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Generalized Riccati equation expansion method and its application to the (3 + 1)-dimensional Jumbo–Miwa equation
Applied Mathematics and Computation 152 (2004) 581–595 www.elsevier.com/locate/amc
Generalized Riccati equation expansion method and its application to the (3 + 1)-dimensional Jumbo–Miwa equation Biao Li a
a,*
, Yong Chen a, Hengnong Xuan b, Hongqing Zhang a
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China b Department of Computer, Nanjing University of Economics, Nanjing 210003, People’s Republic of China
Abstract Based on the computerized symbolic system Maple and a Riccati equation, a generalized Riccati equation expansion method for constructing soliton-like solutions of non-linear evolution equations (NEEs) is presented by a new general ans€ atz. The proposed method is more powerful than most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method. By use of the method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some NEEs. Making use of the method, we study the the (3 + 1)-dimensional Jumbo–Miwa equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functionsÕ soliton-like solutions, singular soliton-like solutions, periodic form solutions. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Generalized Riccati equation expansion method; (3 + 1)-dimensional Jumbo–Miwa equation; Soliton-like solutions; Symbolic computation
*
Corresponding author. E-mail address: [email protected] (B. Li).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00578-2
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1. Introduction In recently years, directly searching for exact solutions of non-linear evolution equations (NEEs) has become more and more attractive, on the one hand, due to their occurrence in many fields of science, in physics as well as in chemistry or biology and the interesting features and rich variety of their solutions, on the other hand, due to the availability of computer systems like Maple or Mathematica which allow us to perform some complicated and tedious algebraic calculation and differential calculation on a computer, at the same time help us to find new exact solution of NEEs. In order to obtain exact solutions of NEEs, many effective methods have been presented, such as, inverse scattering method, B€ acklund transformation, Darboux transformation, Cole– Hopf transformation, Hirota method, Painleve method [1,2], tanh method [3– 8], extended tanh-function method [9–11], modified extended tanh-function method [17], generalized hyperbolic-function method [18,19]. Particularly, various ans€ atz have been proposed in order to obtain new form of solutions. One of most effectively straightforward methods to construct exact solutions of NEEs is tanh method [3–8]. In [9–11], Fan proposed an extended tanhfunction method. Fan [12], Yan [13] and Li and co-workers [14–16] further developed this idea and made it much more lucid and straightforward for a class of NEEs. Recently, Elwakil et al. [17] modified extended tanh-function method and obtained some new exact solutions. Gao and Tian [18,19] presented a generalized hyperbolic-function method by introducing coefficient functions and non-travelling transformation. As we known, when applying direct method, the choice of an appropriate ans€atz is of great importance. In this paper, based on the above work [3–19], by introducing a more general ans€ atz than the ans€ atz in the above methods, we present the generalized Riccati equation expansion method. Then we choose the (3 + 1)-dimensional Jumbo– Miwa equation to illustrate our algorithm and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functionsÕ soliton-like solutions, singular soliton-like solutions, periodic form solutions. 2. Generalized Riccati equation expansion mehtod We now establish the generalized Riccati equation expansion method as follows: (A) 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: We express the solutions of the NEEs by the more general ans€atz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X i i1 2 i uðx; y; tÞ ¼ f þ gi / ðnÞ þ hi / ðnÞ R þ / ðnÞ þ ki / ðnÞ ; i¼1
ð2:1Þ
ð2:2Þ
B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
583
where m is an integer to be determined by balancing the highest order derivative terms with the non-linear terms in (2.1), R is a real constant, while f ¼ f ðx; y; tÞ, gi ¼ gi ðx; y; tÞ, hi ¼ hi ðx; y; tÞ, ki ¼ ki ðx; y; tÞ, ði ¼ 1; . . . ; mÞ, n ¼ nðx; y; tÞ are all arbitrary differentiable functions and /ðnÞ satisfies d/ðnÞ ¼ R þ /2 ðnÞ: dn
ð2:3Þ
(B) Substituting (2.2) along with (2.3) into (2.1), multiplying the most simplify common denominator in the obtained system, setting the coefficients of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin R þ /2 ðnÞ (j ¼ 0; 1; . . .; n ¼ 0; 1) ðNote: where /i ðnÞ denotes i /j ðnÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi power of /ðnÞ and R þ /2 ðnÞ denotes n power of R þ /2 ðnÞÞ to zero, we obtain a set of over-determined partial differential equations with respect to differentiable functions f , gi , hi , ki (i ¼ 1; . . . ; m) and n. (C) Solving the over-determined partial differential equations by use of the PDEtools package of Maple, we would end up with the explicit expressions for f , gi , hi , ki (i ¼ 1; . . . ; m) and n or the constrains among them. (D) It is well-known that the general solutions of Riccati equation (2.3) are 8 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi R tanhð RnÞ; R < 0; > > > pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi > > > R cothð RnÞ; R < 0; > > < 1 ð2:4Þ /ðnÞ ¼ ; R ¼ 0; > n > > p ffiffiffi p ffiffiffi > > > R tanð RnÞ; R > 0; > > pffiffiffi : pffiffiffi R cotð RnÞ; R > 0: Thus according to (2.2), (2.4) and the conclusions in (C), we obtain many general solutions of (2.1). At the same time, it is necessary to point out that, when R ¼ 0, some solutions are missed using the above method. So under this condition, we take the following steps to deal with it: (1) set hi ¼ 0; (2) set the coefficients of the same power of /j ðnÞ (i ¼ 0; 1; . . .) to zero and obtain a set of over-determined partial differential equations with respect to functions f , gi , ki (i ¼ 1; . . . ; m); (3) solve the set of over-determined partial differential equations; (4) write the solutions by use of Eq. (2.4). Remarks 1. Generalization The method proposed here is more general than generalized hyperbolicfunction method [18,19], tanh method [3–8], extended tanh-function method [9–12], 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 of
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x; y; t and the restriction on the coefficients f , gi , hi , ki (i ¼ 1; . . . ; m) as constants are removed. Secondly, compared with the generalized hyperbolicfunction method, setting ki ¼ 0 (i ¼ 1; . . . ; m), we can not only recover the exact soliton-like solutions obtained by generalized hyperbolic-function method for a given NEEs but also we can, with no extra efforts, find singular soliton-like solutions and periodic form solutions. When ki 6¼ 0 (i ¼ 1; . . . ; m), some new formal solutions would be expected for some NEEs. 2. Feasibility 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 or Mathematica 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 (B). As the calculation goes on, in order to drastically simplify the work or make the work feasible, we often choose special function forms for f , gi , hi , ki (i ¼ 1; . . . ; m) and n, on a trial-and-error basis.
3. The (3 + 1)-dimensional Jumbo–Miwa equation In this section, by use of the generalized Riccati equation expansion method, we investigate (3 + 1)-dimensional Jumbo–Miwa equation [20–24] uxxxy þ 3uxy ux þ 3uy uxx þ 2uyt 3uxz ¼ 0;
ð3:1Þ
which comes from the second member of a KP hierarchy. Eq. (3.1) does not pass the Painleve test and hence it is expected to be non-integrable [24]. Also it does not lead to Kac–Moody–Virasoro type subalgebras which typically exist in many of the integrable higher-dimensional equations [21–23]. Some travelling wave solutions for Eq. (3.1) are obtained in [20]. 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 following special form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; y; z; tÞ ¼ f þ g/ðnÞ þ h R þ /2 ðnÞ þ k/1 ðnÞ; ð3:2Þ where f ¼ f ðy; z; tÞ, g ¼ gðy; z; tÞ, h ¼ hðy; z; tÞ, k ¼ kðy; z; tÞ and n ¼ xpðy; z; tÞ þ qðy; z; tÞ are all differentiable functions and /ðnÞ satisfies (2.3). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Substituting (3.2) along with (2.3) into (3.1), multiplying /ðnÞ5 R þ /ðnÞ2 in ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the obtained system, collecting coefficients of monomials of /ðnÞ, q R þ /ðwÞ2 and x (notice that f , g, h, k, p and q are independent of x) with the aid of Maple, then setting each coefficients to zero, we can deduce the following
B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
585
set of over-determined partial differential equations with respect to the unknown functions f , g, h, k, p and q 2Rðhy pt þ hpty þ ht py Þ ¼ 0;
ð3:3Þ
6p2 hy kR3 ¼ 0;
ð3:4Þ
2
3pð2p hy þ 3pgy h þ 6ppy h þ 3phy g þ 2py ghÞ ¼ 0;
ð3:5Þ
3R2 pkð3phy 2py hÞ ¼ 0;
ð3:6Þ
2
2
2
3pð2gy p þ 3pgy g þ 6ppy g þ 3phhy þ py g þ h py Þ ¼ 0;
ð3:7Þ
36p2 gR2 qy h þ 9p2 fy hR 9phqz R þ 6hqy qt R þ 2hty 18p2 hqy kR þ 33p3 hR2 qy ¼ 0;
ð3:8Þ
2
2
3
3Rhð6p py k 3ppz þ 2py qt þ 12p py gR þ 11Rp py þ 2pt qy Þ ¼ 0;
ð3:9Þ
2hpty þ 2ht py þ 2hy pt ¼ 0;
ð3:10Þ
6hpy pt R ¼ 0;
ð3:11Þ
3 2
12R p kqy ð2Rp kÞ ¼ 0;
ð3:12Þ
12R3 p2 py kð2Rp kÞ ¼ 0;
ð3:13Þ
2
2
2
3
2
Rð2hqy qt R þ 2hty þ 3p fy hR þ 6p gR qy h 3phqz R þ 5p hR qy Þ ¼ 0; ð3:14Þ R2 hð6p2 py gR þ 2py qt 3ppz þ 2pt qy þ 5Rp3 py Þ ¼ 0;
ð3:15Þ
2
2hpy pt R ¼ 0;
ð3:16Þ
Rð5p3 hy R þ 3p2 ky h þ 2hqty 3pz h þ 2hy qt 12ppy hk þ 15p2 py hR þ 6ppy gRh 3phz þ 9p2 hy gR þ 6p2 gy hR þ 2ht qy Þ ¼ 0; 3
2
2
ð3:17Þ 2
2hy qt 3phz þ 11p hy R þ 3p ky h þ 15p gy hR 3pz h þ 2hqty 3p hy k þ 33p2 py hR þ 12ppy gRh þ 18p2 hy gR þ 2ht qy 6ppy hk ¼ 0; ð3:18Þ 2hð2pt qy þ 26p3 Rpy 3ppz 6p2 py k þ 27p2 py gR þ 2py qt Þ ¼ 0; 2
3
2
2
ð3:19Þ
2hð27p gRqy þ 26p Rqy 3pqz þ 2qy qt 6p qy k þ 3p fy Þ ¼ 0;
ð3:20Þ
4hpy pt ¼ 0;
ð3:21Þ
2
2
2
3
2R kð12p py k þ 2py qt þ 6p py gR þ 2pt qy þ 20p Rpy 3ppz Þ ¼ 0;
ð3:22Þ
2R2 kð12p2 qy k þ 2qy qt þ 6p2 gRqy þ 20p3 Rqy 3pqz þ 3p2 fy Þ ¼ 0; ð3:23Þ 2
4kR py pt ¼ 0;
ð3:24Þ
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6p2 py kR þ 3p2 gy R2 g 12ppy gRk þ 3p2 hy hR2 3pz gR þ 3ppy k 2 þ 3p2 ky k þ 3ppy g2 R2 þ 6p2 py gR2 3pgz R þ 2p3 gy R2 2p3 ky R þ 2gt Rqy þ 2fty þ 2gRqty 2kqty þ 3pkz 2ky qt þ 2gy Rqt þ 3pz k 2kt qy ¼ 0; ð3:25Þ 2kt py 2kpty þ 2gt Rpy þ 2gRpty 2ky pt þ 2gy Rpt ¼ 0;
ð3:26Þ
3R2 pð2ky Rp2 þ 6ppy kR 3pky k py k 2 Þ ¼ 0; 2 2
2
3
ð3:27Þ
2
2
2
12p k Rqy 6pkRqz þ 6p fy kR þ 16p kR qy þ 2kty þ 12p gR qy k þ 4kRqy qt ¼ 0;
ð3:28Þ
2Rkð6p2 py k þ 2py qt 3ppz þ 6p2 py gR þ 2pt qy þ 8p3 Rpy Þ ¼ 0;
ð3:29Þ
4kRpy pt ¼ 0;
ð3:30Þ
2
2
2
12p hy hR 3pgz þ 2gy qt 6ppy gk þ 6ppy g R þ 2gt qy þ 3p ky g þ 3ppy h2 R þ 24p2 py gR þ 2gqty 3pz g 3p2 gy k þ 12p2 gy Rg þ 8p3 gy R ¼ 0; ð3:31Þ 2Rðkt py þ kpty þ ky pt Þ ¼ 0;
ð3:32Þ
2
3
2
Rð12p ky k þ 6ppy gRk 3pz k þ 8p ky R þ 2ky qt 6ppy k 3pkz þ 24p2 py kR 3Rp2 gy k þ 2kt qy þ 3Rp2 ky ky g þ 2kqty Þ ¼ 0;
ð3:33Þ
16p3 gR2 qy þ 12p2 g2 R2 qy 6pgRqz þ 6p2 fy gR þ 2gty þ 4gRqy qt 12p2 gRqy k þ 6p2 h2 qy R2 ¼ 0;
ð3:34Þ
2Rð8p3 gRpy 6p2 gpy k þ 6p2 g2 Rpy þ 3p2 h2 py R 3pgpz þ 2gqy pt þ 2gpy qt Þ ¼ 0;
ð3:35Þ
4gRpy pt ¼ 0; 2 2
ð3:36Þ 2
2 2
24p g Rpy 12p gpy k 6pgpz þ 4gqy pt þ 18p h py R þ 4gpy qt þ 40p3 gRpy ¼ 0;
ð3:37Þ
4gpy pt ¼ 0; 2 2
ð3:38Þ 2 2
2
2
24p g Rqy þ 18p h qy R þ 6p fy g 6pgqz þ 4gqy qt 12p gqy k þ 40p3 gRqt ¼ 0; 2
3
ð3:39Þ
6p hqy kR ¼ 0;
ð3:40Þ
6p2 hpy kp3 ¼ 0;
ð3:41Þ
2
24p hqy ðp þ gÞ ¼ 0;
ð3:42Þ
24p2 py hðp þ gÞ ¼ 0;
ð3:43Þ
B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
2gpty þ 2gy pt þ 2gt py ¼ 0; 2
2
2
587
ð3:44Þ
12p py ðg þ 2pg þ h Þ ¼ 0;
ð3:45Þ
12p2 qy ðg2 þ 2pg þ h2 Þ ¼ 0:
ð3:46Þ
Using the powerful PDEtools soft package of Maple, solving the set of partial differential equations (3.3)–(3.46), we can obtain the following non-trivial results. (Note: in the rest of this paper, c1 , Ci (i ¼ 1; . . . ; 5) are arbitrary constants and Fi ðtÞ (i ¼ 1; . . . ; 5) denote arbitrary function of t, and so on.) Case 1 g ¼ 2C1 ; h ¼ 0; k ¼ 0; p ¼ C1 ; q ¼ F2 ðy; zÞ þ F4 ðzÞ þ C2 t þ C3 ; Z 1 oF2 ðy; zÞ oF2 ðy; zÞ 3 oF2 ðy; zÞ 2 C2 Þ dy 4C1 R f ¼ þ 3C1 3C12 oy oz oy 1 oF4 ðzÞ y þ F5 ðz; tÞ: þ C1 oz ð3:47Þ Case 2 g ¼ 2C1 ;
h ¼ 0;
k ¼ 0; p ¼ C1 ; Z 2 F3 ðzÞ dz ; q ¼ F3 ðzÞy þ F4 ðzÞ þ F5 t þ 3C1 1 oF3 ðzÞ 2 4 1 oF4 ðzÞ y þ yC1 RF3 ðzÞ þ y þ F6 ðz; tÞ: f ¼ 2C1 oz 3 C1 oz
ð3:48Þ
Case 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c1 z þ C1 eC2 t C3 ; h ¼ 0; k ¼ 0; p ¼ 2c1 z þ C1 C2 eC2 t C3 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 lnð2c1 z þ C1 Þ C2 2c1 z þ C1 F2 t þ q ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2C2 2c1 z þ C1 C2 2 1 4e3C2 t c1 z þ C1 RC33 lnð2c1 z þ C1 Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 yc1 C3 þ 2c1 z þ C1 F1 ðzÞC2 eC2 t ; þ 2 1 oF1 ðzÞ C2 c21 z2 y 2zc21 F1 ðzÞC2 y 4 f ¼ 5=2 oz C2 ð2c1 z þ C1 Þ C3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oF1 ðzÞ 3 C2 C1 c1 y y 2 C3 2c1 z þ C1 c21 þ 4z oz 2 oF1 ðzÞ 2 C1 C2 y þ F3 ðz; tÞ: F1 ðzÞC2 C1 c1 y þ oz g ¼ 2
ð3:49Þ
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B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
Case 4 Z 1 pð4p2 Rpy þ 3pz Þ g ¼ 2p; h ¼ 0; k ¼ 0; q ¼ dt þ F1 ðy;zÞ; 2 py " Z Z 4pz py2 p2 R þ pz2 py þ ppy pzz ppyz pz 1 oF1 ðy; zÞ 3 f¼ dtpy þ 2py 2 2py p oz py # Z 4p2 Rpy3 þ pz py2 þ ppy pyz ppyy pz oF1 ðy; zÞ 3pz dt 2pz dy þ F2 ðz; tÞ; oy py2 ð3:50Þ where p ¼ pðy; z; tÞ satisfies pyz ¼
4py2 ppz R pzz py ; pz
pyy ¼
8pz Rppy3 pzz py2 ; pz2
pt ¼ 0:
Case 5 g ¼ C1 ; h ¼ C1 ; k ¼ 0; p ¼ C1 ; q ¼ F6 ðy; zÞ þ F8 ðzÞ þ C4 t þ C5 ; Z 1 oF6 ðy; zÞ oF6 ðy; zÞ 3 oF6 ðy; zÞ 2 C4 dy C R þ 3C1 f ¼ oy oz oy 3C12 1 1 oF8 ðzÞ y þ F9 ðz; tÞ: þ C1 oz ð3:51Þ Case 6 g ¼ C1 ; h ¼ C1 ; k ¼ 0; p ¼ C1 ; q ¼ F2 ðy; zÞ þ F4 ðzÞ þ C2 t þ C3 ; Z 1 oF2 ðy; zÞ oF2 ðy; zÞ 3 oF2 ðy; zÞ 2 C2 dy C R þ 3C1 f ¼ 3C12 1 oy oz oy 1 oF4 ðzÞ y þ F9 ðz; tÞ: þ C1 oz ð3:52Þ Case 7 g ¼ C1 ;
h ¼ C1 ;
k ¼ 0; p ¼ C1 ; Z 2 q ¼ F8 ðzÞy þ F9 ðzÞ þ F10 t þ F8 ðzÞ dz ; 3C1 1 o 1 1 o ðF8 ðzÞÞy 2 þ yC1 RF8 ðzÞ þ y ðF9 ðzÞÞ þ F11 ðz; tÞ: f ¼ 2C1 oz 3 C1 oz
ð3:53Þ
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589
Case 8 k ¼ 0; p ¼ C1 ; Z 2 F3 ðzÞ dz ; q ¼ F3 ðzÞy þ F4 ðzÞ þ F5 t þ 3C1 1 oF3 ðzÞ 2 1 1 oF4 ðzÞ y þ yC1 RF3 ðzÞ þ y þ F11 ðz; tÞ: f ¼ 2C1 oz 3 C1 oz
g ¼ C1 ;
h ¼ C1 ;
ð3:54Þ
Case 9 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ¼ 2c1 z þ C1 eC2 t C3 ; h ¼ 2c1 z þ C1 eC2 t C3 ; k ¼ 0; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t C3 ; p ¼ 2c1 z þ C1 eC2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 lnð2c1 z þ C1 Þ C2 q ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c1 z þ C1 F4 t þ 2C2 2c1 z þ C1 C2 1 3 3C2 t 2 C3 Re ð2c1 z þ C1 Þ lnð2c1 z þ C1 Þ 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C2 t 3 yc1 C3 þ 2c1 z þ C1 F3 ðzÞC2 ; þe 2 y oFoz3 ðzÞ 3c21 y 2 yc1 F3 ðzÞ þ þ F5 ðz; tÞ: f ¼ 2 1=2 2C2 ð2c1 z þ C1 Þ C3 ð2c1 z þ C1 Þ3=2 C3 ð2c1 z þ C1 Þ ð3:55Þ Case 10 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ¼ 2c1 z þ C1 eC2 t C3 ; h ¼ 2c1 z þ C1 eC2 t C3 ; k ¼ 0; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ 2c1 z þ C1 eC2 t C3 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 4 2c1 z þ C1 F2 t þ lnð2c1 z þ C1 Þ C2 q ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2C2 4 2c1 z þ C1 C2 2 þ C33 Re3C2 t ð2c1 z þ C1 Þ lnð2c1 z þ C1 Þ h i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 6yc1 C3 4 2c1 z þ C1 F1 ðzÞC2 eC2 t ; h i 8 < 2c21 DðF2 Þ t þ 2C1 lnð2c1 z þ C1 Þ z 3c21 y 2 2 þ f ¼ 2 5=2 : 2C2 ð2c1 z þ C1 Þ eC2 t C3 ð2c1 z þ C1Þ C2 h i h i c1 DðF2 Þ t þ 2C1 2 lnð2c1 z þ C1 Þ C1 c1 DðF2 Þ t þ 2C1 2 lnð2c1 z þ C1 Þ þ eC2 t C3 ð2c1 z þ C1 Þ3=2 C2 eC2 t C3 ð2c1 z þ C1 Þ5=2 C2 9 = 1 oF1 ðzÞ 1 þ y þ F5 ðz; tÞ; 1=2 3=2 oz C3 ð2c1 z þ C1 Þ C3 ð2c1 z þ C1 Þ c1 F1 ðzÞ ; ð3:56Þ
590
B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
where DðF2 Þ½t þ ð1=2C2 Þ lnð2c1 z þ C1 Þ denotes the derivative of function F2 ½t þ ð1=2C2 Þ lnð2c1 z þ C1 Þ with respect to variable ½t þ ð1=2C2 Þ lnð2c1 z þ C1 Þ. Case 11 Z 1 pðp2 Rpy þ 3pz Þ g ¼ p; h ¼ p; k ¼ 0; q ¼ dt þ F2 ðy; zÞ; 2 py " Z Z pz py2 p2 R þ pz2 py þ ppy pzz ppyz pz 1 oF2 ðy; zÞ 3 dtpy þ 2py f¼ 2ppy oz py2 # Z 2 3 p Rpy þ pz py2 þ ppy pyz ppyy pz oF2 ðy; zÞ 3pz dt 2pz dy þ F3 ðz; tÞ; oy py2 ð3:57Þ where p ¼ pðy; z; tÞ satisfies pyy ¼
2pz Rppy3 pzz py2 ; pz2
pyz ¼
py2 ppz R þ pzz py ; pz
pt ¼ 0:
Case 12 Z 1 p2 Rpy þ 3pz dt þ F1 ðy; zÞ; k ¼ 0; g ¼ p; h ¼ p; q ¼ 2p py " Z Z pz py2 p2 R þ pz2 py þ ppy pzz ppyz pz 1 oF1 ðy; zÞ 3 dtpy þ 2py f¼ 2ppy oz py2 # Z 2 3 p Rpy þ pz py2 þ ppy pyz ppyy pz oF1 ðy; zÞ dt 2pz 3pz dy þ F3 ðz; tÞ; oy py2 ð3:58Þ where p ¼ pðy; z; tÞ satisfies pyy ¼
2pz Rppy3 pzz py2 ; pz2
pyz ¼
py2 ppz R þ pzz py ; pz
pt ¼ 0:
Case 13 g ¼ 2C1 ; k ¼ 2C1 R; h ¼ 0; p ¼ C1 ; q ¼ F2 ðy; zÞ þ F4 ðzÞ þ C2 t þ C3 ; Z 1 oF2 ðy; zÞ oF2 ðy; zÞ 3 oF2 ðy; zÞ 2 C2 dy f ¼ 16C1 R þ 3C1 3C12 oy oz oy 1 oF4 ðzÞ y þ F5 ðz; tÞ: þ C1 oz
ð3:59Þ
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k ¼ 2C1 R; h ¼ 0; p ¼ C1 ; Z 2 F3 ðzÞ dz ; q ¼ F3 ðzÞy þ F4 ðzÞ þ F5 t þ 3C1 1 oF3 ðzÞ 2 16 1 oF4 ðzÞ y þ yC1 RF3 ðzÞ þ y þ F6 ðz; tÞ: f ¼ 2C1 oz 3 C1 oz
ð3:60Þ
Case 14 g ¼ 2C1 ;
Case 15 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ¼ 2 2c1 z þ C1 eC2 t C3 ; k ¼ 2 2c1 z þ C1 eC2 t C3 R; h ¼ 0; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ 2c1 z þ C1 eC2 t C3 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 z þ C t þ lnð2c z þ C Þ C2 q¼ 2c F 1 1 2 1 1 1=2 2C2 ð2c1 z þ C1 Þ C2 2 1 3C2 t 16e c1 z þ C1 RC33 lnð2c1 z þ C1 Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 yc1 C3 þ 2c1 z þ C1 F1 ðzÞC2 eC2 t ; þ 2 1 oF1 ðzÞ C2 c21 z2 y 2zc21 F1 ðzÞC2 y 4 f ¼ 5=2 oz C2 ð2c1 z þ C1 Þ C3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oF1 ðzÞ 3 C2 C1 c1 y y 2 C3 2c1 z þ C1 c21 þ 4z oz 2 oF1 ðzÞ 2 C1 C2 y þ F3 ðz; tÞ: F1 ðzÞC2 C1 c1 y þ oz
ð3:61Þ
Case 16 Z 1 pð16p2 Rpy þ 3pz Þ k ¼ 2pR; h ¼ 0; q ¼ dt þ F1 ðy; zÞ; 2 py " Z Z 16pz py2 p2 R þ pz2 py þ ppy pzz ppyz pz 1 dtpy f ¼ 3 2ppy py2 Z 16p2 Rpy3 þ pz py2 þ ppy pyz ppyy pz oF1 ðy; zÞ þ 3pz 2py dt oz py2 # oF1 ðy; zÞ dy þ F2 ðz; tÞ; þ 2pz oy g ¼ 2p;
ð3:62Þ where p ¼ pðy; z; tÞ satisfies pyz ¼
16py2 ppz R pzz py ; pz
pyy ¼
32pz Rppy3 pzz py2 ; pz2
pt ¼ 0:
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Case 17 R ¼ 0; h ¼ 0; g ¼ 0; k ¼ F2 ðy; zÞ þ F1 ðz; tÞ; p ¼ F3 ðzÞ; q ¼ q; Z Z Z oF1 ðz; tÞ qdt þ qty F2 ðy; zÞdy dt f ¼ F5 ðy; zÞ þ F4 ðz; tÞ þ ot Z Z Z Z oF2 ðy; zÞ 3 2 qt dy dt F3 ðzÞ F2 ðy; zÞ F1 ðz; tÞdt þ F1 ðz; tÞqt dt þ oy 2 Z Z 3 oF1 ðz; tÞ 3 oF3 ðzÞ dt F1 ðz; tÞdt y þ F3 ðzÞ 2 oz 2 oz Z Z 3 oF2 ðy; zÞ 3 oF3 ðzÞ 3 2 2 dy F2 ðy; zÞdy F3 ðzÞ F2 ðy; zÞ t: þ F3 ðzÞ 2 oz 2 oz 4 ð3:63Þ Case 18 g ¼ 0; k ¼ F1 ðz; tÞ; p ¼ F2 ðz; tÞ; q ¼ q; Z Z oF1 ðz; tÞ q dt f ¼ F4 ðy; zÞ þ F3 ðz; tÞ þ F1 ðz; tÞqt dt þ ot Z Z 3 oF2 ðz; tÞ 3 oF1 ðz; tÞ þ F1 ðz; tÞ dt F2 ðz; tÞ dt y: 2 oz 2 oz
R ¼ 0;
h ¼ 0;
ð3:64Þ
Case 19 1 h ¼ 0; g ¼ C1 ; k ¼ F2 ðy; zÞ þ F4 ðzÞ þ F3 ðtÞ; p ¼ C1 ; 2 Z 1 oF ðy; zÞ oF ðy; zÞ 6 6 72C12 F2 ðy; zÞ f ¼ 24C12 F3 ðtÞ 24C12 oy oy R ¼ 0;
72C12 F4 ðzÞ
oF6 ðy; zÞ oF6 ðy; zÞ oF2 ðy; zÞ þ 64C 2 þ 9C14 F3 ðtÞt oy oy oy
oF2 ðy; zÞ oF2 ðy; zÞ oF2 ðy; zÞ þ 27C14 F4 ðzÞt 24C 2tC12 oy oy oy oF6 ðy; zÞ oF2 ðy; zÞ 3 oF4 ðzÞ 18tC13 y þ F7 ðz; tÞ; þ 48C1 dy þ C1 t oz oz 4 oz þ 27C14 F2 ðy; zÞt
q¼
Z
3 2 3 3 C F3 ðtÞ dt tC12 F2 ðy; zÞ tC12 F4 ðzÞ þ tC2 þ F6 ðy; zÞ: 8 1 8 8 ð3:65Þ
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593
Case 20 1 C1 3C12 t ; q¼ þ F2 ðy; zÞ; R ¼ 0; h ¼ 0; p ¼ 2 F1 ðyÞ 4F1 ðyÞ Z 2 oF2 ðy; zÞ C1 F1 ðyÞ; y þ F3 ðz; tÞ; g ¼ : f ¼ C1 oz F1 ðyÞ
k ¼ F1 ðyÞ;
ð3:66Þ It is necessary to exclaim that the Cases 1–16 is directly obtained by solving the (3.3)–(3.46), the Cases 17–20 is obtained under the condition: R ¼ h ¼ 0 (here we omitted the trivial solutions and very complicated solutions). From (3.2), (2.4) and Cases 1–20, we can obtain the following five types of solutions for (3 + 1)-dimensional Jumbo–Miwa equation. Type I: From Cases 1–4, we can obtain the following types of solutions. hpffiffiffiffiffiffiffi i pffiffiffiffiffiffiffi u11 ¼ f g R tanh Rðxp þ qÞ ; R < 0; ð3:67Þ hpffiffiffiffiffiffiffi i pffiffiffiffiffiffiffi u12 ¼ f g R coth Rðxp þ qÞ ; hpffiffiffi i pffiffiffi u13 ¼ f þ g R tan Rðxp þ qÞ ; hpffiffiffi i pffiffiffi u14 ¼ f g R cot Rðxp þ qÞ ;
ð3:68Þ
R < 0;
R > 0;
ð3:69Þ
R > 0;
ð3:70Þ
where f , g, p, q are determined by (3.47)–(3.50) respectively. Type II: From Cases 5–12, we can obtain the following types of solutions hpffiffiffiffiffiffiffi i hpffiffiffiffiffiffiffi io pffiffiffiffiffiffiffin u21 ¼ f g R tanh Rðxp þ qÞ þ i sech Rðxp þ qÞ ; R < 0; ð3:71Þ
hpffiffiffiffiffiffiffi i hpffiffiffiffiffiffiffi io pffiffiffiffiffiffiffin u22 ¼ f g R coth Rðxp þ qÞ þ csch Rðxp þ qÞ ;
R < 0; ð3:72Þ
hpffiffiffi i hpffiffiffi io pffiffiffin u23 ¼ f þ g R tan Rðxp þ qÞ þ sec Rðxp þ qÞ ;
R > 0; ð3:73Þ
hpffiffiffi i hpffiffiffi io pffiffiffin u24 ¼ f g R tan Rðxp þ qÞ þ cot Rðxp þ qÞ ;
R > 0; ð3:74Þ
where f , g, h, p, q are determined by (3.51)–(3.58) respectively.
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Type III: From Cases 12–16, we can obtain the following types of solutions pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi u31 ¼ f g Rftanh½ Rðxp þ qÞ þ coth½ Rðxp þ qÞg; R < 0; ð3:75Þ pffiffiffi pffiffiffi pffiffiffi u32 ¼ f þ g Rftan½ Rðxp þ qÞ þ cot½ Rðxp þ qÞg;
R > 0;
ð3:76Þ
where f , g, p, q are determined by (3.58)–(3.62) respectively. Type IV: From Cases 17 and 18, we can obtain the following types of solutions u4 ¼ f kðxp þ qÞ;
ð3:77Þ
where f , k, p and q are determined by (3.63) and (3.64) respectively. Type V: From Cases 19 and 20, the following types of solutions is obtained g kðxp þ qÞ; ð3:78Þ u5 ¼ f xp þ q where f , g, k, p, q are determined by (3.65) and (3.66) respectively. Remarks. When setting the arbitrary functions to be equal to special functions or special constants, the travelling wave solutions for the (3+1)-dimensional Jumbo–Miwa equation can be recovered. For example, if setting F4 ðzÞ ¼ 0, F5 ðz; tÞ ¼ a0 , F2 ðy; zÞ ¼ by þ cz, R ¼ k 2 , C2 ¼ 2C13 K 2 þ 2C2b1 c in the solution (3.67) determined by (3.47), the solutions obtained in [20] can be reproduced. But to our knowledge, the other soliton-like solutions, singular soliton-like solutions and periodic form solutions are not found before.
4. Conclusions In summary, based on the computerized symbolic computation, by introducing a more general ans€ atz than the ans€ atz in the extended tanh-function method, modified extended tanh-function method, and generalized hyperbolicfunction method, we have proposed a generalized Riccati equation expansion method for searching for exact soliton-like solutions of NEEs and implemented in computer symbolic systems. Making use of our method and with the aid of maple, we study the (3 + 1)-dimensional Jumbo–Miwa equation and obtain new families of the exact solutions. In our obtained exact solutions the restriction on nðx; y; z; tÞ as merely a linear function x, y, z, t and the restriction on the coefficients f , gi , hi , ki (i ¼ 1; . . . ; m) as constants are removed and, with no extra effects, the singular soliton-like solution and periodic form solutions, even rational solutions could be obtained. To make the work feasible, how to choose the forms of f , gi , hi , ki (i ¼ 1; . . . ; m) and n in the ans€atz would be the key step in the computation of our method. The method, proposed in this
B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
595
paper for single equation, may be extended to find exact soliton-like solutions of coupled equations.
Acknowledgements The author (B. Li) would express his sincerely thanks to Dr. Z.-s. L€ u for his help and guidance in symbolic computation. The work is supported by the National Natural Science Foundation of China under the Grant no. 10072013, the National Key Basic Research Development Project Program under the Grant no. G1998030600.
References [1] M.J. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scatting, Cambridge University Press, New York, 1991. [2] C.H. Gu et al., Soliton Theory ans Its Application, Zhejiang Science and Technology Press, Zhejiang, 1990. [3] E.J. Parkes, B.R. Duffy, Comput. Phys. Commun. 98 (1996) 288. [4] E.J. Parkes, B.R. Duffy, Phys. Lett. A 229 (1997) 217. [5] B.R. Duffy, E.J. Parkes, Phys. Lett. A 214 (1996) 271. [6] Z.B. Li, M.L. Wang, J. Phys. A: Math. Gen. 26 (1993) 6027. [7] S. Lou, G. Huang, H. Ruan, J. Phys. A: Math. Gen. 24 (1991) L584. [8] A.H. Khater, W. Malfiet, D.K. Callebaut, E.S. Kamel, Chaos Soliton. Fract. 14 (2002) 513. [9] E. Fan, Phys. Lett. A 277 (2000) 212. [10] E. Fan, Comput. Math. Appl. 43 (2002) 671. [11] E. Fan, Nuovi Cimento B 116 (2001) 1385. [12] E. Fan, J. Zhang, Y.C. Hon Benny, Phys. Lett. A 291 (2001) 376. [13] Z.Y. Yan, Phys. Lett. A 292 (2001) 100. [14] Y. Chen, B. Li, H.Q. Zhang, Commun. Theor. Phys. 38 (2002) 261. [15] B. Li, Y. Chen, H.Q. Zhang, J Phys A: Math. Gen. 35 (2002) 8253. [16] B. Li, Y. Chen, H.Q. Zhang, Chaos Soliton. Fract. 15 (2003) 647. [17] S.A. Elwakil, S.K. El-labany, M.A. Zahran, R. Sabry, Phys. Lett. A 299 (2002) 179. [18] Y.T. Gao, B. Tian, Comput. Phys. Commun. 133 (2001) 158. [19] B. Tian, Y.T. Gao, Z. Naturforsch. A 57 (2002) 39. [20] M. Senthilvelan, Appl. Math. Comput. 123 (2001) 381. [21] G. Paquin, P. Winternitz, Physica D 46 (1990) 122. [22] M. Faucher, P. Winternitz, Phys. Rev. E 48 (1993) 3066. [23] M. Senthilvelan, M. Winternitz, J. Nonlinear Math. Phys. 5 (1998) 190. [24] B. Dorrizi, B. Grammaticos, A. Ramani, P. Winternitz, J. Math. Phys. 27 (1986) 2848.
Generalized Riccati equation expansion method and its application to the (3 + 1)-dimensional Jumbo–Miwa equation Biao Li a
a,*
, Yong Chen a, Hengnong Xuan b, Hongqing Zhang a
Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, People’s Republic of China b Department of Computer, Nanjing University of Economics, Nanjing 210003, People’s Republic of China
Abstract Based on the computerized symbolic system Maple and a Riccati equation, a generalized Riccati equation expansion method for constructing soliton-like solutions of non-linear evolution equations (NEEs) is presented by a new general ans€ atz. The proposed method is more powerful than most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method. By use of the method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some NEEs. Making use of the method, we study the the (3 + 1)-dimensional Jumbo–Miwa equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functionsÕ soliton-like solutions, singular soliton-like solutions, periodic form solutions. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Generalized Riccati equation expansion method; (3 + 1)-dimensional Jumbo–Miwa equation; Soliton-like solutions; Symbolic computation
*
Corresponding author. E-mail address: [email protected] (B. Li).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00578-2
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1. Introduction In recently years, directly searching for exact solutions of non-linear evolution equations (NEEs) has become more and more attractive, on the one hand, due to their occurrence in many fields of science, in physics as well as in chemistry or biology and the interesting features and rich variety of their solutions, on the other hand, due to the availability of computer systems like Maple or Mathematica which allow us to perform some complicated and tedious algebraic calculation and differential calculation on a computer, at the same time help us to find new exact solution of NEEs. In order to obtain exact solutions of NEEs, many effective methods have been presented, such as, inverse scattering method, B€ acklund transformation, Darboux transformation, Cole– Hopf transformation, Hirota method, Painleve method [1,2], tanh method [3– 8], extended tanh-function method [9–11], modified extended tanh-function method [17], generalized hyperbolic-function method [18,19]. Particularly, various ans€ atz have been proposed in order to obtain new form of solutions. One of most effectively straightforward methods to construct exact solutions of NEEs is tanh method [3–8]. In [9–11], Fan proposed an extended tanhfunction method. Fan [12], Yan [13] and Li and co-workers [14–16] further developed this idea and made it much more lucid and straightforward for a class of NEEs. Recently, Elwakil et al. [17] modified extended tanh-function method and obtained some new exact solutions. Gao and Tian [18,19] presented a generalized hyperbolic-function method by introducing coefficient functions and non-travelling transformation. As we known, when applying direct method, the choice of an appropriate ans€atz is of great importance. In this paper, based on the above work [3–19], by introducing a more general ans€ atz than the ans€ atz in the above methods, we present the generalized Riccati equation expansion method. Then we choose the (3 + 1)-dimensional Jumbo– Miwa equation to illustrate our algorithm and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functionsÕ soliton-like solutions, singular soliton-like solutions, periodic form solutions. 2. Generalized Riccati equation expansion mehtod We now establish the generalized Riccati equation expansion method as follows: (A) 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: We express the solutions of the NEEs by the more general ans€atz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m X i i1 2 i uðx; y; tÞ ¼ f þ gi / ðnÞ þ hi / ðnÞ R þ / ðnÞ þ ki / ðnÞ ; i¼1
ð2:1Þ
ð2:2Þ
B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
583
where m is an integer to be determined by balancing the highest order derivative terms with the non-linear terms in (2.1), R is a real constant, while f ¼ f ðx; y; tÞ, gi ¼ gi ðx; y; tÞ, hi ¼ hi ðx; y; tÞ, ki ¼ ki ðx; y; tÞ, ði ¼ 1; . . . ; mÞ, n ¼ nðx; y; tÞ are all arbitrary differentiable functions and /ðnÞ satisfies d/ðnÞ ¼ R þ /2 ðnÞ: dn
ð2:3Þ
(B) Substituting (2.2) along with (2.3) into (2.1), multiplying the most simplify common denominator in the obtained system, setting the coefficients of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin R þ /2 ðnÞ (j ¼ 0; 1; . . .; n ¼ 0; 1) ðNote: where /i ðnÞ denotes i /j ðnÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi power of /ðnÞ and R þ /2 ðnÞ denotes n power of R þ /2 ðnÞÞ to zero, we obtain a set of over-determined partial differential equations with respect to differentiable functions f , gi , hi , ki (i ¼ 1; . . . ; m) and n. (C) Solving the over-determined partial differential equations by use of the PDEtools package of Maple, we would end up with the explicit expressions for f , gi , hi , ki (i ¼ 1; . . . ; m) and n or the constrains among them. (D) It is well-known that the general solutions of Riccati equation (2.3) are 8 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi R tanhð RnÞ; R < 0; > > > pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi > > > R cothð RnÞ; R < 0; > > < 1 ð2:4Þ /ðnÞ ¼ ; R ¼ 0; > n > > p ffiffiffi p ffiffiffi > > > R tanð RnÞ; R > 0; > > pffiffiffi : pffiffiffi R cotð RnÞ; R > 0: Thus according to (2.2), (2.4) and the conclusions in (C), we obtain many general solutions of (2.1). At the same time, it is necessary to point out that, when R ¼ 0, some solutions are missed using the above method. So under this condition, we take the following steps to deal with it: (1) set hi ¼ 0; (2) set the coefficients of the same power of /j ðnÞ (i ¼ 0; 1; . . .) to zero and obtain a set of over-determined partial differential equations with respect to functions f , gi , ki (i ¼ 1; . . . ; m); (3) solve the set of over-determined partial differential equations; (4) write the solutions by use of Eq. (2.4). Remarks 1. Generalization The method proposed here is more general than generalized hyperbolicfunction method [18,19], tanh method [3–8], extended tanh-function method [9–12], 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 of
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x; y; t and the restriction on the coefficients f , gi , hi , ki (i ¼ 1; . . . ; m) as constants are removed. Secondly, compared with the generalized hyperbolicfunction method, setting ki ¼ 0 (i ¼ 1; . . . ; m), we can not only recover the exact soliton-like solutions obtained by generalized hyperbolic-function method for a given NEEs but also we can, with no extra efforts, find singular soliton-like solutions and periodic form solutions. When ki 6¼ 0 (i ¼ 1; . . . ; m), some new formal solutions would be expected for some NEEs. 2. Feasibility 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 or Mathematica 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 (B). As the calculation goes on, in order to drastically simplify the work or make the work feasible, we often choose special function forms for f , gi , hi , ki (i ¼ 1; . . . ; m) and n, on a trial-and-error basis.
3. The (3 + 1)-dimensional Jumbo–Miwa equation In this section, by use of the generalized Riccati equation expansion method, we investigate (3 + 1)-dimensional Jumbo–Miwa equation [20–24] uxxxy þ 3uxy ux þ 3uy uxx þ 2uyt 3uxz ¼ 0;
ð3:1Þ
which comes from the second member of a KP hierarchy. Eq. (3.1) does not pass the Painleve test and hence it is expected to be non-integrable [24]. Also it does not lead to Kac–Moody–Virasoro type subalgebras which typically exist in many of the integrable higher-dimensional equations [21–23]. Some travelling wave solutions for Eq. (3.1) are obtained in [20]. 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 following special form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; y; z; tÞ ¼ f þ g/ðnÞ þ h R þ /2 ðnÞ þ k/1 ðnÞ; ð3:2Þ where f ¼ f ðy; z; tÞ, g ¼ gðy; z; tÞ, h ¼ hðy; z; tÞ, k ¼ kðy; z; tÞ and n ¼ xpðy; z; tÞ þ qðy; z; tÞ are all differentiable functions and /ðnÞ satisfies (2.3). qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Substituting (3.2) along with (2.3) into (3.1), multiplying /ðnÞ5 R þ /ðnÞ2 in ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the obtained system, collecting coefficients of monomials of /ðnÞ, q R þ /ðwÞ2 and x (notice that f , g, h, k, p and q are independent of x) with the aid of Maple, then setting each coefficients to zero, we can deduce the following
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585
set of over-determined partial differential equations with respect to the unknown functions f , g, h, k, p and q 2Rðhy pt þ hpty þ ht py Þ ¼ 0;
ð3:3Þ
6p2 hy kR3 ¼ 0;
ð3:4Þ
2
3pð2p hy þ 3pgy h þ 6ppy h þ 3phy g þ 2py ghÞ ¼ 0;
ð3:5Þ
3R2 pkð3phy 2py hÞ ¼ 0;
ð3:6Þ
2
2
2
3pð2gy p þ 3pgy g þ 6ppy g þ 3phhy þ py g þ h py Þ ¼ 0;
ð3:7Þ
36p2 gR2 qy h þ 9p2 fy hR 9phqz R þ 6hqy qt R þ 2hty 18p2 hqy kR þ 33p3 hR2 qy ¼ 0;
ð3:8Þ
2
2
3
3Rhð6p py k 3ppz þ 2py qt þ 12p py gR þ 11Rp py þ 2pt qy Þ ¼ 0;
ð3:9Þ
2hpty þ 2ht py þ 2hy pt ¼ 0;
ð3:10Þ
6hpy pt R ¼ 0;
ð3:11Þ
3 2
12R p kqy ð2Rp kÞ ¼ 0;
ð3:12Þ
12R3 p2 py kð2Rp kÞ ¼ 0;
ð3:13Þ
2
2
2
3
2
Rð2hqy qt R þ 2hty þ 3p fy hR þ 6p gR qy h 3phqz R þ 5p hR qy Þ ¼ 0; ð3:14Þ R2 hð6p2 py gR þ 2py qt 3ppz þ 2pt qy þ 5Rp3 py Þ ¼ 0;
ð3:15Þ
2
2hpy pt R ¼ 0;
ð3:16Þ
Rð5p3 hy R þ 3p2 ky h þ 2hqty 3pz h þ 2hy qt 12ppy hk þ 15p2 py hR þ 6ppy gRh 3phz þ 9p2 hy gR þ 6p2 gy hR þ 2ht qy Þ ¼ 0; 3
2
2
ð3:17Þ 2
2hy qt 3phz þ 11p hy R þ 3p ky h þ 15p gy hR 3pz h þ 2hqty 3p hy k þ 33p2 py hR þ 12ppy gRh þ 18p2 hy gR þ 2ht qy 6ppy hk ¼ 0; ð3:18Þ 2hð2pt qy þ 26p3 Rpy 3ppz 6p2 py k þ 27p2 py gR þ 2py qt Þ ¼ 0; 2
3
2
2
ð3:19Þ
2hð27p gRqy þ 26p Rqy 3pqz þ 2qy qt 6p qy k þ 3p fy Þ ¼ 0;
ð3:20Þ
4hpy pt ¼ 0;
ð3:21Þ
2
2
2
3
2R kð12p py k þ 2py qt þ 6p py gR þ 2pt qy þ 20p Rpy 3ppz Þ ¼ 0;
ð3:22Þ
2R2 kð12p2 qy k þ 2qy qt þ 6p2 gRqy þ 20p3 Rqy 3pqz þ 3p2 fy Þ ¼ 0; ð3:23Þ 2
4kR py pt ¼ 0;
ð3:24Þ
586
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6p2 py kR þ 3p2 gy R2 g 12ppy gRk þ 3p2 hy hR2 3pz gR þ 3ppy k 2 þ 3p2 ky k þ 3ppy g2 R2 þ 6p2 py gR2 3pgz R þ 2p3 gy R2 2p3 ky R þ 2gt Rqy þ 2fty þ 2gRqty 2kqty þ 3pkz 2ky qt þ 2gy Rqt þ 3pz k 2kt qy ¼ 0; ð3:25Þ 2kt py 2kpty þ 2gt Rpy þ 2gRpty 2ky pt þ 2gy Rpt ¼ 0;
ð3:26Þ
3R2 pð2ky Rp2 þ 6ppy kR 3pky k py k 2 Þ ¼ 0; 2 2
2
3
ð3:27Þ
2
2
2
12p k Rqy 6pkRqz þ 6p fy kR þ 16p kR qy þ 2kty þ 12p gR qy k þ 4kRqy qt ¼ 0;
ð3:28Þ
2Rkð6p2 py k þ 2py qt 3ppz þ 6p2 py gR þ 2pt qy þ 8p3 Rpy Þ ¼ 0;
ð3:29Þ
4kRpy pt ¼ 0;
ð3:30Þ
2
2
2
12p hy hR 3pgz þ 2gy qt 6ppy gk þ 6ppy g R þ 2gt qy þ 3p ky g þ 3ppy h2 R þ 24p2 py gR þ 2gqty 3pz g 3p2 gy k þ 12p2 gy Rg þ 8p3 gy R ¼ 0; ð3:31Þ 2Rðkt py þ kpty þ ky pt Þ ¼ 0;
ð3:32Þ
2
3
2
Rð12p ky k þ 6ppy gRk 3pz k þ 8p ky R þ 2ky qt 6ppy k 3pkz þ 24p2 py kR 3Rp2 gy k þ 2kt qy þ 3Rp2 ky ky g þ 2kqty Þ ¼ 0;
ð3:33Þ
16p3 gR2 qy þ 12p2 g2 R2 qy 6pgRqz þ 6p2 fy gR þ 2gty þ 4gRqy qt 12p2 gRqy k þ 6p2 h2 qy R2 ¼ 0;
ð3:34Þ
2Rð8p3 gRpy 6p2 gpy k þ 6p2 g2 Rpy þ 3p2 h2 py R 3pgpz þ 2gqy pt þ 2gpy qt Þ ¼ 0;
ð3:35Þ
4gRpy pt ¼ 0; 2 2
ð3:36Þ 2
2 2
24p g Rpy 12p gpy k 6pgpz þ 4gqy pt þ 18p h py R þ 4gpy qt þ 40p3 gRpy ¼ 0;
ð3:37Þ
4gpy pt ¼ 0; 2 2
ð3:38Þ 2 2
2
2
24p g Rqy þ 18p h qy R þ 6p fy g 6pgqz þ 4gqy qt 12p gqy k þ 40p3 gRqt ¼ 0; 2
3
ð3:39Þ
6p hqy kR ¼ 0;
ð3:40Þ
6p2 hpy kp3 ¼ 0;
ð3:41Þ
2
24p hqy ðp þ gÞ ¼ 0;
ð3:42Þ
24p2 py hðp þ gÞ ¼ 0;
ð3:43Þ
B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
2gpty þ 2gy pt þ 2gt py ¼ 0; 2
2
2
587
ð3:44Þ
12p py ðg þ 2pg þ h Þ ¼ 0;
ð3:45Þ
12p2 qy ðg2 þ 2pg þ h2 Þ ¼ 0:
ð3:46Þ
Using the powerful PDEtools soft package of Maple, solving the set of partial differential equations (3.3)–(3.46), we can obtain the following non-trivial results. (Note: in the rest of this paper, c1 , Ci (i ¼ 1; . . . ; 5) are arbitrary constants and Fi ðtÞ (i ¼ 1; . . . ; 5) denote arbitrary function of t, and so on.) Case 1 g ¼ 2C1 ; h ¼ 0; k ¼ 0; p ¼ C1 ; q ¼ F2 ðy; zÞ þ F4 ðzÞ þ C2 t þ C3 ; Z 1 oF2 ðy; zÞ oF2 ðy; zÞ 3 oF2 ðy; zÞ 2 C2 Þ dy 4C1 R f ¼ þ 3C1 3C12 oy oz oy 1 oF4 ðzÞ y þ F5 ðz; tÞ: þ C1 oz ð3:47Þ Case 2 g ¼ 2C1 ;
h ¼ 0;
k ¼ 0; p ¼ C1 ; Z 2 F3 ðzÞ dz ; q ¼ F3 ðzÞy þ F4 ðzÞ þ F5 t þ 3C1 1 oF3 ðzÞ 2 4 1 oF4 ðzÞ y þ yC1 RF3 ðzÞ þ y þ F6 ðz; tÞ: f ¼ 2C1 oz 3 C1 oz
ð3:48Þ
Case 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c1 z þ C1 eC2 t C3 ; h ¼ 0; k ¼ 0; p ¼ 2c1 z þ C1 C2 eC2 t C3 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 lnð2c1 z þ C1 Þ C2 2c1 z þ C1 F2 t þ q ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2C2 2c1 z þ C1 C2 2 1 4e3C2 t c1 z þ C1 RC33 lnð2c1 z þ C1 Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 yc1 C3 þ 2c1 z þ C1 F1 ðzÞC2 eC2 t ; þ 2 1 oF1 ðzÞ C2 c21 z2 y 2zc21 F1 ðzÞC2 y 4 f ¼ 5=2 oz C2 ð2c1 z þ C1 Þ C3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oF1 ðzÞ 3 C2 C1 c1 y y 2 C3 2c1 z þ C1 c21 þ 4z oz 2 oF1 ðzÞ 2 C1 C2 y þ F3 ðz; tÞ: F1 ðzÞC2 C1 c1 y þ oz g ¼ 2
ð3:49Þ
588
B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
Case 4 Z 1 pð4p2 Rpy þ 3pz Þ g ¼ 2p; h ¼ 0; k ¼ 0; q ¼ dt þ F1 ðy;zÞ; 2 py " Z Z 4pz py2 p2 R þ pz2 py þ ppy pzz ppyz pz 1 oF1 ðy; zÞ 3 f¼ dtpy þ 2py 2 2py p oz py # Z 4p2 Rpy3 þ pz py2 þ ppy pyz ppyy pz oF1 ðy; zÞ 3pz dt 2pz dy þ F2 ðz; tÞ; oy py2 ð3:50Þ where p ¼ pðy; z; tÞ satisfies pyz ¼
4py2 ppz R pzz py ; pz
pyy ¼
8pz Rppy3 pzz py2 ; pz2
pt ¼ 0:
Case 5 g ¼ C1 ; h ¼ C1 ; k ¼ 0; p ¼ C1 ; q ¼ F6 ðy; zÞ þ F8 ðzÞ þ C4 t þ C5 ; Z 1 oF6 ðy; zÞ oF6 ðy; zÞ 3 oF6 ðy; zÞ 2 C4 dy C R þ 3C1 f ¼ oy oz oy 3C12 1 1 oF8 ðzÞ y þ F9 ðz; tÞ: þ C1 oz ð3:51Þ Case 6 g ¼ C1 ; h ¼ C1 ; k ¼ 0; p ¼ C1 ; q ¼ F2 ðy; zÞ þ F4 ðzÞ þ C2 t þ C3 ; Z 1 oF2 ðy; zÞ oF2 ðy; zÞ 3 oF2 ðy; zÞ 2 C2 dy C R þ 3C1 f ¼ 3C12 1 oy oz oy 1 oF4 ðzÞ y þ F9 ðz; tÞ: þ C1 oz ð3:52Þ Case 7 g ¼ C1 ;
h ¼ C1 ;
k ¼ 0; p ¼ C1 ; Z 2 q ¼ F8 ðzÞy þ F9 ðzÞ þ F10 t þ F8 ðzÞ dz ; 3C1 1 o 1 1 o ðF8 ðzÞÞy 2 þ yC1 RF8 ðzÞ þ y ðF9 ðzÞÞ þ F11 ðz; tÞ: f ¼ 2C1 oz 3 C1 oz
ð3:53Þ
B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
589
Case 8 k ¼ 0; p ¼ C1 ; Z 2 F3 ðzÞ dz ; q ¼ F3 ðzÞy þ F4 ðzÞ þ F5 t þ 3C1 1 oF3 ðzÞ 2 1 1 oF4 ðzÞ y þ yC1 RF3 ðzÞ þ y þ F11 ðz; tÞ: f ¼ 2C1 oz 3 C1 oz
g ¼ C1 ;
h ¼ C1 ;
ð3:54Þ
Case 9 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ¼ 2c1 z þ C1 eC2 t C3 ; h ¼ 2c1 z þ C1 eC2 t C3 ; k ¼ 0; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t C3 ; p ¼ 2c1 z þ C1 eC2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 lnð2c1 z þ C1 Þ C2 q ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c1 z þ C1 F4 t þ 2C2 2c1 z þ C1 C2 1 3 3C2 t 2 C3 Re ð2c1 z þ C1 Þ lnð2c1 z þ C1 Þ 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C2 t 3 yc1 C3 þ 2c1 z þ C1 F3 ðzÞC2 ; þe 2 y oFoz3 ðzÞ 3c21 y 2 yc1 F3 ðzÞ þ þ F5 ðz; tÞ: f ¼ 2 1=2 2C2 ð2c1 z þ C1 Þ C3 ð2c1 z þ C1 Þ3=2 C3 ð2c1 z þ C1 Þ ð3:55Þ Case 10 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ¼ 2c1 z þ C1 eC2 t C3 ; h ¼ 2c1 z þ C1 eC2 t C3 ; k ¼ 0; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ 2c1 z þ C1 eC2 t C3 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 4 2c1 z þ C1 F2 t þ lnð2c1 z þ C1 Þ C2 q ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2C2 4 2c1 z þ C1 C2 2 þ C33 Re3C2 t ð2c1 z þ C1 Þ lnð2c1 z þ C1 Þ h i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 6yc1 C3 4 2c1 z þ C1 F1 ðzÞC2 eC2 t ; h i 8 < 2c21 DðF2 Þ t þ 2C1 lnð2c1 z þ C1 Þ z 3c21 y 2 2 þ f ¼ 2 5=2 : 2C2 ð2c1 z þ C1 Þ eC2 t C3 ð2c1 z þ C1Þ C2 h i h i c1 DðF2 Þ t þ 2C1 2 lnð2c1 z þ C1 Þ C1 c1 DðF2 Þ t þ 2C1 2 lnð2c1 z þ C1 Þ þ eC2 t C3 ð2c1 z þ C1 Þ3=2 C2 eC2 t C3 ð2c1 z þ C1 Þ5=2 C2 9 = 1 oF1 ðzÞ 1 þ y þ F5 ðz; tÞ; 1=2 3=2 oz C3 ð2c1 z þ C1 Þ C3 ð2c1 z þ C1 Þ c1 F1 ðzÞ ; ð3:56Þ
590
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where DðF2 Þ½t þ ð1=2C2 Þ lnð2c1 z þ C1 Þ denotes the derivative of function F2 ½t þ ð1=2C2 Þ lnð2c1 z þ C1 Þ with respect to variable ½t þ ð1=2C2 Þ lnð2c1 z þ C1 Þ. Case 11 Z 1 pðp2 Rpy þ 3pz Þ g ¼ p; h ¼ p; k ¼ 0; q ¼ dt þ F2 ðy; zÞ; 2 py " Z Z pz py2 p2 R þ pz2 py þ ppy pzz ppyz pz 1 oF2 ðy; zÞ 3 dtpy þ 2py f¼ 2ppy oz py2 # Z 2 3 p Rpy þ pz py2 þ ppy pyz ppyy pz oF2 ðy; zÞ 3pz dt 2pz dy þ F3 ðz; tÞ; oy py2 ð3:57Þ where p ¼ pðy; z; tÞ satisfies pyy ¼
2pz Rppy3 pzz py2 ; pz2
pyz ¼
py2 ppz R þ pzz py ; pz
pt ¼ 0:
Case 12 Z 1 p2 Rpy þ 3pz dt þ F1 ðy; zÞ; k ¼ 0; g ¼ p; h ¼ p; q ¼ 2p py " Z Z pz py2 p2 R þ pz2 py þ ppy pzz ppyz pz 1 oF1 ðy; zÞ 3 dtpy þ 2py f¼ 2ppy oz py2 # Z 2 3 p Rpy þ pz py2 þ ppy pyz ppyy pz oF1 ðy; zÞ dt 2pz 3pz dy þ F3 ðz; tÞ; oy py2 ð3:58Þ where p ¼ pðy; z; tÞ satisfies pyy ¼
2pz Rppy3 pzz py2 ; pz2
pyz ¼
py2 ppz R þ pzz py ; pz
pt ¼ 0:
Case 13 g ¼ 2C1 ; k ¼ 2C1 R; h ¼ 0; p ¼ C1 ; q ¼ F2 ðy; zÞ þ F4 ðzÞ þ C2 t þ C3 ; Z 1 oF2 ðy; zÞ oF2 ðy; zÞ 3 oF2 ðy; zÞ 2 C2 dy f ¼ 16C1 R þ 3C1 3C12 oy oz oy 1 oF4 ðzÞ y þ F5 ðz; tÞ: þ C1 oz
ð3:59Þ
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591
k ¼ 2C1 R; h ¼ 0; p ¼ C1 ; Z 2 F3 ðzÞ dz ; q ¼ F3 ðzÞy þ F4 ðzÞ þ F5 t þ 3C1 1 oF3 ðzÞ 2 16 1 oF4 ðzÞ y þ yC1 RF3 ðzÞ þ y þ F6 ðz; tÞ: f ¼ 2C1 oz 3 C1 oz
ð3:60Þ
Case 14 g ¼ 2C1 ;
Case 15 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g ¼ 2 2c1 z þ C1 eC2 t C3 ; k ¼ 2 2c1 z þ C1 eC2 t C3 R; h ¼ 0; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ 2c1 z þ C1 eC2 t C3 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 z þ C t þ lnð2c z þ C Þ C2 q¼ 2c F 1 1 2 1 1 1=2 2C2 ð2c1 z þ C1 Þ C2 2 1 3C2 t 16e c1 z þ C1 RC33 lnð2c1 z þ C1 Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 yc1 C3 þ 2c1 z þ C1 F1 ðzÞC2 eC2 t ; þ 2 1 oF1 ðzÞ C2 c21 z2 y 2zc21 F1 ðzÞC2 y 4 f ¼ 5=2 oz C2 ð2c1 z þ C1 Þ C3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oF1 ðzÞ 3 C2 C1 c1 y y 2 C3 2c1 z þ C1 c21 þ 4z oz 2 oF1 ðzÞ 2 C1 C2 y þ F3 ðz; tÞ: F1 ðzÞC2 C1 c1 y þ oz
ð3:61Þ
Case 16 Z 1 pð16p2 Rpy þ 3pz Þ k ¼ 2pR; h ¼ 0; q ¼ dt þ F1 ðy; zÞ; 2 py " Z Z 16pz py2 p2 R þ pz2 py þ ppy pzz ppyz pz 1 dtpy f ¼ 3 2ppy py2 Z 16p2 Rpy3 þ pz py2 þ ppy pyz ppyy pz oF1 ðy; zÞ þ 3pz 2py dt oz py2 # oF1 ðy; zÞ dy þ F2 ðz; tÞ; þ 2pz oy g ¼ 2p;
ð3:62Þ where p ¼ pðy; z; tÞ satisfies pyz ¼
16py2 ppz R pzz py ; pz
pyy ¼
32pz Rppy3 pzz py2 ; pz2
pt ¼ 0:
592
B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
Case 17 R ¼ 0; h ¼ 0; g ¼ 0; k ¼ F2 ðy; zÞ þ F1 ðz; tÞ; p ¼ F3 ðzÞ; q ¼ q; Z Z Z oF1 ðz; tÞ qdt þ qty F2 ðy; zÞdy dt f ¼ F5 ðy; zÞ þ F4 ðz; tÞ þ ot Z Z Z Z oF2 ðy; zÞ 3 2 qt dy dt F3 ðzÞ F2 ðy; zÞ F1 ðz; tÞdt þ F1 ðz; tÞqt dt þ oy 2 Z Z 3 oF1 ðz; tÞ 3 oF3 ðzÞ dt F1 ðz; tÞdt y þ F3 ðzÞ 2 oz 2 oz Z Z 3 oF2 ðy; zÞ 3 oF3 ðzÞ 3 2 2 dy F2 ðy; zÞdy F3 ðzÞ F2 ðy; zÞ t: þ F3 ðzÞ 2 oz 2 oz 4 ð3:63Þ Case 18 g ¼ 0; k ¼ F1 ðz; tÞ; p ¼ F2 ðz; tÞ; q ¼ q; Z Z oF1 ðz; tÞ q dt f ¼ F4 ðy; zÞ þ F3 ðz; tÞ þ F1 ðz; tÞqt dt þ ot Z Z 3 oF2 ðz; tÞ 3 oF1 ðz; tÞ þ F1 ðz; tÞ dt F2 ðz; tÞ dt y: 2 oz 2 oz
R ¼ 0;
h ¼ 0;
ð3:64Þ
Case 19 1 h ¼ 0; g ¼ C1 ; k ¼ F2 ðy; zÞ þ F4 ðzÞ þ F3 ðtÞ; p ¼ C1 ; 2 Z 1 oF ðy; zÞ oF ðy; zÞ 6 6 72C12 F2 ðy; zÞ f ¼ 24C12 F3 ðtÞ 24C12 oy oy R ¼ 0;
72C12 F4 ðzÞ
oF6 ðy; zÞ oF6 ðy; zÞ oF2 ðy; zÞ þ 64C 2 þ 9C14 F3 ðtÞt oy oy oy
oF2 ðy; zÞ oF2 ðy; zÞ oF2 ðy; zÞ þ 27C14 F4 ðzÞt 24C 2tC12 oy oy oy oF6 ðy; zÞ oF2 ðy; zÞ 3 oF4 ðzÞ 18tC13 y þ F7 ðz; tÞ; þ 48C1 dy þ C1 t oz oz 4 oz þ 27C14 F2 ðy; zÞt
q¼
Z
3 2 3 3 C F3 ðtÞ dt tC12 F2 ðy; zÞ tC12 F4 ðzÞ þ tC2 þ F6 ðy; zÞ: 8 1 8 8 ð3:65Þ
B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
593
Case 20 1 C1 3C12 t ; q¼ þ F2 ðy; zÞ; R ¼ 0; h ¼ 0; p ¼ 2 F1 ðyÞ 4F1 ðyÞ Z 2 oF2 ðy; zÞ C1 F1 ðyÞ; y þ F3 ðz; tÞ; g ¼ : f ¼ C1 oz F1 ðyÞ
k ¼ F1 ðyÞ;
ð3:66Þ It is necessary to exclaim that the Cases 1–16 is directly obtained by solving the (3.3)–(3.46), the Cases 17–20 is obtained under the condition: R ¼ h ¼ 0 (here we omitted the trivial solutions and very complicated solutions). From (3.2), (2.4) and Cases 1–20, we can obtain the following five types of solutions for (3 + 1)-dimensional Jumbo–Miwa equation. Type I: From Cases 1–4, we can obtain the following types of solutions. hpffiffiffiffiffiffiffi i pffiffiffiffiffiffiffi u11 ¼ f g R tanh Rðxp þ qÞ ; R < 0; ð3:67Þ hpffiffiffiffiffiffiffi i pffiffiffiffiffiffiffi u12 ¼ f g R coth Rðxp þ qÞ ; hpffiffiffi i pffiffiffi u13 ¼ f þ g R tan Rðxp þ qÞ ; hpffiffiffi i pffiffiffi u14 ¼ f g R cot Rðxp þ qÞ ;
ð3:68Þ
R < 0;
R > 0;
ð3:69Þ
R > 0;
ð3:70Þ
where f , g, p, q are determined by (3.47)–(3.50) respectively. Type II: From Cases 5–12, we can obtain the following types of solutions hpffiffiffiffiffiffiffi i hpffiffiffiffiffiffiffi io pffiffiffiffiffiffiffin u21 ¼ f g R tanh Rðxp þ qÞ þ i sech Rðxp þ qÞ ; R < 0; ð3:71Þ
hpffiffiffiffiffiffiffi i hpffiffiffiffiffiffiffi io pffiffiffiffiffiffiffin u22 ¼ f g R coth Rðxp þ qÞ þ csch Rðxp þ qÞ ;
R < 0; ð3:72Þ
hpffiffiffi i hpffiffiffi io pffiffiffin u23 ¼ f þ g R tan Rðxp þ qÞ þ sec Rðxp þ qÞ ;
R > 0; ð3:73Þ
hpffiffiffi i hpffiffiffi io pffiffiffin u24 ¼ f g R tan Rðxp þ qÞ þ cot Rðxp þ qÞ ;
R > 0; ð3:74Þ
where f , g, h, p, q are determined by (3.51)–(3.58) respectively.
594
B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
Type III: From Cases 12–16, we can obtain the following types of solutions pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi u31 ¼ f g Rftanh½ Rðxp þ qÞ þ coth½ Rðxp þ qÞg; R < 0; ð3:75Þ pffiffiffi pffiffiffi pffiffiffi u32 ¼ f þ g Rftan½ Rðxp þ qÞ þ cot½ Rðxp þ qÞg;
R > 0;
ð3:76Þ
where f , g, p, q are determined by (3.58)–(3.62) respectively. Type IV: From Cases 17 and 18, we can obtain the following types of solutions u4 ¼ f kðxp þ qÞ;
ð3:77Þ
where f , k, p and q are determined by (3.63) and (3.64) respectively. Type V: From Cases 19 and 20, the following types of solutions is obtained g kðxp þ qÞ; ð3:78Þ u5 ¼ f xp þ q where f , g, k, p, q are determined by (3.65) and (3.66) respectively. Remarks. When setting the arbitrary functions to be equal to special functions or special constants, the travelling wave solutions for the (3+1)-dimensional Jumbo–Miwa equation can be recovered. For example, if setting F4 ðzÞ ¼ 0, F5 ðz; tÞ ¼ a0 , F2 ðy; zÞ ¼ by þ cz, R ¼ k 2 , C2 ¼ 2C13 K 2 þ 2C2b1 c in the solution (3.67) determined by (3.47), the solutions obtained in [20] can be reproduced. But to our knowledge, the other soliton-like solutions, singular soliton-like solutions and periodic form solutions are not found before.
4. Conclusions In summary, based on the computerized symbolic computation, by introducing a more general ans€ atz than the ans€ atz in the extended tanh-function method, modified extended tanh-function method, and generalized hyperbolicfunction method, we have proposed a generalized Riccati equation expansion method for searching for exact soliton-like solutions of NEEs and implemented in computer symbolic systems. Making use of our method and with the aid of maple, we study the (3 + 1)-dimensional Jumbo–Miwa equation and obtain new families of the exact solutions. In our obtained exact solutions the restriction on nðx; y; z; tÞ as merely a linear function x, y, z, t and the restriction on the coefficients f , gi , hi , ki (i ¼ 1; . . . ; m) as constants are removed and, with no extra effects, the singular soliton-like solution and periodic form solutions, even rational solutions could be obtained. To make the work feasible, how to choose the forms of f , gi , hi , ki (i ¼ 1; . . . ; m) and n in the ans€atz would be the key step in the computation of our method. The method, proposed in this
B. Li et al. / Appl. Math. Comput. 152 (2004) 581–595
595
paper for single equation, may be extended to find exact soliton-like solutions of coupled equations.
Acknowledgements The author (B. Li) would express his sincerely thanks to Dr. Z.-s. L€ u for his help and guidance in symbolic computation. The work is supported by the National Natural Science Foundation of China under the Grant no. 10072013, the National Key Basic Research Development Project Program under the Grant no. G1998030600.
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