomial of }0i(A} + B)j/2}s (i,j = 0,1;

s = 0,1,2,3. . .). Setting their coefficients to zero yields a set of algebraic equations with respect to the unknowns k, k,A, B, g2, g3, a0, ai, bi (i = 1,. . ., n).

Step 4: With the aid of symbolic computation, we solve the set of algebraic equations obtained in Step 3. Finally

we derive the doubly periodic solutions of the given nonlinear equations (A.1) in terms of Weierstrass elliptic

function.

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