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Abundant new travelling wave solutions for the coupled Ito equation
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Chaos, Solitons and Fractals 39 (2009) 393–398 www.elsevier.com/locate/chaos
Abundant new travelling wave solutions for the coupled Ito equation Binlu Feng a, Bo Han a, Huanhe Dong b
b,*
a Department of Mathematics, Harbin Institute of Technology, Harbin 15001, PR China College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266510, China
Abstract Abundant new travelling wave solutions of the coupled Ito equation are obtained by the generalized Jacobi elliptic function method. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Seeking exact solutions of nonlinear physics equations is an important and interesting subject Since the exact solutions to nonlinear wave equations helps to understand the characteristics of the nonlinear equations. In recent years, many powerful methods have been presented, such as homogeneous balance method, Darboux-transformation, Inverse Scattering Transformation and tanh-function method and so on [1–6]. It is well known that many nonlinear evolution equations have been shown to possess the periodic solutions. Recently, some methods were presented to seek the periodic solutions, such as the Jacobi elliptic function expansion method, the algebraic method and the generalized Jacobi elliptic function method[7–11], etc. In this paper, we use the improved generalized Jacobi elliptic function method to obtain abundant new solutions for the coupled Ito equation. For a given nonlinear evolution equation in three independent variables H ðu; ut ; ux ; . . .Þ ¼ 0:
ð1Þ
We seek the following formal travelling wave solutions u ¼ uðx; tÞ ¼ U ðnÞ;
n ¼ kx þ ct;
ð2Þ
where k,c are arbitrary constants. Substituting Eq. (2) into Eq. (1) gives rise to an ordinary equation H 1 ðU ; U 0 ; U 00 ; . . .Þ ¼ 0:
ð3Þ
Then consider the elliptic equation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u0 ¼ r r þ pu2 þ qu4 ;
ð4Þ
*
Corresponding author. E-mail address: [email protected] (H. Dong).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.04.012
394
B. Feng et al. / Chaos, Solitons and Fractals 39 (2009) 393–398
where u0 ¼ du ; r ¼ 1; r; p; q are constants. dn We know that Eq. (4) possess the following solution: Case Case Case Case Case Case Case
1: 2: 3: 4: 5: 6: 7:
if if if if if if if
r = 1, p = (1 + m2), q = m2, then Eq. (4) has solution u1a(n) = sn n, u2a = cd n; r = 1 m2, p = 2m2 1, q = m2, then Eq. (4) has solution u2(n) = cn n, r = m2 1, p = 2 m2, q = 1, then Eq. (4) has solution u3(n) = dn n, r = m2(1 m2), p = 2m2 1, q = 1, then Eq. (4) has solution u4(n) = ds n, r = 1 m2, p = 2 m2, q = 1, then Eq. (4) has solution u5(n) = cs n, 2 2 sn n r ¼ 14 ; p ¼ m 22 ; q ¼ m4 , then Eq. (4) has solution u6 ðnÞ ¼ 1dn ; n m2 m2 2 r ¼ q ¼ 4 ; p ¼ 2 ,then Eq. (4) has solution
dn n u7a ðnÞ ¼ sn n icn n; u7b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; i 1 m2 sn n cn n 2
, then Eq. (4) has solution Case 8: if r ¼ q ¼ 14 ; p ¼ 12m 2 u8a ðnÞ ¼
dn n sn n cn n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; u8b ¼ msn n idn n; u8c ¼ ; u ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 dn n 8d i 1 m2 sn n dn n mcn n i 1 m2 2
2
dn n ; Case 9: if r ¼ q ¼ m 41 ; p ¼ m 2þ1, then Eq. (4) has solution u9 ðnÞ ¼ 1msn n 1m2
2
cn n Case 10: if r ¼ q ¼ 2 4 ; p ¼ m 2þ1, then Eq. (4) has solution u10 ðnÞ ¼ 1sn ; n 2Þ m2 þ1 1 Case 11: if r ¼ ð1m ; p ¼ ; q ¼ , then Eq. (4) has solution u (n) = mcn n ± dn n, 11 4 2 4 2 2
Þ sn n Case 12: if r ¼ 14 ; p ¼ m 2þ1 ; q ¼ ð1m , Eq then Eq. (4) has solution u12 ðnÞ ¼ dn ncn ; 4 n 2 2 cn n 1 m 2 m p ffiffiffiffiffiffiffiffi Case 13: if r ¼ 4 ; p ¼ 2 ; q ¼ 4 , then Eq. (4) has solution u13 ðnÞ ¼ : 2 2
1m dn n
Here, sn n = sn (n, m), m denote the modulus of the Jacobi elliptic functions. Introduce a new independent variable u as follows uðx; tÞ ¼ U ðnÞ ¼ AðuÞ ¼
m¼n X
ðam um Þ
ð5Þ
m¼n
We find that the derivatives with respect to the variable n become the derivatives with respect to the variable u as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d d ¼ r r þ pu2 þ qu4 ; dn du d2 d d2 ¼ ðpu þ 2qu3 Þ þ ðr þ pu2 þ qu4 Þ 2 ; 2 du du dn 3 2 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p d d 3 d 2 4 d 2 þ qu4 ðp þ 6qu2 Þ : ¼ r r þ pu Þ þ ðr þ pu þ qu Þ þ ð3pu þ 6qu du2 du3 du dn3
ð6Þ
Finally, inserting Eq. (5) with Eq. (6) into Eq. (3) yields H 2 ðA; A0 ; A00 ; . . .Þ ¼ 0:
ð7Þ
2. The solution of the coupled Ito equation Consider the coupled Ito equation in Ref.[12] ut ¼ vx ; vt ¼ 2ðvxxx þ 3uvx þ 3vux Þ 12wwx ; wt ¼ wxxx þ 3uwx :
ð8Þ
Let uðx; tÞ ¼ U ðnÞ;
vðx; tÞ ¼ V ðnÞ;
wðx; tÞ ¼ W ðnÞ;
n ¼ kx þ ct:
ð9Þ
B. Feng et al. / Chaos, Solitons and Fractals 39 (2009) 393–398
395
Substituting (9) into (8) yields cU kV ¼ 0; 000
cV 0 þ 2k 3V þ 6kUV 0 þ 6kU 0 V þ 12kWW 0 ¼ 0; 0
cW k
3W 000
ð10Þ
0
3kUW ¼ 0:
Inserting U(n) = E(u), V(n) = F(u), W(n) = G(u) with Eqs. (6) into (10) gives cE kF ¼ 0; cF 0 þ 2k 3 ðp þ 6qu2 ÞF 0 þ 2k 3 ð3pu þ 6qu3 ÞF 00 þ 2k 3 ðr þ pu2 þ qu4 ÞF 000 þ 6kEF 0 þ 6kFE0 þ 6kGG0 ¼ 0;
ð11Þ
cG0 k 3 ðp þ 6qu2 ÞG0 k 3 ð3pu þ 6qu3 ÞG00 k 3 ðr þ pu2 þ qu4 ÞG000 3kEG0 ¼ 0: Pm¼n2 Pm¼n3 P 1 m m m Let EðuÞ ¼ m¼n m¼n1 ðam u Þ; F ðuÞ ¼ m¼n2 ðbm u Þ; GðuÞ ¼ m¼n3 ðd m u Þ. Balancing the linear terms of the highest order with nonlinear terms in Eq. (11), we find that n1 = n2 = n3 = 2. Therefore EðuÞ ¼ a2 u þ a1 u þ a0 þ a1 u þ a2 u; F ðuÞ ¼ b2 u þ b1 u þ b0 þ b1 u þ b2 u; GðuÞ ¼ d 2 u þ d 1 u þ a0 þ d 1 u þ d 2 u:
ð12Þ
where a2, a1, a0, a1, a2, b2, b1, b0, b1, b2, d2, d1, d0, d1, d2 are to be determined later. Substituting Eq. (12) into Eq. (11) yields a set of algebraic equations for ui. Setting the coefficients of ui to zero yields a set of over-determined algebraic equations and we obtain the following results by use of Matlab a1 ¼ a0 ¼ a1 ¼ b2 ¼ b1 ¼ b1 ¼ d 2 ¼ d 1 ¼ d 1 ¼ 0;
1:
b0 ¼
2:
3:
c ðc 4k 3 pÞ; 3k
b2 ¼ 4ckr;
d0 ¼
a2 ¼
1 ðc 4k 3 pÞ; 3k
c pffiffiffiffiffiffiffiffiffiffiffi ð5c þ 8k 3 pÞ; 6k 2ck
d 2
a2 ¼ 4k 2 r; pffiffiffiffiffiffiffiffiffiffiffi ¼ 2rk 2ck :
1 a1 ¼ a0 ¼ a1 ¼ b2 ¼ b1 ¼ b1 ¼ d 1 ¼ d 0 ¼ d 1 ¼ d 2 ¼ 0; a2 ¼ ðc k 3 pÞ; 3k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 1 b0 ¼ ðc k 3 pÞ; b2 ¼ 2ckr; d 2 ¼ 15c2 r þ 12k 3 prc: 3k 3 a1 ¼ a0 ¼ b2 ¼ b1 ¼ d 0 ¼ d 1 ¼ d 2 ¼ 0; a2 ¼ 2k 2 r; b0 ¼
a2 ¼ 2k 2 r;
1 pffiffiffiffiffi ð4c2 q 4k 3 pqc þ 24k 3 cq qrÞ; 12ckq pffiffiffiffiffi ð4c2 q 4k 3 pqc þ 24k 3 cq qrÞ; b1 ¼ 2ckq;
ð13bÞ
a2 ¼
1 12k 2 q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffipffiffiffiffiffi d2 ¼ 12k 3 pqc þ 15qc2 þ 72k 3 qc qr qrÞ; 3q
a1 ¼ 2k 2 q;
ð13aÞ
d1 ¼
1 3
b2 ¼ 2ckr;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 12k 3 pqc þ 15qc2 þ 72k 3 qc qr: ð13cÞ
4:
a1 ¼ a0 ¼ b2 ¼ b1 ¼ d 0 ¼ d 1 ¼ d 2 ¼ 0;
1 pffiffiffiffiffi ð4c2 q 4k 3 pqc 24k 3 cq qrÞ; 12ckq pffiffiffiffiffi ð4c2 q 4k 3 pqc 24k 3 cq qrÞ; b1 ¼ 2ckq; a2 ¼
1 12k 2 q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffipffiffiffiffiffi d2 ¼ 12k 3 pqc þ 15qc2 72k 3 qc qr qrÞ; 3q
a1 ¼ 2k 2 q;
a2 ¼ 2k 2 r;
b0 ¼
d1 ¼
1 3
b2 ¼ 2ckr;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 12k 3 pqc þ 15qc2 72k 3 qc qr: ð13dÞ
Substituting (13a) into (12) and using the solutions of Eq. (4), we obtain the following periodic solutions ðc 4k 3 pÞ 4Ai k 2 r; 3kAi cðc 4k 3 pÞ ; vi ¼ 4Ai ckr þ 3k pffiffiffiffiffiffiffiffiffiffiffi cð5c þ 8k 3 pÞ pffiffiffiffiffiffiffiffiffiffiffi : wi ¼ 2rkAi 2ck 6k 2ck
ui ¼
396
B. Feng et al. / Chaos, Solitons and Fractals 39 (2009) 393–398
where i ¼ 1; 2; . . . ; 11 A1 ¼ A2 ¼ A3 ¼ A4 ¼
1 ; ds2 n
ð14aÞ 1
ðsn n icn nÞ2 1
ð14bÞ
;
ðmsn n idn nÞ2 1
;
; ðmcn n dn nÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 1 m2 dn nÞ2 ; A5 ¼ cn2 n 2 ðe gdn nÞ A6 ¼ ; sn2 n 2 ðe gsn nÞ ; A7 ¼ cn2 n ðe gmsn nÞ2 ; A8 ¼ dn2 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðe 1 m2 sn n gdn nÞ2 A9 ¼ ; cn2 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½eði 1 m2 sn n cn nÞ þ gðmcn n i 1 m2 Þ2 ; A10 ¼ dn2 n ½e þ gdn n ðe þ gÞcnn2 A11 ¼ : sn2 n
ð14cÞ ð14dÞ ð14eÞ ð14fÞ ð14gÞ ð14hÞ ð14iÞ ð14jÞ ð14kÞ
Substituting (13b) into (12) and using the solutions of Eq. (4), we obtain the following periodic solutions ðc 4k 3 pÞ 2Ai11 k 2 r; 3kA11 i cðc k 3 pÞ vi ¼ 2Ai11 ckr þ ; 3k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 15c2 r þ 12k 3 prcÞ : wi ¼ 3Ai11
ui ¼
ð15Þ
where i ¼ 12; 13; . . . ; 22. Substituting (13c) into (12) and using the solutions of Eq. (4), we obtain the following periodic solutions pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ð4c2 q 4k 3 pqc þ 24k 3 cq qrÞ 2k 2 q Ai22 2k 2 rAi22 ; ui ¼ 12ckqAi22 pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 4c2 q 4k 3 pqc þ 24k 3 cq qr ; vi ¼ 2ckq Ai22 2ckrAi22 þ ð16Þ 12k 2 q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffipffiffiffiffiffi 12k 3 pqc þ 15qc2 þ 72k 3 qc qr 12k 3 pqc þ 15qc2 þ 72k 3 qc qr qr pffiffiffiffiffiffiffiffiffiffi wi ¼ : 3qAi22 3 Ai22 where i ¼ 23; 24; . . . ; 33. Substituting (13d) into (12) and using the solutions of Eq. (4), we obtain the following periodic solutions pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ð4c2 q 4k 3 pqc 24k 3 cq qrÞ 2k 2 q Ai33 2k 2 rAi33 ; ui ¼ 12ckqAi33 pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 4c2 q 4k 3 pqc 24k 3 cq qr ; vi ¼ 2ckq Ai33 2ckrAi33 þ 12k 2 q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffipffiffiffiffiffi 12k 3 pqc þ 15qc2 72k 3 qc qr 12k 3 pqc þ 15qc2 72k 3 qc qr qr pffiffiffiffiffiffiffiffiffiffi wi ¼ : 3qAi33 3 Ai33
ð17Þ
B. Feng et al. / Chaos, Solitons and Fractals 39 (2009) 393–398
397
where i ¼ 34; 35; . . . ; 44. where n = kx + ct,e,g are arbitrary elements of {0,1}.
3. Remark In this paper, we have applied an improved generalized Jacobi elliptic function method to find abundant new solutions for the coupled Ito equation. This method can be easily applied to other coupled nonlinear evolution equation.
Appendix In this section, we present over-determined algebraic equations which substituting Eq. (12) into Eq. (11) yields a set of algebraic equations for ui and setting the coefficients of ui to zero 8 ca2 kb2 ¼ 0; > > > > > > > > ca1 kb1 ¼ 0; > > > > ca0 kb0 ¼ 0; > > > > > > > ca1 kb1 ¼ 0; > > > > > > ca2 kb2 ¼ 0; > > > > > > 4k 3 rd 2 þ ka2 d 2 ¼ 0; > > > > > > 2k 3 rd 1 þ ka2 d 1 þ 2ka1 d 2 ¼ 0; > > > > > > 2cd 2 þ 8k 3 pd 2 þ 3ka1 d 1 þ 6ka0 d 2 ¼ 0; > > > > > cd þ k 3 pd þ 3ka d þ 6ka d 3ka d ¼ 0; > > 1 0 1 1 2 2 1 1 > > > > > > > 2ka2 d 2 ka1 d 1 þ ka1 d 1 þ 2ka2 d 2 ¼ 0; > > > > cd 1 k 3 pd 1 3ka0 d 1 6ka1 d 2 þ 3ka2 d 1 ¼ 0; > > < 2cd 2 8k 3 pd 2 3ka1 d 1 6ka0 d 2 ¼ 0; > > > > 2k 3 qd 1 þ ka2 d 1 þ 2ka1 d 2 ¼ 0; > > > > 3 > > 4k qd 2 þ ka2 d 2 ¼ 0; > > > > > > > 2k 3 rb2 þ ka2 b2 þ kd 22 ¼ 0; > > > > > > 2k 3 rb1 þ 3ka2 b1 þ 3ka1 b2 þ 6kd 1 d 2 ¼ 0; > > > > > > cb2 þ 8k 3 pb2 þ 6ka1 b1 þ 6ka0 b2 þ 6ka2 b0 þ 12kd 0 d 2 þ 6kd 21 ¼ 0; > > > > 3 > > > > cb1 þ 2k pb1 þ 6ka0 b1 þ 6ka1 b2 þ 6ka1 b0 þ 6ka2 b1 > > > > þ12kd 0 d 1 þ 12kd 1 d 2 ¼ 0; > > > > > > cb1 þ 2k 3 pb1 þ 6ka0 b1 þ 6ka1 b2 þ 6ka1 b0 þ 6ka2 b1 þ 12kd 0 d 1 þ 12kd 1 d 2 ¼ 0; > > > > > cb þ 8k 3 pb þ 6ka b þ 6ka b þ 6ka b þ 12kd d þ 6kd 2 ¼ 0; > > 2 1 1 0 2 2 0 0 2 2 1 > > > > 3 > 2k qb1 þ 3ka2 b1 þ 3ka1 b2 þ 6kd 1 d 2 ¼ 0; > > > > : 3 2k qb2 þ ka2 b2 þ kd 22 ¼ 0:
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B. Feng et al. / Chaos, Solitons and Fractals 39 (2009) 393–398 Li Y, Ma W, Zhang JE. Phys Lett A. 2000;275:60. Fan E. Phys Lett A 2000;277:212. Fan E, Benny YCH. Phys Lett A 2002;292:335. Fan E. J Phys A Math Gen 2003;36:7009. Yan Q, Zhang Y. Chin Phys 2003;12:131. Chen HT, Zhang YQ. Chaos Solitons & Fractals 2004;20:765. Yao Y. Chaos Solitons & Fractals. 2005;24:301. Guo F. Appl Math 1989;2:112.
Chaos, Solitons and Fractals 39 (2009) 393–398 www.elsevier.com/locate/chaos
Abundant new travelling wave solutions for the coupled Ito equation Binlu Feng a, Bo Han a, Huanhe Dong b
b,*
a Department of Mathematics, Harbin Institute of Technology, Harbin 15001, PR China College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266510, China
Abstract Abundant new travelling wave solutions of the coupled Ito equation are obtained by the generalized Jacobi elliptic function method. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction Seeking exact solutions of nonlinear physics equations is an important and interesting subject Since the exact solutions to nonlinear wave equations helps to understand the characteristics of the nonlinear equations. In recent years, many powerful methods have been presented, such as homogeneous balance method, Darboux-transformation, Inverse Scattering Transformation and tanh-function method and so on [1–6]. It is well known that many nonlinear evolution equations have been shown to possess the periodic solutions. Recently, some methods were presented to seek the periodic solutions, such as the Jacobi elliptic function expansion method, the algebraic method and the generalized Jacobi elliptic function method[7–11], etc. In this paper, we use the improved generalized Jacobi elliptic function method to obtain abundant new solutions for the coupled Ito equation. For a given nonlinear evolution equation in three independent variables H ðu; ut ; ux ; . . .Þ ¼ 0:
ð1Þ
We seek the following formal travelling wave solutions u ¼ uðx; tÞ ¼ U ðnÞ;
n ¼ kx þ ct;
ð2Þ
where k,c are arbitrary constants. Substituting Eq. (2) into Eq. (1) gives rise to an ordinary equation H 1 ðU ; U 0 ; U 00 ; . . .Þ ¼ 0:
ð3Þ
Then consider the elliptic equation pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u0 ¼ r r þ pu2 þ qu4 ;
ð4Þ
*
Corresponding author. E-mail address: [email protected] (H. Dong).
0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.04.012
394
B. Feng et al. / Chaos, Solitons and Fractals 39 (2009) 393–398
where u0 ¼ du ; r ¼ 1; r; p; q are constants. dn We know that Eq. (4) possess the following solution: Case Case Case Case Case Case Case
1: 2: 3: 4: 5: 6: 7:
if if if if if if if
r = 1, p = (1 + m2), q = m2, then Eq. (4) has solution u1a(n) = sn n, u2a = cd n; r = 1 m2, p = 2m2 1, q = m2, then Eq. (4) has solution u2(n) = cn n, r = m2 1, p = 2 m2, q = 1, then Eq. (4) has solution u3(n) = dn n, r = m2(1 m2), p = 2m2 1, q = 1, then Eq. (4) has solution u4(n) = ds n, r = 1 m2, p = 2 m2, q = 1, then Eq. (4) has solution u5(n) = cs n, 2 2 sn n r ¼ 14 ; p ¼ m 22 ; q ¼ m4 , then Eq. (4) has solution u6 ðnÞ ¼ 1dn ; n m2 m2 2 r ¼ q ¼ 4 ; p ¼ 2 ,then Eq. (4) has solution
dn n u7a ðnÞ ¼ sn n icn n; u7b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; i 1 m2 sn n cn n 2
, then Eq. (4) has solution Case 8: if r ¼ q ¼ 14 ; p ¼ 12m 2 u8a ðnÞ ¼
dn n sn n cn n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; u8b ¼ msn n idn n; u8c ¼ ; u ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 dn n 8d i 1 m2 sn n dn n mcn n i 1 m2 2
2
dn n ; Case 9: if r ¼ q ¼ m 41 ; p ¼ m 2þ1, then Eq. (4) has solution u9 ðnÞ ¼ 1msn n 1m2
2
cn n Case 10: if r ¼ q ¼ 2 4 ; p ¼ m 2þ1, then Eq. (4) has solution u10 ðnÞ ¼ 1sn ; n 2Þ m2 þ1 1 Case 11: if r ¼ ð1m ; p ¼ ; q ¼ , then Eq. (4) has solution u (n) = mcn n ± dn n, 11 4 2 4 2 2
Þ sn n Case 12: if r ¼ 14 ; p ¼ m 2þ1 ; q ¼ ð1m , Eq then Eq. (4) has solution u12 ðnÞ ¼ dn ncn ; 4 n 2 2 cn n 1 m 2 m p ffiffiffiffiffiffiffiffi Case 13: if r ¼ 4 ; p ¼ 2 ; q ¼ 4 , then Eq. (4) has solution u13 ðnÞ ¼ : 2 2
1m dn n
Here, sn n = sn (n, m), m denote the modulus of the Jacobi elliptic functions. Introduce a new independent variable u as follows uðx; tÞ ¼ U ðnÞ ¼ AðuÞ ¼
m¼n X
ðam um Þ
ð5Þ
m¼n
We find that the derivatives with respect to the variable n become the derivatives with respect to the variable u as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d d ¼ r r þ pu2 þ qu4 ; dn du d2 d d2 ¼ ðpu þ 2qu3 Þ þ ðr þ pu2 þ qu4 Þ 2 ; 2 du du dn 3 2 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p d d 3 d 2 4 d 2 þ qu4 ðp þ 6qu2 Þ : ¼ r r þ pu Þ þ ðr þ pu þ qu Þ þ ð3pu þ 6qu du2 du3 du dn3
ð6Þ
Finally, inserting Eq. (5) with Eq. (6) into Eq. (3) yields H 2 ðA; A0 ; A00 ; . . .Þ ¼ 0:
ð7Þ
2. The solution of the coupled Ito equation Consider the coupled Ito equation in Ref.[12] ut ¼ vx ; vt ¼ 2ðvxxx þ 3uvx þ 3vux Þ 12wwx ; wt ¼ wxxx þ 3uwx :
ð8Þ
Let uðx; tÞ ¼ U ðnÞ;
vðx; tÞ ¼ V ðnÞ;
wðx; tÞ ¼ W ðnÞ;
n ¼ kx þ ct:
ð9Þ
B. Feng et al. / Chaos, Solitons and Fractals 39 (2009) 393–398
395
Substituting (9) into (8) yields cU kV ¼ 0; 000
cV 0 þ 2k 3V þ 6kUV 0 þ 6kU 0 V þ 12kWW 0 ¼ 0; 0
cW k
3W 000
ð10Þ
0
3kUW ¼ 0:
Inserting U(n) = E(u), V(n) = F(u), W(n) = G(u) with Eqs. (6) into (10) gives cE kF ¼ 0; cF 0 þ 2k 3 ðp þ 6qu2 ÞF 0 þ 2k 3 ð3pu þ 6qu3 ÞF 00 þ 2k 3 ðr þ pu2 þ qu4 ÞF 000 þ 6kEF 0 þ 6kFE0 þ 6kGG0 ¼ 0;
ð11Þ
cG0 k 3 ðp þ 6qu2 ÞG0 k 3 ð3pu þ 6qu3 ÞG00 k 3 ðr þ pu2 þ qu4 ÞG000 3kEG0 ¼ 0: Pm¼n2 Pm¼n3 P 1 m m m Let EðuÞ ¼ m¼n m¼n1 ðam u Þ; F ðuÞ ¼ m¼n2 ðbm u Þ; GðuÞ ¼ m¼n3 ðd m u Þ. Balancing the linear terms of the highest order with nonlinear terms in Eq. (11), we find that n1 = n2 = n3 = 2. Therefore EðuÞ ¼ a2 u þ a1 u þ a0 þ a1 u þ a2 u; F ðuÞ ¼ b2 u þ b1 u þ b0 þ b1 u þ b2 u; GðuÞ ¼ d 2 u þ d 1 u þ a0 þ d 1 u þ d 2 u:
ð12Þ
where a2, a1, a0, a1, a2, b2, b1, b0, b1, b2, d2, d1, d0, d1, d2 are to be determined later. Substituting Eq. (12) into Eq. (11) yields a set of algebraic equations for ui. Setting the coefficients of ui to zero yields a set of over-determined algebraic equations and we obtain the following results by use of Matlab a1 ¼ a0 ¼ a1 ¼ b2 ¼ b1 ¼ b1 ¼ d 2 ¼ d 1 ¼ d 1 ¼ 0;
1:
b0 ¼
2:
3:
c ðc 4k 3 pÞ; 3k
b2 ¼ 4ckr;
d0 ¼
a2 ¼
1 ðc 4k 3 pÞ; 3k
c pffiffiffiffiffiffiffiffiffiffiffi ð5c þ 8k 3 pÞ; 6k 2ck
d 2
a2 ¼ 4k 2 r; pffiffiffiffiffiffiffiffiffiffiffi ¼ 2rk 2ck :
1 a1 ¼ a0 ¼ a1 ¼ b2 ¼ b1 ¼ b1 ¼ d 1 ¼ d 0 ¼ d 1 ¼ d 2 ¼ 0; a2 ¼ ðc k 3 pÞ; 3k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 1 b0 ¼ ðc k 3 pÞ; b2 ¼ 2ckr; d 2 ¼ 15c2 r þ 12k 3 prc: 3k 3 a1 ¼ a0 ¼ b2 ¼ b1 ¼ d 0 ¼ d 1 ¼ d 2 ¼ 0; a2 ¼ 2k 2 r; b0 ¼
a2 ¼ 2k 2 r;
1 pffiffiffiffiffi ð4c2 q 4k 3 pqc þ 24k 3 cq qrÞ; 12ckq pffiffiffiffiffi ð4c2 q 4k 3 pqc þ 24k 3 cq qrÞ; b1 ¼ 2ckq;
ð13bÞ
a2 ¼
1 12k 2 q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffipffiffiffiffiffi d2 ¼ 12k 3 pqc þ 15qc2 þ 72k 3 qc qr qrÞ; 3q
a1 ¼ 2k 2 q;
ð13aÞ
d1 ¼
1 3
b2 ¼ 2ckr;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 12k 3 pqc þ 15qc2 þ 72k 3 qc qr: ð13cÞ
4:
a1 ¼ a0 ¼ b2 ¼ b1 ¼ d 0 ¼ d 1 ¼ d 2 ¼ 0;
1 pffiffiffiffiffi ð4c2 q 4k 3 pqc 24k 3 cq qrÞ; 12ckq pffiffiffiffiffi ð4c2 q 4k 3 pqc 24k 3 cq qrÞ; b1 ¼ 2ckq; a2 ¼
1 12k 2 q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffipffiffiffiffiffi d2 ¼ 12k 3 pqc þ 15qc2 72k 3 qc qr qrÞ; 3q
a1 ¼ 2k 2 q;
a2 ¼ 2k 2 r;
b0 ¼
d1 ¼
1 3
b2 ¼ 2ckr;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 12k 3 pqc þ 15qc2 72k 3 qc qr: ð13dÞ
Substituting (13a) into (12) and using the solutions of Eq. (4), we obtain the following periodic solutions ðc 4k 3 pÞ 4Ai k 2 r; 3kAi cðc 4k 3 pÞ ; vi ¼ 4Ai ckr þ 3k pffiffiffiffiffiffiffiffiffiffiffi cð5c þ 8k 3 pÞ pffiffiffiffiffiffiffiffiffiffiffi : wi ¼ 2rkAi 2ck 6k 2ck
ui ¼
396
B. Feng et al. / Chaos, Solitons and Fractals 39 (2009) 393–398
where i ¼ 1; 2; . . . ; 11 A1 ¼ A2 ¼ A3 ¼ A4 ¼
1 ; ds2 n
ð14aÞ 1
ðsn n icn nÞ2 1
ð14bÞ
;
ðmsn n idn nÞ2 1
;
; ðmcn n dn nÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 1 m2 dn nÞ2 ; A5 ¼ cn2 n 2 ðe gdn nÞ A6 ¼ ; sn2 n 2 ðe gsn nÞ ; A7 ¼ cn2 n ðe gmsn nÞ2 ; A8 ¼ dn2 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðe 1 m2 sn n gdn nÞ2 A9 ¼ ; cn2 n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½eði 1 m2 sn n cn nÞ þ gðmcn n i 1 m2 Þ2 ; A10 ¼ dn2 n ½e þ gdn n ðe þ gÞcnn2 A11 ¼ : sn2 n
ð14cÞ ð14dÞ ð14eÞ ð14fÞ ð14gÞ ð14hÞ ð14iÞ ð14jÞ ð14kÞ
Substituting (13b) into (12) and using the solutions of Eq. (4), we obtain the following periodic solutions ðc 4k 3 pÞ 2Ai11 k 2 r; 3kA11 i cðc k 3 pÞ vi ¼ 2Ai11 ckr þ ; 3k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 15c2 r þ 12k 3 prcÞ : wi ¼ 3Ai11
ui ¼
ð15Þ
where i ¼ 12; 13; . . . ; 22. Substituting (13c) into (12) and using the solutions of Eq. (4), we obtain the following periodic solutions pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ð4c2 q 4k 3 pqc þ 24k 3 cq qrÞ 2k 2 q Ai22 2k 2 rAi22 ; ui ¼ 12ckqAi22 pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 4c2 q 4k 3 pqc þ 24k 3 cq qr ; vi ¼ 2ckq Ai22 2ckrAi22 þ ð16Þ 12k 2 q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffipffiffiffiffiffi 12k 3 pqc þ 15qc2 þ 72k 3 qc qr 12k 3 pqc þ 15qc2 þ 72k 3 qc qr qr pffiffiffiffiffiffiffiffiffiffi wi ¼ : 3qAi22 3 Ai22 where i ¼ 23; 24; . . . ; 33. Substituting (13d) into (12) and using the solutions of Eq. (4), we obtain the following periodic solutions pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ð4c2 q 4k 3 pqc 24k 3 cq qrÞ 2k 2 q Ai33 2k 2 rAi33 ; ui ¼ 12ckqAi33 pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 4c2 q 4k 3 pqc 24k 3 cq qr ; vi ¼ 2ckq Ai33 2ckrAi33 þ 12k 2 q qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffipffiffiffiffiffi 12k 3 pqc þ 15qc2 72k 3 qc qr 12k 3 pqc þ 15qc2 72k 3 qc qr qr pffiffiffiffiffiffiffiffiffiffi wi ¼ : 3qAi33 3 Ai33
ð17Þ
B. Feng et al. / Chaos, Solitons and Fractals 39 (2009) 393–398
397
where i ¼ 34; 35; . . . ; 44. where n = kx + ct,e,g are arbitrary elements of {0,1}.
3. Remark In this paper, we have applied an improved generalized Jacobi elliptic function method to find abundant new solutions for the coupled Ito equation. This method can be easily applied to other coupled nonlinear evolution equation.
Appendix In this section, we present over-determined algebraic equations which substituting Eq. (12) into Eq. (11) yields a set of algebraic equations for ui and setting the coefficients of ui to zero 8 ca2 kb2 ¼ 0; > > > > > > > > ca1 kb1 ¼ 0; > > > > ca0 kb0 ¼ 0; > > > > > > > ca1 kb1 ¼ 0; > > > > > > ca2 kb2 ¼ 0; > > > > > > 4k 3 rd 2 þ ka2 d 2 ¼ 0; > > > > > > 2k 3 rd 1 þ ka2 d 1 þ 2ka1 d 2 ¼ 0; > > > > > > 2cd 2 þ 8k 3 pd 2 þ 3ka1 d 1 þ 6ka0 d 2 ¼ 0; > > > > > cd þ k 3 pd þ 3ka d þ 6ka d 3ka d ¼ 0; > > 1 0 1 1 2 2 1 1 > > > > > > > 2ka2 d 2 ka1 d 1 þ ka1 d 1 þ 2ka2 d 2 ¼ 0; > > > > cd 1 k 3 pd 1 3ka0 d 1 6ka1 d 2 þ 3ka2 d 1 ¼ 0; > > < 2cd 2 8k 3 pd 2 3ka1 d 1 6ka0 d 2 ¼ 0; > > > > 2k 3 qd 1 þ ka2 d 1 þ 2ka1 d 2 ¼ 0; > > > > 3 > > 4k qd 2 þ ka2 d 2 ¼ 0; > > > > > > > 2k 3 rb2 þ ka2 b2 þ kd 22 ¼ 0; > > > > > > 2k 3 rb1 þ 3ka2 b1 þ 3ka1 b2 þ 6kd 1 d 2 ¼ 0; > > > > > > cb2 þ 8k 3 pb2 þ 6ka1 b1 þ 6ka0 b2 þ 6ka2 b0 þ 12kd 0 d 2 þ 6kd 21 ¼ 0; > > > > 3 > > > > cb1 þ 2k pb1 þ 6ka0 b1 þ 6ka1 b2 þ 6ka1 b0 þ 6ka2 b1 > > > > þ12kd 0 d 1 þ 12kd 1 d 2 ¼ 0; > > > > > > cb1 þ 2k 3 pb1 þ 6ka0 b1 þ 6ka1 b2 þ 6ka1 b0 þ 6ka2 b1 þ 12kd 0 d 1 þ 12kd 1 d 2 ¼ 0; > > > > > cb þ 8k 3 pb þ 6ka b þ 6ka b þ 6ka b þ 12kd d þ 6kd 2 ¼ 0; > > 2 1 1 0 2 2 0 0 2 2 1 > > > > 3 > 2k qb1 þ 3ka2 b1 þ 3ka1 b2 þ 6kd 1 d 2 ¼ 0; > > > > : 3 2k qb2 þ ka2 b2 þ kd 22 ¼ 0:
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