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Exact solutions for the higher-order nonlinear Schördinger equation in nonlinear optical fibres
Chaos, Solitons and Fractals 23 (2005) 949–955 www.elsevier.com/locate/chaos
Exact solutions for the higher-order nonlinear Scho¨rdinger equation in nonlinear optical fibres Chunping Liu Institute of Mathematics, Yangzhou University, Yangzhou 225002, PeopleÕs Republic of China Accepted 7 June 2004 Communicated by Prof. M. Wadati
Abstract First, by using the generally projective Riccati equation method, many kinds of exact solutions for the higher-order nonlinear Scho¨rdinger equation in nonlinear optical fibres are obtained in a unified way. Then, some relations among these solutions are revealed. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction The higher-order nonlinear Scho¨rdinger equation describing propagation of ultrashort pulses in nonlinear optical fibres [1] reads oW o2 W o3 W oWjWj2 ojWj2 ¼ ia1 2 þ ia2 WjWj2 þ a3 3 þ a4 þ a5 W ; oz ot ot ot ot
ð1Þ
where W is slowly varying envelop of the electric field, the subscripts z and t are the spatial and temporal partial derivative in retard time coordinates, and a1, a2, a3, a4 and a5 are the real parameters related to the group velocity (GVD), self-phase modulation (SPM), third-order dispersion (TOP), self-steepening, and self-frequency shift arising from stimulated Raman scattering, respectively. In this paper, first, we will apply the generally projective Riccati equation method to solve Eq. (1), and derive twelve kinds of exact solutions of Eq. (1) in a unified way. Then, we will reveal that there are some certain relations among the solutions. The key idea of this method is to introduce a new projective Riccati equation and use its solutions to replace the elementary functions in the projective Riccati equation method [2,3], which simply proceeds as follows. Step 1. For a given nonlinear evolution equation (NLEE), say in two variables x, t, P ðu; ux ; ut ; uxx ; . . .Þ ¼ 0;
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.06.006
ð2Þ
950
C. Liu / Chaos, Solitons and Fractals 23 (2005) 949–955
we consider its travelling wave solutions uðx; tÞ ¼ uðnÞ;
n ¼ x kt þ n0 ;
ð3Þ
then Eq. (2) is reduced to an ordinary differential equation (ODE) Qðu; u0 ; u00 ; . . .Þ ¼ 0;
ð4Þ
where a prime denotes d/dn. Step 2. We introduce two new variables r = r(n), s = s(n) which are the solutions of the following new projective Riccati equation: r0 ðnÞ ¼ erðnÞsðnÞ;
s0 ðnÞ ¼ R þ es2 ðnÞ lrðnÞ;
e ¼ 1:
It is easy to see that Eq. (5) admits the first integral with R 6¼ 0, ðl2 þ dÞ 2 2 s ðnÞ ¼ e R 2lrðnÞ þ r ðnÞ ; d ¼ 1; R
ð5Þ
ð6Þ
where R and l are constants. By virtue of the variables r and s, we assume that Eq. (4) has the following solutions: n X ri1 ðnÞ½Ai rðnÞ þ Bi sðnÞ þ A0 ; ð7Þ uðnÞ ¼ i¼1
where Ai and Bi are constants to be determined later. The parameter n can be determined by balancing the highest-order derivative term with nonlinear term in (2) or (4). (If n is not a positive integer, first make the transformation u = wn.) Step 3. Substituting the expression (7), (5) and (6) into (4) and setting the coefficients of these terms rj(n)si(n) (or l s (n)) to zero yields a set of over-determined algebraic equations, from which the constants Ai, Bi, R, l, k can be found explicitly. Step 4. Note that Eqs. (5) and (6) admits the following solutions: Case 1 When e = 1, d = 1, R 6¼ 0, pffiffiffi R sechð RnÞ pffiffiffi r1 ðnÞ ¼ ; l sechð RnÞ þ 1 Case 2 When e = 1, d = 1, R 6¼ 0, pffiffiffi R cschð RnÞ pffiffiffi ; r2 ðnÞ ¼ l cschð RnÞ þ 1 Case 3 When e = 1, d = 1, R 6¼ 0, pffiffiffi R secð RnÞ pffiffiffi ; r3 ðnÞ ¼ l secð RnÞ þ 1 pffiffiffi R cscð RnÞ pffiffiffi r4 ðnÞ ¼ ; l cscð RnÞ þ 1
pffiffiffi pffiffiffi R tanhð RnÞ pffiffiffi : l sechð RnÞ þ 1
ð8Þ
pffiffiffi pffiffiffi R cothð RnÞ pffiffiffi : s2 ðnÞ ¼ l cschð RnÞ þ 1
ð9Þ
s1 ðnÞ ¼
pffiffiffi pffiffiffi R tanð RnÞ pffiffiffi ; s3 ðnÞ ¼ l secð RnÞ þ 1
ð10Þ
pffiffiffi pffiffiffi R cotð RnÞ pffiffiffi : s4 ðnÞ ¼ l cscð RnÞ þ 1
ð11Þ
Substituting the constants Ai, Bi, R, l, k obtained in Step 3 into (3) and (7) along with (8)–(11), we can then obtain the solutions of Eq. (2).
2. Exact solutions of Eq. (1) Now let us turn to Eq. (1). According to the above steps, to seek travelling wave solutions of Eq. (1), we make the gauge transformation Wðz; tÞ ¼ wðnÞ exp½iðkz wtÞ ;
n ¼ t kz þ n0 ;
ð12Þ
C. Liu / Chaos, Solitons and Fractals 23 (2005) 949–955
951
where k, w, k are constants to be determined later, n0 is an arbitrary constant. Substituting (12) into Eq. (1) yields the real system ða1 3a3 wÞw00 þ ða3 w3 a1 w2 kÞw þ ða2 a4 wÞw3 ¼ 0;
ð13Þ
a3 w000 þ ð2a1 w 3a3 w2 þ kÞw0 þ ð3a4 þ 2a5 Þw2 w0 ¼ 0;
ð14Þ
under the constraint conditions w¼
a1 ð3a4 þ 2a5 Þ 3a2 a3 ; 6a3 ða4 þ a5 Þ
k¼
ð15Þ
1 ½ða1 3a3 wÞð2a1 w 3a3 w2 þ kÞ a1 w2 þ a3 w3 : a3
ð16Þ
Eqs. (13) and (14) becomes w00 þ k 1 w þ k 3 w3 ¼ 0;
ð17Þ
where k1 ¼
2a1 w 3a3 w2 þ k ; a3
k3 ¼
3a4 þ 2a5 : 3a3
ð18Þ 00
By balancing the highest-order derivative term w with the nonlinear term w3 in Eq. (17), we obtain n = 1 in (7), so we assume (17) has the solutions in the form with R 6¼ 0, wðnÞ ¼ A0 þ A1 rðnÞ þ B1 sðnÞ;
ð19Þ
where A0, A1 and B1 are constants to be determined later. r(n) and s(n) satisfy (5) and (6). According to the Step 3, we substitute (19) into Eq. (17) along with (5) and (6). The following algebraic system occur const : k 1 A0 þ k 3 ðA30 3eA0 B21 RÞ ¼ 0;
ð20Þ
rðnÞ : eA1 R þ k 1 A1 þ k 3 ð3A20 A1 þ 6elA0 B21 3eA1 B21 RÞ ¼ 0;
ð21Þ
sðnÞ : k 1 B1 þ k 3 ð3A20 B1 eB31 RÞ ¼ 0;
ð22Þ
2 2 2 2 2 l þd ¼ 0; r ðnÞ : 3elA1 þ k 3 3A0 A1 þ 6elA1 B1 3eA0 B1 R
ð23Þ
rðnÞsðnÞ : elB1 þ k 3 ð6A0 A1 B1 þ 2elB31 Þ ¼ 0;
ð24Þ
r3 ðnÞ : 2eA1
2 l2 þ d l þd þ k 3 A31 3eA1 B21 ¼ 0; R R
ð25Þ
2 2 l þd 2 3 l þd r ðnÞsðnÞ : 2eB1 þ k 3 3A1 B1 eB1 ¼ 0: R R 2
ð26Þ
From which, by using the Wu elimination method [4–6], we have the following solutions: 2 ; k3
R¼
2d ; k1 k3
R¼
ðIÞ A0 ¼ A1 ¼ l ¼ 0;
B21 ¼
ðIIÞ A0 ¼ B1 ¼ l ¼ 0;
A21 ¼
ðIIIÞ A0 ¼ 0;
A21 ¼
l2 þ d ; 4k 1 k 3
B21 ¼
k1 ; 2e
k1 ; e
1 ; 2k 3
e ¼ 1;
e ¼ 1;
R¼
d ¼ 1;
ð27Þ
d ¼ 1;
2k 1 ; e
e ¼ 1;
ð28Þ
d ¼ 1:
ð29Þ
Therefore from (8)–(11) and (I)–(III), we obtain twelve kinds of exact travelling wave solution of Eq. (1) in a unified way
952
C. Liu / Chaos, Solitons and Fractals 23 (2005) 949–955
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 3ð2a1 w 3a3 sw2 þ kÞ 2a1 w 3a3 w2 þ k W1 ¼ tanh ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 2a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 3ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W2 ¼ coth ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 2a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 6ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W3 ¼ sech ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 6ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W4 ¼ csch ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 3ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W5 ¼ tan ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 2a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 3ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W6 ¼ cot ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 2a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 6ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W7 ¼ sec ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 6ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W8 ¼ csc ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 a3
ð30Þ
ð31Þ
ð32Þ
ð33Þ
ð34Þ
ð35Þ
ð36Þ
ð37Þ
ffi hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8 2ð2a1 w3a3 w2 þkÞ >pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðt kz þ n0 Þ a3 3ð2a1 w 3a3 w2 þ kÞ< l 1sech W9 ¼ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i > 3a4 þ 2a5 : l sech 2ð2a1 w3a3 w2 þkÞðt kz þ n Þ þ 1 a3
0
i 9 2ð2a1 w3a3 w þkÞ > = ðt kz þ n e tanh Þ 0 a3 ffi exp½iðkz wtÞ ; þ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 > 3 w þkÞ ; ðt kz þ n Þ þ 1 l sech 2ð2a1 w3a 0 a3 ffi hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
ð38Þ
ffi hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8 2ð2a1 w3a3 w2 þkÞ >pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðt kz þ n0 Þ a3 3ð2a1 w 3a3 w2 þ kÞ< l þ 1csch W10 ¼ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i > 3a4 þ 2a5 : l csch 2ð2a1 w3a3 w2 þkÞðt kz þ n Þ þ 1 a3
0
ffi hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 9 2ð2a1 w3a3 w2 þkÞ > = ðt kz þ n e coth Þ 0 a3 ffi exp½iðkz wtÞ ; þ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 > 3 w þkÞ ; ðt kz þ n Þ þ 1 l csch 2ð2a1 w3a 0 a3
ð39Þ
hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i pffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8 2 > 3 w þkÞ 2 sec < ðt kz þ n0 Þ 1 l 2ð2a1 w3a 2 a3 3ð2a1 w 3a3 w þ kÞ W11 ¼ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 > l sec 3a4 þ 2a5 : 2ð2a1 w3a3 w þkÞðt kz þ n Þ þ 1 a3
hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 9 2 > 3 w þkÞ = ðt kz þ n e tan 2ð2a1 w3a Þ 0 a3 exp½iðkz wtÞ ; þ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 3 w þkÞ ; ðt kz þ n0 Þ þ 1> l sec 2ð2a1 w3a a3
0
ð40Þ
C. Liu / Chaos, Solitons and Fractals 23 (2005) 949–955
953
hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8 2 >pffiffiffiffiffiffiffiffiffiffiffiffiffi2 3 w þkÞ ðt kz þ n0 Þ 2ð2a1 w3a a3 3ð2a1 w 3a3 w2 þ kÞ< 1 l csc W12 ¼ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 > 3a4 þ 2a5 : l csc 2ð2a1 w3a3 w þkÞðt kz þ n Þ þ 1 0
a3
hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 9 2 > 3 w þkÞ = ðt kz þ n e cot 2ð2a1 w3a Þ 0 a3 þ exp½iðkz wtÞ : hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 > 3 w þkÞ ; ðt kz þ n Þ þ 1 l csc 2ð2a1 w3a 0 a3
ð41Þ
Remark 1. In Refs. [7,8], Yan thought (9) satisfied ðl2 1Þ 2 r ðnÞ : s2 ðnÞ ¼ R 2lrðnÞ þ R
ð42Þ
Actually, it is wrong. In addition, (1) Page 762 in Ref. [7], Case 1. l = A0 = A1 = 0, e = 1, B21 ¼
6a3 ; 3a4 þ 2a5
R¼
2a1 w 3a3 w2 þ k ; 2a3
should be B21 ¼
6a3 ; 3a4 þ 2a5
R¼
2a1 w 3a3 w2 þ k : 2a3
(2) Page 763 in Ref. [7] Case 5. A0 = 0, e = 1, A21 ¼
3a23 ð1 l2 Þ ; ð3a4 þ 2a5 Þð2a1 w 3a3 w2 þ kÞ
B21 ¼
6a3 ; 3a4 þ 2a5
R¼
2ð2a1 w 3a3 w2 þ kÞ ; a3
should be A21 ¼
3a23 ðl2 1Þ ; 4ð3a4 þ 2a5 Þð2a1 w 3a3 w2 þ kÞ
B21 ¼
3a3 ; 2ð3a4 þ 2a5 Þ
R¼
2ð2a1 w 3a3 w2 þ kÞ : a3
(3) Page 763 in Ref. [7] Case 6. A0 = 0, e = 1, A21 ¼
3a23 ð1 l2 Þ ; ð3a4 þ 2a5 Þð2a1 w 3a3 w2 þ kÞ
B21 ¼
6a3 ; 3a4 þ 2a5
R¼
2ð2a1 w 3a3 w2 þ kÞ ; a3
should be A21 ¼
3a23 ðl2 1Þ ; 4ð3a4 þ 2a5 Þð2a1 w 3a3 w2 þ kÞ
B21 ¼
3a3 ; 2ð3a4 þ 2a5 Þ
R¼
2ð2a1 w 3a3 w2 þ kÞ : a3
Therefore, those solutions W1, W2, W3, W4, W7, W8, W10, W11, W12 in Ref. [7] (or Ref. [8]) are wrong. Remark 2. When R = 0, we assume Eq. (17) has the solution wðnÞ ¼ A0 þ A1 sðnÞ;
ð43Þ
where A0, A1 are constants to be determined later. s(n) satisfy s 0 (n) = s2(n). Similar to the above process, we can obtain the rational solution of Eq. (1) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6a3 1 exp½iðkz wtÞ : W13 ¼ 3a4 þ 2a5 t ð3a3 w2 2a1 wÞz þ n0
ð44Þ
954
C. Liu / Chaos, Solitons and Fractals 23 (2005) 949–955
3. The relations among the solutions W1 W12 of Eq. (1) Next, we reveal some relations among these solutions W1 W12 of Eq. (1). Notice the identical formula tanh ih ¼
eih eih cos h þ i sin h cos h þ i sin h ¼ ¼ i tan h; eih þ eih cos h þ i sin h þ cos h i sin h
ð45Þ
sech ih ¼
2 2 ¼ sec h; ¼ eih þ eih cos h þ i sin h þ cos h i sin h
ð46Þ
coth ih ¼
eih þ eih cos h þ i sin h þ cos h i sin h ¼ i cot h; ¼ eih eih cos h þ i sin h cos h þ i sin h
ð47Þ
csch ih ¼
2 2 ¼ icsch; ¼ eih eih cos h þ i sin h cos h þ isinh
ð48Þ
pffiffiffiffiffiffiffi here i ¼ 1. Therefore we can obtain directly W5, W6, W7 and W8 from W1, W2, W3 and W4, respectively. We have showed this method in Refs. [9,10]. In addition, we can prove pffiffiffiffiffiffiffiffiffiffiffiffiffi h u1 l2 1sech h tanh h ¼ tanh ; ð49Þ l sech h þ 1 2 pffiffiffiffiffiffiffiffi l2 1 where u1 ¼ tanh1 , l P 1. l pffiffiffiffiffiffiffiffiffiffiffiffiffi h u2 l2 1sech h tanh h ¼ coth ; ð50Þ l sech h þ 1 2 pffiffiffiffiffiffiffiffi l2 1 , l 6 1. where u2 ¼ tanh1 l pffiffiffiffiffiffiffiffiffiffiffiffiffi h þ u3 l2 þ 1cschh þ coth h ¼ coth ; ð51Þ l cschh þ 1 2 l ffi . where u3 ¼ tanh1 pffiffiffiffiffiffiffi l2 þ1 pffiffiffiffiffiffiffiffiffiffiffiffiffi h u4 l2 þ 1cschh coth h ¼ tanh ; ð52Þ l cschh þ 1 2 l ffi . where u4 ¼ tanh1 pffiffiffiffiffiffiffi l2 þ1 pffiffiffiffiffiffiffiffiffiffiffiffiffi h u5 1 l2 sec h tan h ; ð53Þ ¼ tan 2 l sec h þ 1 pffiffiffiffiffiffiffiffi 1l2 , jlj 6 1. where u5 ¼ tan1 l pffiffiffiffiffiffiffiffiffiffiffiffiffi h þ u6 1 l2 csc h þ cot h ; ð54Þ ¼ cot 2 l csc h þ 1 l ffi , jlj 6 1. where u6 ¼ tan1 pffiffiffiffiffiffiffi 2 1l
pffiffiffiffiffiffiffiffiffiffiffiffiffi h u7 1 l2 csc h cot h ¼ tan ; l csc h þ 1 2 l ffi , jlj 6 1. where u7 ¼ tan1 pffiffiffiffiffiffiffi 2
ð55Þ
1l
We take (49) as a sample to show the detailed process, and (50)–(55) can be proved in a similar way. pffiffiffiffiffiffiffiffiffiffiffiffiffi In fact, when l P 1, set l = cosh u1, thus l2 1 ¼ sinh u1 , notice the identical formula hþu hu sinh h þ sinh u ¼ 2 sinh cosh ; 2 2
ð56Þ
C. Liu / Chaos, Solitons and Fractals 23 (2005) 949–955
sinh h sinh u ¼ 2 cosh
hþu hu sinh ; 2 2
cosh h þ cosh u ¼ 2 cosh
hþu hu cosh ; 2 2
cosh h cosh u ¼ 2 sinh
hþu hu sinh ; 2 2
955
ð57Þ
ð58Þ
ð59Þ
we have pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi h u1 l2 1sech h tanh h l2 1 sinh h sinh u1 sinh h ¼ ¼ ¼ tanh : l sech h þ 1 l þ cosh h cosh u1 þ cosh h 2 So, from (49)–(55), we can see that when l P 1, the solutions W9 and W1 are same in waveform and only different in phase, when l 6 1, the solutions W9 and W2 are same in waveform and only different in phase. The solutions W10 and W1 (or W2), W11 and W5, W12 and W5 (or W6) are also in the same way.
4. Conclusions In summary, by using the generally projective Riccati equation method, many kinds of exact solutions of Eq. (1) are obtained in a unified way. And then the relations among these solutions are revealed, namely, the periodic wave solutions W5, W6, W7, W8 of Eq. (1) can be obtained directly from soliton solutions W1, W2, W3, W4 of Eq. (1), respectively. When jlj P 1, the solutions W9 and W1 (or W2) are same in waveform and only different in phase. The solutions W10 and W1 (or W2) are also in the same way. When jlj 6 1, W11 and W5 are same in waveform and only different in phase, W12 and W5 (or W6) are also in the same way.
Acknowledgments This project is supported by the Natural Science Foundation of Jiangsu Education Committee and the Natural Science Foundation of China.
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Exact solutions for the higher-order nonlinear Scho¨rdinger equation in nonlinear optical fibres Chunping Liu Institute of Mathematics, Yangzhou University, Yangzhou 225002, PeopleÕs Republic of China Accepted 7 June 2004 Communicated by Prof. M. Wadati
Abstract First, by using the generally projective Riccati equation method, many kinds of exact solutions for the higher-order nonlinear Scho¨rdinger equation in nonlinear optical fibres are obtained in a unified way. Then, some relations among these solutions are revealed. Ó 2004 Elsevier Ltd. All rights reserved.
1. Introduction The higher-order nonlinear Scho¨rdinger equation describing propagation of ultrashort pulses in nonlinear optical fibres [1] reads oW o2 W o3 W oWjWj2 ojWj2 ¼ ia1 2 þ ia2 WjWj2 þ a3 3 þ a4 þ a5 W ; oz ot ot ot ot
ð1Þ
where W is slowly varying envelop of the electric field, the subscripts z and t are the spatial and temporal partial derivative in retard time coordinates, and a1, a2, a3, a4 and a5 are the real parameters related to the group velocity (GVD), self-phase modulation (SPM), third-order dispersion (TOP), self-steepening, and self-frequency shift arising from stimulated Raman scattering, respectively. In this paper, first, we will apply the generally projective Riccati equation method to solve Eq. (1), and derive twelve kinds of exact solutions of Eq. (1) in a unified way. Then, we will reveal that there are some certain relations among the solutions. The key idea of this method is to introduce a new projective Riccati equation and use its solutions to replace the elementary functions in the projective Riccati equation method [2,3], which simply proceeds as follows. Step 1. For a given nonlinear evolution equation (NLEE), say in two variables x, t, P ðu; ux ; ut ; uxx ; . . .Þ ¼ 0;
E-mail address: [email protected] 0960-0779/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2004.06.006
ð2Þ
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C. Liu / Chaos, Solitons and Fractals 23 (2005) 949–955
we consider its travelling wave solutions uðx; tÞ ¼ uðnÞ;
n ¼ x kt þ n0 ;
ð3Þ
then Eq. (2) is reduced to an ordinary differential equation (ODE) Qðu; u0 ; u00 ; . . .Þ ¼ 0;
ð4Þ
where a prime denotes d/dn. Step 2. We introduce two new variables r = r(n), s = s(n) which are the solutions of the following new projective Riccati equation: r0 ðnÞ ¼ erðnÞsðnÞ;
s0 ðnÞ ¼ R þ es2 ðnÞ lrðnÞ;
e ¼ 1:
It is easy to see that Eq. (5) admits the first integral with R 6¼ 0, ðl2 þ dÞ 2 2 s ðnÞ ¼ e R 2lrðnÞ þ r ðnÞ ; d ¼ 1; R
ð5Þ
ð6Þ
where R and l are constants. By virtue of the variables r and s, we assume that Eq. (4) has the following solutions: n X ri1 ðnÞ½Ai rðnÞ þ Bi sðnÞ þ A0 ; ð7Þ uðnÞ ¼ i¼1
where Ai and Bi are constants to be determined later. The parameter n can be determined by balancing the highest-order derivative term with nonlinear term in (2) or (4). (If n is not a positive integer, first make the transformation u = wn.) Step 3. Substituting the expression (7), (5) and (6) into (4) and setting the coefficients of these terms rj(n)si(n) (or l s (n)) to zero yields a set of over-determined algebraic equations, from which the constants Ai, Bi, R, l, k can be found explicitly. Step 4. Note that Eqs. (5) and (6) admits the following solutions: Case 1 When e = 1, d = 1, R 6¼ 0, pffiffiffi R sechð RnÞ pffiffiffi r1 ðnÞ ¼ ; l sechð RnÞ þ 1 Case 2 When e = 1, d = 1, R 6¼ 0, pffiffiffi R cschð RnÞ pffiffiffi ; r2 ðnÞ ¼ l cschð RnÞ þ 1 Case 3 When e = 1, d = 1, R 6¼ 0, pffiffiffi R secð RnÞ pffiffiffi ; r3 ðnÞ ¼ l secð RnÞ þ 1 pffiffiffi R cscð RnÞ pffiffiffi r4 ðnÞ ¼ ; l cscð RnÞ þ 1
pffiffiffi pffiffiffi R tanhð RnÞ pffiffiffi : l sechð RnÞ þ 1
ð8Þ
pffiffiffi pffiffiffi R cothð RnÞ pffiffiffi : s2 ðnÞ ¼ l cschð RnÞ þ 1
ð9Þ
s1 ðnÞ ¼
pffiffiffi pffiffiffi R tanð RnÞ pffiffiffi ; s3 ðnÞ ¼ l secð RnÞ þ 1
ð10Þ
pffiffiffi pffiffiffi R cotð RnÞ pffiffiffi : s4 ðnÞ ¼ l cscð RnÞ þ 1
ð11Þ
Substituting the constants Ai, Bi, R, l, k obtained in Step 3 into (3) and (7) along with (8)–(11), we can then obtain the solutions of Eq. (2).
2. Exact solutions of Eq. (1) Now let us turn to Eq. (1). According to the above steps, to seek travelling wave solutions of Eq. (1), we make the gauge transformation Wðz; tÞ ¼ wðnÞ exp½iðkz wtÞ ;
n ¼ t kz þ n0 ;
ð12Þ
C. Liu / Chaos, Solitons and Fractals 23 (2005) 949–955
951
where k, w, k are constants to be determined later, n0 is an arbitrary constant. Substituting (12) into Eq. (1) yields the real system ða1 3a3 wÞw00 þ ða3 w3 a1 w2 kÞw þ ða2 a4 wÞw3 ¼ 0;
ð13Þ
a3 w000 þ ð2a1 w 3a3 w2 þ kÞw0 þ ð3a4 þ 2a5 Þw2 w0 ¼ 0;
ð14Þ
under the constraint conditions w¼
a1 ð3a4 þ 2a5 Þ 3a2 a3 ; 6a3 ða4 þ a5 Þ
k¼
ð15Þ
1 ½ða1 3a3 wÞð2a1 w 3a3 w2 þ kÞ a1 w2 þ a3 w3 : a3
ð16Þ
Eqs. (13) and (14) becomes w00 þ k 1 w þ k 3 w3 ¼ 0;
ð17Þ
where k1 ¼
2a1 w 3a3 w2 þ k ; a3
k3 ¼
3a4 þ 2a5 : 3a3
ð18Þ 00
By balancing the highest-order derivative term w with the nonlinear term w3 in Eq. (17), we obtain n = 1 in (7), so we assume (17) has the solutions in the form with R 6¼ 0, wðnÞ ¼ A0 þ A1 rðnÞ þ B1 sðnÞ;
ð19Þ
where A0, A1 and B1 are constants to be determined later. r(n) and s(n) satisfy (5) and (6). According to the Step 3, we substitute (19) into Eq. (17) along with (5) and (6). The following algebraic system occur const : k 1 A0 þ k 3 ðA30 3eA0 B21 RÞ ¼ 0;
ð20Þ
rðnÞ : eA1 R þ k 1 A1 þ k 3 ð3A20 A1 þ 6elA0 B21 3eA1 B21 RÞ ¼ 0;
ð21Þ
sðnÞ : k 1 B1 þ k 3 ð3A20 B1 eB31 RÞ ¼ 0;
ð22Þ
2 2 2 2 2 l þd ¼ 0; r ðnÞ : 3elA1 þ k 3 3A0 A1 þ 6elA1 B1 3eA0 B1 R
ð23Þ
rðnÞsðnÞ : elB1 þ k 3 ð6A0 A1 B1 þ 2elB31 Þ ¼ 0;
ð24Þ
r3 ðnÞ : 2eA1
2 l2 þ d l þd þ k 3 A31 3eA1 B21 ¼ 0; R R
ð25Þ
2 2 l þd 2 3 l þd r ðnÞsðnÞ : 2eB1 þ k 3 3A1 B1 eB1 ¼ 0: R R 2
ð26Þ
From which, by using the Wu elimination method [4–6], we have the following solutions: 2 ; k3
R¼
2d ; k1 k3
R¼
ðIÞ A0 ¼ A1 ¼ l ¼ 0;
B21 ¼
ðIIÞ A0 ¼ B1 ¼ l ¼ 0;
A21 ¼
ðIIIÞ A0 ¼ 0;
A21 ¼
l2 þ d ; 4k 1 k 3
B21 ¼
k1 ; 2e
k1 ; e
1 ; 2k 3
e ¼ 1;
e ¼ 1;
R¼
d ¼ 1;
ð27Þ
d ¼ 1;
2k 1 ; e
e ¼ 1;
ð28Þ
d ¼ 1:
ð29Þ
Therefore from (8)–(11) and (I)–(III), we obtain twelve kinds of exact travelling wave solution of Eq. (1) in a unified way
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C. Liu / Chaos, Solitons and Fractals 23 (2005) 949–955
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 3ð2a1 w 3a3 sw2 þ kÞ 2a1 w 3a3 w2 þ k W1 ¼ tanh ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 2a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 3ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W2 ¼ coth ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 2a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 6ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W3 ¼ sech ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 6ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W4 ¼ csch ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 3ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W5 ¼ tan ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 2a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 3ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W6 ¼ cot ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 2a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 6ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W7 ¼ sec ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 a3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 6ð2a1 w 3a3 w2 þ kÞ 2a1 w 3a3 w2 þ k W8 ¼ csc ðt kz þ n0 Þ exp½iðkz wtÞ ; 3a4 þ 2a5 a3
ð30Þ
ð31Þ
ð32Þ
ð33Þ
ð34Þ
ð35Þ
ð36Þ
ð37Þ
ffi hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8 2ð2a1 w3a3 w2 þkÞ >pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðt kz þ n0 Þ a3 3ð2a1 w 3a3 w2 þ kÞ< l 1sech W9 ¼ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i > 3a4 þ 2a5 : l sech 2ð2a1 w3a3 w2 þkÞðt kz þ n Þ þ 1 a3
0
i 9 2ð2a1 w3a3 w þkÞ > = ðt kz þ n e tanh Þ 0 a3 ffi exp½iðkz wtÞ ; þ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 > 3 w þkÞ ; ðt kz þ n Þ þ 1 l sech 2ð2a1 w3a 0 a3 ffi hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
ð38Þ
ffi hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8 2ð2a1 w3a3 w2 þkÞ >pffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðt kz þ n0 Þ a3 3ð2a1 w 3a3 w2 þ kÞ< l þ 1csch W10 ¼ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i > 3a4 þ 2a5 : l csch 2ð2a1 w3a3 w2 þkÞðt kz þ n Þ þ 1 a3
0
ffi hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 9 2ð2a1 w3a3 w2 þkÞ > = ðt kz þ n e coth Þ 0 a3 ffi exp½iðkz wtÞ ; þ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 > 3 w þkÞ ; ðt kz þ n Þ þ 1 l csch 2ð2a1 w3a 0 a3
ð39Þ
hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i pffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8 2 > 3 w þkÞ 2 sec < ðt kz þ n0 Þ 1 l 2ð2a1 w3a 2 a3 3ð2a1 w 3a3 w þ kÞ W11 ¼ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 > l sec 3a4 þ 2a5 : 2ð2a1 w3a3 w þkÞðt kz þ n Þ þ 1 a3
hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 9 2 > 3 w þkÞ = ðt kz þ n e tan 2ð2a1 w3a Þ 0 a3 exp½iðkz wtÞ ; þ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 3 w þkÞ ; ðt kz þ n0 Þ þ 1> l sec 2ð2a1 w3a a3
0
ð40Þ
C. Liu / Chaos, Solitons and Fractals 23 (2005) 949–955
953
hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8 2 >pffiffiffiffiffiffiffiffiffiffiffiffiffi2 3 w þkÞ ðt kz þ n0 Þ 2ð2a1 w3a a3 3ð2a1 w 3a3 w2 þ kÞ< 1 l csc W12 ¼ hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 > 3a4 þ 2a5 : l csc 2ð2a1 w3a3 w þkÞðt kz þ n Þ þ 1 0
a3
hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 9 2 > 3 w þkÞ = ðt kz þ n e cot 2ð2a1 w3a Þ 0 a3 þ exp½iðkz wtÞ : hqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i 2 > 3 w þkÞ ; ðt kz þ n Þ þ 1 l csc 2ð2a1 w3a 0 a3
ð41Þ
Remark 1. In Refs. [7,8], Yan thought (9) satisfied ðl2 1Þ 2 r ðnÞ : s2 ðnÞ ¼ R 2lrðnÞ þ R
ð42Þ
Actually, it is wrong. In addition, (1) Page 762 in Ref. [7], Case 1. l = A0 = A1 = 0, e = 1, B21 ¼
6a3 ; 3a4 þ 2a5
R¼
2a1 w 3a3 w2 þ k ; 2a3
should be B21 ¼
6a3 ; 3a4 þ 2a5
R¼
2a1 w 3a3 w2 þ k : 2a3
(2) Page 763 in Ref. [7] Case 5. A0 = 0, e = 1, A21 ¼
3a23 ð1 l2 Þ ; ð3a4 þ 2a5 Þð2a1 w 3a3 w2 þ kÞ
B21 ¼
6a3 ; 3a4 þ 2a5
R¼
2ð2a1 w 3a3 w2 þ kÞ ; a3
should be A21 ¼
3a23 ðl2 1Þ ; 4ð3a4 þ 2a5 Þð2a1 w 3a3 w2 þ kÞ
B21 ¼
3a3 ; 2ð3a4 þ 2a5 Þ
R¼
2ð2a1 w 3a3 w2 þ kÞ : a3
(3) Page 763 in Ref. [7] Case 6. A0 = 0, e = 1, A21 ¼
3a23 ð1 l2 Þ ; ð3a4 þ 2a5 Þð2a1 w 3a3 w2 þ kÞ
B21 ¼
6a3 ; 3a4 þ 2a5
R¼
2ð2a1 w 3a3 w2 þ kÞ ; a3
should be A21 ¼
3a23 ðl2 1Þ ; 4ð3a4 þ 2a5 Þð2a1 w 3a3 w2 þ kÞ
B21 ¼
3a3 ; 2ð3a4 þ 2a5 Þ
R¼
2ð2a1 w 3a3 w2 þ kÞ : a3
Therefore, those solutions W1, W2, W3, W4, W7, W8, W10, W11, W12 in Ref. [7] (or Ref. [8]) are wrong. Remark 2. When R = 0, we assume Eq. (17) has the solution wðnÞ ¼ A0 þ A1 sðnÞ;
ð43Þ
where A0, A1 are constants to be determined later. s(n) satisfy s 0 (n) = s2(n). Similar to the above process, we can obtain the rational solution of Eq. (1) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6a3 1 exp½iðkz wtÞ : W13 ¼ 3a4 þ 2a5 t ð3a3 w2 2a1 wÞz þ n0
ð44Þ
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C. Liu / Chaos, Solitons and Fractals 23 (2005) 949–955
3. The relations among the solutions W1 W12 of Eq. (1) Next, we reveal some relations among these solutions W1 W12 of Eq. (1). Notice the identical formula tanh ih ¼
eih eih cos h þ i sin h cos h þ i sin h ¼ ¼ i tan h; eih þ eih cos h þ i sin h þ cos h i sin h
ð45Þ
sech ih ¼
2 2 ¼ sec h; ¼ eih þ eih cos h þ i sin h þ cos h i sin h
ð46Þ
coth ih ¼
eih þ eih cos h þ i sin h þ cos h i sin h ¼ i cot h; ¼ eih eih cos h þ i sin h cos h þ i sin h
ð47Þ
csch ih ¼
2 2 ¼ icsch; ¼ eih eih cos h þ i sin h cos h þ isinh
ð48Þ
pffiffiffiffiffiffiffi here i ¼ 1. Therefore we can obtain directly W5, W6, W7 and W8 from W1, W2, W3 and W4, respectively. We have showed this method in Refs. [9,10]. In addition, we can prove pffiffiffiffiffiffiffiffiffiffiffiffiffi h u1 l2 1sech h tanh h ¼ tanh ; ð49Þ l sech h þ 1 2 pffiffiffiffiffiffiffiffi l2 1 where u1 ¼ tanh1 , l P 1. l pffiffiffiffiffiffiffiffiffiffiffiffiffi h u2 l2 1sech h tanh h ¼ coth ; ð50Þ l sech h þ 1 2 pffiffiffiffiffiffiffiffi l2 1 , l 6 1. where u2 ¼ tanh1 l pffiffiffiffiffiffiffiffiffiffiffiffiffi h þ u3 l2 þ 1cschh þ coth h ¼ coth ; ð51Þ l cschh þ 1 2 l ffi . where u3 ¼ tanh1 pffiffiffiffiffiffiffi l2 þ1 pffiffiffiffiffiffiffiffiffiffiffiffiffi h u4 l2 þ 1cschh coth h ¼ tanh ; ð52Þ l cschh þ 1 2 l ffi . where u4 ¼ tanh1 pffiffiffiffiffiffiffi l2 þ1 pffiffiffiffiffiffiffiffiffiffiffiffiffi h u5 1 l2 sec h tan h ; ð53Þ ¼ tan 2 l sec h þ 1 pffiffiffiffiffiffiffiffi 1l2 , jlj 6 1. where u5 ¼ tan1 l pffiffiffiffiffiffiffiffiffiffiffiffiffi h þ u6 1 l2 csc h þ cot h ; ð54Þ ¼ cot 2 l csc h þ 1 l ffi , jlj 6 1. where u6 ¼ tan1 pffiffiffiffiffiffiffi 2 1l
pffiffiffiffiffiffiffiffiffiffiffiffiffi h u7 1 l2 csc h cot h ¼ tan ; l csc h þ 1 2 l ffi , jlj 6 1. where u7 ¼ tan1 pffiffiffiffiffiffiffi 2
ð55Þ
1l
We take (49) as a sample to show the detailed process, and (50)–(55) can be proved in a similar way. pffiffiffiffiffiffiffiffiffiffiffiffiffi In fact, when l P 1, set l = cosh u1, thus l2 1 ¼ sinh u1 , notice the identical formula hþu hu sinh h þ sinh u ¼ 2 sinh cosh ; 2 2
ð56Þ
C. Liu / Chaos, Solitons and Fractals 23 (2005) 949–955
sinh h sinh u ¼ 2 cosh
hþu hu sinh ; 2 2
cosh h þ cosh u ¼ 2 cosh
hþu hu cosh ; 2 2
cosh h cosh u ¼ 2 sinh
hþu hu sinh ; 2 2
955
ð57Þ
ð58Þ
ð59Þ
we have pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi h u1 l2 1sech h tanh h l2 1 sinh h sinh u1 sinh h ¼ ¼ ¼ tanh : l sech h þ 1 l þ cosh h cosh u1 þ cosh h 2 So, from (49)–(55), we can see that when l P 1, the solutions W9 and W1 are same in waveform and only different in phase, when l 6 1, the solutions W9 and W2 are same in waveform and only different in phase. The solutions W10 and W1 (or W2), W11 and W5, W12 and W5 (or W6) are also in the same way.
4. Conclusions In summary, by using the generally projective Riccati equation method, many kinds of exact solutions of Eq. (1) are obtained in a unified way. And then the relations among these solutions are revealed, namely, the periodic wave solutions W5, W6, W7, W8 of Eq. (1) can be obtained directly from soliton solutions W1, W2, W3, W4 of Eq. (1), respectively. When jlj P 1, the solutions W9 and W1 (or W2) are same in waveform and only different in phase. The solutions W10 and W1 (or W2) are also in the same way. When jlj 6 1, W11 and W5 are same in waveform and only different in phase, W12 and W5 (or W6) are also in the same way.
Acknowledgments This project is supported by the Natural Science Foundation of Jiangsu Education Committee and the Natural Science Foundation of China.
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