- Email: [email protected]
Soliton solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled MKdV equation
9 April 2001
Physics Letters A 282 (2001) 18–22 www.elsevier.nl/locate/pla
Soliton solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled MKdV equation Engui Fan Institute of Mathematics, Fudan University, Shanghai 200433, PR China Received 20 November 2000; received in revised form 22 February 2001; accepted 23 February 2001 Communicated by A.R. Bishop
Abstract We make use of an extended tanh-function method and symbolic computation to obtain respectively four kinds of soliton solutions for a new generalized Hirota–Satsuma coupled KdV equation and a new coupled MKdV equation, which were introduced recently by Wu et al. (Phys. Lett. A 255 (1999) 259). 2001 Elsevier Science B.V. All rights reserved. Keywords: Coupled MKdV equation; Soliton solution; Riccati equation; Symbolic computation
1. Introduction Recently, by introducing a 4 × 4 matrix spectral problem with three potentials, Wu et al. derived a new hierarchy of nonlinear evolution equations [1]. Two typical equations in the hierarchy are a new generalized Hirota–Satsuma coupled KdV equation 1 ut = uxxx − 3uux + 3(vw)x , 2 vt = −vxxx + 3uvx , wt = −wxxx + 3uwx ,
(1)
and a new coupled MKdV equation 1 3 ut = uxxx − 3u2 ux + vxx + 3(uv)x − 3λux , 2 2 vt = −vxxx − 3vvx − 3ux vx + 3u2 vx + 3λvx .
H (u, ux , ut , uxx , . . .) = 0, (2)
With w = v ∗ and w = v, Eq. (1) reduces respectively to a new complex coupled KdV equation [1] and E-mail address: [email protected] (E. Fan).
the Hirota–Satsuma equation [2,3]. Eq. (2) becomes a generalized KdV equation for u = 0 and the MKdV equation for v = 0, respectively. The soliton solutions for these two equations are still unknown. The aim of this Letter is to construct four kinds of soliton solutions for them by using an extended tanh-function method and symbolic computation [4]. The key idea of this method is to take full advantages of a Riccati equation involving a parameter and use its solutions to replace the tanh-function in the tanh-function method [5–7], which simply proceeds as follows. For a given partial differential equation, say, in two independent variables, (3)
we first consider its travelling solutions u(x, t) = u(ξ ) = x + βt, then Eq. (3) becomes an ordinary differential equation. The next crucial step is that the solution we are looking for is expressed in the form u(ξ ) =
m i=0
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 1 6 1 - X
ai ϕ i ,
(4)
E. Fan / Physics Letters A 282 (2001) 18–22
with
19
and
ϕ = k 1 − ϕ2 ,
(5)
where k is a parameter, ϕ = ϕ(ξ ), ϕ = dϕ/dξ . The parameter m can be found by balancing the highestorder linear term with the nonlinear terms. Substituting (4) and (5) into the relevant ordinary differential equation will yield a set of algebraic equations with respect to ai , k, β because the coefficients of ϕ i have to vanish. From these relations ai , k, β can be determined. The Riccati equation (5) has the general solutions ϕ = tanh(kξ ),
ϕ = coth(kξ ).
(6)
The algorithm presented here is also a computerizable method, in which generating an algebraic system from Eq. (3) and solving it are two key procedures and laborious to do by hand. But they can be implemented on a computer with the help of Mathematica. The outputs of solving the algebraic system from a computer comprise a list of the form {β, k, a0, . . .}. In general, if β or any of the parameters is left unspecified, then it is to be regard as arbitrary for the solution of Eq. (3). For simplification, singular coth-type soliton solutions are omitted in this Letter, since they always appear in pairs with tanh-type solutions according to (6).
u = a0 + a1 ϕ + a2 ϕ 2 , v = b0 + b1 ϕ, w = c0 + c1 ϕ.
(9)
Substituting (8) into Eq. (7) and using Mathematica yields a set of algebraic system for ai , bi , ci (i = 0, 1, 2), k and β, namely βka1 + k 3 a1 + 3ka0 a1 − 3kb1c0 − 3kb0 c1 = 0, 3ka12 + 2βka2 + 8k 3 a2 + 6ka0 a2 − 6kb2 c0 − 6kb1c1 − 6kb0 c2 = 0, −βka1 − 4k 3 a1 − 3ka0 a1 + 9ka1 a2 + 3kb1 c0 + 3kb0c1 − 9kb2 c1 − 9kb1 c2 = 0, −3ka12 − 2βka2 − 20k 3a2 − 6ka0a2 + 6ka22 + 6kb2c0 + 6kb1 c1 + 6kb0 c2 − 12kb2c2 = 0, 3k 3 a1 − 9ka1a2 + 9kb2c1 + 9kb1c2 = 0, 12k 3a2 − 6ka22 + 12kb2c2 = 0, βkb1 − 2k 3 b1 − 3ka0b1 = 0, −3ka1b1 + 2βkb2 − 16k 3 b2 − 6ka0 b2 = 0, −βkb1 + 8k 3 β1 + 3ka0b1 − 3ka2 b1 − 6ka1 b2 = 0, 3ka1b1 − 2βkb2 + 40k 3b2 + 6ka0 b2 − 6ka2 b2 = 0, −6kb1 + 3ka2b1 + 6ka1b2 = 0, −24k 3b2 + 6ka2 b2 = 0,
2. Soliton solutions for the generalized Hirota–Satsuma coupled KdV equation
βkc1 − 2k 3 c1 − 3ka0 c1 = 0,
To look for the travelling wave solution of Eq. (1), we make the transformation u(x, t) = u(ξ ), v(x, t) = v(ξ ), w(x, t) = w(ξ ), ξ = x + βt and change Eq. (1) into the form 1 βu = u − 3uu + 3(vw) , 2 βv = −v + 3uv , βw = −w + 3uw .
−3ka1c1 + 2βkc2 − 16k 3 c2 − 6ka0 c2 = 0, −βkc1 + 8k 3 c1 + 3ka0 c1 − 3ka2 c1 − 6ka1 c2 = 0, 3ka1c1 − 2βkc2 + 40k 3 c2 + 6ka0 c2 − 6ka2c2 − 6k 3c1 + 3ka2 c1 + 6ka1c2 = 0, for which, with the aid of Mathematica, we find
(7)
Balancing the highest-order linear terms and nonlinear terms in Eq. (7) gives the following two ansätze: u = a0 + a1 ϕ + a2 ϕ 2 , v = b0 + b1 ϕ + b2 ϕ 2 , w = c0 + c1 ϕ + c2 ϕ 2 ,
−24k 3c2 + 6ka2 c2 = 0,
(8)
1 β − 8k 2 , a2 = 4k 2 , 3 4(3k 4c0 − 2βk 2 c2 + 4k 4 c2 ) , b0 = − 3c22 a1 = b1 = c1 = 0,
a0 =
b2 =
4k 4 , c2 (10)
where c0 , k, β and c2 = 0 are arbitrary constants. Then from (6), (8) and (10) we obtain the soliton solutions
20
E. Fan / Physics Letters A 282 (2001) 18–22
of Eq. (1) of bell-type for all u, v and w: 1 u = β − 8k 2 + 4k 2 tanh2 k(x + βt) , 3 4(3k 4c0 − 2βk 2 c2 + 4k 4 c2 ) v=− 3c22
3. Soliton solutions for the coupled MKdV equation
4k 2 tanh2 k(x + βt) , c2 w = c0 + c2 tanh2 k(x + βt) .
1 3 βu = u − 3u2 u + v + 3(uv) − 3λu , 2 2 βv = −v − 3vv − 3u v + 3u2 v + 3λv .
Let u(x, t) = u(ξ ), v(x, t) = v(ξ ), ξ = x + βt, then Eq. (2) becomes
+
Similarly, substituting ansatz (9) into (7) yields βka1 + k 3 a1 + 3ka0a1 − 3kb1 c0 − 3kb0c1 = 0,
Balancing the highest-order linear terms and nonlinear terms leads to the following two ansätze:
3ka12 + 2βka2 + 8k 3 a2 + 6ka0a2 − 6kb1c1 = 0,
u = a0 + a1 ϕ,
−βka1 − 4k a1 − 3ka0a1 + 9ka1a2 + 3kb1 c0
and
3
+ 3kb0c1 = 0, −3ka12 − 2βka2 − 20k 3 a2 − 6ka0 a2 + 6ka22 + 6kb1c1 = 0,
u = a0 + a1 ϕ,
v = b0 + b1 ϕ,
(12)
v = b0 + b1 ϕ + b2 ϕ 2 .
(13)
Substituting (12) into (11) gives
3k a1 − 9ka1 a2 = 0, 2
12k
3
a2 − 6ka22
= 0,
βkb1 − 2k b1 − 3ka0 b1 = 0, 3
−βkb1 + 8k 3 b1 + 3ka0b1 − 3ka2b1 = 0, 3ka1 b1 = 0,
3ka1c1 = 0,
−6k 3 b1 + 3ka2 b1 = 0, βkc1 − 2k 3 c1 − 3ka0c1 = 0,
βka1 + k 3 a1 + 3kλa1 + 3ka02a1 − 3ka1b0 − 3ka0b1 = 0, 6ka0a12 + 3k 2 b1 − 6ka1 b1 = 0, −βka1 − 4k 3 a1 − 3kλa1 − 3ka02 a1 + 3ka13 + 3ka1b0 + 3ka0b1 = 0,
βkc1 + 8k 3 c1 + 3ka0c1 − 3ka2c1 = 0,
−6ka0a12 − 3k 2 b1 + 6ka1 b1 = 0,
−6k 3 c1 + 3ka2 c1 = 0,
3k 3 a1 − 3ka13 = 0,
which has solutions 1 a1 = 0, a0 = β − 2k 2 , a2 = 2k 2 , 3 4k 2c0 (β + k 2 ) 4k 2 (β + k 2 ) , b = , b0 = − 1 3c1 3c12
−3kλa1 + βkb1 − 2k 3 b1 − 3ka02 b1 + 3k 2 a1 b1
where c0 , β, k and c1 = 0 are arbitrary constants. In this way, we also find another soliton solution of Eq. (1) of bell-type for u but kink-type for v, w: 1 u = β − 2k 2 + 2k 2 tanh2 k(x + βt) , 3 4k 2 c0 (β + k 2 ) v=− 3c12 4k 2 (β + k 2 ) tanh k(x + βt) , 3c1 w = c0 + c1 tanh k(x + βt) . +
(11)
+ 3kb0b1 = 0, −6ka0a1 b1 + 3kb12 = 0,
6ka0 a1 b1 − 3kb12 = 0,
3kλa1 − βkb1 + 8k 3 b1 + 3k02 b1 − 6k 2 a1 b1 − 3ka12b1 − 3kb0 b1 = 0, −6k 3b1 + 3k 2 a1 b1 + 3ka12b1 = 0, for which, with the aid of Mathematica, we find k b1 λ a1 = k, 1+ , a0 = , b0 = 2k 2 b1 1 6kλ 3b12 2 β= (14) −4k − 6λ + + 2 , 4 b1 k where b0 = 0, k = 0 are arbitrary constants. By using (12) and (14), we get kink-type soliton solutions of
E. Fan / Physics Letters A 282 (2001) 18–22
Eq. (2) for both u and v:
type for u but bell-type for v:
b1 + k tanh(kξ ), 2k k λ 1+ + b1 tanh(kξ ), v= 2 b1
u = k tanh(kξ ), 1 v = 4k 2 + λ − 2k 2 tanh2 (kξ ), 2 with 1 ξ = x + −2k 2 − 3λ t. 2
u=
with ξ =x+
6kλ 3b12 1 + 2 t. −4k 2 − 6λ + 4 b1 k
For ansatz (13), with help of Mathematica, we have βka1 + k 3 a1 + 3kλa1 + 3ka02a1 − 3ka1 b0 − 3ka0b1 − 3k 2 b2 = 0, 6ka0 a12 + 3k 2 b1 − 6ka1b1 − 6ka0b2 = 0, −βka1 − 4k 3 a1 − 3kλa1 − 3ka02a1 + 3ka13 + 3ka1b0 + 3ka0b1 + 12k 2 b2 − 9ka1 b2 = 0, −6ka0a12 − 3k 2 b1 + 6ka1b1 + 6ka0 b2 = 0, 3k 3 a1 − 3ka13 − 9k 2 b2 + 9ka1 b2 = 0, −3kλa1 + βkb1 − 2k 3 b1 − 3ka02b1 + 3k 2 a1 b1 + 3kb0b1 = 0, −6ka0a1 b1 + 3kb12 + 2βkb2 − 16k 3 b2 − 6ka02b2 + 6k 2a1 b2 + 6kb0b2 = 0,
21
Remark. We have found four kinds of soliton solutions for the new generalized Hirota–Satsuma coupled KdV equation (1) and the new coupled KdV equation (2) by using a Riccati equation and symbolic computation. Two kinds of them are singular soliton solutions. Such solutions develop a singularity at a finite point, i.e., for any fixed t = t0 , there exist x0 at which these solutions blow up. There is much current interest in the formation of so-called “hot spots” or “blow up” of solutions [8–10]. It appears that these singular solutions will model this physical phenomena. The method used in this Letter has some merits in contrast with the tanh-function method. It not only uses a more simple algorithm to produce an algebraic system but also can pick up singular soliton solutions with no extra effort. In addition, this method is also computerizable, which allow us to perform complicated and tedious algebraic calculation on a computer.
3kλa1 − βkb1 + 8k 3 b1 + 3ka02b1 − 6k 2 a1 b1 − 3ka12b1 − 3kb0b1 − 12ka0a1 b2 + 9kb1b2 = 0,
Acknowledgements
6ka0 a1 b1 − 3kb12 − 2βkb2 + 40k 3b2 + 6ka02 b2 − 12k 2a1 b2 − 6ka12 b2 − 6kb0b2 + 6kb22 = 0, −6k 3 b1 + 3k 2 a1 b1 + 3ka12b1 + 12ka0a1 b2 − 9kb1b2 = 0, −24k 3b2 + 6k 2 a1 b2 + 6ka 2b2 − 6kb22 = 0. Solving these equations by means of Mathematica gives a1 = k, a0 = 0, 1 b0 = 4k 2 + λ , b1 = 0, b2 = −2k 2 , 2 1 β = −2k 2 − 3λ , 2 where k is an arbitrary constant. Then we find that another kind of soliton solution for Eq. (2) is of kink-
I am grateful to Professor Gu Chaohao, Professor Hu Hesheng and Professor Zhou Zixiang for their enthusiastic guidance and help. I also would like to express my sincere thanks to the referees for their helpful suggestions. This work has been supported by Chinese Basic Research Plan “Mathematics Mechanization and a Platform for Automated Reasoning”, the Postdoctoral Science Foundation of China and the Shanghai Postdoctoral Science Foundation of China.
References [1] Y.T. Wu, X.G. Geng, X.B. Hu, S.M. Zhu, Phys. Lett. A 255 (1999) 259. [2] R. Hirota, J. Satsuma, Phys. Lett. A 85 (1981) 407. [3] J. Satsuma, R. Hirota, J. Phys. Soc. Jpn. 51 (1982) 332.
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E. Fan / Physics Letters A 282 (2001) 18–22
[4] E.G. Fan, H.Q. Zhang, Phys. Lett. A 246 (1998) 403. [5] W. Malfliet, Am. J. Phys. 60 (1992) 650. [6] E.J. Parkes, B.R. Duffy, Comput. Phys. Commun. 98 (1996) 288.
[7] [8] [9] [10]
E.J. Parkes, B.R. Duffy, Phys. Lett. A 229 (1997) 217. N.F. Smyth, J. Austral. Math. Soc. Ser. B 33 (1992) 403. P.A. Clarkson, E.L. Mansfield, Physica D 70 (1993) 250. N.A. Kudryashov, D. Zargayan, J. Phys. A 29 (1996) 8067.
Physics Letters A 282 (2001) 18–22 www.elsevier.nl/locate/pla
Soliton solutions for a generalized Hirota–Satsuma coupled KdV equation and a coupled MKdV equation Engui Fan Institute of Mathematics, Fudan University, Shanghai 200433, PR China Received 20 November 2000; received in revised form 22 February 2001; accepted 23 February 2001 Communicated by A.R. Bishop
Abstract We make use of an extended tanh-function method and symbolic computation to obtain respectively four kinds of soliton solutions for a new generalized Hirota–Satsuma coupled KdV equation and a new coupled MKdV equation, which were introduced recently by Wu et al. (Phys. Lett. A 255 (1999) 259). 2001 Elsevier Science B.V. All rights reserved. Keywords: Coupled MKdV equation; Soliton solution; Riccati equation; Symbolic computation
1. Introduction Recently, by introducing a 4 × 4 matrix spectral problem with three potentials, Wu et al. derived a new hierarchy of nonlinear evolution equations [1]. Two typical equations in the hierarchy are a new generalized Hirota–Satsuma coupled KdV equation 1 ut = uxxx − 3uux + 3(vw)x , 2 vt = −vxxx + 3uvx , wt = −wxxx + 3uwx ,
(1)
and a new coupled MKdV equation 1 3 ut = uxxx − 3u2 ux + vxx + 3(uv)x − 3λux , 2 2 vt = −vxxx − 3vvx − 3ux vx + 3u2 vx + 3λvx .
H (u, ux , ut , uxx , . . .) = 0, (2)
With w = v ∗ and w = v, Eq. (1) reduces respectively to a new complex coupled KdV equation [1] and E-mail address: [email protected] (E. Fan).
the Hirota–Satsuma equation [2,3]. Eq. (2) becomes a generalized KdV equation for u = 0 and the MKdV equation for v = 0, respectively. The soliton solutions for these two equations are still unknown. The aim of this Letter is to construct four kinds of soliton solutions for them by using an extended tanh-function method and symbolic computation [4]. The key idea of this method is to take full advantages of a Riccati equation involving a parameter and use its solutions to replace the tanh-function in the tanh-function method [5–7], which simply proceeds as follows. For a given partial differential equation, say, in two independent variables, (3)
we first consider its travelling solutions u(x, t) = u(ξ ) = x + βt, then Eq. (3) becomes an ordinary differential equation. The next crucial step is that the solution we are looking for is expressed in the form u(ξ ) =
m i=0
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 1 6 1 - X
ai ϕ i ,
(4)
E. Fan / Physics Letters A 282 (2001) 18–22
with
19
and
ϕ = k 1 − ϕ2 ,
(5)
where k is a parameter, ϕ = ϕ(ξ ), ϕ = dϕ/dξ . The parameter m can be found by balancing the highestorder linear term with the nonlinear terms. Substituting (4) and (5) into the relevant ordinary differential equation will yield a set of algebraic equations with respect to ai , k, β because the coefficients of ϕ i have to vanish. From these relations ai , k, β can be determined. The Riccati equation (5) has the general solutions ϕ = tanh(kξ ),
ϕ = coth(kξ ).
(6)
The algorithm presented here is also a computerizable method, in which generating an algebraic system from Eq. (3) and solving it are two key procedures and laborious to do by hand. But they can be implemented on a computer with the help of Mathematica. The outputs of solving the algebraic system from a computer comprise a list of the form {β, k, a0, . . .}. In general, if β or any of the parameters is left unspecified, then it is to be regard as arbitrary for the solution of Eq. (3). For simplification, singular coth-type soliton solutions are omitted in this Letter, since they always appear in pairs with tanh-type solutions according to (6).
u = a0 + a1 ϕ + a2 ϕ 2 , v = b0 + b1 ϕ, w = c0 + c1 ϕ.
(9)
Substituting (8) into Eq. (7) and using Mathematica yields a set of algebraic system for ai , bi , ci (i = 0, 1, 2), k and β, namely βka1 + k 3 a1 + 3ka0 a1 − 3kb1c0 − 3kb0 c1 = 0, 3ka12 + 2βka2 + 8k 3 a2 + 6ka0 a2 − 6kb2 c0 − 6kb1c1 − 6kb0 c2 = 0, −βka1 − 4k 3 a1 − 3ka0 a1 + 9ka1 a2 + 3kb1 c0 + 3kb0c1 − 9kb2 c1 − 9kb1 c2 = 0, −3ka12 − 2βka2 − 20k 3a2 − 6ka0a2 + 6ka22 + 6kb2c0 + 6kb1 c1 + 6kb0 c2 − 12kb2c2 = 0, 3k 3 a1 − 9ka1a2 + 9kb2c1 + 9kb1c2 = 0, 12k 3a2 − 6ka22 + 12kb2c2 = 0, βkb1 − 2k 3 b1 − 3ka0b1 = 0, −3ka1b1 + 2βkb2 − 16k 3 b2 − 6ka0 b2 = 0, −βkb1 + 8k 3 β1 + 3ka0b1 − 3ka2 b1 − 6ka1 b2 = 0, 3ka1b1 − 2βkb2 + 40k 3b2 + 6ka0 b2 − 6ka2 b2 = 0, −6kb1 + 3ka2b1 + 6ka1b2 = 0, −24k 3b2 + 6ka2 b2 = 0,
2. Soliton solutions for the generalized Hirota–Satsuma coupled KdV equation
βkc1 − 2k 3 c1 − 3ka0 c1 = 0,
To look for the travelling wave solution of Eq. (1), we make the transformation u(x, t) = u(ξ ), v(x, t) = v(ξ ), w(x, t) = w(ξ ), ξ = x + βt and change Eq. (1) into the form 1 βu = u − 3uu + 3(vw) , 2 βv = −v + 3uv , βw = −w + 3uw .
−3ka1c1 + 2βkc2 − 16k 3 c2 − 6ka0 c2 = 0, −βkc1 + 8k 3 c1 + 3ka0 c1 − 3ka2 c1 − 6ka1 c2 = 0, 3ka1c1 − 2βkc2 + 40k 3 c2 + 6ka0 c2 − 6ka2c2 − 6k 3c1 + 3ka2 c1 + 6ka1c2 = 0, for which, with the aid of Mathematica, we find
(7)
Balancing the highest-order linear terms and nonlinear terms in Eq. (7) gives the following two ansätze: u = a0 + a1 ϕ + a2 ϕ 2 , v = b0 + b1 ϕ + b2 ϕ 2 , w = c0 + c1 ϕ + c2 ϕ 2 ,
−24k 3c2 + 6ka2 c2 = 0,
(8)
1 β − 8k 2 , a2 = 4k 2 , 3 4(3k 4c0 − 2βk 2 c2 + 4k 4 c2 ) , b0 = − 3c22 a1 = b1 = c1 = 0,
a0 =
b2 =
4k 4 , c2 (10)
where c0 , k, β and c2 = 0 are arbitrary constants. Then from (6), (8) and (10) we obtain the soliton solutions
20
E. Fan / Physics Letters A 282 (2001) 18–22
of Eq. (1) of bell-type for all u, v and w: 1 u = β − 8k 2 + 4k 2 tanh2 k(x + βt) , 3 4(3k 4c0 − 2βk 2 c2 + 4k 4 c2 ) v=− 3c22
3. Soliton solutions for the coupled MKdV equation
4k 2 tanh2 k(x + βt) , c2 w = c0 + c2 tanh2 k(x + βt) .
1 3 βu = u − 3u2 u + v + 3(uv) − 3λu , 2 2 βv = −v − 3vv − 3u v + 3u2 v + 3λv .
Let u(x, t) = u(ξ ), v(x, t) = v(ξ ), ξ = x + βt, then Eq. (2) becomes
+
Similarly, substituting ansatz (9) into (7) yields βka1 + k 3 a1 + 3ka0a1 − 3kb1 c0 − 3kb0c1 = 0,
Balancing the highest-order linear terms and nonlinear terms leads to the following two ansätze:
3ka12 + 2βka2 + 8k 3 a2 + 6ka0a2 − 6kb1c1 = 0,
u = a0 + a1 ϕ,
−βka1 − 4k a1 − 3ka0a1 + 9ka1a2 + 3kb1 c0
and
3
+ 3kb0c1 = 0, −3ka12 − 2βka2 − 20k 3 a2 − 6ka0 a2 + 6ka22 + 6kb1c1 = 0,
u = a0 + a1 ϕ,
v = b0 + b1 ϕ,
(12)
v = b0 + b1 ϕ + b2 ϕ 2 .
(13)
Substituting (12) into (11) gives
3k a1 − 9ka1 a2 = 0, 2
12k
3
a2 − 6ka22
= 0,
βkb1 − 2k b1 − 3ka0 b1 = 0, 3
−βkb1 + 8k 3 b1 + 3ka0b1 − 3ka2b1 = 0, 3ka1 b1 = 0,
3ka1c1 = 0,
−6k 3 b1 + 3ka2 b1 = 0, βkc1 − 2k 3 c1 − 3ka0c1 = 0,
βka1 + k 3 a1 + 3kλa1 + 3ka02a1 − 3ka1b0 − 3ka0b1 = 0, 6ka0a12 + 3k 2 b1 − 6ka1 b1 = 0, −βka1 − 4k 3 a1 − 3kλa1 − 3ka02 a1 + 3ka13 + 3ka1b0 + 3ka0b1 = 0,
βkc1 + 8k 3 c1 + 3ka0c1 − 3ka2c1 = 0,
−6ka0a12 − 3k 2 b1 + 6ka1 b1 = 0,
−6k 3 c1 + 3ka2 c1 = 0,
3k 3 a1 − 3ka13 = 0,
which has solutions 1 a1 = 0, a0 = β − 2k 2 , a2 = 2k 2 , 3 4k 2c0 (β + k 2 ) 4k 2 (β + k 2 ) , b = , b0 = − 1 3c1 3c12
−3kλa1 + βkb1 − 2k 3 b1 − 3ka02 b1 + 3k 2 a1 b1
where c0 , β, k and c1 = 0 are arbitrary constants. In this way, we also find another soliton solution of Eq. (1) of bell-type for u but kink-type for v, w: 1 u = β − 2k 2 + 2k 2 tanh2 k(x + βt) , 3 4k 2 c0 (β + k 2 ) v=− 3c12 4k 2 (β + k 2 ) tanh k(x + βt) , 3c1 w = c0 + c1 tanh k(x + βt) . +
(11)
+ 3kb0b1 = 0, −6ka0a1 b1 + 3kb12 = 0,
6ka0 a1 b1 − 3kb12 = 0,
3kλa1 − βkb1 + 8k 3 b1 + 3k02 b1 − 6k 2 a1 b1 − 3ka12b1 − 3kb0 b1 = 0, −6k 3b1 + 3k 2 a1 b1 + 3ka12b1 = 0, for which, with the aid of Mathematica, we find k b1 λ a1 = k, 1+ , a0 = , b0 = 2k 2 b1 1 6kλ 3b12 2 β= (14) −4k − 6λ + + 2 , 4 b1 k where b0 = 0, k = 0 are arbitrary constants. By using (12) and (14), we get kink-type soliton solutions of
E. Fan / Physics Letters A 282 (2001) 18–22
Eq. (2) for both u and v:
type for u but bell-type for v:
b1 + k tanh(kξ ), 2k k λ 1+ + b1 tanh(kξ ), v= 2 b1
u = k tanh(kξ ), 1 v = 4k 2 + λ − 2k 2 tanh2 (kξ ), 2 with 1 ξ = x + −2k 2 − 3λ t. 2
u=
with ξ =x+
6kλ 3b12 1 + 2 t. −4k 2 − 6λ + 4 b1 k
For ansatz (13), with help of Mathematica, we have βka1 + k 3 a1 + 3kλa1 + 3ka02a1 − 3ka1 b0 − 3ka0b1 − 3k 2 b2 = 0, 6ka0 a12 + 3k 2 b1 − 6ka1b1 − 6ka0b2 = 0, −βka1 − 4k 3 a1 − 3kλa1 − 3ka02a1 + 3ka13 + 3ka1b0 + 3ka0b1 + 12k 2 b2 − 9ka1 b2 = 0, −6ka0a12 − 3k 2 b1 + 6ka1b1 + 6ka0 b2 = 0, 3k 3 a1 − 3ka13 − 9k 2 b2 + 9ka1 b2 = 0, −3kλa1 + βkb1 − 2k 3 b1 − 3ka02b1 + 3k 2 a1 b1 + 3kb0b1 = 0, −6ka0a1 b1 + 3kb12 + 2βkb2 − 16k 3 b2 − 6ka02b2 + 6k 2a1 b2 + 6kb0b2 = 0,
21
Remark. We have found four kinds of soliton solutions for the new generalized Hirota–Satsuma coupled KdV equation (1) and the new coupled KdV equation (2) by using a Riccati equation and symbolic computation. Two kinds of them are singular soliton solutions. Such solutions develop a singularity at a finite point, i.e., for any fixed t = t0 , there exist x0 at which these solutions blow up. There is much current interest in the formation of so-called “hot spots” or “blow up” of solutions [8–10]. It appears that these singular solutions will model this physical phenomena. The method used in this Letter has some merits in contrast with the tanh-function method. It not only uses a more simple algorithm to produce an algebraic system but also can pick up singular soliton solutions with no extra effort. In addition, this method is also computerizable, which allow us to perform complicated and tedious algebraic calculation on a computer.
3kλa1 − βkb1 + 8k 3 b1 + 3ka02b1 − 6k 2 a1 b1 − 3ka12b1 − 3kb0b1 − 12ka0a1 b2 + 9kb1b2 = 0,
Acknowledgements
6ka0 a1 b1 − 3kb12 − 2βkb2 + 40k 3b2 + 6ka02 b2 − 12k 2a1 b2 − 6ka12 b2 − 6kb0b2 + 6kb22 = 0, −6k 3 b1 + 3k 2 a1 b1 + 3ka12b1 + 12ka0a1 b2 − 9kb1b2 = 0, −24k 3b2 + 6k 2 a1 b2 + 6ka 2b2 − 6kb22 = 0. Solving these equations by means of Mathematica gives a1 = k, a0 = 0, 1 b0 = 4k 2 + λ , b1 = 0, b2 = −2k 2 , 2 1 β = −2k 2 − 3λ , 2 where k is an arbitrary constant. Then we find that another kind of soliton solution for Eq. (2) is of kink-
I am grateful to Professor Gu Chaohao, Professor Hu Hesheng and Professor Zhou Zixiang for their enthusiastic guidance and help. I also would like to express my sincere thanks to the referees for their helpful suggestions. This work has been supported by Chinese Basic Research Plan “Mathematics Mechanization and a Platform for Automated Reasoning”, the Postdoctoral Science Foundation of China and the Shanghai Postdoctoral Science Foundation of China.
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