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Symbolic computation and construction of soliton-like solutions for a breaking soliton equation
Chaos, Solitons and Fractals 17 (2003) 885–893 www.elsevier.com/locate/chaos
Symbolic computation and construction of soliton-like solutions for a breaking soliton equation Biao Li a
a,*
, Yong Chen a, Hengnong Xuan b, Hongqing Zhang
a
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, PeopleÕs Republic of China b Department of Computer, Nanjing University of Economics, Nanjing 210003, PeopleÕs Republic of China Accepted 20 November 2002 Communicated by M. Wadati
Abstract Based on the symbolic computation system––Maple and a Riccati equation, by introducing a new more general ans€ atz than the ans€atz in the tanh method, extended tanh-function method, modified extended tanh-function method, generalized tanh method and generalized hyperbolic-function method, we propose a generalized Riccati equation expansion method for searching for exact soliton-like solutions of nonlinear evolution equations and implemented in computer symbolic system––Maple. Making use of our method, we study a typical breaking soliton equation and obtain new families of exact solutions, which include the nontravelling waveÕ and coefficient functionÕ soliton-like solutions, singular soliton-like solutions and periodic solutions. The arbitrary functions of some solutions are taken to be some special constants or functions, the known solutions of this equation can be recovered. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction Many powerful methods for searching for exact solutions to nonlinear evolution equations (NEE) have been proposed. Among these are inverse scattering method, B€acklund transformation, Darboux transformation, Hirota method. All these methods are described in Ref. [1–7]. In recently years, some other ans€ atz method have been developed, such as, tanh method [8–10], extended tanh-function method [11,12], modified extended tanh-function method [17], generalized hyperbolic-function method [20,21], variable separation method [24,25]. Tanh method [8–10] is one of most effectively straightforward methods to construct exact travelling wave solutions of NEEs. Recently, Fan [11,12] has proposed an extended tanh-function method. More recently, Fan et al. [13], Yan et al. [14–16] and Li et al. [22,23] further developed this idea and made it much more lucid and straightforward for a class of NEEs. Most recently, Elwakil et al. [17] modified extended tanh-function method and obtain some new formal exact solutions. To obtain the soliton-like solutions for NEES, Gao and Tian [18–21] developed a generalized hyperbolic-function method. As we known, when applying directed method, the choice of an appropriate ans€ atz is of great importance. In this paper, based on the above work [8–21], by introducing a new more general ans€ atz than the ans€ atz in the above methods, we present a generalized Riccati equation expansion method. Then we choose a typical breaking soliton equation to illustrate our algorithm and obtain rich new families of the exact solutions, including the nontravelling waveÕ and coefficient functionÕ soliton-like solutions, singular soliton-like solutions, triangular functions solutions.
*
Corresponding author. E-mail address: [email protected] (B. Li).
0960-0779/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0960-0779(02)00570-2
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B. Li et al. / Chaos, Solitons and Fractals 17 (2003) 885–893
2. Generalized Riccati equation expansion method We now describe the generalized Riccati equation expansion method, as follows: For a given NEEs with one physical field uðx; y; tÞ in three variables x; y; t H ðu; ut ; ux ; uy ; uxx ; uxt ; uxy ; uyt ; . . . ; Þ ¼ 0: Step 1. We express the solutions of the NEEs (2.1) by the new more general ans€ atz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X uðx; y; tÞ ¼ a0 þ ai /i ðnÞ þ bi /i1 ðnÞ R þ /2 ðnÞ þ ki /i ðnÞ ;
ð2:1Þ
ð2:2Þ
i¼1
where m is a integer to be determined by balancing the highest order derivative terms with the nonlinear terms in (2.1), R is a real constant, where a0 ¼ a0 ðx; y; tÞ, ai ¼ ai ðx; y; tÞ, bi ¼ bi ðx; y; tÞ, ki ¼ ki ðx; y; tÞ, ði ¼ 1; . . . ; mÞ, n ¼ nðx; y; tÞ are all differential functions and new variable /ðnÞ satisfies d/ðnÞ ¼ R þ /2 ðnÞ: dn
ð2:3Þ
Step 2. Substituting (2.2) along with (2.3) into (2.1), multiplying the most simplify common denominator in the obqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tained system, setting the coefficients of /j ðnÞð R þ /2 ðnÞÞn ðj ¼ 0; 1; . . . ; n ¼ 0; 1Þ to zero, we obtain a set of over-determined partial differential equations with regard to differential functions a0 ;ai ;bi ;ki ði ¼ 1;...;mÞ and n. Step 3. Solving the over-determined partial differential equations by use of the PDEtools package of Maple, we would end up the explicit expressions for a0 ; ai ; bi ; ki ði ¼ 1; . . . ; mÞ and n or the constrains among them. Step 4. It is well-known that the general solutions of Riccati equation (2.3) are 8 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi R tanhðpffiffiffiffiffiffiffi RnÞ; R < 0; > > > > > R cothð RnÞ; R < 0; > < 1 ð2:4Þ /ðnÞ ¼ ; R ¼ 0; > n > p ffiffiffi p ffiffiffi > > > R tanð p RnÞ; R > 0; > ffiffiffi : pffiffiffi R cotð RnÞ; R > 0: Thus according to (2.2), (2.4) and the conclusions in Step 3, the soliton-like solutions of (2.1) can be obtained. Remark 1. The method proposed here is more general than generalized hyperbolic-function method [20,21], tanh method [8–10], extended tanh-function method [11–16], modified extended tanh-function method [17]. Firstly, compared with the tanh method, extended tanh-function, as well as the modified extended tanh-function method, the restriction on nðx; y; tÞ as merely a linear function x; y; t and the restriction on the coefficients a0 ; ai ; bi ; ki ði ¼ 1; . . . ; mÞ as constants are removed. Secondly, compared with the generalized hyperbolic-function method, setting ki ¼ 0 ði ¼ 1; . . . ; mÞ, we can not only recovered the exact soliton-like solutions obtained by generalized hyperbolicfunction method for a given NEEs but also we can, with no extra effect, find singular soliton-like solutions and periodic formal solutions. When ki 6¼ 0 ði ¼ 1; . . . ; mÞ, some new formal solutions would be expected for some equations. Remark 2. For the generalization of the ans€atz, naturally more complicated computation is expected than ever before. Even if the availability of computer symbolic systems like Maple allows us to perform the complicated and tedious algebraic calculation and differential calculation on a computer. In general, it is very difficult, sometime impossible, to solve the set of over-determined partial differential equations in Step 2. As the calculation goes on, in order to drastically simplify the work or make the work feasible, we often choose special function forms for a0 ; ai ; bi ; ki ði ¼ 1; . . . ; mÞ and n, on a trial-and-error basis. 3. The soliton-like solutions for a breaking soliton equation In this section, by use of the generalized Riccati equation expansion method, we investigate a typical breaking equation [18] uxt 4ux uxy 2uy uxx þ uxxxy ¼ 0:
ð3:1Þ
B. Li et al. / Chaos, Solitons and Fractals 17 (2003) 885–893
887
By balancing the highest-order contributions from both the linear and nonlinear terms in Eq. (3.1), we obtain m ¼ 1 in (2.2). Therefore we assume the solutions of Eq. (3.1) in the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 ; ð3:2Þ uðx; y; tÞ ¼ a0 þ a1 /ðnÞ þ b1 R þ /2 ðnÞ þ /ðnÞ where a0 ¼ a0 ðy; tÞ, a1 ¼ a1 ðy; tÞ, b1 ¼ b1 ðy; tÞ, k1 ¼ k1 ðy; tÞ, and n ¼ xpðy; tÞ þ qðy; tÞ are all differential functions and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /ðnÞ satisfies (2.3). Substituting (3.2) along with q (2.3) into (3.1), multiplying /ðnÞ5 R þ /ðnÞ2 in the obtained system, collecting coffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
efficients of monomials of /ðnÞ; R þ /ðnÞ2 and x ðNote: a0 ; a1 ; b1 ; k1 ; p; q are independent of x), then setting each coefficient to zero, we can deduce the following set of over-determined partial differential equations with respect to the unknown derivative functions f ; g1 ; g2 ; h1 ; h2 ; k1 ; k2 ; p and q (Note: in the rest of this paper a1y denotes oa1 ðy; tÞ=oy, and so on). 12qy p2 ðb21 þ a21 2pa1 Þ ¼ 0; 12py p
2
ðb21
þ
a21
ð3:3Þ
2pa1 Þ ¼ 0;
ð3:4Þ
3pb1 Rð11qy Rp2 12pa1 qy R þ 6pk1 qy 2pa0y þ qt Þ ¼ 0;
ð3:5Þ
2
3pb1 Rð11py Rp 12pa1 py R þ 6pk1 py þ pt Þ ¼ 0;
ð3:6Þ
4p2 a1 Rk1y 2p3 k1y R þ 2p3 a1y R2 6p2 py k1 R þ pt a1 R þ pa1t R 4pk12 py 4p2 a1 R2 a1y 2p2 b1y b1 R2 4p2 k1 k1y þ 6p2 py a1 R2 þ 16pa1 Rpy k1 þ 4p2 k1 Ra1y pt k1 4pa21 R2 py pk1t ¼ 0; 2
2
3
2
ð3:7Þ
2
Rð15p py b1 R 6p a1y b1 R þ 5p b1y R 8p a1 b1y R 8pa1 py b1 R þ pb1t þ 2p k1y b1 þ pt b1 þ 16pb1 py k1 þ 4p2 k1 b1y Þ ¼ 0;
ð3:8Þ
pb1t 14p2 a1y b1 R þ pt b1 þ 33p2 py b1 R 16pa1 py b1 R 16p2 a1 b1y R þ 4p2 k1 b1y þ 11p3 b1y R þ 8pb1 py k1 ¼ 0; 2
ð3:9Þ
2pð3p b1y þ 4pa1 b1y þ 4pa1y b1 9pb1 py þ 4a1 py b1 Þ ¼ 0;
ð3:10Þ
Rð8p3 k1y R þ 24p2 py k1 R 4p2 a1 Rk1y 8pa1 Rpy k1 þ 8pk12 py þ pk1t þ 12p2 k1 k1y þ pt k1 Þ ¼ 0;
ð3:11Þ
2
2pð3a1y p þ 4pa1 a1y 9pa1 py þ 4pb1 b1y þ
2py b21
þ
2a21 py Þ
¼ 0;
ð3:12Þ
4p2 k1 a1y 4pb21 py R 10p2 b1y b1 R þ 8pa1 py k1 þ 8p3 a1y R þ 24p2 py a1 R 12p2 a1 Ra1y þ pt a1 8pa21 Rpy þ pa1t ¼ 0; ð3:13Þ 2R2 pð2pb1y k1 þ pk1y b1 þ 4b1 py k1 Þ ¼ 0;
ð3:14Þ
4p2 b1y k1 R3 ¼ 0;
ð3:15Þ
2
2b1 pð26py Rp 27pa1 py R þ 6pk1 py þ pt Þ ¼ 0;
ð3:16Þ
2b1 pð26qy Rp2 þ 6pk1 qy 2pa0y 27pa1 qy R þ qt Þ ¼ 0;
ð3:17Þ
2
2
2k1 R pð20py Rp 6pa1 py R þ 12pk1 py þ pt Þ ¼ 0;
ð3:18Þ
pR2 b1 ð5py Rp2 þ 6pa1 py R pt Þ ¼ 0;
ð3:19Þ
2
2pRð8p py a1 R
3pb21 py R
6pa21 Rpy
þ 6pa1 py k1 þ pt a1 Þ ¼ 0;
ð3:20Þ
2pRð8a1 qy Rp2 3pRb21 qy 6pRa21 qy þ 6pa1 k1 qy 2pa0y a1 þ a1 qt Þ ¼ 0;
ð3:21Þ
2pð20p2 py a1 R 6pa1 py k1 þ 12pa21 Rpy þ 9pb21 py R pt a1 Þ ¼ 0;
ð3:22Þ
2pð2pa0y a1 6pa1 k1 qy a1 qt þ
9pRb21 qy
þ
12pRa21 qy
2
20a1 qy Rp Þ ¼ 0;
ð3:23Þ
2pR2 ð9Rppy k1 þ 3Rp2 k1y þ 2k12 py þ 4pk1 k1y Þ ¼ 0;
ð3:24Þ
2pk1 R2 ð20Rqy p2 12qy k1 p þ 6pa1 qy R þ 2pa0y qt Þ ¼ 0;
ð3:25Þ
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B. Li et al. / Chaos, Solitons and Fractals 17 (2003) 885–893
2pk1 Rð8py Rp2 6ppy k1 þ 6pa1 Rpy pt Þ ¼ 0; 2
ð3:26Þ
2
pb1 R ð5Rqy p þ 6pa1 qy R þ 2pa0y qt Þ ¼ 0;
ð3:27Þ
6p2 k1 R3 b1 qy ¼ 0;
ð3:28Þ
2
3
6p k1 R b1 py ¼ 0;
ð3:29Þ
2pk1 Rð8Rqy p2 6qy k1 p þ 6pa1 qy R þ 2pa0y qt Þ ¼ 0;
ð3:30Þ
12p2 k1 R3 qy ðk1 þ 2pRÞ ¼ 0;
ð3:31Þ
12p2 py k1 R3 ðk1 þ 2pRÞ ¼ 0;
ð3:32Þ
2
24p b1 qy ðp þ a1 Þ ¼ 0;
ð3:33Þ
24p2 b1 py ðp þ a1 Þ ¼ 0:
ð3:34Þ
Using the powerful PDEtools package of Maple, solving the set of partial differential equations (3.3)–(3.34), we can obtain the following nontrivial solutions. Note: 1. In the following Case 1–10, q ¼ q denotes that q is an arbitrary function with respect to y; t. 2. In the rest of this paper, F1 ðtÞ and F2 ðtÞ denote arbitrary differential function with regard to t. 3. We omit the solutions with p ¼ 0 and 46 cases of solutions with q ¼ WðtÞ, where WðtÞ is an arbitrary function with respect to t. Case 1 b1 ¼ k1 ¼ 0;
a0 ¼
Z
2Rqy C12 þ 12 qt dy þ F1 ðtÞ; C1
a1 ¼ 2C1 ;
p ¼ C1 ;
q ¼ q:
ð3:35Þ
Case 2 b1 ¼ k1 ¼ 0;
a0 ¼
Z
2Rqy RootOfð4F1 ðZÞRZ 2 þ 4tRZ 2 þ yÞ2 þ 12 qt dy þ F2 ðtÞ; RootOfð4F1 ðZÞRZ 2 þ 4tRZ 2 þ yÞ
a1 ¼ 2RootOfð4F1 ðZÞRZ 2 þ 4tRZ 2 þ yÞ;
p ¼ RootOfð4F1 ðZÞRZ 2 þ 4tRZ 2 þ yÞ;
q ¼ q;
ð3:36Þ
where F ðZÞ denotes arbitrary function with respect to Z and RootOfð4F1 ðZÞRZ 2 þ 4tRZ 2 þ yÞ denotes all the roots of the equation 4F1 ðZÞRZ 2 þ 4tRZ 2 þ y ¼ 0 with respect to one variable Z. Case 3 k1 ¼ 0;
a0 ¼
Z
1 Rqy C12 qt dy þ F1 ðtÞ; 2 C1
a 1 ¼ C1 ;
b1 ¼ C1 ;
1 Rqy C12 qt dy þ F1 ðtÞ; 2 C1
a 1 ¼ C1 ;
b1 ¼ C1 ;
1 Rqy RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ2 qt dy þ F2 ðtÞ; 2 RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ
p ¼ C1 ;
q ¼ q:
ð3:37Þ
Case 4 k1 ¼ 0;
a0 ¼
Z
p ¼ C1 ;
q ¼ q:
ð3:38Þ
Case 5 k1 ¼ 0;
a0 ¼
Z
a1 ¼ RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ; p ¼ RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ;
b1 ¼ RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ; q ¼ q:
ð3:39Þ
B. Li et al. / Chaos, Solitons and Fractals 17 (2003) 885–893
889
Case 6 k1 ¼ 0;
a0 ¼
Z
1 Rqy RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ2 qt dy þ F2 ðtÞ; 2 RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ
a1 ¼ RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ; 2
2
p ¼ RootOfðF1 ðZÞRZ þ tRZ þ yÞ;
b1 ¼ RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ; q ¼ q:
ð3:40Þ
Case 7 a1 ¼ b1 ¼ 0;
a0 ¼
Z
qt R þ qy C12 dy þ F1 ðtÞ; C1
k1 ¼ C1 ;
p¼
1 C1 ; 2 R
q ¼ q:
ð3:41Þ
Case 8 Z
ðqt R þ qy RootOfðF1 ðZÞZ 2 tZ 2 RyÞ2 Þ dy þ F2 ðtÞ; RootOfðF1 ðZÞZ 2 tZ 2 RyÞ 1 RootOfðF1 ðZÞZ 2 tZ 2 RyÞ ; k1 ¼ RootOfðF1 ðZÞZ 2 tZ 2 RyÞ; p ¼ 2 R
a1 ¼ b1 ¼ 0;
a0 ¼
q ¼ q:
ð3:42Þ
Case 9 b1 ¼ 0;
a0 ¼
Z
4qy C12 qt R dy þ F1 ðtÞ; C1
a1 ¼
C1 ; R
k1 ¼ C1 ;
p¼
1 C1 ; 2 R
q ¼ q:
ð3:43Þ
Case 10 Z
4qy RootOfð4F1 ðZÞZ 2 þ 4tZ 2 þ RyÞ2 qt R dy þ F2 ðtÞ; RootOfð4F1 ðZÞZ2 þ 4tZ2 þ RyÞ RootOfð4F1 ðZÞZ 2 þ 4tZ 2 þ RyÞ ; k1 ¼ RootOfð4F1 ðZÞZ 2 þ 4tZ 2 þ RyÞ; a1 ¼ R 1 RootOfð4F1 ðZÞZ 2 þ 4tZ 2 þ RyÞ ; q ¼ q: p¼ 2 R
b1 ¼ 0;
a0 ¼
ð3:44Þ
From (3.2), (2.4) and Case 1–10, we can obtain the following three types of solutions for the breaking soliton equation. Type 1: From Case 1, we can obtain the following solutions: when R < 0, Z npffiffiffiffiffiffiffi o pffiffiffiffiffiffiffi ð2Rqy C12 þ 12 qt Þ u11 ¼ dy þ F1 ðtÞ 2C1 R tanh R½xC1 þ q ; C1 Z npffiffiffiffiffiffiffi o pffiffiffiffiffiffiffi ð2Rqy C12 þ 12 qt Þ u12 ¼ dy þ F1 ðtÞ 2C1 R coth R½xC1 þ q ; C1
ð3:45Þ ð3:46Þ
when R > 0, Z
npffiffiffi o pffiffiffi ð2Rqy C12 þ 12 qt Þ dy þ F1 ðtÞ þ 2C1 R tan R½xC1 þ q ; C1 Z npffiffiffi o pffiffiffi ð2Rqy C12 þ 12 qt Þ u14 ¼ dy þ F1 ðtÞ 2C1 R cot R½xC1 þ q : C1
u13 ¼
ð3:47Þ ð3:48Þ
Type 2: From Case 2, we can obtain the following solutions: when R < 0, npffiffiffiffiffiffiffi o pffiffiffiffiffiffiffi R½xp þ q ; u21 ¼ a0 a1 R tanh
ð3:49Þ
npffiffiffiffiffiffiffi o pffiffiffiffiffiffiffi u22 ¼ a0 a1 R coth R½xp þ q ;
ð3:50Þ
890
B. Li et al. / Chaos, Solitons and Fractals 17 (2003) 885–893
when R > 0, npffiffiffi o pffiffiffi u23 ¼ a0 þ a1 R tan R½xp þ q ;
ð3:51Þ
npffiffiffi o pffiffiffi R½xp þ q ; u24 ¼ a0 a1 R cot
ð3:52Þ
where a0 ; a1 ; p and q are determined by (3.36). Type 3: From Case 3 and 4, we can obtain the following solutions: when R < 0, Z pffiffiffiffiffiffiffih pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i 1 Rqy C12 qt dy þ F1 ðtÞ C1 R tanhð RnÞ isechð RnÞ ; u31 ¼ 2 C1 Z pffiffiffiffiffiffiffih pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i 1 Rqy C12 qt dy þ F1 ðtÞ C1 R cothð RnÞ cschð RnÞ ; u32 ¼ 2 C1 when R > 0, Z pffiffiffih pffiffiffi pffiffiffi i 1 Rqy C12 qt dy þ F1 ðtÞ þ C1 R tanð RnÞ secð RnÞ ; u33 ¼ C1 2 Z pffiffiffih pffiffiffi pffiffiffi i 1 Rqy C12 qt dy þ F1 ðtÞ C1 R cotð RnÞ cscð RnÞ ; u33 ¼ 2 C1
ð3:53Þ ð3:54Þ
ð3:55Þ ð3:56Þ
where n ¼ xC1 þ q. Type 4: From Case 5 and 6, we can obtain the following solutions: when R < 0, pffiffiffiffiffiffiffih pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i u41 ¼ a0 a1 R tanhð RnÞ isechð RnÞ ;
ð3:57Þ
pffiffiffiffiffiffiffih pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i u42 ¼ a0 a1 R cothð RnÞ cschð RnÞ ;
ð3:58Þ
when R > 0, pffiffiffiffiffiffiffih pffiffiffi pffiffiffi i u43 ¼ a0 þ a1 R tanð RnÞ secð RnÞ ;
ð3:59Þ
pffiffiffiffiffiffiffih pffiffiffi pffiffiffi i u44 ¼ a0 a1 R cotð RnÞ cscð RnÞ ;
ð3:60Þ
where n ¼ xp þ q and a0 ; a1 ; p; q are determined by (3.39) or (3.40). Type 5: From Case 7, we can obtain the following solutions: when R < 0, Z pffiffiffiffiffiffiffi qt R þ qy C12 C1 1 C1 x þ q ; dy þ F1 ðtÞ pffiffiffiffiffiffiffi coth R½ u51 ¼ C1 2 R R Z 2 pffiffiffiffiffiffiffi qt R þ qy C1 C1 1 C1 u52 ¼ x þ q ; dy þ F1 ðtÞ pffiffiffiffiffiffiffi tanh R½ C1 2 R R
ð3:61Þ ð3:62Þ
when R > 0, pffiffiffi qt R þ qy C12 C1 1 C1 x þ q ; u53 ¼ dy þ F1 ðtÞ þ pffiffiffi cot R½ 2 R C1 R Z pffiffiffi qt R þ qy C12 C1 1 C1 x þ q : dy þ F1 ðtÞ pffiffiffi tan R½ u53 ¼ 2 R C1 R Z
ð3:63Þ ð3:64Þ
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891
Type 6: From Case 8, we can obtain the following solutions: when R < 0, npffiffiffiffiffiffiffi o k1 R½xp þ q ; u61 ¼ a0 pffiffiffiffiffiffiffi coth R npffiffiffiffiffiffiffi o k1 u62 ¼ a0 pffiffiffiffiffiffiffi tanh R½xp þ q ; R
ð3:65Þ ð3:66Þ
when R > 0, npffiffiffi o k1 u63 ¼ a0 þ pffiffiffi cot R½xp þ q ; R npffiffiffi o k1 u64 ¼ a0 pffiffiffi tan R½xp þ q ; R
ð3:67Þ ð3:68Þ
where a0 ; k1 ; p and q are determined by (3.42). Type 7: From Case 9, we can obtain the following solutions: u71 ¼ u72 ¼
Z Z
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i 4qy C12 qt R C1 h dy þ F1 ðtÞ pffiffiffiffiffiffiffi tanhð RnÞ cothð RnÞ ; C1 R pffiffiffi pffiffiffi i 4qy C12 qt R C1 h dy þ F1 ðtÞ pffiffiffi tanð RnÞ cotð RnÞ ; C1 R
R < 0;
R > 0;
where n ¼ ð1=2ÞðC1 =RÞx þ q. Type 8: From Case 10, we can obtain the following solutions: pffiffiffiffiffiffiffih pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i u81 ¼ a0 a1 R tanhð RnÞ cothð RnÞ ; R < 0; pffiffiffih pffiffiffi pffiffiffi i u72 ¼ a0 a1 R tanð RnÞ cotð RnÞ ;
R > 0;
ð3:69Þ ð3:70Þ
ð3:71Þ ð3:72Þ
where n ¼ xp þ q and a0 ; a1 ; p; q are determined by (3.44). In order to further understand the solutions obtained in this paper, we only take the solutions in Type 2 as a simple explanation. Setting F ðZÞ ¼ k0 ;
q ¼ ðk1 y þ k2 ÞhðtÞ;
where k0 ; k1 ; k2 are arbitrary constants and hðtÞ is arbitrary function with respect to t, we can obtain rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 y : RootOfð4k0 RZ 2 þ 4tRZ 2 þ yÞ ¼
2 Rðt k0 Þ
ð3:73Þ
ð3:74Þ
Therefore, from (3.49) in Type 2, we are able to find an explicit solutions, as follows: " # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k1 RhðtÞ u¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rðt k0 Þ y 2k2 h0 ðtÞ Rðt k0 Þy 1=2 þ F2 ðtÞ þ C 3 Rðt k0 Þ
rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi y x y tanh
þ Rðk1 y þ k2 ÞhðtÞ ; t k0 2 t k0
ð3:75Þ
where C is an arbitrary constant. It is easy to see that, when setting k0 ¼ k1 ¼ 0; k2 ¼ R ¼ 1, F2 ðtÞ ¼ CðtÞ in the solution (3.75), the solution (12) in Ref. [18] is recovered. Therefore, when taking the arbitrary functions q; F1 ðtÞ and F2 ðtÞ as special constants or functions, we can obtain rich explicit exact solutions for Eq. (3.1).
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4. Conclusions In summary, based on the computerized symbolic computation and a Riccati equation, a generalized Riccati equation expansion method for searching for exact soliton-like solutions of NEEs is proposed by introducing a new more general ans€atz than the ans€atz in the extended tanh-function method, modified extended tanh-function method, and generalized hyperbolic-function method. Making use of the method, we study a typical breaking soliton equation and obtain many families of the exact solutions and implement in computer symbolic computation system––Maple. From the solutions obtained the restriction on nðx; y; tÞ as merely a linear function x; y; t and the restriction on the coefficients a0 ; a1 ; b1 ; k1 as constants are removed and, with no extra effect, the singular soliton-like solution and triangular function solutions are obtained. To make the work feasible (or to obtain explicit exact soliton-like solutions), how to choose the forms for a0 ; ai ; bi ; ki ði ¼ 1; . . . ; mÞ and n in the ans€ atz would be the key step in the computation of our method. The method, proposed in this paper for single equation, may be extended to find exact soliton-like solutions of other NEEs and coupled NEEs.
Acknowledgements The work is supported by the National Natural Science Foundation of China under the Grant No. 1007201, the National Key Basic Research Development Project Program under the Grant No. G1998030600.
References [1] Ablowitz MJ, Clarkson PA. Soliton, nonlinear evolution equations and inverse scatting. New York: Cambridge University Press; 1991. [2] Gu CH et al. Soliton theory and its application. Zhejiang: Zhejiang Science and Technology Press; 1990. [3] Wadati M. Wave propagation in nonlinear lattice I. J Phys Soc Jpn 1975;38:673–80. [4] Wadati M. Wave propagation in nonlinear lattice II. J Phys Soc Jpn 1975;38:681–6. [5] Yajima T, Wadati M. Soliton solution and its property of unstable nonlinear Schr€ odinger equation. J Phys Soc Jpn 1990;59:41–7. [6] Yajima T, Wadati M. Solitons in electron beam plasma. J Phys Soc Jpn 1990;59:3237–48. [7] El-Kalaawy OH. Exact soliton solutions for some nonlinear partial differential equations. Chaos, Solitons & Fractals 2002;14:547– 52. [8] Parkes EJ, Duffy BR. An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations. Comput Phys commun 1996;98:288–300. [9] Parkes EJ, Duffy BR. Travelling solitary wave solutions to a compound KdV-Burgers equation. Phys Lett A 1997;229: 217–20. [10] Khater AH, Malfiet W, Callebaut DK, Kamel ES. The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction-diffusion equations. Chaos, Solitons & Fractals 2002;14:513–22. [11] Fan E. Extended tanh-function method and its applications to nonlinear equations. Phys Lett A 2000;277:212–8. [12] Fan E. Travelling wave solutions for generalized Hirota–Satsumma coupled KdV systems. Z Naturforsch A 2001;56:312–8. [13] Fan E, Zhang J, Benny H. A new complex line soliton for the two-dimensional KdV-Burgers equation. Phys Lett A 2001;291:376– 80. [14] Yan ZY. New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations. Phys Lett A 2001;292:100–6. [15] Yan ZY, Zhang HQ. New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in mathematical physics. Phys Lett A 2001;285:355–62. [16] Chen Y, Yan YZ, Li B, Zhang HQ. New explicit solitary wave solutions and periodic wave solutions for the generalized coupled Hirota–Satsuma KdV system. Commun Theor Phys (Beijing, China) 2002;38:261–6. [17] Elwakil SA, El-labany SK, Zahran MA, Sabry R. Modified extended tanh-function method for solving nonlinear partial differential equations. Phys Lett A 2002;299:179–88. [18] Tian B, Zhao K, Gao YT. Symbolic computation in engineering: application to a breaking soliton equation. Int J Engng 1997;35:1081–3. [19] Tian B, Gao YT. Extending the generalized tanh method to the generalized Hamiltonian equations: new soliton-like solutions. Appl Math Lett 1997;10:125–7. [20] Gao YT, Tian B. Generalized hyperbolic-function method with computerized symbolic computation to construct the solitonic solutions to nonlinear equations of mathematical physics. Comput Phys Commun 2001;133:158–64. [21] Tian B, Gao YT. Observable solitonic features of the generalized reaction Duffing Model. Z Naturforsch A 2002;57:39–44. [22] Li B, Chen Y, Zhang HQ. Explicit exact solutions for new general two-dimensional KdV-type and two-dimensional KdV-Burgerstype equations with nonlinear terms of any order. J Phys A: Math Gen 2002;35:8253–65.
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[23] Li B, Chen Y, Zhang HQ. Explicit exact solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order. Chaos, Solitons & Fractals 2003;15:647. [24] Tang X-Y, Lou S-Y. Abundant structures of the dispersive long-wave equation in (2 þ 1)-dimensional spaces. Chaos, Solitons & Fractals 2002;14:1451–6. [25] Tang X-Y, Lou S-Y. Localized excitations in (2 þ 1)-dimensional systems. Phys Rev E 2002;66:046601–17.
Symbolic computation and construction of soliton-like solutions for a breaking soliton equation Biao Li a
a,*
, Yong Chen a, Hengnong Xuan b, Hongqing Zhang
a
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, PeopleÕs Republic of China b Department of Computer, Nanjing University of Economics, Nanjing 210003, PeopleÕs Republic of China Accepted 20 November 2002 Communicated by M. Wadati
Abstract Based on the symbolic computation system––Maple and a Riccati equation, by introducing a new more general ans€ atz than the ans€atz in the tanh method, extended tanh-function method, modified extended tanh-function method, generalized tanh method and generalized hyperbolic-function method, we propose a generalized Riccati equation expansion method for searching for exact soliton-like solutions of nonlinear evolution equations and implemented in computer symbolic system––Maple. Making use of our method, we study a typical breaking soliton equation and obtain new families of exact solutions, which include the nontravelling waveÕ and coefficient functionÕ soliton-like solutions, singular soliton-like solutions and periodic solutions. The arbitrary functions of some solutions are taken to be some special constants or functions, the known solutions of this equation can be recovered. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction Many powerful methods for searching for exact solutions to nonlinear evolution equations (NEE) have been proposed. Among these are inverse scattering method, B€acklund transformation, Darboux transformation, Hirota method. All these methods are described in Ref. [1–7]. In recently years, some other ans€ atz method have been developed, such as, tanh method [8–10], extended tanh-function method [11,12], modified extended tanh-function method [17], generalized hyperbolic-function method [20,21], variable separation method [24,25]. Tanh method [8–10] is one of most effectively straightforward methods to construct exact travelling wave solutions of NEEs. Recently, Fan [11,12] has proposed an extended tanh-function method. More recently, Fan et al. [13], Yan et al. [14–16] and Li et al. [22,23] further developed this idea and made it much more lucid and straightforward for a class of NEEs. Most recently, Elwakil et al. [17] modified extended tanh-function method and obtain some new formal exact solutions. To obtain the soliton-like solutions for NEES, Gao and Tian [18–21] developed a generalized hyperbolic-function method. As we known, when applying directed method, the choice of an appropriate ans€ atz is of great importance. In this paper, based on the above work [8–21], by introducing a new more general ans€ atz than the ans€ atz in the above methods, we present a generalized Riccati equation expansion method. Then we choose a typical breaking soliton equation to illustrate our algorithm and obtain rich new families of the exact solutions, including the nontravelling waveÕ and coefficient functionÕ soliton-like solutions, singular soliton-like solutions, triangular functions solutions.
*
Corresponding author. E-mail address: [email protected] (B. Li).
0960-0779/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0960-0779(02)00570-2
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B. Li et al. / Chaos, Solitons and Fractals 17 (2003) 885–893
2. Generalized Riccati equation expansion method We now describe the generalized Riccati equation expansion method, as follows: For a given NEEs with one physical field uðx; y; tÞ in three variables x; y; t H ðu; ut ; ux ; uy ; uxx ; uxt ; uxy ; uyt ; . . . ; Þ ¼ 0: Step 1. We express the solutions of the NEEs (2.1) by the new more general ans€ atz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X uðx; y; tÞ ¼ a0 þ ai /i ðnÞ þ bi /i1 ðnÞ R þ /2 ðnÞ þ ki /i ðnÞ ;
ð2:1Þ
ð2:2Þ
i¼1
where m is a integer to be determined by balancing the highest order derivative terms with the nonlinear terms in (2.1), R is a real constant, where a0 ¼ a0 ðx; y; tÞ, ai ¼ ai ðx; y; tÞ, bi ¼ bi ðx; y; tÞ, ki ¼ ki ðx; y; tÞ, ði ¼ 1; . . . ; mÞ, n ¼ nðx; y; tÞ are all differential functions and new variable /ðnÞ satisfies d/ðnÞ ¼ R þ /2 ðnÞ: dn
ð2:3Þ
Step 2. Substituting (2.2) along with (2.3) into (2.1), multiplying the most simplify common denominator in the obqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tained system, setting the coefficients of /j ðnÞð R þ /2 ðnÞÞn ðj ¼ 0; 1; . . . ; n ¼ 0; 1Þ to zero, we obtain a set of over-determined partial differential equations with regard to differential functions a0 ;ai ;bi ;ki ði ¼ 1;...;mÞ and n. Step 3. Solving the over-determined partial differential equations by use of the PDEtools package of Maple, we would end up the explicit expressions for a0 ; ai ; bi ; ki ði ¼ 1; . . . ; mÞ and n or the constrains among them. Step 4. It is well-known that the general solutions of Riccati equation (2.3) are 8 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi R tanhðpffiffiffiffiffiffiffi RnÞ; R < 0; > > > > > R cothð RnÞ; R < 0; > < 1 ð2:4Þ /ðnÞ ¼ ; R ¼ 0; > n > p ffiffiffi p ffiffiffi > > > R tanð p RnÞ; R > 0; > ffiffiffi : pffiffiffi R cotð RnÞ; R > 0: Thus according to (2.2), (2.4) and the conclusions in Step 3, the soliton-like solutions of (2.1) can be obtained. Remark 1. The method proposed here is more general than generalized hyperbolic-function method [20,21], tanh method [8–10], extended tanh-function method [11–16], modified extended tanh-function method [17]. Firstly, compared with the tanh method, extended tanh-function, as well as the modified extended tanh-function method, the restriction on nðx; y; tÞ as merely a linear function x; y; t and the restriction on the coefficients a0 ; ai ; bi ; ki ði ¼ 1; . . . ; mÞ as constants are removed. Secondly, compared with the generalized hyperbolic-function method, setting ki ¼ 0 ði ¼ 1; . . . ; mÞ, we can not only recovered the exact soliton-like solutions obtained by generalized hyperbolicfunction method for a given NEEs but also we can, with no extra effect, find singular soliton-like solutions and periodic formal solutions. When ki 6¼ 0 ði ¼ 1; . . . ; mÞ, some new formal solutions would be expected for some equations. Remark 2. For the generalization of the ans€atz, naturally more complicated computation is expected than ever before. Even if the availability of computer symbolic systems like Maple allows us to perform the complicated and tedious algebraic calculation and differential calculation on a computer. In general, it is very difficult, sometime impossible, to solve the set of over-determined partial differential equations in Step 2. As the calculation goes on, in order to drastically simplify the work or make the work feasible, we often choose special function forms for a0 ; ai ; bi ; ki ði ¼ 1; . . . ; mÞ and n, on a trial-and-error basis. 3. The soliton-like solutions for a breaking soliton equation In this section, by use of the generalized Riccati equation expansion method, we investigate a typical breaking equation [18] uxt 4ux uxy 2uy uxx þ uxxxy ¼ 0:
ð3:1Þ
B. Li et al. / Chaos, Solitons and Fractals 17 (2003) 885–893
887
By balancing the highest-order contributions from both the linear and nonlinear terms in Eq. (3.1), we obtain m ¼ 1 in (2.2). Therefore we assume the solutions of Eq. (3.1) in the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 ; ð3:2Þ uðx; y; tÞ ¼ a0 þ a1 /ðnÞ þ b1 R þ /2 ðnÞ þ /ðnÞ where a0 ¼ a0 ðy; tÞ, a1 ¼ a1 ðy; tÞ, b1 ¼ b1 ðy; tÞ, k1 ¼ k1 ðy; tÞ, and n ¼ xpðy; tÞ þ qðy; tÞ are all differential functions and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /ðnÞ satisfies (2.3). Substituting (3.2) along with q (2.3) into (3.1), multiplying /ðnÞ5 R þ /ðnÞ2 in the obtained system, collecting coffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
efficients of monomials of /ðnÞ; R þ /ðnÞ2 and x ðNote: a0 ; a1 ; b1 ; k1 ; p; q are independent of x), then setting each coefficient to zero, we can deduce the following set of over-determined partial differential equations with respect to the unknown derivative functions f ; g1 ; g2 ; h1 ; h2 ; k1 ; k2 ; p and q (Note: in the rest of this paper a1y denotes oa1 ðy; tÞ=oy, and so on). 12qy p2 ðb21 þ a21 2pa1 Þ ¼ 0; 12py p
2
ðb21
þ
a21
ð3:3Þ
2pa1 Þ ¼ 0;
ð3:4Þ
3pb1 Rð11qy Rp2 12pa1 qy R þ 6pk1 qy 2pa0y þ qt Þ ¼ 0;
ð3:5Þ
2
3pb1 Rð11py Rp 12pa1 py R þ 6pk1 py þ pt Þ ¼ 0;
ð3:6Þ
4p2 a1 Rk1y 2p3 k1y R þ 2p3 a1y R2 6p2 py k1 R þ pt a1 R þ pa1t R 4pk12 py 4p2 a1 R2 a1y 2p2 b1y b1 R2 4p2 k1 k1y þ 6p2 py a1 R2 þ 16pa1 Rpy k1 þ 4p2 k1 Ra1y pt k1 4pa21 R2 py pk1t ¼ 0; 2
2
3
2
ð3:7Þ
2
Rð15p py b1 R 6p a1y b1 R þ 5p b1y R 8p a1 b1y R 8pa1 py b1 R þ pb1t þ 2p k1y b1 þ pt b1 þ 16pb1 py k1 þ 4p2 k1 b1y Þ ¼ 0;
ð3:8Þ
pb1t 14p2 a1y b1 R þ pt b1 þ 33p2 py b1 R 16pa1 py b1 R 16p2 a1 b1y R þ 4p2 k1 b1y þ 11p3 b1y R þ 8pb1 py k1 ¼ 0; 2
ð3:9Þ
2pð3p b1y þ 4pa1 b1y þ 4pa1y b1 9pb1 py þ 4a1 py b1 Þ ¼ 0;
ð3:10Þ
Rð8p3 k1y R þ 24p2 py k1 R 4p2 a1 Rk1y 8pa1 Rpy k1 þ 8pk12 py þ pk1t þ 12p2 k1 k1y þ pt k1 Þ ¼ 0;
ð3:11Þ
2
2pð3a1y p þ 4pa1 a1y 9pa1 py þ 4pb1 b1y þ
2py b21
þ
2a21 py Þ
¼ 0;
ð3:12Þ
4p2 k1 a1y 4pb21 py R 10p2 b1y b1 R þ 8pa1 py k1 þ 8p3 a1y R þ 24p2 py a1 R 12p2 a1 Ra1y þ pt a1 8pa21 Rpy þ pa1t ¼ 0; ð3:13Þ 2R2 pð2pb1y k1 þ pk1y b1 þ 4b1 py k1 Þ ¼ 0;
ð3:14Þ
4p2 b1y k1 R3 ¼ 0;
ð3:15Þ
2
2b1 pð26py Rp 27pa1 py R þ 6pk1 py þ pt Þ ¼ 0;
ð3:16Þ
2b1 pð26qy Rp2 þ 6pk1 qy 2pa0y 27pa1 qy R þ qt Þ ¼ 0;
ð3:17Þ
2
2
2k1 R pð20py Rp 6pa1 py R þ 12pk1 py þ pt Þ ¼ 0;
ð3:18Þ
pR2 b1 ð5py Rp2 þ 6pa1 py R pt Þ ¼ 0;
ð3:19Þ
2
2pRð8p py a1 R
3pb21 py R
6pa21 Rpy
þ 6pa1 py k1 þ pt a1 Þ ¼ 0;
ð3:20Þ
2pRð8a1 qy Rp2 3pRb21 qy 6pRa21 qy þ 6pa1 k1 qy 2pa0y a1 þ a1 qt Þ ¼ 0;
ð3:21Þ
2pð20p2 py a1 R 6pa1 py k1 þ 12pa21 Rpy þ 9pb21 py R pt a1 Þ ¼ 0;
ð3:22Þ
2pð2pa0y a1 6pa1 k1 qy a1 qt þ
9pRb21 qy
þ
12pRa21 qy
2
20a1 qy Rp Þ ¼ 0;
ð3:23Þ
2pR2 ð9Rppy k1 þ 3Rp2 k1y þ 2k12 py þ 4pk1 k1y Þ ¼ 0;
ð3:24Þ
2pk1 R2 ð20Rqy p2 12qy k1 p þ 6pa1 qy R þ 2pa0y qt Þ ¼ 0;
ð3:25Þ
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B. Li et al. / Chaos, Solitons and Fractals 17 (2003) 885–893
2pk1 Rð8py Rp2 6ppy k1 þ 6pa1 Rpy pt Þ ¼ 0; 2
ð3:26Þ
2
pb1 R ð5Rqy p þ 6pa1 qy R þ 2pa0y qt Þ ¼ 0;
ð3:27Þ
6p2 k1 R3 b1 qy ¼ 0;
ð3:28Þ
2
3
6p k1 R b1 py ¼ 0;
ð3:29Þ
2pk1 Rð8Rqy p2 6qy k1 p þ 6pa1 qy R þ 2pa0y qt Þ ¼ 0;
ð3:30Þ
12p2 k1 R3 qy ðk1 þ 2pRÞ ¼ 0;
ð3:31Þ
12p2 py k1 R3 ðk1 þ 2pRÞ ¼ 0;
ð3:32Þ
2
24p b1 qy ðp þ a1 Þ ¼ 0;
ð3:33Þ
24p2 b1 py ðp þ a1 Þ ¼ 0:
ð3:34Þ
Using the powerful PDEtools package of Maple, solving the set of partial differential equations (3.3)–(3.34), we can obtain the following nontrivial solutions. Note: 1. In the following Case 1–10, q ¼ q denotes that q is an arbitrary function with respect to y; t. 2. In the rest of this paper, F1 ðtÞ and F2 ðtÞ denote arbitrary differential function with regard to t. 3. We omit the solutions with p ¼ 0 and 46 cases of solutions with q ¼ WðtÞ, where WðtÞ is an arbitrary function with respect to t. Case 1 b1 ¼ k1 ¼ 0;
a0 ¼
Z
2Rqy C12 þ 12 qt dy þ F1 ðtÞ; C1
a1 ¼ 2C1 ;
p ¼ C1 ;
q ¼ q:
ð3:35Þ
Case 2 b1 ¼ k1 ¼ 0;
a0 ¼
Z
2Rqy RootOfð4F1 ðZÞRZ 2 þ 4tRZ 2 þ yÞ2 þ 12 qt dy þ F2 ðtÞ; RootOfð4F1 ðZÞRZ 2 þ 4tRZ 2 þ yÞ
a1 ¼ 2RootOfð4F1 ðZÞRZ 2 þ 4tRZ 2 þ yÞ;
p ¼ RootOfð4F1 ðZÞRZ 2 þ 4tRZ 2 þ yÞ;
q ¼ q;
ð3:36Þ
where F ðZÞ denotes arbitrary function with respect to Z and RootOfð4F1 ðZÞRZ 2 þ 4tRZ 2 þ yÞ denotes all the roots of the equation 4F1 ðZÞRZ 2 þ 4tRZ 2 þ y ¼ 0 with respect to one variable Z. Case 3 k1 ¼ 0;
a0 ¼
Z
1 Rqy C12 qt dy þ F1 ðtÞ; 2 C1
a 1 ¼ C1 ;
b1 ¼ C1 ;
1 Rqy C12 qt dy þ F1 ðtÞ; 2 C1
a 1 ¼ C1 ;
b1 ¼ C1 ;
1 Rqy RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ2 qt dy þ F2 ðtÞ; 2 RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ
p ¼ C1 ;
q ¼ q:
ð3:37Þ
Case 4 k1 ¼ 0;
a0 ¼
Z
p ¼ C1 ;
q ¼ q:
ð3:38Þ
Case 5 k1 ¼ 0;
a0 ¼
Z
a1 ¼ RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ; p ¼ RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ;
b1 ¼ RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ; q ¼ q:
ð3:39Þ
B. Li et al. / Chaos, Solitons and Fractals 17 (2003) 885–893
889
Case 6 k1 ¼ 0;
a0 ¼
Z
1 Rqy RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ2 qt dy þ F2 ðtÞ; 2 RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ
a1 ¼ RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ; 2
2
p ¼ RootOfðF1 ðZÞRZ þ tRZ þ yÞ;
b1 ¼ RootOfðF1 ðZÞRZ 2 þ tRZ 2 þ yÞ; q ¼ q:
ð3:40Þ
Case 7 a1 ¼ b1 ¼ 0;
a0 ¼
Z
qt R þ qy C12 dy þ F1 ðtÞ; C1
k1 ¼ C1 ;
p¼
1 C1 ; 2 R
q ¼ q:
ð3:41Þ
Case 8 Z
ðqt R þ qy RootOfðF1 ðZÞZ 2 tZ 2 RyÞ2 Þ dy þ F2 ðtÞ; RootOfðF1 ðZÞZ 2 tZ 2 RyÞ 1 RootOfðF1 ðZÞZ 2 tZ 2 RyÞ ; k1 ¼ RootOfðF1 ðZÞZ 2 tZ 2 RyÞ; p ¼ 2 R
a1 ¼ b1 ¼ 0;
a0 ¼
q ¼ q:
ð3:42Þ
Case 9 b1 ¼ 0;
a0 ¼
Z
4qy C12 qt R dy þ F1 ðtÞ; C1
a1 ¼
C1 ; R
k1 ¼ C1 ;
p¼
1 C1 ; 2 R
q ¼ q:
ð3:43Þ
Case 10 Z
4qy RootOfð4F1 ðZÞZ 2 þ 4tZ 2 þ RyÞ2 qt R dy þ F2 ðtÞ; RootOfð4F1 ðZÞZ2 þ 4tZ2 þ RyÞ RootOfð4F1 ðZÞZ 2 þ 4tZ 2 þ RyÞ ; k1 ¼ RootOfð4F1 ðZÞZ 2 þ 4tZ 2 þ RyÞ; a1 ¼ R 1 RootOfð4F1 ðZÞZ 2 þ 4tZ 2 þ RyÞ ; q ¼ q: p¼ 2 R
b1 ¼ 0;
a0 ¼
ð3:44Þ
From (3.2), (2.4) and Case 1–10, we can obtain the following three types of solutions for the breaking soliton equation. Type 1: From Case 1, we can obtain the following solutions: when R < 0, Z npffiffiffiffiffiffiffi o pffiffiffiffiffiffiffi ð2Rqy C12 þ 12 qt Þ u11 ¼ dy þ F1 ðtÞ 2C1 R tanh R½xC1 þ q ; C1 Z npffiffiffiffiffiffiffi o pffiffiffiffiffiffiffi ð2Rqy C12 þ 12 qt Þ u12 ¼ dy þ F1 ðtÞ 2C1 R coth R½xC1 þ q ; C1
ð3:45Þ ð3:46Þ
when R > 0, Z
npffiffiffi o pffiffiffi ð2Rqy C12 þ 12 qt Þ dy þ F1 ðtÞ þ 2C1 R tan R½xC1 þ q ; C1 Z npffiffiffi o pffiffiffi ð2Rqy C12 þ 12 qt Þ u14 ¼ dy þ F1 ðtÞ 2C1 R cot R½xC1 þ q : C1
u13 ¼
ð3:47Þ ð3:48Þ
Type 2: From Case 2, we can obtain the following solutions: when R < 0, npffiffiffiffiffiffiffi o pffiffiffiffiffiffiffi R½xp þ q ; u21 ¼ a0 a1 R tanh
ð3:49Þ
npffiffiffiffiffiffiffi o pffiffiffiffiffiffiffi u22 ¼ a0 a1 R coth R½xp þ q ;
ð3:50Þ
890
B. Li et al. / Chaos, Solitons and Fractals 17 (2003) 885–893
when R > 0, npffiffiffi o pffiffiffi u23 ¼ a0 þ a1 R tan R½xp þ q ;
ð3:51Þ
npffiffiffi o pffiffiffi R½xp þ q ; u24 ¼ a0 a1 R cot
ð3:52Þ
where a0 ; a1 ; p and q are determined by (3.36). Type 3: From Case 3 and 4, we can obtain the following solutions: when R < 0, Z pffiffiffiffiffiffiffih pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i 1 Rqy C12 qt dy þ F1 ðtÞ C1 R tanhð RnÞ isechð RnÞ ; u31 ¼ 2 C1 Z pffiffiffiffiffiffiffih pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i 1 Rqy C12 qt dy þ F1 ðtÞ C1 R cothð RnÞ cschð RnÞ ; u32 ¼ 2 C1 when R > 0, Z pffiffiffih pffiffiffi pffiffiffi i 1 Rqy C12 qt dy þ F1 ðtÞ þ C1 R tanð RnÞ secð RnÞ ; u33 ¼ C1 2 Z pffiffiffih pffiffiffi pffiffiffi i 1 Rqy C12 qt dy þ F1 ðtÞ C1 R cotð RnÞ cscð RnÞ ; u33 ¼ 2 C1
ð3:53Þ ð3:54Þ
ð3:55Þ ð3:56Þ
where n ¼ xC1 þ q. Type 4: From Case 5 and 6, we can obtain the following solutions: when R < 0, pffiffiffiffiffiffiffih pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i u41 ¼ a0 a1 R tanhð RnÞ isechð RnÞ ;
ð3:57Þ
pffiffiffiffiffiffiffih pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i u42 ¼ a0 a1 R cothð RnÞ cschð RnÞ ;
ð3:58Þ
when R > 0, pffiffiffiffiffiffiffih pffiffiffi pffiffiffi i u43 ¼ a0 þ a1 R tanð RnÞ secð RnÞ ;
ð3:59Þ
pffiffiffiffiffiffiffih pffiffiffi pffiffiffi i u44 ¼ a0 a1 R cotð RnÞ cscð RnÞ ;
ð3:60Þ
where n ¼ xp þ q and a0 ; a1 ; p; q are determined by (3.39) or (3.40). Type 5: From Case 7, we can obtain the following solutions: when R < 0, Z pffiffiffiffiffiffiffi qt R þ qy C12 C1 1 C1 x þ q ; dy þ F1 ðtÞ pffiffiffiffiffiffiffi coth R½ u51 ¼ C1 2 R R Z 2 pffiffiffiffiffiffiffi qt R þ qy C1 C1 1 C1 u52 ¼ x þ q ; dy þ F1 ðtÞ pffiffiffiffiffiffiffi tanh R½ C1 2 R R
ð3:61Þ ð3:62Þ
when R > 0, pffiffiffi qt R þ qy C12 C1 1 C1 x þ q ; u53 ¼ dy þ F1 ðtÞ þ pffiffiffi cot R½ 2 R C1 R Z pffiffiffi qt R þ qy C12 C1 1 C1 x þ q : dy þ F1 ðtÞ pffiffiffi tan R½ u53 ¼ 2 R C1 R Z
ð3:63Þ ð3:64Þ
B. Li et al. / Chaos, Solitons and Fractals 17 (2003) 885–893
891
Type 6: From Case 8, we can obtain the following solutions: when R < 0, npffiffiffiffiffiffiffi o k1 R½xp þ q ; u61 ¼ a0 pffiffiffiffiffiffiffi coth R npffiffiffiffiffiffiffi o k1 u62 ¼ a0 pffiffiffiffiffiffiffi tanh R½xp þ q ; R
ð3:65Þ ð3:66Þ
when R > 0, npffiffiffi o k1 u63 ¼ a0 þ pffiffiffi cot R½xp þ q ; R npffiffiffi o k1 u64 ¼ a0 pffiffiffi tan R½xp þ q ; R
ð3:67Þ ð3:68Þ
where a0 ; k1 ; p and q are determined by (3.42). Type 7: From Case 9, we can obtain the following solutions: u71 ¼ u72 ¼
Z Z
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i 4qy C12 qt R C1 h dy þ F1 ðtÞ pffiffiffiffiffiffiffi tanhð RnÞ cothð RnÞ ; C1 R pffiffiffi pffiffiffi i 4qy C12 qt R C1 h dy þ F1 ðtÞ pffiffiffi tanð RnÞ cotð RnÞ ; C1 R
R < 0;
R > 0;
where n ¼ ð1=2ÞðC1 =RÞx þ q. Type 8: From Case 10, we can obtain the following solutions: pffiffiffiffiffiffiffih pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi i u81 ¼ a0 a1 R tanhð RnÞ cothð RnÞ ; R < 0; pffiffiffih pffiffiffi pffiffiffi i u72 ¼ a0 a1 R tanð RnÞ cotð RnÞ ;
R > 0;
ð3:69Þ ð3:70Þ
ð3:71Þ ð3:72Þ
where n ¼ xp þ q and a0 ; a1 ; p; q are determined by (3.44). In order to further understand the solutions obtained in this paper, we only take the solutions in Type 2 as a simple explanation. Setting F ðZÞ ¼ k0 ;
q ¼ ðk1 y þ k2 ÞhðtÞ;
where k0 ; k1 ; k2 are arbitrary constants and hðtÞ is arbitrary function with respect to t, we can obtain rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 y : RootOfð4k0 RZ 2 þ 4tRZ 2 þ yÞ ¼
2 Rðt k0 Þ
ð3:73Þ
ð3:74Þ
Therefore, from (3.49) in Type 2, we are able to find an explicit solutions, as follows: " # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2k1 RhðtÞ u¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rðt k0 Þ y 2k2 h0 ðtÞ Rðt k0 Þy 1=2 þ F2 ðtÞ þ C 3 Rðt k0 Þ
rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi y x y tanh
þ Rðk1 y þ k2 ÞhðtÞ ; t k0 2 t k0
ð3:75Þ
where C is an arbitrary constant. It is easy to see that, when setting k0 ¼ k1 ¼ 0; k2 ¼ R ¼ 1, F2 ðtÞ ¼ CðtÞ in the solution (3.75), the solution (12) in Ref. [18] is recovered. Therefore, when taking the arbitrary functions q; F1 ðtÞ and F2 ðtÞ as special constants or functions, we can obtain rich explicit exact solutions for Eq. (3.1).
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B. Li et al. / Chaos, Solitons and Fractals 17 (2003) 885–893
4. Conclusions In summary, based on the computerized symbolic computation and a Riccati equation, a generalized Riccati equation expansion method for searching for exact soliton-like solutions of NEEs is proposed by introducing a new more general ans€atz than the ans€atz in the extended tanh-function method, modified extended tanh-function method, and generalized hyperbolic-function method. Making use of the method, we study a typical breaking soliton equation and obtain many families of the exact solutions and implement in computer symbolic computation system––Maple. From the solutions obtained the restriction on nðx; y; tÞ as merely a linear function x; y; t and the restriction on the coefficients a0 ; a1 ; b1 ; k1 as constants are removed and, with no extra effect, the singular soliton-like solution and triangular function solutions are obtained. To make the work feasible (or to obtain explicit exact soliton-like solutions), how to choose the forms for a0 ; ai ; bi ; ki ði ¼ 1; . . . ; mÞ and n in the ans€ atz would be the key step in the computation of our method. The method, proposed in this paper for single equation, may be extended to find exact soliton-like solutions of other NEEs and coupled NEEs.
Acknowledgements The work is supported by the National Natural Science Foundation of China under the Grant No. 1007201, the National Key Basic Research Development Project Program under the Grant No. G1998030600.
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