- Email: [email protected]
New Weierstrass elliptic function solutions of the N-coupled nonlinear Klein–Gordon equations
Chaos, Solitons and Fractals 26 (2005) 393–398 www.elsevier.com/locate/chaos
New Weierstrass elliptic function solutions of the N-coupled nonlinear Klein–Gordon equations Yong Chen b
a,*
, Zhenya Yan
b
a Department of Mathematics, Ningbo University, Ningbo 315211, China 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-coupled 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. Ó 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 solitary wave solutions, for example [1] 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 ordinary differential equation }02 ðn; g2 ; g3 Þ ¼ 4}3 ðn; g2 ; g3 Þ g2 }ðn; g2 ; g3 Þ g3 ;
*
Corresponding author. E-mail address: [email protected] (Y. Chen).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.01.003
ð3aÞ
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 1 }00 ðn; g2 ; g3 Þ ¼ 6}2 ðn; g2 ; g3 Þ g2 ; 2
ð3bÞ
Therefore it is of important significance to investigate Weierstrass elliptic function solutions of nonlinear wave equations. 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 ! N X o2 ws o2 ws 2 2 ws þ 2 wj þ q ws ¼ 0; ð4Þ ox2 ot j¼1 N oq oq o X 2 w2 ¼ 0; ox ot ot j¼1 j
ð5Þ
where s = 1,2,. . .,N. They used the Hirota bilinear method to investigate one-soliton solutions of (4) and (5). In addition, 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 extended 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 ws ðx; tÞ ¼ Ws ðnÞ;
s ¼ 1; 2; . . . ; N ;
qðx; tÞ ¼ QðnÞ;
n ¼ kðx þ ktÞ;
ð6Þ
where k, k are constants. Therefore (4) and (5) reduce to the set of N + 1 nonlinear ordinary differential equations " # N 2 X 2 d Ws ðnÞ 2 2 Ws ðnÞ þ 2 Wj ðnÞ þ QðnÞ Ws ðnÞ ¼ 0; s ¼ 1; 2; . . . ; N; ð7Þ k ð1 k Þ dn2 j¼1 ð1 kÞ
N dQðnÞ d X 2k W2 ðnÞ ¼ 0: dn dn j¼1 j
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 " # N 2 X 2 d WðnÞ 2 2 WðnÞ þ 2 lj W ðnÞ þ QðnÞ WðnÞ ¼ 0; ð9Þ k ð1 k Þ dn2 j¼1 ð1 kÞ
N dQðnÞ d X l W2 ðnÞ ¼ 0; 2k dn dn j¼1 j
where lsÕs are constants and W(n) the new variable, s = 1,2,. . .,N. We have the relationship from (10) P 2k Nj¼1 lj 2 W ðnÞ þ C; k 6¼ 1; QðnÞ ¼ 1k when k = 1, we only obtain trivial solutions for variables W or Q. The substitution of (11) into (9) yields " # P N 2 X 4k Nj¼1 lj 3 2 d WðnÞ 2 W ðnÞ ¼ 0: þ ð2C 1ÞWðnÞ þ 2 lj þ k ð1 k Þ 1k dn2 j¼1
ð10Þ
ð11Þ
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
WðnÞ ¼ W½}ðn; g2 ; g3 Þ ¼ a0 þ a1 ½A}ðn; g2 ; g3 Þ þ B 1=2 þ b1 ½A}ðn; g2 ; g3 Þ þ B 1=2 ;
395
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) d2 WðnÞ 1 8a1 A3 }ðn; g2 ; g3 Þ4 þ 20a1 A2 B}ðn; g2 ; g3 Þ3 þ ð12Aa1 B2 12Ab1 BÞ}ðn; g2 ; g3 Þ2 ¼ dn2 4ðA}ðn; g2 ; g3 Þ þ BÞ5=2 þða1 A2 g2 B þ a1 A3 g3 2b1 A2 g2 Þ}ðn; g2 ; g3 Þ þ a1 A2 g3 B Aa1 B2 g2 3b1 A2 g3 þ Ab1 Bg2 :
ð14Þ
With the aidpof symbolic computation (Maple), we substitute (13) and (14) into (12) and equate the coefficients of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi these terms }j ð A} þ BÞi ði ¼ 0; 1; j ¼ 0; 1; 2; 3; 4; 5Þ, we get the set of nonlinear algebraic equations with respect to unknowns k,k,a0, a1,b1,A,B (Here we omit them, since it is complicated). By solving the set of nonlinear algebraic equations, we can determine these unknowns as follows: Case 1 a0 ¼ b1 ¼ 0;
B¼ 3a21
g3 ¼
ð2C 1Þ½4ð2C 1Þ2 þ 9k 4 ð1 k2 Þ2 g2 27k 6 ð1 k2 Þ3
;
A¼
2k 2 ð1 k2 Þ PN ;
4k lj P a21 2 j ¼ 1N lj þ 1kj¼1
2ð2C 1Þ PN : 4k lj PN 2 j¼1 lj þ 1kj¼1
ð15Þ
Case 2 PN
4k lj P 2ð2C 1Þ 2ð2C 1ÞB þ b21 2 Nj¼1 lj þ 1kj¼1 a0 ¼ a1 ¼ 0;
g2 ¼
3Bk 4 ð1 k2 Þ2 PN
4k lj P þ 4Bð2C 1Þ 2ð2C 1Þ2 3b21 2 Nj¼1 lj þ 1kj¼1
g3 ¼
6
3
27Bk ð2C 1Þ
;
A¼
3k 2 ð1 k2 ÞB : 2C 1
ð16Þ
Case 3 PN
4k lj P 2ð2C 1Þ þ 3a21 B 2 Nj¼1 lj þ 1kj¼1 2k ð1 k Þ A¼
; PN ; b1 ¼ PN
4k lj 4k lj P P a21 2 Nj¼1 lj þ 1kj¼1 a1 2 Nj¼1 lj þ 1kj¼1 2 3 !2 ! P P N N X X 4k Nj¼1 lj 4k Nj¼1 lj 1 2 4 2 2 499a B 2 5 g2 ¼ ; l þ 4ð2C 1Þ Bð2C 1Þ 2 l þ 12a j j 1 1 1k 1k 36k 4 ð1 k2 Þ2 j¼1 j¼1 !2 !2 P P N N X X 4k Nj¼1 lj 4k Nj¼1 lj a21 B 4 2 4 63a1 B 2 g3 ¼ 2 lj þ lj þ 1k 1k 72k 6 ð1 k2 Þ3 j¼1 j¼1 3 ! P N X 4k Nj¼1 lj ð17Þ 4ð2C 1Þ2 5: lj þ 12a21 Bð2C 1Þ 2 1 k j¼1 2
2
Therefore according to (11), (13) and (15)–(17), we get three families of Weierstrass elliptic function solutions of (4) and (5). Family 1 ws ðx; tÞ ¼ ls
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 1 k 2 ð1 k2 Þ}ðn; g2 ; g3 Þ þ ð2C 1Þ ; PN 3 ðk þ 1Þ j¼1 lj
s ¼ 1; 2; . . . ; N ;
ð18Þ
396
Y. Chen, Z. Yan / Chaos, Solitons and Fractals 26 (2005) 393–398
qðx; tÞ ¼
2k 1 k 2 ð1 k2 Þ}ðn; g2 ; g3 Þ þ ð2C 1Þ þ C; kþ1 3
ð19Þ
where n = k(x + kt), g2,g3 are defined by (15). Family 2 b1 ls ffi; ws ðx; tÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3k 2 ð1 k ÞBð2C 1Þ1 }ðn; g2 ; g3 Þ þ B qðx; tÞ ¼
2kb1
s ¼ 1; 2; . . . ; N ;
ð20Þ
PN
j¼1 lj
ðk 1Þ½3k 2 ð1 k2 ÞBð2C 1Þ1 }ðn; g2 ; g3 Þ þ B
þ C;
ð21Þ
where n = k(x + kt), g2, g3 are determined by (16). Family 3 ð1 þ kÞ½k 2 ðk2 1Þ}ðn; g2 ; g3 Þ þ 1 2C 2a21 Bð1 þ kÞ ls rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ws ¼ PN P 1 ðk þ 1Þ j¼1 lj N ð1 kÞ2 k 2 }ðn; g2 ; g3 Þ þ Ba21 j¼1 lj
PN
j¼1 lj
;
s ¼ 1; 2; . . . ; N;
n o2 PN 2 2 2 ð1 þ kÞ½k ðk 1Þ}ðn; g ; g Þ þ 1 2C 2a Bð1 þ kÞ l 2 3 j 1 j¼1 2k q¼ ; P ð1 kÞ2 k 2 }ðn; g2 ; g3 Þ þ Ba21 Nj¼1 lj ð1 kÞðk þ 1Þ2
ð22Þ
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 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi }ðn; g2 ; g3 Þ ¼ e2 ðe2 e3 Þcn2 ð e1 e3 n; mÞ; ð24Þ in terms of the Jacobi elliptic cosine function, where m2 = (e2 e3)/ (e1 e3) is the modulus of the Jacobi elliptic function, ei(i = 1,2,3;e1 P e2 P e3) are three roots of the cubic equation 4y 3 g2 y
ð2C 1Þ½4ð2C 1Þ2 þ 9k 4 ð1 k2 Þ2 g2 27k 6 ð1 k2 Þ3
¼ 0:
ð25Þ
Therefore solutions (18) and (19) are rewritten as ws ¼ lk
q¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 1 2 2 2 ð e e n; mÞ þ ð2C 1Þ : ½ k ð1 k Þ e ðe e Þcn P 2 2 3 1 3 3 ðk þ 1Þ Nj¼1 lj
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k k 2 ð1 k2 Þ e2 ðe2 e3 Þcn2 ð e1 e3 n; mÞ þ ð2C 1Þ þ C: kþ1 3
ð26Þ
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 be written as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 k1 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ws ¼ lk k ð1 k Þ e2 ðe2 e3 Þsech ð e1 e3 nÞ þ ð2C 1Þ : ð28Þ P 3 ðk þ 1Þ Nj¼1 lj q¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2k k 2 ð1 k2 Þ e2 ðe2 e3 Þsech2 ð e1 e3 nÞ þ ð2C 1Þ þ C: kþ1 3
ð29Þ
Similarly, we also write the solutions (13) and (14) as other forms in terms of Jacobi elliptic function or the hyperbolic 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 differential equation 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 n X ai ½A}ðn; g2 ; g3 Þ þ B i=2 þ bi ½A}ðn; g2 ; g3 Þ þ B i=2 ; uðnÞ ¼ uð}ðn; g2 ; g3 ÞÞ ¼ a0 þ
ðA:3Þ
i¼1
where n, A 5 0, B, a0, ai, bi are parameters to be determined later, and }(n;g2,g3) the Weierstrass elliptic function satisfying }02 ðn; g2 ; g3 Þ ¼ 4}3 ðn; g2 ; g3 Þ g2 }ðn; g2 ; g3 Þ g3 ; 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
s q d uð}Þ ¼ np þ qðn þ sÞ: D up ð}Þ dns
ðA:4Þ
ðA:5Þ
Therefore we can determine n in (A.3) by balancing the highest degree linear term and nonlinear terms. 0 Step 3: The substitution of (A.3) into (A.2) along with (A.4) leads to a polynomial of } i(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.
References [1] [2] [3] [4]
Patrick DV. Elliptic function and elliptic curves. Cambridge: Cambridge University Press; 1973. Lawden DF. Elliptic functions and applications. New York: Springer-Verlag; 1989. Porubov AV. Phys Lett A 1996;221:391. Porubov AV, Velarde MG. J Math Phys 1999;40:884.
398 [5] [6] [7] [8] [9]
Y. Chen, Z. Yan / Chaos, Solitons and Fractals 26 (2005) 393–398
Yan ZY. Chaos, Solitons & Fractals 2004;21:1013. Yan ZY. Z Naturforsch A 2004;59:29. Alagesan T et al. Chaos, Solitons & Fractals 2004;21:879. Hirota H, Ohta Y. J Phys Soc Jpn 1991;60:798. Porsezian K, Alagesan T. Phys Lett A 1995;198:378.
New Weierstrass elliptic function solutions of the N-coupled nonlinear Klein–Gordon equations Yong Chen b
a,*
, Zhenya Yan
b
a Department of Mathematics, Ningbo University, Ningbo 315211, China 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-coupled 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. Ó 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 solitary wave solutions, for example [1] 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 ordinary differential equation }02 ðn; g2 ; g3 Þ ¼ 4}3 ðn; g2 ; g3 Þ g2 }ðn; g2 ; g3 Þ g3 ;
*
Corresponding author. E-mail address: [email protected] (Y. Chen).
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.01.003
ð3aÞ
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 1 }00 ðn; g2 ; g3 Þ ¼ 6}2 ðn; g2 ; g3 Þ g2 ; 2
ð3bÞ
Therefore it is of important significance to investigate Weierstrass elliptic function solutions of nonlinear wave equations. 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 ! N X o2 ws o2 ws 2 2 ws þ 2 wj þ q ws ¼ 0; ð4Þ ox2 ot j¼1 N oq oq o X 2 w2 ¼ 0; ox ot ot j¼1 j
ð5Þ
where s = 1,2,. . .,N. They used the Hirota bilinear method to investigate one-soliton solutions of (4) and (5). In addition, 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 extended 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 ws ðx; tÞ ¼ Ws ðnÞ;
s ¼ 1; 2; . . . ; N ;
qðx; tÞ ¼ QðnÞ;
n ¼ kðx þ ktÞ;
ð6Þ
where k, k are constants. Therefore (4) and (5) reduce to the set of N + 1 nonlinear ordinary differential equations " # N 2 X 2 d Ws ðnÞ 2 2 Ws ðnÞ þ 2 Wj ðnÞ þ QðnÞ Ws ðnÞ ¼ 0; s ¼ 1; 2; . . . ; N; ð7Þ k ð1 k Þ dn2 j¼1 ð1 kÞ
N dQðnÞ d X 2k W2 ðnÞ ¼ 0: dn dn j¼1 j
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 " # N 2 X 2 d WðnÞ 2 2 WðnÞ þ 2 lj W ðnÞ þ QðnÞ WðnÞ ¼ 0; ð9Þ k ð1 k Þ dn2 j¼1 ð1 kÞ
N dQðnÞ d X l W2 ðnÞ ¼ 0; 2k dn dn j¼1 j
where lsÕs are constants and W(n) the new variable, s = 1,2,. . .,N. We have the relationship from (10) P 2k Nj¼1 lj 2 W ðnÞ þ C; k 6¼ 1; QðnÞ ¼ 1k when k = 1, we only obtain trivial solutions for variables W or Q. The substitution of (11) into (9) yields " # P N 2 X 4k Nj¼1 lj 3 2 d WðnÞ 2 W ðnÞ ¼ 0: þ ð2C 1ÞWðnÞ þ 2 lj þ k ð1 k Þ 1k dn2 j¼1
ð10Þ
ð11Þ
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
WðnÞ ¼ W½}ðn; g2 ; g3 Þ ¼ a0 þ a1 ½A}ðn; g2 ; g3 Þ þ B 1=2 þ b1 ½A}ðn; g2 ; g3 Þ þ B 1=2 ;
395
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) d2 WðnÞ 1 8a1 A3 }ðn; g2 ; g3 Þ4 þ 20a1 A2 B}ðn; g2 ; g3 Þ3 þ ð12Aa1 B2 12Ab1 BÞ}ðn; g2 ; g3 Þ2 ¼ dn2 4ðA}ðn; g2 ; g3 Þ þ BÞ5=2 þða1 A2 g2 B þ a1 A3 g3 2b1 A2 g2 Þ}ðn; g2 ; g3 Þ þ a1 A2 g3 B Aa1 B2 g2 3b1 A2 g3 þ Ab1 Bg2 :
ð14Þ
With the aidpof symbolic computation (Maple), we substitute (13) and (14) into (12) and equate the coefficients of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi these terms }j ð A} þ BÞi ði ¼ 0; 1; j ¼ 0; 1; 2; 3; 4; 5Þ, we get the set of nonlinear algebraic equations with respect to unknowns k,k,a0, a1,b1,A,B (Here we omit them, since it is complicated). By solving the set of nonlinear algebraic equations, we can determine these unknowns as follows: Case 1 a0 ¼ b1 ¼ 0;
B¼ 3a21
g3 ¼
ð2C 1Þ½4ð2C 1Þ2 þ 9k 4 ð1 k2 Þ2 g2 27k 6 ð1 k2 Þ3
;
A¼
2k 2 ð1 k2 Þ PN ;
4k lj P a21 2 j ¼ 1N lj þ 1kj¼1
2ð2C 1Þ PN : 4k lj PN 2 j¼1 lj þ 1kj¼1
ð15Þ
Case 2 PN
4k lj P 2ð2C 1Þ 2ð2C 1ÞB þ b21 2 Nj¼1 lj þ 1kj¼1 a0 ¼ a1 ¼ 0;
g2 ¼
3Bk 4 ð1 k2 Þ2 PN
4k lj P þ 4Bð2C 1Þ 2ð2C 1Þ2 3b21 2 Nj¼1 lj þ 1kj¼1
g3 ¼
6
3
27Bk ð2C 1Þ
;
A¼
3k 2 ð1 k2 ÞB : 2C 1
ð16Þ
Case 3 PN
4k lj P 2ð2C 1Þ þ 3a21 B 2 Nj¼1 lj þ 1kj¼1 2k ð1 k Þ A¼
; PN ; b1 ¼ PN
4k lj 4k lj P P a21 2 Nj¼1 lj þ 1kj¼1 a1 2 Nj¼1 lj þ 1kj¼1 2 3 !2 ! P P N N X X 4k Nj¼1 lj 4k Nj¼1 lj 1 2 4 2 2 499a B 2 5 g2 ¼ ; l þ 4ð2C 1Þ Bð2C 1Þ 2 l þ 12a j j 1 1 1k 1k 36k 4 ð1 k2 Þ2 j¼1 j¼1 !2 !2 P P N N X X 4k Nj¼1 lj 4k Nj¼1 lj a21 B 4 2 4 63a1 B 2 g3 ¼ 2 lj þ lj þ 1k 1k 72k 6 ð1 k2 Þ3 j¼1 j¼1 3 ! P N X 4k Nj¼1 lj ð17Þ 4ð2C 1Þ2 5: lj þ 12a21 Bð2C 1Þ 2 1 k j¼1 2
2
Therefore according to (11), (13) and (15)–(17), we get three families of Weierstrass elliptic function solutions of (4) and (5). Family 1 ws ðx; tÞ ¼ ls
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 1 k 2 ð1 k2 Þ}ðn; g2 ; g3 Þ þ ð2C 1Þ ; PN 3 ðk þ 1Þ j¼1 lj
s ¼ 1; 2; . . . ; N ;
ð18Þ
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Y. Chen, Z. Yan / Chaos, Solitons and Fractals 26 (2005) 393–398
qðx; tÞ ¼
2k 1 k 2 ð1 k2 Þ}ðn; g2 ; g3 Þ þ ð2C 1Þ þ C; kþ1 3
ð19Þ
where n = k(x + kt), g2,g3 are defined by (15). Family 2 b1 ls ffi; ws ðx; tÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3k 2 ð1 k ÞBð2C 1Þ1 }ðn; g2 ; g3 Þ þ B qðx; tÞ ¼
2kb1
s ¼ 1; 2; . . . ; N ;
ð20Þ
PN
j¼1 lj
ðk 1Þ½3k 2 ð1 k2 ÞBð2C 1Þ1 }ðn; g2 ; g3 Þ þ B
þ C;
ð21Þ
where n = k(x + kt), g2, g3 are determined by (16). Family 3 ð1 þ kÞ½k 2 ðk2 1Þ}ðn; g2 ; g3 Þ þ 1 2C 2a21 Bð1 þ kÞ ls rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ws ¼ PN P 1 ðk þ 1Þ j¼1 lj N ð1 kÞ2 k 2 }ðn; g2 ; g3 Þ þ Ba21 j¼1 lj
PN
j¼1 lj
;
s ¼ 1; 2; . . . ; N;
n o2 PN 2 2 2 ð1 þ kÞ½k ðk 1Þ}ðn; g ; g Þ þ 1 2C 2a Bð1 þ kÞ l 2 3 j 1 j¼1 2k q¼ ; P ð1 kÞ2 k 2 }ðn; g2 ; g3 Þ þ Ba21 Nj¼1 lj ð1 kÞðk þ 1Þ2
ð22Þ
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 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi }ðn; g2 ; g3 Þ ¼ e2 ðe2 e3 Þcn2 ð e1 e3 n; mÞ; ð24Þ in terms of the Jacobi elliptic cosine function, where m2 = (e2 e3)/ (e1 e3) is the modulus of the Jacobi elliptic function, ei(i = 1,2,3;e1 P e2 P e3) are three roots of the cubic equation 4y 3 g2 y
ð2C 1Þ½4ð2C 1Þ2 þ 9k 4 ð1 k2 Þ2 g2 27k 6 ð1 k2 Þ3
¼ 0:
ð25Þ
Therefore solutions (18) and (19) are rewritten as ws ¼ lk
q¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 1 2 2 2 ð e e n; mÞ þ ð2C 1Þ : ½ k ð1 k Þ e ðe e Þcn P 2 2 3 1 3 3 ðk þ 1Þ Nj¼1 lj
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k k 2 ð1 k2 Þ e2 ðe2 e3 Þcn2 ð e1 e3 n; mÞ þ ð2C 1Þ þ C: kþ1 3
ð26Þ
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 be written as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 k1 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ws ¼ lk k ð1 k Þ e2 ðe2 e3 Þsech ð e1 e3 nÞ þ ð2C 1Þ : ð28Þ P 3 ðk þ 1Þ Nj¼1 lj q¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2k k 2 ð1 k2 Þ e2 ðe2 e3 Þsech2 ð e1 e3 nÞ þ ð2C 1Þ þ C: kþ1 3
ð29Þ
Similarly, we also write the solutions (13) and (14) as other forms in terms of Jacobi elliptic function or the hyperbolic 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 differential equation 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 n X ai ½A}ðn; g2 ; g3 Þ þ B i=2 þ bi ½A}ðn; g2 ; g3 Þ þ B i=2 ; uðnÞ ¼ uð}ðn; g2 ; g3 ÞÞ ¼ a0 þ
ðA:3Þ
i¼1
where n, A 5 0, B, a0, ai, bi are parameters to be determined later, and }(n;g2,g3) the Weierstrass elliptic function satisfying }02 ðn; g2 ; g3 Þ ¼ 4}3 ðn; g2 ; g3 Þ g2 }ðn; g2 ; g3 Þ g3 ; 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
s q d uð}Þ ¼ np þ qðn þ sÞ: D up ð}Þ dns
ðA:4Þ
ðA:5Þ
Therefore we can determine n in (A.3) by balancing the highest degree linear term and nonlinear terms. 0 Step 3: The substitution of (A.3) into (A.2) along with (A.4) leads to a polynomial of } i(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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