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New exact solutions for two generalized Hirota–Satsuma coupled KdV systems
Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1120–1127 www.elsevier.com/locate/cnsns
Short communication
New exact solutions for two generalized Hirota–Satsuma coupled KdV systems Huiqun Zhang Department of Mathematics, Qingdao University, Qingdao, Shandong 266071, China Received 19 November 2005; received in revised form 6 January 2006; accepted 6 January 2006 Available online 9 March 2006
Abstract By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact solutions for nonlinear evolution equations. By this method two generalized Hirota–Satsuma coupled KdV systems are investigated and new exact solutions are explicitly obtained with the aid of symbolic computation. Ó 2006 Elsevier B.V. All rights reserved. PACS: 03.40.Kf; 02.30.Jr Keywords: Generalized Hirota–Satsuma coupled KdV systems; Exact solutions; Algebraic method; Symbolic computation
1. Introduction It is important to seek more exact solutions of nonlinear partial differential equations (NLPDEs) in mathematical physics. Many powerful methods have been presented such as Backlund transformation [1], Darboux transformation [2], the extended tanh-function method [3], the F-expansion method [4], and so on. For a given partial differential equation H ðu; ux ; ut ; uxx ; . . .Þ ¼ 0;
ð1Þ
these methods are applied to seek exact solutions by using the following transformation: uðx; tÞ ¼ F ð/1 ; /2 ; . . .Þ;
Gð/1 ; /2 ; . . .Þ ¼ 0.
ð2Þ
If one can determine the functions /i from G(/1, /2 , . . .) = 0, then he can obtain the solutions from the transformation u = F(/1, /2 , . . .). That is to say, the key idea of these generalized methods is to use the solutions of an auxiliary ordinary differential equation (or auxiliary ordinary differential equations) to construct the solutions of NLPDEs.
E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2006.01.011
H. Zhang / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1120–1127
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In a travelling frame of reference u(x, t) = u(n), n = k(x ct), we may transform Eq. (1) into an ordinary differential equation (ODE) H ðu; ku0 ; kcu0 ; k 2 u00 ; . . .Þ ¼ 0.
ð3Þ
We expand the solution of Eq. (3) as the following series: uðnÞ ¼
n X
ai F i ðnÞ;
ð4Þ
i¼0
where F = F(n) is a solution of the following first-order ODE: m X 2 qj F j ðnÞ. F0 ¼
ð5Þ
j¼0
Balancing the highest derivative terms with the nonlinear terms in Eq. (3) will give a relation for the positive integers n and m, from which the different possible values of n and m can be determined. These values lead to a series expansions of the exact solutions for Eq. (3). For example, in the case of KdV equation ut þ 6uux þ uxxx ¼ 0;
ð6Þ
we have m ¼ n þ 2.
ð7Þ
If we take m = 4 and n = 2 in (7), we obtain the following expansion of an exact solution of the KdV Eq. (6) as: F 02 ¼ q4 F 4 þ q3 F 3 þ q2 F 2 þ q1 F þ q0 :
u ¼ a2 F 2 þ a 1 F þ a 0 ;
ð8Þ
Especially, when we take q3 = q1 = 0 in (8), we obtain the F-expansion method. In the present paper, we take m = 6 then n = 4, q5 = q3 = q1 = q0 = 0 in (5) to construct exact solutions for two generalized Hirota–Satsuma coupled KdV systems. In other words, we shall introduce the following new auxiliary ordinary differential equation 2 dF ¼ q6 F 6 ðnÞ þ q4 F 4 ðnÞ þ q2 F 2 ðnÞ; ð9Þ dn where q6, q4, q2 are real parameters, and study two generalized Hirota–Satsuma coupled KdV equations by using the following solutions of Eq. (9) 8h i12 pffiffiffiffiffi > q2 > ð1 tanhð q ; q2 > 0; q4 < 0; q24 ¼ 4q2 q6 ; ðaÞ nÞÞ > 2 q4 > > > > > rffiffiffiffiffiffiffiffiffiffiffi > 912 8 > > q2 > > > p ffiffiffi 2 2 > > > 2 sech ð q2 nÞ > = < > q2 4q2 q6 > 4 > > ; q2 > 0; q24 4q2 q6 > 0; > > > > > q4 p ffiffiffi > > < :2þ 1pffiffiffiffiffiffiffiffiffiffiffi sech2 ð q2 nÞ; q2 4q2 q6 4 ð10Þ F ðnÞ ¼ > > 4 2 > where q4 þ 3q4 ¼ q2 q4 q6 q2 q6 ðbÞ > > > > > 912 8 > > > > > > > > > = < > > > 1 > r ffiffiffiffiffiffiffiffiffiffiffi ; q2 < 0; q24 4q2 q6 > 0. ðcÞ > > > > q2 4q2 q6 q4 pffiffiffiffiffiffi > > > > 4 sinð2 q2 nÞ; : :2q2 4q2 2
As a result, we indeed successfully find some new exact solutions to two generalized Hirota–Satsuma coupled KdV equations.
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2. Exact solutions In this section, we consider two generalized Hirota–Satsuma coupled KdV equations, 8 u ¼ 1 u 3uux þ 3ðvwÞx ; > < t 2 xxx vt ¼ vxxx þ 3uvx ; > : wt ¼ wxxx þ 3uwx
ð11Þ
and 8 u ¼ 1 u 3uux þ 3ðv2 þ wÞx ; > < t 4 xxx vt ¼ 12 vxxx 3uvx ; > : wt ¼ 12 wxxx 3uwx .
ð12Þ
System (11) is a new generalized Hirota–Satsuma coupled KdV system given by Wu et al. recently [5]. It was shown that a new complex coupled KdV system and Hirota–Satsuma equation can be reduced from system (11) with w = v* and w = v, respectively. System (12) was proposed by Satsuma and Hirota. They found its three-soliton solutions and showed thatpthe equation is a special case of system (12) with ffiffiffi Hirota–Satsuma pffiffiffi w = 0 and scaling transformation x ! 2x; t ! 2t. First we consider the system (11). Making the transformation u(x, t) = u(n), v(x, t) = v(n), w(x, t) = w(n), n = k(x ct), system (5) becomes 8 0 > cu0 ¼ 12 k 2 u000 3uu0 þ 3ðvwÞ > < ð13Þ cv0 ¼ k 2 v000 þ 3uv0 ; > > : 2 000 0 0 cw ¼ k w þ 3uw ; where k and c are the wave number and wave speed, respectively. Balancing the highest-order linear terms with nonlinear terms in (13) admits the following ansatz: 8 uðnÞ ¼ a4 F 4 þ a3 F 3 þ a2 F 2 þ a1 F þ a0 ; > < ð14Þ vðnÞ ¼ b4 F 4 þ b3 F 3 þ b2 F 2 þ b1 F þ b0 ; > : 4 3 2 wðnÞ ¼ c4 F þ c3 F þ c2 F þ c1 F þ c0 ; where ai, bi, ci (i = 0, 1, 2, 3, 4) are constants to be determined later, and F = F(n) expresses the solutions (10a)– (10c) of Eq. (9). It is easy to deduce that u0 ¼ 4a4 F 3 F 0 þ 3a3 F 2 F 0 þ 2a2 FF 0 þ a1 F 0 ;
ð15Þ
uu0 ¼ ½4a4 F 7 þ 10a3 a4 F 6 þ ð6a2 a4 þ 3a23 ÞF 5 þ 5ða1 a4 þ a2 a3 ÞF 4 þ ð4a1 a3 þ 2a22 þ 4a0 a4 ÞF 3 þ ð3a1 a2 þ 3a0 a3 ÞF 2 þ ða21 þ 2a0 a2 ÞF þ a0 a1 F 0 ; 000
7
6
ð16Þ
5
u ¼ ½192a4 q6 F þ 105a3 q6 F þ 6ð8a2 q6 þ 20a4 q4 ÞF þ 5ð12a3 q4 þ 3a1 q6 ÞF
4
þ 4ð16a4 q2 þ 6a2 q4 ÞF 3 þ 3ð9a3 q2 þ 2a1 q4 ÞF 2 þ 8a2 q2 F þ a1 q2 F 0 ; 0
7
6
5
ðvwÞ ¼ ½8b4 c4 F þ 7ðb4 c3 þ b3 c4 ÞF þ 6ðb4 c2 þ b3 c3 þ b2 c4 ÞF þ 5ðb4 c1 þ b3 c2 þ b2 c3 þ b1 c4 ÞF
ð17Þ 4
þ 4ðb4 c0 þ b3 c1 þ b2 c2 þ b1 c3 þ b0 c4 ÞF 3 þ 3ðb3 c0 þ b2 c1 þ b1 c2 þ b0 c3 ÞF 2 þ 2ðb2 c0 þ b1 c1 þ b0 c2 ÞF þ ðb1 c0 þ b0 c1 ÞF 0 ; 0
3
0
2
0
0
0
v ¼ 4b4 F F þ 3b3 F F þ 2b2 FF þ b1 F ;
ð18Þ ð19Þ
v000 ¼ ½192b4 q6 F 7 þ 105b3 q6 F 6 þ 6ð8b2 q6 þ 20b4 q4 ÞF 5 þ 5ð12b3 q4 þ 3b1 q6 ÞF 4 þ 4ð16b4 q2 þ 6b2 q4 ÞF 3 þ 3ð9b3 q2 þ 2b1 q4 ÞF 2 þ 8b2 q2 F þ b1 q2 F 0 ;
ð20Þ
H. Zhang / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1120–1127
1123
uv0 ¼ ½4a4 b4 F 7 þ ð3a4 b3 þ 4a3 b4 ÞF 6 þ ð2a4 b2 þ 3a3 b3 þ 4a2 b4 ÞF 5 þ ða4 b1 þ 2a3 b2 þ 3a2 b3 þ 4a1 b4 ÞF 4 þ ða3 b1 þ 2a2 b2 þ 3a1 b3 þ 4a0 b4 ÞF 3 þ ða2 b1 þ 2a1 b2 þ 3a0 b3 ÞF 2 þ ða1 b1 þ 2a0 b2 ÞF þ a0 b1 F 0 ; ð21Þ w0 ¼ 4c4 F 3 F 0 þ 3c3 F 2 F 0 þ 2c2 FF 0 þ c1 F 0 ;
ð22Þ
w000 ¼ ½192c4 q6 F 7 þ 105c3 q6 F 6 þ 6ð8c2 q6 þ 20c4 q4 ÞF 5 þ 5ð12c3 q4 þ 3c1 q6 ÞF 4 þ 4ð16c4 q2 þ 6c2 q4 ÞF 3 þ 3ð9c3 q2 þ 2c1 q4 ÞF 2 þ 8c2 q2 F þ c1 q2 F 0 ; 0
7
6
ð23Þ 5
uw ¼ ½4a4 c4 F þ ð3a4 c3 þ 4a3 c4 ÞF þ ð2a4 c2 þ 3a3 c3 þ 4a2 c4 ÞF þ ða4 c1 þ 2a3 c2 þ 3a2 c3 þ 4a1 c4 ÞF þ ða3 c1 þ 2a2 c2 þ 3a1 c3 þ 4a0 c4 ÞF 3 þ ða2 c1 þ 2a1 c2 þ 3a0 c3 ÞF 2 þ ða1 c1 þ 2a0 c2 ÞF þ a0 c1 F 0 .
4
ð24Þ
0
Substituting (15)–(24) into (13), and setting each coefficient of FiF (i = 0, 1, 2, 3, 4, 5, 6, 7) to zero, yields a set of equations for ai, bi, ci, k, c 2b4 c4 þ 8k 2 a4 q6 a24 ¼ 0;
ð25Þ
35 2 k a3 q6 10a3 a4 þ 7ðb4 c3 þ b3 c4 Þ ¼ 0; 2 3ðb4 c2 þ b2 c4 Þ þ k 2 ð10a4 q4 þ 4a2 q6 Þ 3a2 a4 ¼ 0; 1 2 k ð4a3 q4 þ a1 q6 Þ a1 a4 a2 a3 þ b4 c1 þ b3 c2 þ b2 c3 þ b1 c4 ¼ 0; 2 6ðb4 c0 þ b2 c2 þ b0 c4 Þ þ k 2 ð16a4 q2 þ 6a2 q4 Þ 3ða22 þ 2a0 a4 Þ þ 2ca4 ¼ 0; 1 ca3 ¼ k 2 ð9a3 q2 þ 2a1 q4 Þ 3ða1 a2 þ a0 a3 Þ þ 3ðb3 c0 þ b2 c1 þ b1 c2 þ b0 c3 Þ; 2 3ðb2 c0 þ b0 c2 Þ þ 2k 2 a2 q2 3a0 a2 þ ca2 ¼ 0;
ð26Þ ð27Þ ð28Þ ð29Þ ð30Þ ð31Þ
1 ca1 ¼ k 2 a1 q2 3a0 a1 þ 3b1 c0 þ b0 c1 ; 2 a4 b4 ¼ 16k 2 b4 q6 ;
ð33Þ
3a4 b3 þ 4a3 b4 35k 2 b3 q6 ¼ 0;
ð34Þ
2
ð32Þ
a4 b2 þ 2a2 b4 ¼ k ð20b4 q4 þ 8b2 q6 Þ;
ð35Þ
2
a4 b1 þ 2a3 b2 þ 3a2 b3 þ 4a1 b4 5k ð4b3 q4 þ b1 q6 Þ ¼ 0;
ð36Þ
3ða2 b2 þ 2a0 b4 Þ 2k 2 ð16b4 q2 þ 6b2 q4 Þ þ 2cb4 ¼ 0;
ð37Þ
2
ð38Þ
a2 b1 þ 2a1 b2 þ 3a0 b3 k ð9b3 q2 þ 2b1 q4 Þ þ cb3 ¼ 0; 2
3a0 b2 4k b2 q2 þ cb2 ¼ 0;
ð39Þ
3a0 b1 k 2 b1 q2 þ cb1 ¼ 0;
ð40Þ
2
a4 c4 ¼ 16k c4 q6 ;
ð41Þ 2
3a4 c3 þ 4a3 c4 35k c3 q6 ¼ 0;
ð42Þ
a4 c2 þ 2a2 c4 ¼ k 2 ð20c4 q4 þ 8c2 q6 Þ;
ð43Þ
2
a4 c1 þ 2a3 c2 þ 3a2 c3 þ 4a1 c4 5k ð4c3 q4 þ c1 q6 Þ ¼ 0;
ð44Þ
3ða2 c2 þ 2a0 c4 Þ 2k62ð16c4 q2 þ 6c2 q4 Þ þ 2cc4 ¼ 0;
ð45Þ
2
a2 c1 þ 2a1 c2 þ 3a0 c3 k ð9c3 q2 þ 2c1 q4 Þ þ cc3 ¼ 0; 2
3a0 c2 4k c2 q2 þ cc2 ¼ 0; 2
3a0 c1 k c1 q2 þ cc1 ¼ 0.
ð46Þ ð47Þ ð48Þ
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H. Zhang / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1120–1127
Solving above algebraic equations , we obtain A a4 ¼ 8k 2 q6 ;
1 a0 ¼ ð4k 2 q2 cÞ; 3
a2 ¼ 4k 2 q4 ;
a3 ¼ a1 ¼ b4 ¼ b3 ¼ b1 ¼ c4 ¼ c3 ¼ c1 ¼ 0;
ð49Þ
32k 4 q2 q6 12k 4 q24 þ 16k 2 q6 c þ 3b2 c2 ¼ 0;
ð50Þ
8k 4 q2 q4 þ 8k 2 q4 c þ 3b2 c0 þ 3b0 c2 ¼ 0.
ð51Þ
B a4 ¼ 16k 2 q6 ;
a2 ¼ 8k 2 q4 ;
b4 ¼ c4 ¼ 8k 2 q6 ;
1 a0 ¼ ð4k 2 q2 cÞ; 3
b2 ¼ c2 ¼ 4k 2 q4 ;
a3 ¼ a1 ¼ b3 ¼ b1 ¼ c3 ¼ c1 ¼ 0;
4 b0 þ c0 ¼ ðk 2 q2 cÞ; 3
ð52Þ
q24 ¼ 4q2 q6 .
ð53Þ
C a4 ¼ 16k 2 q6 ;
a2 ¼ 8k 2 q4 ;
b4 ¼ c4 ¼ 8k 2 q6 ;
1 a0 ¼ ð4k 2 q2 cÞ; 3
b2 ¼ c2 ¼ 4k 2 q4 ;
a3 ¼ a1 ¼ b3 ¼ b1 ¼ c3 ¼ c1 ¼ 0;
4 b0 þ c0 ¼ ðc k 2 q2 Þ; 3
ð54Þ
q24 ¼ 4q2 q6 .
ð55Þ
Substituting above three of solutions with (10a)–(10c) into (14) we obtain exact solutions of system (13) as follows: A
uðnÞ ¼ 32k 2 q6
rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi q22 sech2 ð q2 nÞ q2 4q q
8 > > <
92 > > =
2 6
4
rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi q22 sech2 ð q2 nÞ q2 4q q
8 > > <
4
2 6
9 > > =
þ 8k 2 q4 > > > > q4 q4 2 pffiffiffiffiffi 2 pffiffiffiffiffi > > > > ; ; :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi : p ffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 1 sech sech nÞ nÞ ð q ð q 2 2 2 2 q4 4q2 q6
q4 4q2 q6
1 þ ð4k 2 q2 cÞ; 3 rffiffiffiffiffiffiffiffiffiffiffiffiffi 9 8 q22 2 pffiffiffiffiffi > > > > ð q sech nÞ = < 2 2 q4 4q2 q6 þ b0 ; vðnÞ ¼ 2b2 > pffiffiffiffiffi > q4 > ; :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi sech2 ð q2 nÞ > 2
ð56Þ
ð57Þ
q4 4q2 q6
wðnÞ ¼ 2c2
rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi q22 sech2 ð q2 nÞ q2 4q q
8 > > <
4
9 > > =
2 6
þ c0 ; > > q4 2 pffiffiffiffiffi > ; :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi sech ð q2 nÞ > 2
ð58Þ
q4 4q2 q6
where b2 and c2 are determined by (50) and (51), c, b0, c0 are arbitrary constants, and q2 > 0, q24 4q2 q6 > 0, q44 þ 3q34 ¼ q2 q4 q6 q2 q6 , or
uðnÞ ¼ 8k 2 q6
8 > > < > > : q4 2q 2
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi q24 4q2 q6 4q22
92 > > = pffiffiffiffiffiffiffiffi > ; sinð2 q2 nÞ>
þ 4k 2 q4
8 > > < > > : q4 2q 2
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi q24 4q2 q6 4q22
9 > > = pffiffiffiffiffiffiffiffi > ; sinð2 q2 nÞ>
1 þ ð4k 2 q2 cÞ; 3
ð59Þ
H. Zhang / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1120–1127
8 > > <
1 rffiffiffiffiffiffiffiffiffiffiffiffiffi
9 > > =
þ b0 ; pffiffiffiffiffiffiffiffi > > sinð2 q2 nÞ; 4q22 9 8 > > > > = < 1 rffiffiffiffiffiffiffiffiffiffiffiffiffi þ c0 ; wðnÞ ¼ c2 2 > > > > ; : q4 q4 4q22 q6 sinð2pffiffiffiffiffiffiffiffi q2 nÞ 2q 4q
vðnÞ ¼ b2
> > :
q4 2q2
2
1125
ð60Þ
q24 4q2 q6
ð61Þ
2
where b2 and c2 are determined by (50) and (51), k, c, b0, c0 are arbitrary constants, and q2 < 0, q24 4q2 q6 > 0, or 2 1 pffiffiffiffiffi uðnÞ ¼ 2k 2 q2 tanh2 ð q2 nÞ k 2 q2 c; 3 3 2 2 4k q4 ðc k q2 Þ 8k 2 q4 c2 ðk 2 q2 cÞ 4k 2 c0 ðk 2 q24 4q6 cÞ pffiffiffiffiffi vðnÞ ¼ ½1 tanhð q2 nÞ þ ; 3c2 3c22 q c2 pffiffiffiffiffi wðnÞ ¼ 2 ½1 tanhð q2 nÞ þ c0 ; q4 where c2, c0, k, c are arbitrary constants, and q2 > 0, q24 ¼ 4q2 q6 . B 8 c pffiffiffiffiffi uðnÞ ¼ 4k 2 q2 tanh2 ð q2 nÞ k 2 q2 ; 3 3 2 4 pffiffiffiffiffi vðnÞ ¼ 2k 2 q2 tanh2 ð q2 nÞ k 2 q2 c0 ; 3 3 2 pffiffiffiffiffi 2 2 wðnÞ ¼ 2k q2 tanh ð q2 nÞ 2k q2 þ c0 ; where c0, k, c are arbitrary constants, and q2 > 0, q24 ¼ 4q2 q6 . C 8 c pffiffiffiffiffi uðnÞ ¼ 4k 2 q2 tanh2 ð q2 nÞ k 2 q2 ; 3 3 2 4 pffiffiffiffiffi vðnÞ ¼ 2k 2 q2 tanh2 ð q2 nÞ þ k 2 q2 þ c0 ; 3 3 pffiffiffiffiffi wðnÞ ¼ 2k 2 q2 tanh2 ð q2 nÞ þ 2k 2 q2 þ c0 ; where c0, k, c are arbitrary constants, and q2 > 0, q24 ¼ 4q2 q6 . Remark. It is notable that the solutions (56)–(58) and (59)–(61) to systems (11) are not shown in the previous literature. Next we consider system (12). After making the transformation u(x, t) = u(n), v(x, t) = v(n), w(x, t) = w(n), n = k(x ct), system (12) becomes 8 0 > cu0 ¼ 14 k 2 u000 þ 3uu0 þ 3ðw v2 Þ > > < cv0 ¼ 12 k 2 v000 3uv0 ; > > > : cw0 ¼ 12 k 2 w000 3uw0 .
ð62Þ
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In a similar way, we also can obtain new exact solutions for system (62): rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi 92 9 8 8 q22 q22 2 pffiffiffiffiffi 2 pffiffiffiffiffi > > > > > > > > 2 q2 4q q sech ð q2 nÞ 2 q2 4q q sech ð q2 nÞ = = < < 2 6 2 6 4 4 2 2 2k q4 uðnÞ ¼ 4k q6 > > pffiffiffiffiffi > pffiffiffiffiffi > q4 q4 > > ; ; :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi sech2 ð q2 nÞ > sech2 ð q2 nÞ > 2 2 q4 4q2 q6
q4 4q2 q6
1 ð2k 2 q2 cÞ; 3 rffiffiffiffiffiffiffiffiffiffiffiffiffi 9 8 pffiffiffiffiffi q2 > > > > 2 q2 4q2 q sech2 ð q2 nÞ = < 2 6 4 þ b0 ; vðnÞ ¼ b2 > pffiffiffiffiffi > q4 > ; :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi sech2 ð q2 nÞ > 2
ð63Þ
ð64Þ
q4 4q2 q6
wðnÞ ¼ c2
8 > > <
rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi q2 2 q2 4q2 q sech2 ð q2 nÞ 4
2 6
9 > > =
þ c0 ; > > q4 2 pffiffiffiffiffi > ; :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi sech ð q2 nÞ > 2
ð65Þ
q4 4q2 q6
where b2 and c2 are determined by 8k 4 q2 q6 8k 2 q6 c þ 3k 4 q24 3b22 ¼ 0; 4
ð66Þ
2
4k q2 q4 þ 4k q4 c 3c2 þ 6b0 b2 ¼ 0;
ð67Þ
k, c, b0, c0 are arbitrary constants, and q2 > 0, q24 4q2 q6 > 0, q44 þ 3q34 ¼ q2 q4 q6 q2 q6 , or
uðnÞ ¼ 4k 2 q6
8 > > < > > :
q4 2q2
1 rffiffiffiffiffiffiffiffiffiffiffiffiffi q24 4q2 q6
4q22
92 > > = pffiffiffiffiffiffiffiffi > ; sinð2 q2 nÞ>
1 ð4k 2 q2 cÞ; 3 9 8 > > > > = < 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi r þ b0 ; vðnÞ ¼ b2 2 > > > > ; : q4 q4 4q22 q6 sinð2pffiffiffiffiffiffiffiffi q nÞ 2 2q2 4q2 9 8 > > > > = < 1 rffiffiffiffiffiffiffiffiffiffiffiffiffi þ c0 ; wðnÞ ¼ c2 2 > > > > ; : q4 q4 4q22 q6 sinð2pffiffiffiffiffiffiffiffi q2 nÞ 2q 4q 2
2k 2 q4
8 > > < > > :
q4 2q2
1 rffiffiffiffiffiffiffiffiffiffiffiffiffi
q24 4q2 q6 4q22
9 > > = pffiffiffiffiffiffiffiffi > ; sinð2 q2 nÞ> ð68Þ
ð69Þ
ð70Þ
2
where b2 and c2 are determined by (66) and (67), c, b0, c0 are arbitrary constants, and q2 < 0, q24 4q2 q6 > 0. Remark. Of course we could also use the solutions (10a) of Eq. (9) to construct exact solutions for system (12). But this kind of solutions had been obtained, we omit them here for simplicity.
3. Conclusions We have proposed an approach for finding exact solutions for nonlinear evolution equations by using the solutions (10a)–(10c) of the auxiliary ordinary differential equation (9). By this method and computerized
H. Zhang / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1120–1127
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symbolic computation, we have found some new types of exact solutions for two generalized Hirota–Satsuma coupled KdV systems. More importantly, our method is actually applicable to find new solutions to various kinds of nonlinear evolution equations, such as KdV equation, mKdV equation, Boussinesq equation, the coupled Klein–Gordon–Schrodiner equations, etc. References [1] Wadati M, Sanuki H, Konno K. Relationships among inverse method, Ba¨cklund transformation and an infinite number of conservation laws. Prog Theor Phys 1975;53:419–36. [2] Matveev VA, Salle MA. Darboux transformations and solitons. Berlin, Heidelberg: Springer-Verlag; 1991. [3] Fan EG. Extended tanh-function method and its applications to nonlinear equations. Phys Lett A 2000;277:212–8. [4] Zhou YB, Wang ML, Wang YM. Periodic wave solutions to a coupled KdV equations with variable coefficients. Phys Lett A 2003;308:31–6. [5] Wu YT, Geng XG, Hu XB, Zhu SM. A generalized Hirota–Satsuma coupled Korteweg-de Vries equation and Miura transformations. Phys Lett A 1999;255:259–64.
Short communication
New exact solutions for two generalized Hirota–Satsuma coupled KdV systems Huiqun Zhang Department of Mathematics, Qingdao University, Qingdao, Shandong 266071, China Received 19 November 2005; received in revised form 6 January 2006; accepted 6 January 2006 Available online 9 March 2006
Abstract By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct the exact solutions for nonlinear evolution equations. By this method two generalized Hirota–Satsuma coupled KdV systems are investigated and new exact solutions are explicitly obtained with the aid of symbolic computation. Ó 2006 Elsevier B.V. All rights reserved. PACS: 03.40.Kf; 02.30.Jr Keywords: Generalized Hirota–Satsuma coupled KdV systems; Exact solutions; Algebraic method; Symbolic computation
1. Introduction It is important to seek more exact solutions of nonlinear partial differential equations (NLPDEs) in mathematical physics. Many powerful methods have been presented such as Backlund transformation [1], Darboux transformation [2], the extended tanh-function method [3], the F-expansion method [4], and so on. For a given partial differential equation H ðu; ux ; ut ; uxx ; . . .Þ ¼ 0;
ð1Þ
these methods are applied to seek exact solutions by using the following transformation: uðx; tÞ ¼ F ð/1 ; /2 ; . . .Þ;
Gð/1 ; /2 ; . . .Þ ¼ 0.
ð2Þ
If one can determine the functions /i from G(/1, /2 , . . .) = 0, then he can obtain the solutions from the transformation u = F(/1, /2 , . . .). That is to say, the key idea of these generalized methods is to use the solutions of an auxiliary ordinary differential equation (or auxiliary ordinary differential equations) to construct the solutions of NLPDEs.
E-mail address: [email protected] 1007-5704/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2006.01.011
H. Zhang / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1120–1127
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In a travelling frame of reference u(x, t) = u(n), n = k(x ct), we may transform Eq. (1) into an ordinary differential equation (ODE) H ðu; ku0 ; kcu0 ; k 2 u00 ; . . .Þ ¼ 0.
ð3Þ
We expand the solution of Eq. (3) as the following series: uðnÞ ¼
n X
ai F i ðnÞ;
ð4Þ
i¼0
where F = F(n) is a solution of the following first-order ODE: m X 2 qj F j ðnÞ. F0 ¼
ð5Þ
j¼0
Balancing the highest derivative terms with the nonlinear terms in Eq. (3) will give a relation for the positive integers n and m, from which the different possible values of n and m can be determined. These values lead to a series expansions of the exact solutions for Eq. (3). For example, in the case of KdV equation ut þ 6uux þ uxxx ¼ 0;
ð6Þ
we have m ¼ n þ 2.
ð7Þ
If we take m = 4 and n = 2 in (7), we obtain the following expansion of an exact solution of the KdV Eq. (6) as: F 02 ¼ q4 F 4 þ q3 F 3 þ q2 F 2 þ q1 F þ q0 :
u ¼ a2 F 2 þ a 1 F þ a 0 ;
ð8Þ
Especially, when we take q3 = q1 = 0 in (8), we obtain the F-expansion method. In the present paper, we take m = 6 then n = 4, q5 = q3 = q1 = q0 = 0 in (5) to construct exact solutions for two generalized Hirota–Satsuma coupled KdV systems. In other words, we shall introduce the following new auxiliary ordinary differential equation 2 dF ¼ q6 F 6 ðnÞ þ q4 F 4 ðnÞ þ q2 F 2 ðnÞ; ð9Þ dn where q6, q4, q2 are real parameters, and study two generalized Hirota–Satsuma coupled KdV equations by using the following solutions of Eq. (9) 8h i12 pffiffiffiffiffi > q2 > ð1 tanhð q ; q2 > 0; q4 < 0; q24 ¼ 4q2 q6 ; ðaÞ nÞÞ > 2 q4 > > > > > rffiffiffiffiffiffiffiffiffiffiffi > 912 8 > > q2 > > > p ffiffiffi 2 2 > > > 2 sech ð q2 nÞ > = < > q2 4q2 q6 > 4 > > ; q2 > 0; q24 4q2 q6 > 0; > > > > > q4 p ffiffiffi > > < :2þ 1pffiffiffiffiffiffiffiffiffiffiffi sech2 ð q2 nÞ; q2 4q2 q6 4 ð10Þ F ðnÞ ¼ > > 4 2 > where q4 þ 3q4 ¼ q2 q4 q6 q2 q6 ðbÞ > > > > > 912 8 > > > > > > > > > = < > > > 1 > r ffiffiffiffiffiffiffiffiffiffiffi ; q2 < 0; q24 4q2 q6 > 0. ðcÞ > > > > q2 4q2 q6 q4 pffiffiffiffiffiffi > > > > 4 sinð2 q2 nÞ; : :2q2 4q2 2
As a result, we indeed successfully find some new exact solutions to two generalized Hirota–Satsuma coupled KdV equations.
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2. Exact solutions In this section, we consider two generalized Hirota–Satsuma coupled KdV equations, 8 u ¼ 1 u 3uux þ 3ðvwÞx ; > < t 2 xxx vt ¼ vxxx þ 3uvx ; > : wt ¼ wxxx þ 3uwx
ð11Þ
and 8 u ¼ 1 u 3uux þ 3ðv2 þ wÞx ; > < t 4 xxx vt ¼ 12 vxxx 3uvx ; > : wt ¼ 12 wxxx 3uwx .
ð12Þ
System (11) is a new generalized Hirota–Satsuma coupled KdV system given by Wu et al. recently [5]. It was shown that a new complex coupled KdV system and Hirota–Satsuma equation can be reduced from system (11) with w = v* and w = v, respectively. System (12) was proposed by Satsuma and Hirota. They found its three-soliton solutions and showed thatpthe equation is a special case of system (12) with ffiffiffi Hirota–Satsuma pffiffiffi w = 0 and scaling transformation x ! 2x; t ! 2t. First we consider the system (11). Making the transformation u(x, t) = u(n), v(x, t) = v(n), w(x, t) = w(n), n = k(x ct), system (5) becomes 8 0 > cu0 ¼ 12 k 2 u000 3uu0 þ 3ðvwÞ > < ð13Þ cv0 ¼ k 2 v000 þ 3uv0 ; > > : 2 000 0 0 cw ¼ k w þ 3uw ; where k and c are the wave number and wave speed, respectively. Balancing the highest-order linear terms with nonlinear terms in (13) admits the following ansatz: 8 uðnÞ ¼ a4 F 4 þ a3 F 3 þ a2 F 2 þ a1 F þ a0 ; > < ð14Þ vðnÞ ¼ b4 F 4 þ b3 F 3 þ b2 F 2 þ b1 F þ b0 ; > : 4 3 2 wðnÞ ¼ c4 F þ c3 F þ c2 F þ c1 F þ c0 ; where ai, bi, ci (i = 0, 1, 2, 3, 4) are constants to be determined later, and F = F(n) expresses the solutions (10a)– (10c) of Eq. (9). It is easy to deduce that u0 ¼ 4a4 F 3 F 0 þ 3a3 F 2 F 0 þ 2a2 FF 0 þ a1 F 0 ;
ð15Þ
uu0 ¼ ½4a4 F 7 þ 10a3 a4 F 6 þ ð6a2 a4 þ 3a23 ÞF 5 þ 5ða1 a4 þ a2 a3 ÞF 4 þ ð4a1 a3 þ 2a22 þ 4a0 a4 ÞF 3 þ ð3a1 a2 þ 3a0 a3 ÞF 2 þ ða21 þ 2a0 a2 ÞF þ a0 a1 F 0 ; 000
7
6
ð16Þ
5
u ¼ ½192a4 q6 F þ 105a3 q6 F þ 6ð8a2 q6 þ 20a4 q4 ÞF þ 5ð12a3 q4 þ 3a1 q6 ÞF
4
þ 4ð16a4 q2 þ 6a2 q4 ÞF 3 þ 3ð9a3 q2 þ 2a1 q4 ÞF 2 þ 8a2 q2 F þ a1 q2 F 0 ; 0
7
6
5
ðvwÞ ¼ ½8b4 c4 F þ 7ðb4 c3 þ b3 c4 ÞF þ 6ðb4 c2 þ b3 c3 þ b2 c4 ÞF þ 5ðb4 c1 þ b3 c2 þ b2 c3 þ b1 c4 ÞF
ð17Þ 4
þ 4ðb4 c0 þ b3 c1 þ b2 c2 þ b1 c3 þ b0 c4 ÞF 3 þ 3ðb3 c0 þ b2 c1 þ b1 c2 þ b0 c3 ÞF 2 þ 2ðb2 c0 þ b1 c1 þ b0 c2 ÞF þ ðb1 c0 þ b0 c1 ÞF 0 ; 0
3
0
2
0
0
0
v ¼ 4b4 F F þ 3b3 F F þ 2b2 FF þ b1 F ;
ð18Þ ð19Þ
v000 ¼ ½192b4 q6 F 7 þ 105b3 q6 F 6 þ 6ð8b2 q6 þ 20b4 q4 ÞF 5 þ 5ð12b3 q4 þ 3b1 q6 ÞF 4 þ 4ð16b4 q2 þ 6b2 q4 ÞF 3 þ 3ð9b3 q2 þ 2b1 q4 ÞF 2 þ 8b2 q2 F þ b1 q2 F 0 ;
ð20Þ
H. Zhang / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1120–1127
1123
uv0 ¼ ½4a4 b4 F 7 þ ð3a4 b3 þ 4a3 b4 ÞF 6 þ ð2a4 b2 þ 3a3 b3 þ 4a2 b4 ÞF 5 þ ða4 b1 þ 2a3 b2 þ 3a2 b3 þ 4a1 b4 ÞF 4 þ ða3 b1 þ 2a2 b2 þ 3a1 b3 þ 4a0 b4 ÞF 3 þ ða2 b1 þ 2a1 b2 þ 3a0 b3 ÞF 2 þ ða1 b1 þ 2a0 b2 ÞF þ a0 b1 F 0 ; ð21Þ w0 ¼ 4c4 F 3 F 0 þ 3c3 F 2 F 0 þ 2c2 FF 0 þ c1 F 0 ;
ð22Þ
w000 ¼ ½192c4 q6 F 7 þ 105c3 q6 F 6 þ 6ð8c2 q6 þ 20c4 q4 ÞF 5 þ 5ð12c3 q4 þ 3c1 q6 ÞF 4 þ 4ð16c4 q2 þ 6c2 q4 ÞF 3 þ 3ð9c3 q2 þ 2c1 q4 ÞF 2 þ 8c2 q2 F þ c1 q2 F 0 ; 0
7
6
ð23Þ 5
uw ¼ ½4a4 c4 F þ ð3a4 c3 þ 4a3 c4 ÞF þ ð2a4 c2 þ 3a3 c3 þ 4a2 c4 ÞF þ ða4 c1 þ 2a3 c2 þ 3a2 c3 þ 4a1 c4 ÞF þ ða3 c1 þ 2a2 c2 þ 3a1 c3 þ 4a0 c4 ÞF 3 þ ða2 c1 þ 2a1 c2 þ 3a0 c3 ÞF 2 þ ða1 c1 þ 2a0 c2 ÞF þ a0 c1 F 0 .
4
ð24Þ
0
Substituting (15)–(24) into (13), and setting each coefficient of FiF (i = 0, 1, 2, 3, 4, 5, 6, 7) to zero, yields a set of equations for ai, bi, ci, k, c 2b4 c4 þ 8k 2 a4 q6 a24 ¼ 0;
ð25Þ
35 2 k a3 q6 10a3 a4 þ 7ðb4 c3 þ b3 c4 Þ ¼ 0; 2 3ðb4 c2 þ b2 c4 Þ þ k 2 ð10a4 q4 þ 4a2 q6 Þ 3a2 a4 ¼ 0; 1 2 k ð4a3 q4 þ a1 q6 Þ a1 a4 a2 a3 þ b4 c1 þ b3 c2 þ b2 c3 þ b1 c4 ¼ 0; 2 6ðb4 c0 þ b2 c2 þ b0 c4 Þ þ k 2 ð16a4 q2 þ 6a2 q4 Þ 3ða22 þ 2a0 a4 Þ þ 2ca4 ¼ 0; 1 ca3 ¼ k 2 ð9a3 q2 þ 2a1 q4 Þ 3ða1 a2 þ a0 a3 Þ þ 3ðb3 c0 þ b2 c1 þ b1 c2 þ b0 c3 Þ; 2 3ðb2 c0 þ b0 c2 Þ þ 2k 2 a2 q2 3a0 a2 þ ca2 ¼ 0;
ð26Þ ð27Þ ð28Þ ð29Þ ð30Þ ð31Þ
1 ca1 ¼ k 2 a1 q2 3a0 a1 þ 3b1 c0 þ b0 c1 ; 2 a4 b4 ¼ 16k 2 b4 q6 ;
ð33Þ
3a4 b3 þ 4a3 b4 35k 2 b3 q6 ¼ 0;
ð34Þ
2
ð32Þ
a4 b2 þ 2a2 b4 ¼ k ð20b4 q4 þ 8b2 q6 Þ;
ð35Þ
2
a4 b1 þ 2a3 b2 þ 3a2 b3 þ 4a1 b4 5k ð4b3 q4 þ b1 q6 Þ ¼ 0;
ð36Þ
3ða2 b2 þ 2a0 b4 Þ 2k 2 ð16b4 q2 þ 6b2 q4 Þ þ 2cb4 ¼ 0;
ð37Þ
2
ð38Þ
a2 b1 þ 2a1 b2 þ 3a0 b3 k ð9b3 q2 þ 2b1 q4 Þ þ cb3 ¼ 0; 2
3a0 b2 4k b2 q2 þ cb2 ¼ 0;
ð39Þ
3a0 b1 k 2 b1 q2 þ cb1 ¼ 0;
ð40Þ
2
a4 c4 ¼ 16k c4 q6 ;
ð41Þ 2
3a4 c3 þ 4a3 c4 35k c3 q6 ¼ 0;
ð42Þ
a4 c2 þ 2a2 c4 ¼ k 2 ð20c4 q4 þ 8c2 q6 Þ;
ð43Þ
2
a4 c1 þ 2a3 c2 þ 3a2 c3 þ 4a1 c4 5k ð4c3 q4 þ c1 q6 Þ ¼ 0;
ð44Þ
3ða2 c2 þ 2a0 c4 Þ 2k62ð16c4 q2 þ 6c2 q4 Þ þ 2cc4 ¼ 0;
ð45Þ
2
a2 c1 þ 2a1 c2 þ 3a0 c3 k ð9c3 q2 þ 2c1 q4 Þ þ cc3 ¼ 0; 2
3a0 c2 4k c2 q2 þ cc2 ¼ 0; 2
3a0 c1 k c1 q2 þ cc1 ¼ 0.
ð46Þ ð47Þ ð48Þ
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H. Zhang / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1120–1127
Solving above algebraic equations , we obtain A a4 ¼ 8k 2 q6 ;
1 a0 ¼ ð4k 2 q2 cÞ; 3
a2 ¼ 4k 2 q4 ;
a3 ¼ a1 ¼ b4 ¼ b3 ¼ b1 ¼ c4 ¼ c3 ¼ c1 ¼ 0;
ð49Þ
32k 4 q2 q6 12k 4 q24 þ 16k 2 q6 c þ 3b2 c2 ¼ 0;
ð50Þ
8k 4 q2 q4 þ 8k 2 q4 c þ 3b2 c0 þ 3b0 c2 ¼ 0.
ð51Þ
B a4 ¼ 16k 2 q6 ;
a2 ¼ 8k 2 q4 ;
b4 ¼ c4 ¼ 8k 2 q6 ;
1 a0 ¼ ð4k 2 q2 cÞ; 3
b2 ¼ c2 ¼ 4k 2 q4 ;
a3 ¼ a1 ¼ b3 ¼ b1 ¼ c3 ¼ c1 ¼ 0;
4 b0 þ c0 ¼ ðk 2 q2 cÞ; 3
ð52Þ
q24 ¼ 4q2 q6 .
ð53Þ
C a4 ¼ 16k 2 q6 ;
a2 ¼ 8k 2 q4 ;
b4 ¼ c4 ¼ 8k 2 q6 ;
1 a0 ¼ ð4k 2 q2 cÞ; 3
b2 ¼ c2 ¼ 4k 2 q4 ;
a3 ¼ a1 ¼ b3 ¼ b1 ¼ c3 ¼ c1 ¼ 0;
4 b0 þ c0 ¼ ðc k 2 q2 Þ; 3
ð54Þ
q24 ¼ 4q2 q6 .
ð55Þ
Substituting above three of solutions with (10a)–(10c) into (14) we obtain exact solutions of system (13) as follows: A
uðnÞ ¼ 32k 2 q6
rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi q22 sech2 ð q2 nÞ q2 4q q
8 > > <
92 > > =
2 6
4
rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi q22 sech2 ð q2 nÞ q2 4q q
8 > > <
4
2 6
9 > > =
þ 8k 2 q4 > > > > q4 q4 2 pffiffiffiffiffi 2 pffiffiffiffiffi > > > > ; ; :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi : p ffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 1 sech sech nÞ nÞ ð q ð q 2 2 2 2 q4 4q2 q6
q4 4q2 q6
1 þ ð4k 2 q2 cÞ; 3 rffiffiffiffiffiffiffiffiffiffiffiffiffi 9 8 q22 2 pffiffiffiffiffi > > > > ð q sech nÞ = < 2 2 q4 4q2 q6 þ b0 ; vðnÞ ¼ 2b2 > pffiffiffiffiffi > q4 > ; :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi sech2 ð q2 nÞ > 2
ð56Þ
ð57Þ
q4 4q2 q6
wðnÞ ¼ 2c2
rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi q22 sech2 ð q2 nÞ q2 4q q
8 > > <
4
9 > > =
2 6
þ c0 ; > > q4 2 pffiffiffiffiffi > ; :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi sech ð q2 nÞ > 2
ð58Þ
q4 4q2 q6
where b2 and c2 are determined by (50) and (51), c, b0, c0 are arbitrary constants, and q2 > 0, q24 4q2 q6 > 0, q44 þ 3q34 ¼ q2 q4 q6 q2 q6 , or
uðnÞ ¼ 8k 2 q6
8 > > < > > : q4 2q 2
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi q24 4q2 q6 4q22
92 > > = pffiffiffiffiffiffiffiffi > ; sinð2 q2 nÞ>
þ 4k 2 q4
8 > > < > > : q4 2q 2
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi q24 4q2 q6 4q22
9 > > = pffiffiffiffiffiffiffiffi > ; sinð2 q2 nÞ>
1 þ ð4k 2 q2 cÞ; 3
ð59Þ
H. Zhang / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1120–1127
8 > > <
1 rffiffiffiffiffiffiffiffiffiffiffiffiffi
9 > > =
þ b0 ; pffiffiffiffiffiffiffiffi > > sinð2 q2 nÞ; 4q22 9 8 > > > > = < 1 rffiffiffiffiffiffiffiffiffiffiffiffiffi þ c0 ; wðnÞ ¼ c2 2 > > > > ; : q4 q4 4q22 q6 sinð2pffiffiffiffiffiffiffiffi q2 nÞ 2q 4q
vðnÞ ¼ b2
> > :
q4 2q2
2
1125
ð60Þ
q24 4q2 q6
ð61Þ
2
where b2 and c2 are determined by (50) and (51), k, c, b0, c0 are arbitrary constants, and q2 < 0, q24 4q2 q6 > 0, or 2 1 pffiffiffiffiffi uðnÞ ¼ 2k 2 q2 tanh2 ð q2 nÞ k 2 q2 c; 3 3 2 2 4k q4 ðc k q2 Þ 8k 2 q4 c2 ðk 2 q2 cÞ 4k 2 c0 ðk 2 q24 4q6 cÞ pffiffiffiffiffi vðnÞ ¼ ½1 tanhð q2 nÞ þ ; 3c2 3c22 q c2 pffiffiffiffiffi wðnÞ ¼ 2 ½1 tanhð q2 nÞ þ c0 ; q4 where c2, c0, k, c are arbitrary constants, and q2 > 0, q24 ¼ 4q2 q6 . B 8 c pffiffiffiffiffi uðnÞ ¼ 4k 2 q2 tanh2 ð q2 nÞ k 2 q2 ; 3 3 2 4 pffiffiffiffiffi vðnÞ ¼ 2k 2 q2 tanh2 ð q2 nÞ k 2 q2 c0 ; 3 3 2 pffiffiffiffiffi 2 2 wðnÞ ¼ 2k q2 tanh ð q2 nÞ 2k q2 þ c0 ; where c0, k, c are arbitrary constants, and q2 > 0, q24 ¼ 4q2 q6 . C 8 c pffiffiffiffiffi uðnÞ ¼ 4k 2 q2 tanh2 ð q2 nÞ k 2 q2 ; 3 3 2 4 pffiffiffiffiffi vðnÞ ¼ 2k 2 q2 tanh2 ð q2 nÞ þ k 2 q2 þ c0 ; 3 3 pffiffiffiffiffi wðnÞ ¼ 2k 2 q2 tanh2 ð q2 nÞ þ 2k 2 q2 þ c0 ; where c0, k, c are arbitrary constants, and q2 > 0, q24 ¼ 4q2 q6 . Remark. It is notable that the solutions (56)–(58) and (59)–(61) to systems (11) are not shown in the previous literature. Next we consider system (12). After making the transformation u(x, t) = u(n), v(x, t) = v(n), w(x, t) = w(n), n = k(x ct), system (12) becomes 8 0 > cu0 ¼ 14 k 2 u000 þ 3uu0 þ 3ðw v2 Þ > > < cv0 ¼ 12 k 2 v000 3uv0 ; > > > : cw0 ¼ 12 k 2 w000 3uw0 .
ð62Þ
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In a similar way, we also can obtain new exact solutions for system (62): rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi 92 9 8 8 q22 q22 2 pffiffiffiffiffi 2 pffiffiffiffiffi > > > > > > > > 2 q2 4q q sech ð q2 nÞ 2 q2 4q q sech ð q2 nÞ = = < < 2 6 2 6 4 4 2 2 2k q4 uðnÞ ¼ 4k q6 > > pffiffiffiffiffi > pffiffiffiffiffi > q4 q4 > > ; ; :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi sech2 ð q2 nÞ > sech2 ð q2 nÞ > 2 2 q4 4q2 q6
q4 4q2 q6
1 ð2k 2 q2 cÞ; 3 rffiffiffiffiffiffiffiffiffiffiffiffiffi 9 8 pffiffiffiffiffi q2 > > > > 2 q2 4q2 q sech2 ð q2 nÞ = < 2 6 4 þ b0 ; vðnÞ ¼ b2 > pffiffiffiffiffi > q4 > ; :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi sech2 ð q2 nÞ > 2
ð63Þ
ð64Þ
q4 4q2 q6
wðnÞ ¼ c2
8 > > <
rffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi q2 2 q2 4q2 q sech2 ð q2 nÞ 4
2 6
9 > > =
þ c0 ; > > q4 2 pffiffiffiffiffi > ; :2 þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi sech ð q2 nÞ > 2
ð65Þ
q4 4q2 q6
where b2 and c2 are determined by 8k 4 q2 q6 8k 2 q6 c þ 3k 4 q24 3b22 ¼ 0; 4
ð66Þ
2
4k q2 q4 þ 4k q4 c 3c2 þ 6b0 b2 ¼ 0;
ð67Þ
k, c, b0, c0 are arbitrary constants, and q2 > 0, q24 4q2 q6 > 0, q44 þ 3q34 ¼ q2 q4 q6 q2 q6 , or
uðnÞ ¼ 4k 2 q6
8 > > < > > :
q4 2q2
1 rffiffiffiffiffiffiffiffiffiffiffiffiffi q24 4q2 q6
4q22
92 > > = pffiffiffiffiffiffiffiffi > ; sinð2 q2 nÞ>
1 ð4k 2 q2 cÞ; 3 9 8 > > > > = < 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi r þ b0 ; vðnÞ ¼ b2 2 > > > > ; : q4 q4 4q22 q6 sinð2pffiffiffiffiffiffiffiffi q nÞ 2 2q2 4q2 9 8 > > > > = < 1 rffiffiffiffiffiffiffiffiffiffiffiffiffi þ c0 ; wðnÞ ¼ c2 2 > > > > ; : q4 q4 4q22 q6 sinð2pffiffiffiffiffiffiffiffi q2 nÞ 2q 4q 2
2k 2 q4
8 > > < > > :
q4 2q2
1 rffiffiffiffiffiffiffiffiffiffiffiffiffi
q24 4q2 q6 4q22
9 > > = pffiffiffiffiffiffiffiffi > ; sinð2 q2 nÞ> ð68Þ
ð69Þ
ð70Þ
2
where b2 and c2 are determined by (66) and (67), c, b0, c0 are arbitrary constants, and q2 < 0, q24 4q2 q6 > 0. Remark. Of course we could also use the solutions (10a) of Eq. (9) to construct exact solutions for system (12). But this kind of solutions had been obtained, we omit them here for simplicity.
3. Conclusions We have proposed an approach for finding exact solutions for nonlinear evolution equations by using the solutions (10a)–(10c) of the auxiliary ordinary differential equation (9). By this method and computerized
H. Zhang / Communications in Nonlinear Science and Numerical Simulation 12 (2007) 1120–1127
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symbolic computation, we have found some new types of exact solutions for two generalized Hirota–Satsuma coupled KdV systems. More importantly, our method is actually applicable to find new solutions to various kinds of nonlinear evolution equations, such as KdV equation, mKdV equation, Boussinesq equation, the coupled Klein–Gordon–Schrodiner equations, etc. References [1] Wadati M, Sanuki H, Konno K. Relationships among inverse method, Ba¨cklund transformation and an infinite number of conservation laws. Prog Theor Phys 1975;53:419–36. [2] Matveev VA, Salle MA. Darboux transformations and solitons. Berlin, Heidelberg: Springer-Verlag; 1991. [3] Fan EG. Extended tanh-function method and its applications to nonlinear equations. Phys Lett A 2000;277:212–8. [4] Zhou YB, Wang ML, Wang YM. Periodic wave solutions to a coupled KdV equations with variable coefficients. Phys Lett A 2003;308:31–6. [5] Wu YT, Geng XG, Hu XB, Zhu SM. A generalized Hirota–Satsuma coupled Korteweg-de Vries equation and Miura transformations. Phys Lett A 1999;255:259–64.