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Bäcklund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation
Applied Mathematics and Computation 187 (2007) 1286–1297 www.elsevier.com/locate/amc
Ba¨cklund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation Yadong Shang School of Mathematics and Information Science, Guangzhou University, No 230 Waihuanxilu Xiaoguwei, Guangzhou 510006, China
Abstract In this paper we consider the Hirota–Satsuma equation for shallow water waves. We first obtain the Ba¨cklund transformation and Lax pairs by using the extended homogeneous balance method. Then we find some explicit exact solutions by means of Ba¨cklund transformation and the extended hyperbolic function method. These solutions include the solitary wave solution of rational function, soliton solutions, double-soliton solutions, N-soliton solutions, the multiple solitary wave solutions, singular solutions, and the periodic wave solutions of triangle function type. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Hirota–Satsuma equation; Ba¨cklund transformation; Lax pairs; Explicit exact solutions; The extended hyperbolic function method
1. Introduction The following Hirota–Satsuma equation for shallow water waves: Z 1 rx rxxt 3rrt þ 3rx rt dx þ rt ¼ 0
ð1Þ
x
has been proposed by Hirota and Satsuma [1–3]. Eq. (1) also arises as a reduction of several higher-dimensional partial differential equations which have been discussed in the literature (see [4] and the references therein). The Ba¨cklund transformation in bilinear form to the Hirota–Satsuma equation was first introduced by Satsuma and Kaup [5]. In 1991, Musette and Conte obtained its Lax pairs by Painleve´ analysis [6]. Zhang and Chen obtained the novel multi-solitons for it and pointed out the solutions have singularity in a recent short note [7]. Zhang and Yang considered the Darboux transformation of Hirota–Satsuma equation and obtained some interesting solutions of the equation by means of the Darboux transformation [8]. More recently, Zhang and Chen presented a new Ba¨cklund transformation in the bilinear form for the shallow water waves equation (1) in [9]. The Ba¨cklund transformation is not only a useful method to obtain exact solutions of some soliton equation from a trivial ‘seed’ but also related to infinite conservation laws and inverse scattering
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.038
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method [10]. In [11–13], Wang proposed the homogeneous balance method—an effective method solving nonlinear partial differential equations. Fan and Zhang extended the homogeneous balance method and proposed an approach to obtain Ba¨cklund transformation for the nonlinear evolution equations [14]. In this paper, we first obtain the Ba¨cklund transformation and a Lax pair by using an extended homogeneous balance method. Then we obtain some new explicit exact solutions by means of the Ba¨cklund transformation and the extension of the hyperbolic function method presented in [19]. These solutions include the solitary wave solution of rational function, soliton solutions, double-soliton solutions, N-soliton solutions, singular solutions, and the periodic wave solutions of triangle function type. 2. Ba¨cklund transformation and Lax pairs for the Hirota–Satsuma shallow water waves equation Let r(x, t) = ux(x, t), the shallow water waves equation (1) can be transformed into the following form: ut uxxt 3ux ut þ ux ¼ a;
ð2Þ
where a is an arbitrary integral constant. When a ¼ 13, Musette and Conte obtained its Lax pair [6] uxxx ¼ ð1 3ux Þux þ ku; 1 ut uxx þ uxt ux : kut ¼ 3
ð3Þ ð4Þ
According to the extended homogeneous balance method, we suppose that the solution of Eq. (2) is of the form uðx; tÞ ¼ f 0 ð/Þ/x þ u1 ðx; tÞ;
ð5Þ
where f, / are two functions to be determined and u1(x, t) is a solution of Eq. (2). From (5), we have ut ¼ f 00 ð/Þ/x /t þ f 0 ð/Þ/xt þ u1t ;
ð6Þ
ux ¼ f ð/Þ/2x þ f 0 ð/Þ/xx þ u1x ; uxx ¼ f 000 ð/Þ/3x þ 3f 00 ð/Þ/x /xx þ f 0 ð/Þ/xxx þ u1xx ; uxxt ¼ f ð4Þ ð/Þ/t /3x þ 3f 000 ð/Þ/2x /xt þ 3f 000 ð/Þ/x /t /xx þ f 00 ð/Þ/t /xxx þ f 0 ð/Þ/xxxt þ u1xxt :
ð7Þ
00
ð8Þ þ 3f 00 ð/Þ/xt /xx þ 3f 00 ð/Þ/x /xxt ð9Þ
Substituting (6)–(9) into the left side of Eq. (2), and collecting all terms with /3x /t , we obtain ut uxxt 3ux ut þ ux a ¼ 3ðf 00 Þ2 f ð4Þ /3x /t þ ð3f 000 /2x /xt 3f 000 /x /t /xx 3f 0 f 00 /2x /xt 2
3f 0 f 00 /x /t /xx Þ þ ð3f 00 /xt /xx 3f 00 /x /xxt f 00 /t /xxx 3ðf 0 Þ /xx /xt þ f 00 /x /t þ f 00 /2x 3f 00 /2x u1t 3f 00 /x /t u1x Þ þ ð/xt þ /xx /xxxt 3/xx u1t 3/xt u1x Þf 0 þ ðu1t u1xxt 3u1x u1t þ u1x aÞ ¼ 0:
ð10Þ
Setting the coefficient of /3x /t in (10) to be zero, we obtain an ordinary differential equation for f 2
3ðf 00 Þ f ð4Þ ¼ 0;
ð11Þ
which has solution f ð/Þ ¼ 2 lnð/Þ
ð12Þ
and therefore, we have f 0 f 00 ¼ f 000 ;
2
ðf 0 Þ ¼ 2f 00 :
ð13Þ
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Y. Shang / Applied Mathematics and Computation 187 (2007) 1286–1297
In view of (11)–(13), then (10) can be rewritten as ut uxxt 3ux ut þ ux a ¼ /x /t þ /2x 3/x /xxt /t /xxx þ 3/xx /xt 3/2x u1t 3/x /t u1x f 00 þ ð/xt þ /xx /xxxt 3/xx u1t 3/xt u1x Þf 0 þ ðu1t u1xxt 3u1x u1t þ u1x aÞ ¼ 0: ð14Þ
It is easy to see from (14) that setting the coefficients of f00 , f 0 to be zero respectively, we have /x /t þ /2x 3/x /xxt /t /xxx þ 3/xx /xt 3/2x u1t 3/x /t u1x ¼ 0;
ð15Þ
/xt þ /xx /xxxt 3/xx u1t 3/xt u1x ¼ 0; u1t u1xxt 3u1x u1t þ u1x ¼ a:
ð16Þ ð17Þ
By (5) and (12), we obtain an auto-Ba¨cklund transformation of Eq. (2) / uðx; tÞ ¼ 2ðln /Þx þ u1 ðx; tÞ ¼ 2 x þ u1 ; /
ð18Þ
where / satisfies the Eqs. (15) and (16), u1(x, t) is a solution of Eq. (2). We can easily obtain Lax pair of the Hirota–Satsuma equation for shallow water waves from Eqs. (15) and (16) as follows: /xxx ¼ ð1 3u1x Þ/x þ k/; k/t ¼ ð1 3u1t Þ/xx þ 3u1xt /x :
ð19Þ ð20Þ
3. Explicit and exact solutions of the Hirota–Satsuma equation for shallow water waves From the Ba¨cklund transformation (18), we can construct a new solution of (2) from a trivial one. For u1(x, t) = ax + c (c is an arbitrary constant), Eqs. (15) and (16) can be rewritten as ð1 3aÞ/x /t þ /2x 3/x /xxt /t /xxx þ 3/xx /xt ¼ 0;
ð21Þ
ð1 3aÞ/xt þ /xx /xxxt ¼ 0:
ð22Þ
So we have found the Ba¨cklund transformation between (2) and Eqs. (21) and (22): uðx; tÞ ¼ 2
/x þ ax þ c: /
ð23Þ
For u1(x, t) = at + c (c is an arbitrary constant), Eqs. (15) and (16) can be rewritten as /x /t þ ð1 3aÞ/2x 3/x /xxt /t /xxx þ 3/xx /xt ¼ 0; /xt þ ð1 3aÞ/xx /xxxt ¼ 0:
ð24Þ ð25Þ
We also find the Ba¨cklund transformation between (2) and Eqs. (24) and (25): uðx; tÞ ¼ 2
/x þ at þ c: /
ð26Þ
In order to get some nontrivial solutions of Eq. (2), we need only to solve Eqs. (21), (22) and (24), (25). Obviously, / = (3a 1)vx + vt + n0, with arbitrary constants v, n0, is a solution of (21) and (22). While / = kx + (3a 1)kt + n0, with arbitrary constants k, n0, is a solution of Eqs. (24) and (25). By (23), we know that Eq. (2) has a solution in the form uðx; tÞ ¼
2ð3a 1Þ þ ax þ c; ð3a 1Þx þ t þ n0
ð27Þ
where n0 is an arbitrary constant. From (26) we obtain a solution of Eq. (2) in the form uðx; tÞ ¼
2 þ at þ c; x þ ð3a 1Þt þ n0
ð28Þ
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where n0 is an arbitrary constant. Therefore, we obtain the solitary wave solutions of the Hirota–Satsuma equation (1) for shallow water waves in the rational function form: rðx; tÞ ¼ a
2ð3a 1Þ2 ½ð3a 1Þx þ t þ n0
ð29Þ
2
and rðx; tÞ ¼
1 ½x þ ð3a 1Þt þ n0
2
ð30Þ
:
Note that the homogeneous property of Eqs. (21), (22) and (24), (25), we can expect that / in (21) and (22) is of the exponential function form: /ðx; tÞ ¼ A þ B expðkx þ vt þ n0 Þ;
ð31Þ
where the constants A, B, k, v, and n0 to be determined. We assume that k 5 0 for obtaining an nontrivial solution. Substituting (31) into (21) and (22), we find that (31) satisfies Eqs. (21) and (22), provided that k v ¼ 3a1þk 2 , A, B and k are arbitrary constants. So we obtain the explicit exact solutions of (2) given by uðx; tÞ ¼
2Bk expðkx þ vt þ n0 Þ þ ax þ c; A þ B expðkx þ vt þ n0 Þ
ð32Þ
k where v ¼ 3a1þk 2 ; A; B; k; n0 ; c are arbitrary constants. And thus we obtain the exact and explicit solutions of the Hirota–Satsuma equation (1) for shallow water waves
rðx; tÞ ¼
2ABk 2 expðkx þ vt þ n0 Þ 2
½A þ B expðkx þ vt þ n0 Þ
þ a;
ð33Þ
k where v ¼ 3a1þk 2 ; A; B; k; n0 ; c are arbitrary constants. Taking A = B in (33), we obtain the exact and explicit soliton solutions of the Hirota–Satsuma equation (1) for shallow water waves
k2 1 sech2 ½kx þ vt þ n0 þ a; ð34Þ 2 2 k where v ¼ 3a1þk 2 ; k; n0 are arbitrary constants. While taking A = B in (33), we obtain the exact and explicit singular travelling wave solutions of the Hirota–Satsuma equation (1) for shallow water waves rðx; tÞ ¼
k2 1 csch2 ½kx þ vt þ n0 þ a; ð35Þ 2 2 k where v ¼ 3a1þk 2 ; k; n0 are arbitrary constants. Analogously, substituting (31) into (24) and (25), we find that (31) satisfies Eqs. (24) and (25), provided that v ¼ ð3a1Þk ; k 6¼ 1 when a 6¼ 13 or k = ±1, v = arbitrary constant when a ¼ 13. So when a 6¼ 13 we obtain the 1k 2 explicit exact solutions of (2): rðx; tÞ ¼
uðx; tÞ ¼
2Bk expðkx þ vt þ n0 Þ þ at þ c; A þ B expðkx þ vt þ n0 Þ
ð36Þ
where v ¼ ð3a1Þk ; k 6¼ 1; A; B; c; n0 are arbitrary constants. When a ¼ 13, we obtain the explicit exact solutions 1k 2 of (2) uðx; tÞ ¼
2B expðx þ vt þ n0 Þ þ at þ c; A þ B expðx þ vt þ n0 Þ
ð37Þ
where A, B, v, c, n0 are arbitrary constants. From (36) we find that the explicit exact solutions of the Hirota– Satsuma equation (1) for shallow water waves given as rðx; tÞ ¼
2BAk 2 expðkx þ vt þ n0 Þ 2
½A þ B expðkx þ vt þ n0 Þ
;
ð38Þ
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Y. Shang / Applied Mathematics and Computation 187 (2007) 1286–1297
where v ¼ ð3a1Þk ; k 6¼ 1; a 6¼ 13 ; A; B; c; n0 are arbitrary constants. From (37) we find that the Hirota–Satsuma 1k 2 equation (1) for shallow water waves has the explicit exact solutions given as rðx; tÞ ¼
2BA expðvt x þ n0 Þ ½A þ B expðvt x þ n0 Þ2
;
ð39Þ
where A, B, v, c, n0 are arbitrary constants. In the case of A = B in (38) and (39), respectively, we can find the Hirota–Satsuma equation (1) for shallow water waves has exact soliton solutions rðx; tÞ ¼
k2 1 sech2 ½kx þ vt þ n0 ; 2 2
ð40Þ
; k 6¼ 1; a 6¼ 13 ; A; B; c; n0 are arbitrary constants, and where v ¼ ð3a1Þk 1k 2 1 rðx; tÞ ¼ sech2 ½vt x þ n0 ; ð41Þ 2 where A, B, v, c, n0 are arbitrary constants. And in the case of A = B in (38) and (39), respectively, we obtain the exact and explicit singular travelling wave solutions of the Hirota–Satsuma equation (1) for shallow water waves k2 1 csch2 ½kx þ vt þ n0 ; 2 2 ; k ¼ 6 1; a 6¼ 13 ; A; B; c; n0 are arbitrary constants, and where v ¼ ð3a1Þk 1k 2 rðx; tÞ ¼
ð42Þ
1 rðx; tÞ ¼ csch2 ½vt x þ n0 ; 2
ð43Þ
where A, B, v, c, n0 are arbitrary constants. The soliton solution (34) of the Hirito–Satsuma equation (1) for shallow water waves has nonzero asymptotic value. Note the fact that sech(ikn) = sec(kn), csch(ikn) = icsc(kn), we obtain the explicit exact periodic wave solutions of the Hirito–Satsuma equation (1) for shallow water waves in triangle function type k2 1 sec2 ½kx þ vt þ n0 þ a; 2 2 k2 2 1 rðx; tÞ ¼ csc ½kx þ vt þ n0 þ a; 2 2 k where v ¼ 3a1þk2 and k = arbitrary constant, rðx; tÞ ¼
k2 1 sec2 ½kx þ vt þ n0 ; 2 2 2 k 1 rðx; tÞ ¼ csc2 ½kx þ vt þ n0 ; 2 2 ð3a1Þk where v ¼ 1k2 and a ¼ 13, k = arbitrary constant. When a ¼ 13 in Eqs. (24) and (25), we have rðx; tÞ ¼
/x /t 3/x /xxt /t /xxx þ 3/xx /xt ¼ 0; /xt /xxxt ¼ 0:
ð44Þ ð45Þ
ð46Þ ð47Þ
ð48Þ ð49Þ
Solving Eqs. (48) and (49), we obtain /1;2 ðx; tÞ ¼ 1 Aext chx
ð50Þ
/3;4 ðx; tÞ ¼ 1 Aext shx;
ð51Þ
and
where A, x are two arbitrary constants. By (26) and (50) we find Eq. (2) have solution uðx; tÞ ¼ 2
Aext shx t þ þc 1 þ Aext chx 3
ð52Þ
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and thus the Hirota–Satsuma equation (1) for shallow water waves has the explicit exact solution rðx; tÞ ¼ 2Aext
Aext chx ð1 þ Aext chxÞ
2
ð53Þ
:
Analogously, from (26) and (51), we obtain the explicit exact solutions of Eq. (2) uðx; tÞ ¼
2Aext chx t þ þc xt 1 Ae shx 3
ð54Þ
and thus the explicit exact solutions of the Hirota–Satsuma equation (1) for shallow water waves rðx; tÞ ¼ 2Aext
Aext shx ð1 Aext shxÞ
2
ð55Þ
:
The solutions (53) and (55) are the exact solutions of the Hirota–Satsuma equation (1) for shallow water waves that are not travelling wave solutions. Now we seek for the double-soliton solutions of the Hirota–Satsuma equation (1) for shallow water waves. We suppose that Eqs. (21) and (22) have solutions in the form / ¼ 1 þ expðk 1 x þ v1 t þ n1 Þ þ expðk 2 x þ v2 t þ n2 Þ;
ð56Þ
where ki, vi (i = 1, 2) are constants to be determined and ni (i = 1, 2) are arbitrary constants. Substituting (56) into (21) and (22), we obtain a system of algebras equations for k1, v1, k2, v2 ð1 3aÞk 1 v1 þ k 21 k 31 v1 ¼ 0; ð1 3aÞk 2 v2 þ k 22 k 32 v2 ¼ 0; ð1 3aÞðk 1 v2 þ k 2 v1 Þ þ 2k 1 k 2
ð57Þ 3k 2 k 21 v1
3k 1 k 22 v2
k 31 v2
k 32 v1
þ
3k 2 k 21 v2
þ
3k 1 k 22 v1
¼ 0:
When k1 = v1 = 0 or k2 = v2 = 0, we obtain the solution (32) of Eq. (2) and thus the soliton solution (34) of Hirota–Satsuma equation (1) for shallow water waves. By (23) we obtain the double-soliton solution of Eq. (2) in the form expðkx þ vt þ n1 Þ þ expðkx þ vt þ n2 Þ þ ax þ c: ð58Þ uðx; tÞ ¼ 2k 1 þ expðkx þ vt þ n1 Þ þ expðkx þ vt þ n2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 k2 k When v ¼ 3a1þk , v2 ¼ 3a1þk 2 ; k is an arbitrary constant. While v1 ¼ 2, k1 ¼ 3a1þk 21 2 double-soliton solution of Eq. (2) in the form
uðx; tÞ ¼ 2
k2
36a123k 22 , 2
we obtain the
k 1 expðk 1 x þ v1 t þ n1 Þ þ k 2 expðk 2 x þ v2 t þ n2 Þ þ ax þ c: 1 þ expðk 1 x þ v1 t þ n1 Þ þ expðk 2 x þ v2 t þ n2 Þ
ð59Þ
So the Hirota–Satsuma equation (1) for shallow water waves possesses the double-soliton solutions rðx; tÞ ¼ 2k 2
expðkx þ vt þ n1 Þ þ expðkx þ vt þ n2 Þ 2
½1 þ expðkx þ vt þ n2 Þ þ expðkx þ vt þ n2 Þ
þ a;
ð60Þ
2
rðx; tÞ ¼ 2
k 21 expðg1 Þ þ k 22 expðg2 Þ þ ðk 1 k 2 Þ expðg1 Þ expðg2 Þ ½1 þ expðg1 Þ þ expðg1 Þ2
þ a;
ð61Þ
ki k ; i ¼ 1; 2, where v ¼ 3a1þk 2 ; k is an arbitrary constant and g1 = k1x + v1t + n1, g2 = k2x + v2t + n2, vi ¼ 3a1þk 2i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi k 36a123k 2 . k1 ¼ 2 2 Analogously, when a 6¼ 13, we obtain Eq. (2) has double-soliton solutions
expðkx þ vt þ n0 Þ þ expðkx þ vt þ n0 Þ þ at þ c; 1 þ expðkx þ vt þ n0 Þ þ expðkx þ vt þ n0 Þ k 1 expðg1 Þ þ k 2 expðg2 Þ uðx; tÞ ¼ 2 þ at þ c; 1 þ expðg1 Þ þ expðg2 Þ
uðx; tÞ ¼ 2k
ð62Þ ð63Þ
1 , k 6¼ and g1 = k1x + v1t + n1, g2 = k2x + v2t + n2, vi ¼ ð3a1Þk ; i ¼ 1; 2; k 1 ¼ where v ¼ ð3a1Þk 1k 2 1k 2 i
k2
pffiffiffiffiffiffiffiffiffiffi2ffi
123k 2 . 2
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So we also obtain the double-soliton solutions of Hirota–Satsuma equation (1) for shallow water waves in the form rðx; tÞ ¼ 2k 2
expðkx þ vt þ n1 Þ þ expðkx þ vt þ n2 Þ ½1 þ expðkx þ vt þ n2 Þ þ expðkx þ vt þ n2 Þ2
ð64Þ
;
; k 6¼ 1. The Hirota–Satsuma equation (1) for shallow water waves also has double-soliton where v ¼ ð3a1Þk 1k 2 solutions rðx; tÞ ¼ 2
k 21 expðg1 Þ þ k 22 expðg2 Þ þ ðk 1 k 2 Þ2 expðg1 Þ expðg2 Þ ½1 þ expðg1 Þ þ expðg1 Þ
2
1 where g1 = k1x + v1t + n1, g2 = k2x + v2t + n2, vi ¼ ð3a1Þk ; i ¼ 1; 2; k 1 ¼ 1k 2i 1 While a ¼ 3, Eq. (2) possesses double-soliton solution
uðx; tÞ ¼ 2
ð65Þ
; k2
pffiffiffiffiffiffiffiffiffiffi2ffi
123k 2 . 2
expðx þ v1 t þ n1 Þ þ expðx þ v2 t þ n2 Þ 1 þ t þ c; 1 þ expðx þ v1 t þ n1 Þ þ expðx þ v2 t þ n2 Þ 3
ð66Þ
where v1, v2 are arbitrary constants. Eq. (2) also has double-soliton solution uðx; tÞ ¼ 2
expðx þ v1 t þ n1 Þ þ expðx þ v2 t þ n2 Þ 1 þ t þ c; 1 þ expðx þ v1 t þ n1 Þ þ expðx þ v2 t þ n2 Þ 3
ð67Þ
where v1, v2 are arbitrary constants. When a ¼ 13, we also obtain the double-soliton solution of Eq. (2) in the form uðx; tÞ ¼ 2
expðx þ vt þ n1 Þ þ expðx þ vt þ n2 Þ 1 þ t þ c; 1 þ expðx þ vt þ n1 Þ þ expðx þ vt þ n2 Þ 3
ð68Þ
where v is an arbitrary constant. Therefore, we obtain the double-soliton solutions of the Hirota–Satsuma equation for shallow water waves given as rðx; tÞ ¼ 2
expðx þ v1 t þ n1 Þ þ expðx þ v2 t þ n2 Þ ½1 þ expðx þ v1 t þ n1 Þ þ expðx þ v2 t þ n2 Þ
2
;
ð69Þ
where v1, v2 are arbitrary constants. We also obtain the double-soliton solution of the Hirota–Satsuma equation (1) for shallow water waves in form of rðx; tÞ ¼ 2
expðx þ vt þ n1 Þ þ expðx þ vt þ n2 Þ ½1 þ expðx þ vt þ n1 Þ þ expðx þ vt þ n2 Þ2
;
where v is an arbitrary constant. We assume that (21), (22) and (24), (25) have solutions in the form n X expðk i x þ vi t þ ni Þ; /¼1þ
ð70Þ
ð71Þ
i¼1
where ki, vi (i = 1, 2, . . . , n) are constants to be determined and ni (i = 1, 2, . . . , n) are arbitrary constants. Substituting (71) into (21), (22) and (24), (25), respectively, we can get the N-soliton solutions Eq. (2) Pn k i expðk i x þ vi t þ ni Þ þ ax þ c ð72Þ uðx; tÞ ¼ 2 i¼1 1 þ expðk i x þ vi t þ ni Þ and
Pn uðx; tÞ ¼ 2
expðk i x þ vi t þ ni Þ þ at þ c; 1 þ expðk i x þ vi t þ ni Þ i¼1 k i
ð73Þ
Y. Shang / Applied Mathematics and Computation 187 (2007) 1286–1297
1293
where ki, vi (i = 1, 2, . . . , n) are constants satisfy certain condition equations. By using r(x, t) = ux(x, t) we can obtain from (72) and (73) the N-solitons of the Hirota–Satsuma equation (1) for shallow water waves. We omitted their concrete form here. In 1992, Conte and Musette presented an indirect method to find more new solitary wave solutions of nonlinear partial differential equations that can be expressed as a polynomial in two elementary functions which satisfy a projective Riccati equation [15]. In [16], Zhang et al. proposed the hyperbolic function method based upon the fact that many solitary wave solutions have the format of hyperbolic functions. They expressed the solitary wave solutions of the nonlinear wave equations as the combination of hyperbolic functions and obtained many new exact solitary wave solutions. In [17], Yan presented the generally projective Riccati equation method. More recently, Chen and Ding improved the projective Riccati equation method in [18] and obtained some new solitary wave solutions to the nonlinear evolution equation. In order to obtain some other exact solutions of the Hirota–Satsuma equation for shallow water waves, we employ an extension of the hyperbolic function method proposed in [19] by the author to find many more exact solutions of the Hirota–Satsuma equation for shallow water waves. We suppose that the Hirota–Satsuma equation (2) for shallow water waves has travelling wave solution uðx; tÞ ¼ uðnÞ ¼ uðkx xt þ n0 Þ;
ð74Þ
where k, x are constants to be determined and n0 is an arbitrary constant. Substituting (74) into (2) we get 2
xu0 ðnÞ þ k 2 xu000 ðnÞ þ 3kxðu0 ðnÞÞ þ ku0 ðnÞ ¼ a:
ð75Þ
Let u 0 (n) = u(n), it infers that from (75) k 2 xu00 ðnÞ þ ðk xÞuðnÞ þ 3kxu2 ðnÞ ¼ a:
ð76Þ
We assume that the solutions of the ODE (76) is of the form uðnÞ ¼ a0 þ a1 f ðnÞ þ a2 f 2 ðnÞ þ b1 gðnÞ þ b2 f ðnÞgðnÞ;
ð77Þ
where a0, a1, a2, b1 and b2 are constants to be determined. The functions f and g satisfy the coupled Riccati equations f 0 ðnÞ ¼ f ðnÞgðnÞ;
g0 ðnÞ ¼ 1 rf ðnÞ g2 ðnÞ;
ð78Þ
f 0 ðnÞ ¼ f ðnÞgðnÞ;
g0 ðnÞ ¼ 1 þ rf ðnÞ g2 ðnÞ;
ð79Þ
respectively. Furthermore, we can obtain their first integrals as given g2 ðnÞ ¼ 1 2rf ðnÞ þ ðb2 a2 þ r2 Þf 2 ðnÞ;
ð80Þ
g2 ðnÞ ¼ 1 þ 2rf ðnÞ þ ðb2 þ a2 r2 Þf 2 ðnÞ;
ð81Þ
respectively. The ODEs (78) and (79) have the following special solutions: f ðnÞ ¼
1 ; a cosh n þ b sinh n þ r
f ðnÞ ¼
1 ; a cos n þ b sin n þ r
gðnÞ ¼ gðnÞ ¼
a sinh n þ b cosh n ; a cosh n þ b sinh n þ r
b cos n a sin n ; a cos n þ b sin n þ r
ð82Þ ð83Þ
respectively. Substituting (77) into (76) and using (78) and (80), we obtain a set of nonlinear algebraic equations for a0, a1, a2, b1, b2, k, and x by setting the coefficients of power f4, f3g, f3, f2g, f2, fg, f and g to be zero
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6a2 k 2 xðb2 a2 þ r2 Þ þ 3kx½a22 þ b22 ðb2 a2 þ r2 Þ ¼ 0; 6b2 k 2 xðb2 a2 þ r2 Þ þ 6kxa2 b2 ¼ 0;
k 2 x½2a1 ðb2 a2 þ r2 Þ 10a2 r þ 6kx a1 a2 þ b1 b2 ðb2 a2 þ r2 Þ b22 r ¼ 0; 6k 2 xb2 r þ 2k 2 xb1 ðb2 a2 þ r2 Þ þ 6kxða1 b2 þ a2 b1 Þ ¼ 0;
k 2 xð4a2 3a1 rÞ þ 3kx½a21 þ 2a0 a2 þ b22 4b1 b2 r þ b21 ðb2 a2 þ r2 Þ þ ðk xÞa2 ¼ 0;
ð84Þ
k 2 xðb2 b1 rÞ þ 6kxða1 b1 þ a0 b2 Þ þ ðk xÞb2 ¼ 0; k 2 xa1 þ 6kxða0 a1 þ b1 b2 b21 rÞ þ ðk xÞa1 ¼ 0; 6kxa0 b1 þ ðk xÞb1 ¼ 0; 3kxða20 þ b21 Þ þ ðk xÞa0 ¼ a: With the aid of the computer program Mathematica or Maple 4 [20], Solving (81), we obtain the following five set of solutions: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k a0 ¼ ; a1 ¼ k a2 b2 ; a2 ¼ 0; b1 ¼ b2 ¼ 0; r ¼ a2 b2 ; k 6¼ 0; x 6¼ 0; 6k 6x 6 ð85Þ where a P b are two arbitrary parameters, 1 1 ; a1 ¼ 0; a2 ¼ 0; b1 6¼ 0; b2 ¼ 0; r ¼ 0; b ¼ a; k 6¼ 0; x 6¼ 0; 6k 6x 1 1 k ; a1 ¼ 2kr; a2 ¼ 2kr2 ; b1 ¼ b2 ¼ 0; b ¼ a; r 6¼ 0; k 6¼ 0; x 6¼ 0; a0 ¼ 6k 6x 6 1 1 2k ; a1 ¼ 0; a2 ¼ 2kða2 b2 Þ; r ¼ 0; b1 ¼ 0; b2 ¼ 0; k 6¼ 0; x 6¼ 0; a0 ¼ 6k 6x 3
a0 ¼
ð86Þ ð87Þ ð88Þ
and 1 1 k ; 6k 6x 6 k 6¼ 0; x 6¼ 0;
a0 ¼
a1 ¼ kr;
a2 ¼ kðb2 a2 þ r2 Þ;
b1 ¼ 0;
b2 ¼ k
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 a2 þ r 2 ; ð89Þ
where a, b and r are three arbitrary parameters satisfy b2 a2 + r2 > 0. According to (77), (82) and (85), we obtain that the Hirota–Satsuma equation for shallow water wave (1) have explicit exact solitary solution as given pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k k2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; k 2 a2 b2 ð90Þ rðx; tÞ ¼ 6 6x 6 a cosh n þ b sinh n a2 b2 where a P b are two arbitrary parameters and n = kx x t + n0, k 5 0, x 5 0. From (77), (82) and (87) we obtain that the Hirota–Satsuma equation for shallow water wave (1) have explicit exact solitary solution as given rðx; tÞ ¼
1 k k2 2k 2 r 2k 2 r2 þ ; 6 6x 6 a cosh n þ b sinh n þ r ða cosh n þ b sinh n þ rÞ2
ð91Þ
where a = ±b is arbitrary parameter and n = kx xt + n0, r 5 0, k 5 0, x 5 0. By (77), (82) and (88), we obtain that the Hirota–Satsuma equation for shallow water wave (1) have explicit exact solitary solution as given rðx; tÞ ¼
1 k 2k 2 2k 2 ða2 b2 Þ þ ; 2 6 6x 3 ða cosh n þ b sinh nÞ
where a 5 ±b are two arbitrary parameters and n = kx x t + n0, k 5 0, x 5 0,
ð92Þ
Y. Shang / Applied Mathematics and Computation 187 (2007) 1286–1297
1295
Analogously, from (77), (82) and (89), we also obtain that the Hirota–Satsuma equation for shallow water wave (1) have explicit exact solitary solution as given 1 k k2 k2r k 2 ðb2 a2 þ r2 Þ þ rðx; tÞ ¼ 6 6x 6 a cosh n þ b sinh n þ r ða cosh n þ b sinh n þ rÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a sinh n þ b cosh n k 2 b2 a2 þ r 2 ; 2 ða cosh n þ b sinh n þ rÞ
ð93Þ
where a, b and r are three arbitrary parameters satisfy b2 a2 + r2 P 0 and n = kx xt + n0, k 5 0, x 5 0. We now seek for the exact periodic wave solutions of the Hirota–Satsuma equation for shallow water waves. Substituting (77) into (76) and using (79) and (81), we obtain a set of nonlinear algebraic equations for a0, a1, a2, b1, b2, k, and x by setting the coefficients of power f4, f3g, f3, f2g, f2, fg, f and g to be zero 6a2 k 2 xðb2 þ a2 r2 Þ þ 3kx½a22 þ b22 ðb2 þ a2 r2 Þ ¼ 0; 6b2 k 2 xðb2 þ a2 r2 Þ þ 6kxa2 b2 ¼ 0; k 2 x½2a1 ðb2 þ a2 r2 Þ þ 10a2 r þ 6kx½a1 a2 þ b1 b2 ðb2 þ a2 r2 Þ þ b22 r ¼ 0; 6k 2 xb2 r þ 2k 2 xb1 ðb2 þ a2 r2 Þ þ 6kxða1 b2 þ a2 b1 Þ ¼ 0; k 2 xð4a2 þ 3a1 rÞ þ 3kx½a21 þ 2a0 a2 b22 þ 4b1 b2 r þ b21 ðb2 þ a2 r2 Þ þ ðk xÞa2 ¼ 0;
ð94Þ
k 2 xðb2 þ b1 rÞ þ 6kxða1 b1 þ a0 b2 Þ þ ðk xÞb2 ¼ 0; k 2 xa1 þ 6kxða0 a1 b1 b2 þ b21 rÞ þ ðk xÞa1 ¼ 0; 6kxa0 b1 þ ðk xÞb1 ¼ 0; 3kxða20 b21 Þ þ ðk xÞa0 ¼ a: With the aid of the computer program Mathematica or Maple 4 [20], Solving (91), we obtain the following five set of solutions: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k þ ; a1 ¼ k a2 þ b2 ; a2 ¼ 0; b1 ¼ b2 ¼ 0; r ¼ a2 þ b2 ; k 6¼ 0; x 6¼ 0; a0 ¼ 6k 6x 6 ð95Þ where a, b are two arbitrary parameters satisfy a2 + b2 5 0, 1 1 ; a1 ¼ 0; r ¼ 0; b ¼ a ¼ 0; b2 ¼ 0; k 6¼ 0; 6k 6x 1 1 k þ ; a1 ¼ 2kr; a2 ¼ 2kr2 ; b1 ¼ 0; b2 ¼ 0; a0 ¼ 6k 6x 6
a0 ¼
x 6¼ 0;
ð96Þ
b1 arbitrary
a ¼ b ¼ 0;
k 6¼ 0;
x 6¼ 0;
ð97Þ
where r is an arbitrary parameter, a0 ¼
1 1 2k þ ; 6k 6x 3
a1 ¼ 0;
a2 ¼ 2kða2 þ b2 Þ;
b1 ¼ b2 ¼ 0;
r ¼ 0;
k 6¼ 0;
x 6¼ 0;
b2 ¼ k
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 r 2 ;
ð98Þ
where a, b are two arbitrary parameters satisfy a2 + b2 > 0, and 1 1 k þ ; a1 ¼ kr; 6k 6x 6 k 6¼ 0; x 6¼ 0;
a0 ¼
a2 ¼ kða2 þ b2 r2 Þ;
b1 ¼ 0;
ð99Þ
where a, b are two arbitrary parameters satisfy a2 + b2 r2 > 0 and n = kx + xt + n0, k 5 0, x 5 0. By (77), (83), (95)–(98), we obtain the Hirota–Satsuma equation for shallow water waves (1) have multiple nontrivial exact periodic travelling wave solutions pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k k2 k 2 a2 þ b2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; þ ð100Þ rðx; tÞ ¼ 6 6x 6 a cos n þ b sin n a2 þ b2
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Y. Shang / Applied Mathematics and Computation 187 (2007) 1286–1297
where a, b are two arbitrary parameters satisfy a2 + b2 > 0 and n = kx + xt + n0, k 5 0, x 5 0, rðx; tÞ ¼
1 k 2k 2 2k 2 ða2 þ b2 Þ þ ; 2 6 6x 3 ða cos n þ b sin nÞ
ð101Þ
where a, b are two arbitrary parameters satisfy a2 + b2 > 0 and n = kx + xt + n0, k 5 0, x 5 0, 1 k k2 k2r k 2 ða2 þ b2 r2 Þ þ rðx; tÞ ¼ 6 6x 6 a cos n þ b sin n þ r ða cos n þ b sin n þ rÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b cos n a sin n k 2 a2 þ b2 r 2 ; 2 ða cos n þ b sin n þ rÞ
ð102Þ
where a, b and r are three arbitrary parameters satisfy a2 + b2 r2 > 0 and n = kx + xt + n0, k 5 0, x 5 0. 4. Summary and conclusions In summary, we first obtain the Ba¨cklund transformation and Lax pairs by using the extended homogeneous balance method. Then we obtain some multiple exact explicit solutions of the Hirota–Satsuma equation for shallow water waves by the extended homogeneous balance method and an extension of the hyperbolic functions method. These solutions include the solitary wave solution of rational function, soliton solutions, double-soliton solutions, N-soliton solutions, the multiple solitary wave solutions of a compound of the bell-type and the kink-type solitary wave solutions, singular solutions, and the periodic wave solutions of triangle function type. By the Ba¨cklund transformation (18) and Eqs. (15) and (16), we can obtain even more new solutions of the Hirota–Satsuma equation from the solutions (27), (28), (32), (36), (37), (52), (54), (58), (59), (62), (63), (66)–(68), (72) and (73) obtained in this paper. Acknowledgement The authors are grateful to Professor Wang Mingliang for his suggestions and helps. References [1] R. Hirota, J. Satsuma, N-soliton solutions of model equations for shallow water waves, J. Phys. Soc. Jpn. 40 (1976) 611–614. [2] R. Hirota, J. Satsuma, Nonlinear equations generated from the Backlund transformation for the Boussinesq equation, Prog. Theor. Phys. 57 (1977) 797–800. [3] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. [4] P.A. Clarkson, E.L. Mansfield, On a shallow water wave equation, Nonlinearity 7 (1994) 975–1000. [5] J. Satsuma, D.J. Kaup, Ba¨cklund transformation in bilinear form to the Hirota–Satsuma equation, J. Phys. Soc. Jpn. 43 (1978) 692. [6] M. Musette, R. Conte, Algorithmic method for deriving Lax pairs from the invariant Painleve analysis of nonlinear partial differential equations, J. Math. Phys. 32 (6) (1991) 1450–1457. [7] Yi Zhang, S.F. Deng, Dengyuan Chen, The novel multi-soliton solutions of equation for shallow water waves, J. Phys. Soc. Jpn. 72 (3) (2003) 763–764. [8] Jinshun Zhang, Yunping Yang, Darboux transformation of Hirota–Satsuma equation and its solutions, J. Zhengzhou Univ. (Sci.) 35 (3) (2003) 1–4. [9] Yi Zhang, Dengyuan Chen, Ba¨cklund transformation and soliton solutions for the shallow water wave equation, Chaos, Solitons and Fractals 20 (2004) 343–351. [10] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991. [11] Mingliang Wang et al., Applications of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996) 67–74. [12] Mingliang Wang, Exact solutions for a compound KdV–Burgers equation, Phys. Lett. A 213 (1996) 279–287. [13] Mingliang Wang, Yubin Zhou, Huiqun Zhang, A nonlinear transformation of the shallow water wave equations and its application, Adv. Math. (China) 28 (1) (1999) 72–75. [14] Engui Fan, Hongqin Zhang, A new approach to Backlund transformations of nonlinear evolution equations, Appl. Math. Mech. 19 (7) (1998) 645–650. [15] R. Conte, M. Musette, Link between solitary waves and projective Riccati equations, J. Phys. A: Math. Gen. 25 (1992) 5609–5612.
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[16] Guixu Zhang, Zhibin Li, Yishi Duan, Exact solitary wave solutions of nonlinear exact solitary wave solutions of nonlinear wave equations, Sci. China 44 (3) (2001) 396–401. [17] Zhenya Yan, Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres, Chaos, Solitons and Fractals 16 (2003) 759–766. [18] Yanze Chen, Xinwei Ding, Exact travelling wave solutions of nonlinear evolution equations in (1 + 1) and (2 + 1) dimensions, Nonlinear Anal. 61 (2005) 1005–1013. [19] Yadong Shang, The extended hyperbolic functions method and exact solutions to the long–short wave resonance equations, Chaos, Solitons and Fractals, in press, doi:10.1016/j.chaos.2006.07.007. [20] Zhibin Li, Yinping Liu, RAEEM: A Maple package for finding a series of exact travelling wave solutions for nonlinear evolution equations, Comput. Phys. Commun. 163 (2004) 191–201.
Ba¨cklund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation Yadong Shang School of Mathematics and Information Science, Guangzhou University, No 230 Waihuanxilu Xiaoguwei, Guangzhou 510006, China
Abstract In this paper we consider the Hirota–Satsuma equation for shallow water waves. We first obtain the Ba¨cklund transformation and Lax pairs by using the extended homogeneous balance method. Then we find some explicit exact solutions by means of Ba¨cklund transformation and the extended hyperbolic function method. These solutions include the solitary wave solution of rational function, soliton solutions, double-soliton solutions, N-soliton solutions, the multiple solitary wave solutions, singular solutions, and the periodic wave solutions of triangle function type. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Hirota–Satsuma equation; Ba¨cklund transformation; Lax pairs; Explicit exact solutions; The extended hyperbolic function method
1. Introduction The following Hirota–Satsuma equation for shallow water waves: Z 1 rx rxxt 3rrt þ 3rx rt dx þ rt ¼ 0
ð1Þ
x
has been proposed by Hirota and Satsuma [1–3]. Eq. (1) also arises as a reduction of several higher-dimensional partial differential equations which have been discussed in the literature (see [4] and the references therein). The Ba¨cklund transformation in bilinear form to the Hirota–Satsuma equation was first introduced by Satsuma and Kaup [5]. In 1991, Musette and Conte obtained its Lax pairs by Painleve´ analysis [6]. Zhang and Chen obtained the novel multi-solitons for it and pointed out the solutions have singularity in a recent short note [7]. Zhang and Yang considered the Darboux transformation of Hirota–Satsuma equation and obtained some interesting solutions of the equation by means of the Darboux transformation [8]. More recently, Zhang and Chen presented a new Ba¨cklund transformation in the bilinear form for the shallow water waves equation (1) in [9]. The Ba¨cklund transformation is not only a useful method to obtain exact solutions of some soliton equation from a trivial ‘seed’ but also related to infinite conservation laws and inverse scattering
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.038
Y. Shang / Applied Mathematics and Computation 187 (2007) 1286–1297
1287
method [10]. In [11–13], Wang proposed the homogeneous balance method—an effective method solving nonlinear partial differential equations. Fan and Zhang extended the homogeneous balance method and proposed an approach to obtain Ba¨cklund transformation for the nonlinear evolution equations [14]. In this paper, we first obtain the Ba¨cklund transformation and a Lax pair by using an extended homogeneous balance method. Then we obtain some new explicit exact solutions by means of the Ba¨cklund transformation and the extension of the hyperbolic function method presented in [19]. These solutions include the solitary wave solution of rational function, soliton solutions, double-soliton solutions, N-soliton solutions, singular solutions, and the periodic wave solutions of triangle function type. 2. Ba¨cklund transformation and Lax pairs for the Hirota–Satsuma shallow water waves equation Let r(x, t) = ux(x, t), the shallow water waves equation (1) can be transformed into the following form: ut uxxt 3ux ut þ ux ¼ a;
ð2Þ
where a is an arbitrary integral constant. When a ¼ 13, Musette and Conte obtained its Lax pair [6] uxxx ¼ ð1 3ux Þux þ ku; 1 ut uxx þ uxt ux : kut ¼ 3
ð3Þ ð4Þ
According to the extended homogeneous balance method, we suppose that the solution of Eq. (2) is of the form uðx; tÞ ¼ f 0 ð/Þ/x þ u1 ðx; tÞ;
ð5Þ
where f, / are two functions to be determined and u1(x, t) is a solution of Eq. (2). From (5), we have ut ¼ f 00 ð/Þ/x /t þ f 0 ð/Þ/xt þ u1t ;
ð6Þ
ux ¼ f ð/Þ/2x þ f 0 ð/Þ/xx þ u1x ; uxx ¼ f 000 ð/Þ/3x þ 3f 00 ð/Þ/x /xx þ f 0 ð/Þ/xxx þ u1xx ; uxxt ¼ f ð4Þ ð/Þ/t /3x þ 3f 000 ð/Þ/2x /xt þ 3f 000 ð/Þ/x /t /xx þ f 00 ð/Þ/t /xxx þ f 0 ð/Þ/xxxt þ u1xxt :
ð7Þ
00
ð8Þ þ 3f 00 ð/Þ/xt /xx þ 3f 00 ð/Þ/x /xxt ð9Þ
Substituting (6)–(9) into the left side of Eq. (2), and collecting all terms with /3x /t , we obtain ut uxxt 3ux ut þ ux a ¼ 3ðf 00 Þ2 f ð4Þ /3x /t þ ð3f 000 /2x /xt 3f 000 /x /t /xx 3f 0 f 00 /2x /xt 2
3f 0 f 00 /x /t /xx Þ þ ð3f 00 /xt /xx 3f 00 /x /xxt f 00 /t /xxx 3ðf 0 Þ /xx /xt þ f 00 /x /t þ f 00 /2x 3f 00 /2x u1t 3f 00 /x /t u1x Þ þ ð/xt þ /xx /xxxt 3/xx u1t 3/xt u1x Þf 0 þ ðu1t u1xxt 3u1x u1t þ u1x aÞ ¼ 0:
ð10Þ
Setting the coefficient of /3x /t in (10) to be zero, we obtain an ordinary differential equation for f 2
3ðf 00 Þ f ð4Þ ¼ 0;
ð11Þ
which has solution f ð/Þ ¼ 2 lnð/Þ
ð12Þ
and therefore, we have f 0 f 00 ¼ f 000 ;
2
ðf 0 Þ ¼ 2f 00 :
ð13Þ
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In view of (11)–(13), then (10) can be rewritten as ut uxxt 3ux ut þ ux a ¼ /x /t þ /2x 3/x /xxt /t /xxx þ 3/xx /xt 3/2x u1t 3/x /t u1x f 00 þ ð/xt þ /xx /xxxt 3/xx u1t 3/xt u1x Þf 0 þ ðu1t u1xxt 3u1x u1t þ u1x aÞ ¼ 0: ð14Þ
It is easy to see from (14) that setting the coefficients of f00 , f 0 to be zero respectively, we have /x /t þ /2x 3/x /xxt /t /xxx þ 3/xx /xt 3/2x u1t 3/x /t u1x ¼ 0;
ð15Þ
/xt þ /xx /xxxt 3/xx u1t 3/xt u1x ¼ 0; u1t u1xxt 3u1x u1t þ u1x ¼ a:
ð16Þ ð17Þ
By (5) and (12), we obtain an auto-Ba¨cklund transformation of Eq. (2) / uðx; tÞ ¼ 2ðln /Þx þ u1 ðx; tÞ ¼ 2 x þ u1 ; /
ð18Þ
where / satisfies the Eqs. (15) and (16), u1(x, t) is a solution of Eq. (2). We can easily obtain Lax pair of the Hirota–Satsuma equation for shallow water waves from Eqs. (15) and (16) as follows: /xxx ¼ ð1 3u1x Þ/x þ k/; k/t ¼ ð1 3u1t Þ/xx þ 3u1xt /x :
ð19Þ ð20Þ
3. Explicit and exact solutions of the Hirota–Satsuma equation for shallow water waves From the Ba¨cklund transformation (18), we can construct a new solution of (2) from a trivial one. For u1(x, t) = ax + c (c is an arbitrary constant), Eqs. (15) and (16) can be rewritten as ð1 3aÞ/x /t þ /2x 3/x /xxt /t /xxx þ 3/xx /xt ¼ 0;
ð21Þ
ð1 3aÞ/xt þ /xx /xxxt ¼ 0:
ð22Þ
So we have found the Ba¨cklund transformation between (2) and Eqs. (21) and (22): uðx; tÞ ¼ 2
/x þ ax þ c: /
ð23Þ
For u1(x, t) = at + c (c is an arbitrary constant), Eqs. (15) and (16) can be rewritten as /x /t þ ð1 3aÞ/2x 3/x /xxt /t /xxx þ 3/xx /xt ¼ 0; /xt þ ð1 3aÞ/xx /xxxt ¼ 0:
ð24Þ ð25Þ
We also find the Ba¨cklund transformation between (2) and Eqs. (24) and (25): uðx; tÞ ¼ 2
/x þ at þ c: /
ð26Þ
In order to get some nontrivial solutions of Eq. (2), we need only to solve Eqs. (21), (22) and (24), (25). Obviously, / = (3a 1)vx + vt + n0, with arbitrary constants v, n0, is a solution of (21) and (22). While / = kx + (3a 1)kt + n0, with arbitrary constants k, n0, is a solution of Eqs. (24) and (25). By (23), we know that Eq. (2) has a solution in the form uðx; tÞ ¼
2ð3a 1Þ þ ax þ c; ð3a 1Þx þ t þ n0
ð27Þ
where n0 is an arbitrary constant. From (26) we obtain a solution of Eq. (2) in the form uðx; tÞ ¼
2 þ at þ c; x þ ð3a 1Þt þ n0
ð28Þ
Y. Shang / Applied Mathematics and Computation 187 (2007) 1286–1297
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where n0 is an arbitrary constant. Therefore, we obtain the solitary wave solutions of the Hirota–Satsuma equation (1) for shallow water waves in the rational function form: rðx; tÞ ¼ a
2ð3a 1Þ2 ½ð3a 1Þx þ t þ n0
ð29Þ
2
and rðx; tÞ ¼
1 ½x þ ð3a 1Þt þ n0
2
ð30Þ
:
Note that the homogeneous property of Eqs. (21), (22) and (24), (25), we can expect that / in (21) and (22) is of the exponential function form: /ðx; tÞ ¼ A þ B expðkx þ vt þ n0 Þ;
ð31Þ
where the constants A, B, k, v, and n0 to be determined. We assume that k 5 0 for obtaining an nontrivial solution. Substituting (31) into (21) and (22), we find that (31) satisfies Eqs. (21) and (22), provided that k v ¼ 3a1þk 2 , A, B and k are arbitrary constants. So we obtain the explicit exact solutions of (2) given by uðx; tÞ ¼
2Bk expðkx þ vt þ n0 Þ þ ax þ c; A þ B expðkx þ vt þ n0 Þ
ð32Þ
k where v ¼ 3a1þk 2 ; A; B; k; n0 ; c are arbitrary constants. And thus we obtain the exact and explicit solutions of the Hirota–Satsuma equation (1) for shallow water waves
rðx; tÞ ¼
2ABk 2 expðkx þ vt þ n0 Þ 2
½A þ B expðkx þ vt þ n0 Þ
þ a;
ð33Þ
k where v ¼ 3a1þk 2 ; A; B; k; n0 ; c are arbitrary constants. Taking A = B in (33), we obtain the exact and explicit soliton solutions of the Hirota–Satsuma equation (1) for shallow water waves
k2 1 sech2 ½kx þ vt þ n0 þ a; ð34Þ 2 2 k where v ¼ 3a1þk 2 ; k; n0 are arbitrary constants. While taking A = B in (33), we obtain the exact and explicit singular travelling wave solutions of the Hirota–Satsuma equation (1) for shallow water waves rðx; tÞ ¼
k2 1 csch2 ½kx þ vt þ n0 þ a; ð35Þ 2 2 k where v ¼ 3a1þk 2 ; k; n0 are arbitrary constants. Analogously, substituting (31) into (24) and (25), we find that (31) satisfies Eqs. (24) and (25), provided that v ¼ ð3a1Þk ; k 6¼ 1 when a 6¼ 13 or k = ±1, v = arbitrary constant when a ¼ 13. So when a 6¼ 13 we obtain the 1k 2 explicit exact solutions of (2): rðx; tÞ ¼
uðx; tÞ ¼
2Bk expðkx þ vt þ n0 Þ þ at þ c; A þ B expðkx þ vt þ n0 Þ
ð36Þ
where v ¼ ð3a1Þk ; k 6¼ 1; A; B; c; n0 are arbitrary constants. When a ¼ 13, we obtain the explicit exact solutions 1k 2 of (2) uðx; tÞ ¼
2B expðx þ vt þ n0 Þ þ at þ c; A þ B expðx þ vt þ n0 Þ
ð37Þ
where A, B, v, c, n0 are arbitrary constants. From (36) we find that the explicit exact solutions of the Hirota– Satsuma equation (1) for shallow water waves given as rðx; tÞ ¼
2BAk 2 expðkx þ vt þ n0 Þ 2
½A þ B expðkx þ vt þ n0 Þ
;
ð38Þ
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where v ¼ ð3a1Þk ; k 6¼ 1; a 6¼ 13 ; A; B; c; n0 are arbitrary constants. From (37) we find that the Hirota–Satsuma 1k 2 equation (1) for shallow water waves has the explicit exact solutions given as rðx; tÞ ¼
2BA expðvt x þ n0 Þ ½A þ B expðvt x þ n0 Þ2
;
ð39Þ
where A, B, v, c, n0 are arbitrary constants. In the case of A = B in (38) and (39), respectively, we can find the Hirota–Satsuma equation (1) for shallow water waves has exact soliton solutions rðx; tÞ ¼
k2 1 sech2 ½kx þ vt þ n0 ; 2 2
ð40Þ
; k 6¼ 1; a 6¼ 13 ; A; B; c; n0 are arbitrary constants, and where v ¼ ð3a1Þk 1k 2 1 rðx; tÞ ¼ sech2 ½vt x þ n0 ; ð41Þ 2 where A, B, v, c, n0 are arbitrary constants. And in the case of A = B in (38) and (39), respectively, we obtain the exact and explicit singular travelling wave solutions of the Hirota–Satsuma equation (1) for shallow water waves k2 1 csch2 ½kx þ vt þ n0 ; 2 2 ; k ¼ 6 1; a 6¼ 13 ; A; B; c; n0 are arbitrary constants, and where v ¼ ð3a1Þk 1k 2 rðx; tÞ ¼
ð42Þ
1 rðx; tÞ ¼ csch2 ½vt x þ n0 ; 2
ð43Þ
where A, B, v, c, n0 are arbitrary constants. The soliton solution (34) of the Hirito–Satsuma equation (1) for shallow water waves has nonzero asymptotic value. Note the fact that sech(ikn) = sec(kn), csch(ikn) = icsc(kn), we obtain the explicit exact periodic wave solutions of the Hirito–Satsuma equation (1) for shallow water waves in triangle function type k2 1 sec2 ½kx þ vt þ n0 þ a; 2 2 k2 2 1 rðx; tÞ ¼ csc ½kx þ vt þ n0 þ a; 2 2 k where v ¼ 3a1þk2 and k = arbitrary constant, rðx; tÞ ¼
k2 1 sec2 ½kx þ vt þ n0 ; 2 2 2 k 1 rðx; tÞ ¼ csc2 ½kx þ vt þ n0 ; 2 2 ð3a1Þk where v ¼ 1k2 and a ¼ 13, k = arbitrary constant. When a ¼ 13 in Eqs. (24) and (25), we have rðx; tÞ ¼
/x /t 3/x /xxt /t /xxx þ 3/xx /xt ¼ 0; /xt /xxxt ¼ 0:
ð44Þ ð45Þ
ð46Þ ð47Þ
ð48Þ ð49Þ
Solving Eqs. (48) and (49), we obtain /1;2 ðx; tÞ ¼ 1 Aext chx
ð50Þ
/3;4 ðx; tÞ ¼ 1 Aext shx;
ð51Þ
and
where A, x are two arbitrary constants. By (26) and (50) we find Eq. (2) have solution uðx; tÞ ¼ 2
Aext shx t þ þc 1 þ Aext chx 3
ð52Þ
Y. Shang / Applied Mathematics and Computation 187 (2007) 1286–1297
1291
and thus the Hirota–Satsuma equation (1) for shallow water waves has the explicit exact solution rðx; tÞ ¼ 2Aext
Aext chx ð1 þ Aext chxÞ
2
ð53Þ
:
Analogously, from (26) and (51), we obtain the explicit exact solutions of Eq. (2) uðx; tÞ ¼
2Aext chx t þ þc xt 1 Ae shx 3
ð54Þ
and thus the explicit exact solutions of the Hirota–Satsuma equation (1) for shallow water waves rðx; tÞ ¼ 2Aext
Aext shx ð1 Aext shxÞ
2
ð55Þ
:
The solutions (53) and (55) are the exact solutions of the Hirota–Satsuma equation (1) for shallow water waves that are not travelling wave solutions. Now we seek for the double-soliton solutions of the Hirota–Satsuma equation (1) for shallow water waves. We suppose that Eqs. (21) and (22) have solutions in the form / ¼ 1 þ expðk 1 x þ v1 t þ n1 Þ þ expðk 2 x þ v2 t þ n2 Þ;
ð56Þ
where ki, vi (i = 1, 2) are constants to be determined and ni (i = 1, 2) are arbitrary constants. Substituting (56) into (21) and (22), we obtain a system of algebras equations for k1, v1, k2, v2 ð1 3aÞk 1 v1 þ k 21 k 31 v1 ¼ 0; ð1 3aÞk 2 v2 þ k 22 k 32 v2 ¼ 0; ð1 3aÞðk 1 v2 þ k 2 v1 Þ þ 2k 1 k 2
ð57Þ 3k 2 k 21 v1
3k 1 k 22 v2
k 31 v2
k 32 v1
þ
3k 2 k 21 v2
þ
3k 1 k 22 v1
¼ 0:
When k1 = v1 = 0 or k2 = v2 = 0, we obtain the solution (32) of Eq. (2) and thus the soliton solution (34) of Hirota–Satsuma equation (1) for shallow water waves. By (23) we obtain the double-soliton solution of Eq. (2) in the form expðkx þ vt þ n1 Þ þ expðkx þ vt þ n2 Þ þ ax þ c: ð58Þ uðx; tÞ ¼ 2k 1 þ expðkx þ vt þ n1 Þ þ expðkx þ vt þ n2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 k2 k When v ¼ 3a1þk , v2 ¼ 3a1þk 2 ; k is an arbitrary constant. While v1 ¼ 2, k1 ¼ 3a1þk 21 2 double-soliton solution of Eq. (2) in the form
uðx; tÞ ¼ 2
k2
36a123k 22 , 2
we obtain the
k 1 expðk 1 x þ v1 t þ n1 Þ þ k 2 expðk 2 x þ v2 t þ n2 Þ þ ax þ c: 1 þ expðk 1 x þ v1 t þ n1 Þ þ expðk 2 x þ v2 t þ n2 Þ
ð59Þ
So the Hirota–Satsuma equation (1) for shallow water waves possesses the double-soliton solutions rðx; tÞ ¼ 2k 2
expðkx þ vt þ n1 Þ þ expðkx þ vt þ n2 Þ 2
½1 þ expðkx þ vt þ n2 Þ þ expðkx þ vt þ n2 Þ
þ a;
ð60Þ
2
rðx; tÞ ¼ 2
k 21 expðg1 Þ þ k 22 expðg2 Þ þ ðk 1 k 2 Þ expðg1 Þ expðg2 Þ ½1 þ expðg1 Þ þ expðg1 Þ2
þ a;
ð61Þ
ki k ; i ¼ 1; 2, where v ¼ 3a1þk 2 ; k is an arbitrary constant and g1 = k1x + v1t + n1, g2 = k2x + v2t + n2, vi ¼ 3a1þk 2i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi k 36a123k 2 . k1 ¼ 2 2 Analogously, when a 6¼ 13, we obtain Eq. (2) has double-soliton solutions
expðkx þ vt þ n0 Þ þ expðkx þ vt þ n0 Þ þ at þ c; 1 þ expðkx þ vt þ n0 Þ þ expðkx þ vt þ n0 Þ k 1 expðg1 Þ þ k 2 expðg2 Þ uðx; tÞ ¼ 2 þ at þ c; 1 þ expðg1 Þ þ expðg2 Þ
uðx; tÞ ¼ 2k
ð62Þ ð63Þ
1 , k 6¼ and g1 = k1x + v1t + n1, g2 = k2x + v2t + n2, vi ¼ ð3a1Þk ; i ¼ 1; 2; k 1 ¼ where v ¼ ð3a1Þk 1k 2 1k 2 i
k2
pffiffiffiffiffiffiffiffiffiffi2ffi
123k 2 . 2
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So we also obtain the double-soliton solutions of Hirota–Satsuma equation (1) for shallow water waves in the form rðx; tÞ ¼ 2k 2
expðkx þ vt þ n1 Þ þ expðkx þ vt þ n2 Þ ½1 þ expðkx þ vt þ n2 Þ þ expðkx þ vt þ n2 Þ2
ð64Þ
;
; k 6¼ 1. The Hirota–Satsuma equation (1) for shallow water waves also has double-soliton where v ¼ ð3a1Þk 1k 2 solutions rðx; tÞ ¼ 2
k 21 expðg1 Þ þ k 22 expðg2 Þ þ ðk 1 k 2 Þ2 expðg1 Þ expðg2 Þ ½1 þ expðg1 Þ þ expðg1 Þ
2
1 where g1 = k1x + v1t + n1, g2 = k2x + v2t + n2, vi ¼ ð3a1Þk ; i ¼ 1; 2; k 1 ¼ 1k 2i 1 While a ¼ 3, Eq. (2) possesses double-soliton solution
uðx; tÞ ¼ 2
ð65Þ
; k2
pffiffiffiffiffiffiffiffiffiffi2ffi
123k 2 . 2
expðx þ v1 t þ n1 Þ þ expðx þ v2 t þ n2 Þ 1 þ t þ c; 1 þ expðx þ v1 t þ n1 Þ þ expðx þ v2 t þ n2 Þ 3
ð66Þ
where v1, v2 are arbitrary constants. Eq. (2) also has double-soliton solution uðx; tÞ ¼ 2
expðx þ v1 t þ n1 Þ þ expðx þ v2 t þ n2 Þ 1 þ t þ c; 1 þ expðx þ v1 t þ n1 Þ þ expðx þ v2 t þ n2 Þ 3
ð67Þ
where v1, v2 are arbitrary constants. When a ¼ 13, we also obtain the double-soliton solution of Eq. (2) in the form uðx; tÞ ¼ 2
expðx þ vt þ n1 Þ þ expðx þ vt þ n2 Þ 1 þ t þ c; 1 þ expðx þ vt þ n1 Þ þ expðx þ vt þ n2 Þ 3
ð68Þ
where v is an arbitrary constant. Therefore, we obtain the double-soliton solutions of the Hirota–Satsuma equation for shallow water waves given as rðx; tÞ ¼ 2
expðx þ v1 t þ n1 Þ þ expðx þ v2 t þ n2 Þ ½1 þ expðx þ v1 t þ n1 Þ þ expðx þ v2 t þ n2 Þ
2
;
ð69Þ
where v1, v2 are arbitrary constants. We also obtain the double-soliton solution of the Hirota–Satsuma equation (1) for shallow water waves in form of rðx; tÞ ¼ 2
expðx þ vt þ n1 Þ þ expðx þ vt þ n2 Þ ½1 þ expðx þ vt þ n1 Þ þ expðx þ vt þ n2 Þ2
;
where v is an arbitrary constant. We assume that (21), (22) and (24), (25) have solutions in the form n X expðk i x þ vi t þ ni Þ; /¼1þ
ð70Þ
ð71Þ
i¼1
where ki, vi (i = 1, 2, . . . , n) are constants to be determined and ni (i = 1, 2, . . . , n) are arbitrary constants. Substituting (71) into (21), (22) and (24), (25), respectively, we can get the N-soliton solutions Eq. (2) Pn k i expðk i x þ vi t þ ni Þ þ ax þ c ð72Þ uðx; tÞ ¼ 2 i¼1 1 þ expðk i x þ vi t þ ni Þ and
Pn uðx; tÞ ¼ 2
expðk i x þ vi t þ ni Þ þ at þ c; 1 þ expðk i x þ vi t þ ni Þ i¼1 k i
ð73Þ
Y. Shang / Applied Mathematics and Computation 187 (2007) 1286–1297
1293
where ki, vi (i = 1, 2, . . . , n) are constants satisfy certain condition equations. By using r(x, t) = ux(x, t) we can obtain from (72) and (73) the N-solitons of the Hirota–Satsuma equation (1) for shallow water waves. We omitted their concrete form here. In 1992, Conte and Musette presented an indirect method to find more new solitary wave solutions of nonlinear partial differential equations that can be expressed as a polynomial in two elementary functions which satisfy a projective Riccati equation [15]. In [16], Zhang et al. proposed the hyperbolic function method based upon the fact that many solitary wave solutions have the format of hyperbolic functions. They expressed the solitary wave solutions of the nonlinear wave equations as the combination of hyperbolic functions and obtained many new exact solitary wave solutions. In [17], Yan presented the generally projective Riccati equation method. More recently, Chen and Ding improved the projective Riccati equation method in [18] and obtained some new solitary wave solutions to the nonlinear evolution equation. In order to obtain some other exact solutions of the Hirota–Satsuma equation for shallow water waves, we employ an extension of the hyperbolic function method proposed in [19] by the author to find many more exact solutions of the Hirota–Satsuma equation for shallow water waves. We suppose that the Hirota–Satsuma equation (2) for shallow water waves has travelling wave solution uðx; tÞ ¼ uðnÞ ¼ uðkx xt þ n0 Þ;
ð74Þ
where k, x are constants to be determined and n0 is an arbitrary constant. Substituting (74) into (2) we get 2
xu0 ðnÞ þ k 2 xu000 ðnÞ þ 3kxðu0 ðnÞÞ þ ku0 ðnÞ ¼ a:
ð75Þ
Let u 0 (n) = u(n), it infers that from (75) k 2 xu00 ðnÞ þ ðk xÞuðnÞ þ 3kxu2 ðnÞ ¼ a:
ð76Þ
We assume that the solutions of the ODE (76) is of the form uðnÞ ¼ a0 þ a1 f ðnÞ þ a2 f 2 ðnÞ þ b1 gðnÞ þ b2 f ðnÞgðnÞ;
ð77Þ
where a0, a1, a2, b1 and b2 are constants to be determined. The functions f and g satisfy the coupled Riccati equations f 0 ðnÞ ¼ f ðnÞgðnÞ;
g0 ðnÞ ¼ 1 rf ðnÞ g2 ðnÞ;
ð78Þ
f 0 ðnÞ ¼ f ðnÞgðnÞ;
g0 ðnÞ ¼ 1 þ rf ðnÞ g2 ðnÞ;
ð79Þ
respectively. Furthermore, we can obtain their first integrals as given g2 ðnÞ ¼ 1 2rf ðnÞ þ ðb2 a2 þ r2 Þf 2 ðnÞ;
ð80Þ
g2 ðnÞ ¼ 1 þ 2rf ðnÞ þ ðb2 þ a2 r2 Þf 2 ðnÞ;
ð81Þ
respectively. The ODEs (78) and (79) have the following special solutions: f ðnÞ ¼
1 ; a cosh n þ b sinh n þ r
f ðnÞ ¼
1 ; a cos n þ b sin n þ r
gðnÞ ¼ gðnÞ ¼
a sinh n þ b cosh n ; a cosh n þ b sinh n þ r
b cos n a sin n ; a cos n þ b sin n þ r
ð82Þ ð83Þ
respectively. Substituting (77) into (76) and using (78) and (80), we obtain a set of nonlinear algebraic equations for a0, a1, a2, b1, b2, k, and x by setting the coefficients of power f4, f3g, f3, f2g, f2, fg, f and g to be zero
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Y. Shang / Applied Mathematics and Computation 187 (2007) 1286–1297
6a2 k 2 xðb2 a2 þ r2 Þ þ 3kx½a22 þ b22 ðb2 a2 þ r2 Þ ¼ 0; 6b2 k 2 xðb2 a2 þ r2 Þ þ 6kxa2 b2 ¼ 0;
k 2 x½2a1 ðb2 a2 þ r2 Þ 10a2 r þ 6kx a1 a2 þ b1 b2 ðb2 a2 þ r2 Þ b22 r ¼ 0; 6k 2 xb2 r þ 2k 2 xb1 ðb2 a2 þ r2 Þ þ 6kxða1 b2 þ a2 b1 Þ ¼ 0;
k 2 xð4a2 3a1 rÞ þ 3kx½a21 þ 2a0 a2 þ b22 4b1 b2 r þ b21 ðb2 a2 þ r2 Þ þ ðk xÞa2 ¼ 0;
ð84Þ
k 2 xðb2 b1 rÞ þ 6kxða1 b1 þ a0 b2 Þ þ ðk xÞb2 ¼ 0; k 2 xa1 þ 6kxða0 a1 þ b1 b2 b21 rÞ þ ðk xÞa1 ¼ 0; 6kxa0 b1 þ ðk xÞb1 ¼ 0; 3kxða20 þ b21 Þ þ ðk xÞa0 ¼ a: With the aid of the computer program Mathematica or Maple 4 [20], Solving (81), we obtain the following five set of solutions: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k a0 ¼ ; a1 ¼ k a2 b2 ; a2 ¼ 0; b1 ¼ b2 ¼ 0; r ¼ a2 b2 ; k 6¼ 0; x 6¼ 0; 6k 6x 6 ð85Þ where a P b are two arbitrary parameters, 1 1 ; a1 ¼ 0; a2 ¼ 0; b1 6¼ 0; b2 ¼ 0; r ¼ 0; b ¼ a; k 6¼ 0; x 6¼ 0; 6k 6x 1 1 k ; a1 ¼ 2kr; a2 ¼ 2kr2 ; b1 ¼ b2 ¼ 0; b ¼ a; r 6¼ 0; k 6¼ 0; x 6¼ 0; a0 ¼ 6k 6x 6 1 1 2k ; a1 ¼ 0; a2 ¼ 2kða2 b2 Þ; r ¼ 0; b1 ¼ 0; b2 ¼ 0; k 6¼ 0; x 6¼ 0; a0 ¼ 6k 6x 3
a0 ¼
ð86Þ ð87Þ ð88Þ
and 1 1 k ; 6k 6x 6 k 6¼ 0; x 6¼ 0;
a0 ¼
a1 ¼ kr;
a2 ¼ kðb2 a2 þ r2 Þ;
b1 ¼ 0;
b2 ¼ k
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 a2 þ r 2 ; ð89Þ
where a, b and r are three arbitrary parameters satisfy b2 a2 + r2 > 0. According to (77), (82) and (85), we obtain that the Hirota–Satsuma equation for shallow water wave (1) have explicit exact solitary solution as given pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k k2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; k 2 a2 b2 ð90Þ rðx; tÞ ¼ 6 6x 6 a cosh n þ b sinh n a2 b2 where a P b are two arbitrary parameters and n = kx x t + n0, k 5 0, x 5 0. From (77), (82) and (87) we obtain that the Hirota–Satsuma equation for shallow water wave (1) have explicit exact solitary solution as given rðx; tÞ ¼
1 k k2 2k 2 r 2k 2 r2 þ ; 6 6x 6 a cosh n þ b sinh n þ r ða cosh n þ b sinh n þ rÞ2
ð91Þ
where a = ±b is arbitrary parameter and n = kx xt + n0, r 5 0, k 5 0, x 5 0. By (77), (82) and (88), we obtain that the Hirota–Satsuma equation for shallow water wave (1) have explicit exact solitary solution as given rðx; tÞ ¼
1 k 2k 2 2k 2 ða2 b2 Þ þ ; 2 6 6x 3 ða cosh n þ b sinh nÞ
where a 5 ±b are two arbitrary parameters and n = kx x t + n0, k 5 0, x 5 0,
ð92Þ
Y. Shang / Applied Mathematics and Computation 187 (2007) 1286–1297
1295
Analogously, from (77), (82) and (89), we also obtain that the Hirota–Satsuma equation for shallow water wave (1) have explicit exact solitary solution as given 1 k k2 k2r k 2 ðb2 a2 þ r2 Þ þ rðx; tÞ ¼ 6 6x 6 a cosh n þ b sinh n þ r ða cosh n þ b sinh n þ rÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a sinh n þ b cosh n k 2 b2 a2 þ r 2 ; 2 ða cosh n þ b sinh n þ rÞ
ð93Þ
where a, b and r are three arbitrary parameters satisfy b2 a2 + r2 P 0 and n = kx xt + n0, k 5 0, x 5 0. We now seek for the exact periodic wave solutions of the Hirota–Satsuma equation for shallow water waves. Substituting (77) into (76) and using (79) and (81), we obtain a set of nonlinear algebraic equations for a0, a1, a2, b1, b2, k, and x by setting the coefficients of power f4, f3g, f3, f2g, f2, fg, f and g to be zero 6a2 k 2 xðb2 þ a2 r2 Þ þ 3kx½a22 þ b22 ðb2 þ a2 r2 Þ ¼ 0; 6b2 k 2 xðb2 þ a2 r2 Þ þ 6kxa2 b2 ¼ 0; k 2 x½2a1 ðb2 þ a2 r2 Þ þ 10a2 r þ 6kx½a1 a2 þ b1 b2 ðb2 þ a2 r2 Þ þ b22 r ¼ 0; 6k 2 xb2 r þ 2k 2 xb1 ðb2 þ a2 r2 Þ þ 6kxða1 b2 þ a2 b1 Þ ¼ 0; k 2 xð4a2 þ 3a1 rÞ þ 3kx½a21 þ 2a0 a2 b22 þ 4b1 b2 r þ b21 ðb2 þ a2 r2 Þ þ ðk xÞa2 ¼ 0;
ð94Þ
k 2 xðb2 þ b1 rÞ þ 6kxða1 b1 þ a0 b2 Þ þ ðk xÞb2 ¼ 0; k 2 xa1 þ 6kxða0 a1 b1 b2 þ b21 rÞ þ ðk xÞa1 ¼ 0; 6kxa0 b1 þ ðk xÞb1 ¼ 0; 3kxða20 b21 Þ þ ðk xÞa0 ¼ a: With the aid of the computer program Mathematica or Maple 4 [20], Solving (91), we obtain the following five set of solutions: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k þ ; a1 ¼ k a2 þ b2 ; a2 ¼ 0; b1 ¼ b2 ¼ 0; r ¼ a2 þ b2 ; k 6¼ 0; x 6¼ 0; a0 ¼ 6k 6x 6 ð95Þ where a, b are two arbitrary parameters satisfy a2 + b2 5 0, 1 1 ; a1 ¼ 0; r ¼ 0; b ¼ a ¼ 0; b2 ¼ 0; k 6¼ 0; 6k 6x 1 1 k þ ; a1 ¼ 2kr; a2 ¼ 2kr2 ; b1 ¼ 0; b2 ¼ 0; a0 ¼ 6k 6x 6
a0 ¼
x 6¼ 0;
ð96Þ
b1 arbitrary
a ¼ b ¼ 0;
k 6¼ 0;
x 6¼ 0;
ð97Þ
where r is an arbitrary parameter, a0 ¼
1 1 2k þ ; 6k 6x 3
a1 ¼ 0;
a2 ¼ 2kða2 þ b2 Þ;
b1 ¼ b2 ¼ 0;
r ¼ 0;
k 6¼ 0;
x 6¼ 0;
b2 ¼ k
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 r 2 ;
ð98Þ
where a, b are two arbitrary parameters satisfy a2 + b2 > 0, and 1 1 k þ ; a1 ¼ kr; 6k 6x 6 k 6¼ 0; x 6¼ 0;
a0 ¼
a2 ¼ kða2 þ b2 r2 Þ;
b1 ¼ 0;
ð99Þ
where a, b are two arbitrary parameters satisfy a2 + b2 r2 > 0 and n = kx + xt + n0, k 5 0, x 5 0. By (77), (83), (95)–(98), we obtain the Hirota–Satsuma equation for shallow water waves (1) have multiple nontrivial exact periodic travelling wave solutions pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k k2 k 2 a2 þ b2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; þ ð100Þ rðx; tÞ ¼ 6 6x 6 a cos n þ b sin n a2 þ b2
1296
Y. Shang / Applied Mathematics and Computation 187 (2007) 1286–1297
where a, b are two arbitrary parameters satisfy a2 + b2 > 0 and n = kx + xt + n0, k 5 0, x 5 0, rðx; tÞ ¼
1 k 2k 2 2k 2 ða2 þ b2 Þ þ ; 2 6 6x 3 ða cos n þ b sin nÞ
ð101Þ
where a, b are two arbitrary parameters satisfy a2 + b2 > 0 and n = kx + xt + n0, k 5 0, x 5 0, 1 k k2 k2r k 2 ða2 þ b2 r2 Þ þ rðx; tÞ ¼ 6 6x 6 a cos n þ b sin n þ r ða cos n þ b sin n þ rÞ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b cos n a sin n k 2 a2 þ b2 r 2 ; 2 ða cos n þ b sin n þ rÞ
ð102Þ
where a, b and r are three arbitrary parameters satisfy a2 + b2 r2 > 0 and n = kx + xt + n0, k 5 0, x 5 0. 4. Summary and conclusions In summary, we first obtain the Ba¨cklund transformation and Lax pairs by using the extended homogeneous balance method. Then we obtain some multiple exact explicit solutions of the Hirota–Satsuma equation for shallow water waves by the extended homogeneous balance method and an extension of the hyperbolic functions method. These solutions include the solitary wave solution of rational function, soliton solutions, double-soliton solutions, N-soliton solutions, the multiple solitary wave solutions of a compound of the bell-type and the kink-type solitary wave solutions, singular solutions, and the periodic wave solutions of triangle function type. By the Ba¨cklund transformation (18) and Eqs. (15) and (16), we can obtain even more new solutions of the Hirota–Satsuma equation from the solutions (27), (28), (32), (36), (37), (52), (54), (58), (59), (62), (63), (66)–(68), (72) and (73) obtained in this paper. Acknowledgement The authors are grateful to Professor Wang Mingliang for his suggestions and helps. References [1] R. Hirota, J. Satsuma, N-soliton solutions of model equations for shallow water waves, J. Phys. Soc. Jpn. 40 (1976) 611–614. [2] R. Hirota, J. Satsuma, Nonlinear equations generated from the Backlund transformation for the Boussinesq equation, Prog. Theor. Phys. 57 (1977) 797–800. [3] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. [4] P.A. Clarkson, E.L. Mansfield, On a shallow water wave equation, Nonlinearity 7 (1994) 975–1000. [5] J. Satsuma, D.J. Kaup, Ba¨cklund transformation in bilinear form to the Hirota–Satsuma equation, J. Phys. Soc. Jpn. 43 (1978) 692. [6] M. Musette, R. Conte, Algorithmic method for deriving Lax pairs from the invariant Painleve analysis of nonlinear partial differential equations, J. Math. Phys. 32 (6) (1991) 1450–1457. [7] Yi Zhang, S.F. Deng, Dengyuan Chen, The novel multi-soliton solutions of equation for shallow water waves, J. Phys. Soc. Jpn. 72 (3) (2003) 763–764. [8] Jinshun Zhang, Yunping Yang, Darboux transformation of Hirota–Satsuma equation and its solutions, J. Zhengzhou Univ. (Sci.) 35 (3) (2003) 1–4. [9] Yi Zhang, Dengyuan Chen, Ba¨cklund transformation and soliton solutions for the shallow water wave equation, Chaos, Solitons and Fractals 20 (2004) 343–351. [10] M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991. [11] Mingliang Wang et al., Applications of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996) 67–74. [12] Mingliang Wang, Exact solutions for a compound KdV–Burgers equation, Phys. Lett. A 213 (1996) 279–287. [13] Mingliang Wang, Yubin Zhou, Huiqun Zhang, A nonlinear transformation of the shallow water wave equations and its application, Adv. Math. (China) 28 (1) (1999) 72–75. [14] Engui Fan, Hongqin Zhang, A new approach to Backlund transformations of nonlinear evolution equations, Appl. Math. Mech. 19 (7) (1998) 645–650. [15] R. Conte, M. Musette, Link between solitary waves and projective Riccati equations, J. Phys. A: Math. Gen. 25 (1992) 5609–5612.
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