- Email: [email protected]
Exact soliton solutions of some nonlinear physical models1
17 May 1999
Physics Letters A 255 Ž1999. 249–252
Exact soliton solutions of some nonlinear physical models Hongjun Lu a , Mingxin Wang
1
b,2
a
b
Transportation Academy, Huaiyin Industry College, Huaiyin 223300, PR China Department of Applied Mathematics, Southeast UniÕersity, Nanjing 210018, PR China
Received 10 December 1998; received in revised form 2 March 1999; accepted 10 March 1999 Communicated by A.R. Bishop
Abstract We find the analytical soliton solutions to some physical models by using special truncated expansions. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction
2. Water wave equations
It is well known that many important dynamics processes can be described by specific nonlinear partial differential equations. When a nonlinear partial differential equation is used to characterize a physical parameter indicating some kinds of propagation or aggregation properties, it is of fundamental physical interest to solve the partial differential equations in closed form with a certain type of traveling wave solutions. In the past several decades both mathematicians and physicists have made many attempts in this direction w1–17x. In the present paper we shall obtain explicit analytical solutions for the water wave equations, the generalized KdV equations and the coupled KdV equations by using special truncated expansions.
In this section we consider the exact soliton solutions of the following water wave equations which come from the water wave problem for long, small amplitude waves over shallow horizontal bottom Žsee w13x.: u t q u xqc1 uu xqc2 u x x xqc 3 u x u x x q c 4 uu x x x q c5 u x x x x x s 0,
Ž 1.
where the coefficients c1 and c 2 are independent of the depth in which the horizontal velocity uŽP. is measure. Let uŽ x,t . s uŽ z ., z s x q ct, where c is the wave speed, the Eq. Ž1. becomes
Ž 1 q c . uXqc1 uuXqc2 uXXXqc3 uX uXX q c 4 uuXXX q c5 uŽ5. s 0,
Ž 2.
X
1
This work was supported by PRC grants NSFC 19771015, 19831060. 2 Corresponding author. E-mail: [email protected]
where ’’ ’’ stands for drdz. We always consider that uŽy`. s 0. Integrating the Eq. Ž2. we have c1 c3 y c4 X 2 Ž 1 q c . uq u 2qc2 uXXq Žu . 2 2 q c 4 uuXX q c5 uŽ4. s 0.
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 1 9 2 - 9
Ž 3.
H. Lu, M. Wangr Physics Letters A 255 (1999) 249–252
250
Let B1 s 1 q c, B2 s c1r2, B3 s Ž c3 y c 4 .r2, B4 s c 2 , B5 s c 4 , B6 s c5 , then Ž3. becomes
With the relations among c,c i and Bi , we have the final results:
2
B1 u q B2 u 2 q B3 Ž uX . q B4 uXX q B5 uuXX q B6 uŽ4. s 0. Ž 4. Žsee Method 1. We make the following ansatze ¨ w15,16x.: uŽ z . s
A 00 q A 01 y q A 02 y 2
Ž1qy.
2
,
Ž 5.
where y s e k z , and k, A 00 , A 01 and A 02 are undetermined constants. So that uŽ n. s
k
n
ks"
Anm y m ,
Ž 6.
y
c4
2 c2
c2 c3
y
5c5
5c 4 c5
1r2
/
,
A 01 s
60 c5 k 2 c3 q 2 c4
c s y1 y c 2 k 2 y c5 k 4 .
,
Ž 11 .
Thus we have the following theorem. Theorem 1. If 5c1 c5 y c 2 c5 y 2 c2 c 4 and c 4 c5 are of the same sign, then Eq. Ž1. has exact soliton solutions: uŽ z . s
nq1
Ý
ž
c1
A 01 e k z
Ž1qek z .
2
,
z s x q ct ,
where A n0 s A nŽ nq2. s 0, Ž n G 1., and
where A 01 ,k and c satisfy Ž11.. These solutions have the properties: uŽy`. s uŽq`. s 0, i.e., they are the bell-shaped waves.
A n m s mAŽ ny1. m y Ž n y m q 2 . AŽ ny1.Ž my1.
Method 2. Let
Ž1qy.
nq2
ms1
= Ž 1 F m F n q 1. .
Ž 7.
Substituting Ž6. into Ž4., and then comparing the coefficients of the same powers of y we know that only when one of the following holds: Ža. A 00 s A 02 s 0; Žb. A 00 s 0, A 02 s yB1rB2 ; Žc. A 00 s yB1rB2 , A 02 s 0; Žd. A 00 s A 02 s yB1rB2 , Eq. Ž4. has solutions of the form Ž5.. We consider only the first case, i.e. A 00 s A 02 s 0. Substituting Ž6. into Ž4., by using Ž7. and collecting terms with the same powers of y, we get 2
4
B1 q B4 k q B6 k s 0,
Ž B2 q B3 k
2
q B5 k
2
Ž 8.
. A 01 2
4
s y4B1 q 2 B4 k q 26 B6 k ,
Ž B2 y B3 k
2
y 2 B5 k
2
Ž 9.
. A 01 Ž 10 .
k s
A 01 s
5B5 B6 60 B6 k 2 2 B3 q 3 B5
,
then uXX s 6 b 2 u 2 y 5abu 3r2 q a 2 u, uŽ4. s 120b 4 u 3 y 168ab 3 u 5r2 65 3 2 2 2 3r2 q 165 q a4 u. 2 a b u y 4 a bu
Substituting uXX ,uŽ4. and Ž12. into Ž3., we have c 3 q 2 c 4 q 60 b 2 c5 s 0,
Ž 13 .
2 c 3 q 3c 4 q 168b 2 c5 s 0, 2
2
2
Ž 14 . 2
Ž c3 q c4 . a q 165a b c5 q 12 b c2 q c1 s 0, Ž 15 . 4 c 2 q 13a 2 c5 s 0,
Ž 16 .
1 q c q a 2 c 2 q a 4 c5 s 0.
Ž 17 .
4 c 3 q 13c 4 s 0,
26 c1 c5 y 3c 2 c 4 s 0,
Ž 18 .
from Ž13. – Ž17. we get
Solving Eqs. Ž8. – Ž10. we have 10 B2 B6 y 2 B3 B4 y 3 B4 B5
Ž 12 .
When c i satisfy the relative expressions:
s y3B1 q 3 B4 k 2 y 33 B6 k 4 .
2
uX s 2 bu 3r2 y au,
,
B1 s yB4 k 2 y B6 k 4 .
°a s y y4c rŽ 13c . 2
~b s
c4r Ž 48c5 . 36 c 22
¢c s 169c
5
y 1.
5
1r2
1r2
Ž c 2 c5 - 0 . ,
Ž c 4 c5 ) 0 . ,
Ž 19 .
H. Lu, M. Wangr Physics Letters A 255 (1999) 249–252
Solving Ž12., it yields uŽ z . s
a2
Ž 2 b q e Ž a r2. z .
2
.
Ž 20 .
Thus we have Theorem 2. If c 2 c5 - 0,c 4 c5 ) 0 and c i satisfy the relations Ž18., then Eq. Ž1. has exact wave front solutions Ž20. with z s x q ct, where a,b and c satisfy Ž19..
3. Generalized KdV equations In this section we consider the exact soliton solutions of the generalized KdV equations w13x
251
which describes interactions of two long waves with different dispersion relation. They showed that for all values of a and b, this system possesses three conservation laws and a solitary wave solution w8x, and for all values of b, but only a s 1r2, this system is a special case of the four-reduced KP hierarchy which is included in the general theory of t functions w9x. For all values of b and a s 1r2, several authors also obtained the Lax pair of Ž22. w2,9,17x. Feng w6x discussed the global well-posedness of the initial value problem of Ž22.. Furthermore, B. Q. Lu w10x proved that when a s y1, Ž22. has N y soliton solutions for all values of b. For the following HS system: u t q 6 a uu x y 6 ÕÕ x q a u x x x s 0, Õt q 3 a uÕ x s a Õ x x x s 0.
Ž 23 .
u t y 30 u 2 u x q 20 u x u x x q 10uu x x x y u x x x x x s 0. Ž 21 .
The authors of w4,5,14x obtained the exact solitary solutions which take the following forms:
Let uŽ x,t . s uŽ z ., z s x q ct. Under the boundary condition uŽy`. s 0, Ž21. becomes
u s c12 sech2
3
X
2
XX
Õ s "Ž 2 a .
1r2
c12 sech2
1 2 1
c Ž x y c12 t .
qlnc 2 y ln c 3 ,
Let 2
c Ž x y c12 t . q ln c 2 y ln c 3 ,
Ž4.
cu y 10 u q 5 Ž u . q 10uu y u s 0.
u Ž z . s Myr Ž 1 q y . ,
1 2 1
and ysek z,
½ ž
where k, M and c are undetermined constants. Similar to the method 1 of §1, we have
u s 2 k 2 sech2 k x y 4a k 2 y
Theorem 3. For any constants k, Eq. Ž21. has exact soliton solutions
Õ s ctanh k x y 4a k 2 y
uŽ z . s
y2 k 2 e k z kz 2
,
z s x q ct
Ž1qe .
with wave speed c s k 4 . The solutions are bellshaped waves.
½ ž
2k2
2k2
/5
/5 t
t
,
,
where c1 ,c 2 ,c 3 ,k,c and c / 0 being arbitrary constants. In this section, we shall give some exact soliton solutions of Ž22. , which differ from the one given by w8x, for all values of b provided that a / y1r2 and ab - 0. Let uŽ z . s A q BÕ Ž z . with 2
Õ Ž z . s Myr Ž 1 q y . ,
4. Coupled KdV equations
3c 2
3c 2
ysek z,
Ž 24 .
In 1981, Hirota and Satsuma introduced the system of equations w8x
where k is a arbitrary constant, z s x q ct, and A, B, M and c are undetermined constants. By u s A q BÕ and Ž22. one has
u t y a Ž u x x x q 6 uu x . s 2 bÕÕ x ,
Ž c y 6 aA . BÕ y aBÕXX y Ž b q 3aB 2 . Õ 2 s E1 ,
Õt q Õ x x x q 3uÕ x s 0,
Ž 22 .
Ž c q 3 A . Õ q ÕXX q Ž 3 Br2. Õ 2 s E2 ,
Ž 25 .
H. Lu, M. Wangr Physics Letters A 255 (1999) 249–252
252
where E1 , E2 are integration constants. By Ž24. one gets E1 s E2 s 0. Now using ansatze ¨ Ž24. in Ž25., one can easily gets the relations among the physical parameters A, B, M,c and k: A s yŽ1 q a. k 2rŽ3 q 6 a., B s Žy2 br3a.1r2 , and c s yak 2r Ž 1 q 2 a . ,
M s Ž y24 arb .
1r2
k2.
Ž 26 . Thus we have Theorem 4. If a / y1r2, and ab - 0, then the coupled KdV Eqs. Ž22. has exact soliton solutions uŽ z . s y
ÕŽ z. s
1qa 3q6a
k2q4k2
ek z
Ž1qek z .
2
,
Me k z
Ž1qek z .
2
with z s x q ct, where k is a arbitrary constant, M and c satisfy Ž26.. These solutions are the bell-shaped waves.
References w1x M.A. Abdelkaser, J. Math. Anal. Appl. 85 Ž1982. 287. w2x R. Dodd, A. Fordy, Phys. Lett. A 89 Ž1982. 168. w3x P.G. Drazin, R.S. Johnson, Soliton: an introduction ŽCambridge Univ. Press, Cambridge, 1989.. w4x E.G. Fan & H.Q. Zhang, Phys. Lett. A 245 Ž1998. 389. w5x E.G. Fan & H.Q. Zhang, Phys. Lett. A 246 Ž1998. 403. w6x X. Feng, Math. Manus. 84 Ž1994. 361. w7x X. Feng, Phys. Lett. A 213 Ž1996. 167. w8x R. Hirota, J. Satsuma, Phys. Lett. A 85 Ž1981. 407. w9x R. Hirota, J. Satsuma, J. Phys. Soc. Japan 51 Ž1982. 3390. w10x B.Q. Lu, Phys. Lett. A 189 Ž1994. 25. w11x B.Q. Lu et al., Phys. Lett. A 180 Ž1993. 61. w12x J.D. Murray, Mathematical biology ŽSpringer, New York, 1989.. w13x P.L. Olver, Hamiltonian and non-Hamiltonian models for water waves, in Lecture Notes in Physics, No.195, pp.273290, Springer-Verlag, New York, 1984. w14x M.L. Wang et al., Phys. Lett. A 216 Ž1996. 67. w15x M.X. Wang et al., J. Math. Anal. Appl. 182 Ž1994. 705. w16x M.X. Wang, Nonlinear equations of parabolic type ŽScience Press, Beijing, 1994.win Chinesex. w17x G. Wilson, Phys. Lett. A 89 Ž1982. 332.
Physics Letters A 255 Ž1999. 249–252
Exact soliton solutions of some nonlinear physical models Hongjun Lu a , Mingxin Wang
1
b,2
a
b
Transportation Academy, Huaiyin Industry College, Huaiyin 223300, PR China Department of Applied Mathematics, Southeast UniÕersity, Nanjing 210018, PR China
Received 10 December 1998; received in revised form 2 March 1999; accepted 10 March 1999 Communicated by A.R. Bishop
Abstract We find the analytical soliton solutions to some physical models by using special truncated expansions. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction
2. Water wave equations
It is well known that many important dynamics processes can be described by specific nonlinear partial differential equations. When a nonlinear partial differential equation is used to characterize a physical parameter indicating some kinds of propagation or aggregation properties, it is of fundamental physical interest to solve the partial differential equations in closed form with a certain type of traveling wave solutions. In the past several decades both mathematicians and physicists have made many attempts in this direction w1–17x. In the present paper we shall obtain explicit analytical solutions for the water wave equations, the generalized KdV equations and the coupled KdV equations by using special truncated expansions.
In this section we consider the exact soliton solutions of the following water wave equations which come from the water wave problem for long, small amplitude waves over shallow horizontal bottom Žsee w13x.: u t q u xqc1 uu xqc2 u x x xqc 3 u x u x x q c 4 uu x x x q c5 u x x x x x s 0,
Ž 1.
where the coefficients c1 and c 2 are independent of the depth in which the horizontal velocity uŽP. is measure. Let uŽ x,t . s uŽ z ., z s x q ct, where c is the wave speed, the Eq. Ž1. becomes
Ž 1 q c . uXqc1 uuXqc2 uXXXqc3 uX uXX q c 4 uuXXX q c5 uŽ5. s 0,
Ž 2.
X
1
This work was supported by PRC grants NSFC 19771015, 19831060. 2 Corresponding author. E-mail: [email protected]
where ’’ ’’ stands for drdz. We always consider that uŽy`. s 0. Integrating the Eq. Ž2. we have c1 c3 y c4 X 2 Ž 1 q c . uq u 2qc2 uXXq Žu . 2 2 q c 4 uuXX q c5 uŽ4. s 0.
0375-9601r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 9 9 . 0 0 1 9 2 - 9
Ž 3.
H. Lu, M. Wangr Physics Letters A 255 (1999) 249–252
250
Let B1 s 1 q c, B2 s c1r2, B3 s Ž c3 y c 4 .r2, B4 s c 2 , B5 s c 4 , B6 s c5 , then Ž3. becomes
With the relations among c,c i and Bi , we have the final results:
2
B1 u q B2 u 2 q B3 Ž uX . q B4 uXX q B5 uuXX q B6 uŽ4. s 0. Ž 4. Žsee Method 1. We make the following ansatze ¨ w15,16x.: uŽ z . s
A 00 q A 01 y q A 02 y 2
Ž1qy.
2
,
Ž 5.
where y s e k z , and k, A 00 , A 01 and A 02 are undetermined constants. So that uŽ n. s
k
n
ks"
Anm y m ,
Ž 6.
y
c4
2 c2
c2 c3
y
5c5
5c 4 c5
1r2
/
,
A 01 s
60 c5 k 2 c3 q 2 c4
c s y1 y c 2 k 2 y c5 k 4 .
,
Ž 11 .
Thus we have the following theorem. Theorem 1. If 5c1 c5 y c 2 c5 y 2 c2 c 4 and c 4 c5 are of the same sign, then Eq. Ž1. has exact soliton solutions: uŽ z . s
nq1
Ý
ž
c1
A 01 e k z
Ž1qek z .
2
,
z s x q ct ,
where A n0 s A nŽ nq2. s 0, Ž n G 1., and
where A 01 ,k and c satisfy Ž11.. These solutions have the properties: uŽy`. s uŽq`. s 0, i.e., they are the bell-shaped waves.
A n m s mAŽ ny1. m y Ž n y m q 2 . AŽ ny1.Ž my1.
Method 2. Let
Ž1qy.
nq2
ms1
= Ž 1 F m F n q 1. .
Ž 7.
Substituting Ž6. into Ž4., and then comparing the coefficients of the same powers of y we know that only when one of the following holds: Ža. A 00 s A 02 s 0; Žb. A 00 s 0, A 02 s yB1rB2 ; Žc. A 00 s yB1rB2 , A 02 s 0; Žd. A 00 s A 02 s yB1rB2 , Eq. Ž4. has solutions of the form Ž5.. We consider only the first case, i.e. A 00 s A 02 s 0. Substituting Ž6. into Ž4., by using Ž7. and collecting terms with the same powers of y, we get 2
4
B1 q B4 k q B6 k s 0,
Ž B2 q B3 k
2
q B5 k
2
Ž 8.
. A 01 2
4
s y4B1 q 2 B4 k q 26 B6 k ,
Ž B2 y B3 k
2
y 2 B5 k
2
Ž 9.
. A 01 Ž 10 .
k s
A 01 s
5B5 B6 60 B6 k 2 2 B3 q 3 B5
,
then uXX s 6 b 2 u 2 y 5abu 3r2 q a 2 u, uŽ4. s 120b 4 u 3 y 168ab 3 u 5r2 65 3 2 2 2 3r2 q 165 q a4 u. 2 a b u y 4 a bu
Substituting uXX ,uŽ4. and Ž12. into Ž3., we have c 3 q 2 c 4 q 60 b 2 c5 s 0,
Ž 13 .
2 c 3 q 3c 4 q 168b 2 c5 s 0, 2
2
2
Ž 14 . 2
Ž c3 q c4 . a q 165a b c5 q 12 b c2 q c1 s 0, Ž 15 . 4 c 2 q 13a 2 c5 s 0,
Ž 16 .
1 q c q a 2 c 2 q a 4 c5 s 0.
Ž 17 .
4 c 3 q 13c 4 s 0,
26 c1 c5 y 3c 2 c 4 s 0,
Ž 18 .
from Ž13. – Ž17. we get
Solving Eqs. Ž8. – Ž10. we have 10 B2 B6 y 2 B3 B4 y 3 B4 B5
Ž 12 .
When c i satisfy the relative expressions:
s y3B1 q 3 B4 k 2 y 33 B6 k 4 .
2
uX s 2 bu 3r2 y au,
,
B1 s yB4 k 2 y B6 k 4 .
°a s y y4c rŽ 13c . 2
~b s
c4r Ž 48c5 . 36 c 22
¢c s 169c
5
y 1.
5
1r2
1r2
Ž c 2 c5 - 0 . ,
Ž c 4 c5 ) 0 . ,
Ž 19 .
H. Lu, M. Wangr Physics Letters A 255 (1999) 249–252
Solving Ž12., it yields uŽ z . s
a2
Ž 2 b q e Ž a r2. z .
2
.
Ž 20 .
Thus we have Theorem 2. If c 2 c5 - 0,c 4 c5 ) 0 and c i satisfy the relations Ž18., then Eq. Ž1. has exact wave front solutions Ž20. with z s x q ct, where a,b and c satisfy Ž19..
3. Generalized KdV equations In this section we consider the exact soliton solutions of the generalized KdV equations w13x
251
which describes interactions of two long waves with different dispersion relation. They showed that for all values of a and b, this system possesses three conservation laws and a solitary wave solution w8x, and for all values of b, but only a s 1r2, this system is a special case of the four-reduced KP hierarchy which is included in the general theory of t functions w9x. For all values of b and a s 1r2, several authors also obtained the Lax pair of Ž22. w2,9,17x. Feng w6x discussed the global well-posedness of the initial value problem of Ž22.. Furthermore, B. Q. Lu w10x proved that when a s y1, Ž22. has N y soliton solutions for all values of b. For the following HS system: u t q 6 a uu x y 6 ÕÕ x q a u x x x s 0, Õt q 3 a uÕ x s a Õ x x x s 0.
Ž 23 .
u t y 30 u 2 u x q 20 u x u x x q 10uu x x x y u x x x x x s 0. Ž 21 .
The authors of w4,5,14x obtained the exact solitary solutions which take the following forms:
Let uŽ x,t . s uŽ z ., z s x q ct. Under the boundary condition uŽy`. s 0, Ž21. becomes
u s c12 sech2
3
X
2
XX
Õ s "Ž 2 a .
1r2
c12 sech2
1 2 1
c Ž x y c12 t .
qlnc 2 y ln c 3 ,
Let 2
c Ž x y c12 t . q ln c 2 y ln c 3 ,
Ž4.
cu y 10 u q 5 Ž u . q 10uu y u s 0.
u Ž z . s Myr Ž 1 q y . ,
1 2 1
and ysek z,
½ ž
where k, M and c are undetermined constants. Similar to the method 1 of §1, we have
u s 2 k 2 sech2 k x y 4a k 2 y
Theorem 3. For any constants k, Eq. Ž21. has exact soliton solutions
Õ s ctanh k x y 4a k 2 y
uŽ z . s
y2 k 2 e k z kz 2
,
z s x q ct
Ž1qe .
with wave speed c s k 4 . The solutions are bellshaped waves.
½ ž
2k2
2k2
/5
/5 t
t
,
,
where c1 ,c 2 ,c 3 ,k,c and c / 0 being arbitrary constants. In this section, we shall give some exact soliton solutions of Ž22. , which differ from the one given by w8x, for all values of b provided that a / y1r2 and ab - 0. Let uŽ z . s A q BÕ Ž z . with 2
Õ Ž z . s Myr Ž 1 q y . ,
4. Coupled KdV equations
3c 2
3c 2
ysek z,
Ž 24 .
In 1981, Hirota and Satsuma introduced the system of equations w8x
where k is a arbitrary constant, z s x q ct, and A, B, M and c are undetermined constants. By u s A q BÕ and Ž22. one has
u t y a Ž u x x x q 6 uu x . s 2 bÕÕ x ,
Ž c y 6 aA . BÕ y aBÕXX y Ž b q 3aB 2 . Õ 2 s E1 ,
Õt q Õ x x x q 3uÕ x s 0,
Ž 22 .
Ž c q 3 A . Õ q ÕXX q Ž 3 Br2. Õ 2 s E2 ,
Ž 25 .
H. Lu, M. Wangr Physics Letters A 255 (1999) 249–252
252
where E1 , E2 are integration constants. By Ž24. one gets E1 s E2 s 0. Now using ansatze ¨ Ž24. in Ž25., one can easily gets the relations among the physical parameters A, B, M,c and k: A s yŽ1 q a. k 2rŽ3 q 6 a., B s Žy2 br3a.1r2 , and c s yak 2r Ž 1 q 2 a . ,
M s Ž y24 arb .
1r2
k2.
Ž 26 . Thus we have Theorem 4. If a / y1r2, and ab - 0, then the coupled KdV Eqs. Ž22. has exact soliton solutions uŽ z . s y
ÕŽ z. s
1qa 3q6a
k2q4k2
ek z
Ž1qek z .
2
,
Me k z
Ž1qek z .
2
with z s x q ct, where k is a arbitrary constant, M and c satisfy Ž26.. These solutions are the bell-shaped waves.
References w1x M.A. Abdelkaser, J. Math. Anal. Appl. 85 Ž1982. 287. w2x R. Dodd, A. Fordy, Phys. Lett. A 89 Ž1982. 168. w3x P.G. Drazin, R.S. Johnson, Soliton: an introduction ŽCambridge Univ. Press, Cambridge, 1989.. w4x E.G. Fan & H.Q. Zhang, Phys. Lett. A 245 Ž1998. 389. w5x E.G. Fan & H.Q. Zhang, Phys. Lett. A 246 Ž1998. 403. w6x X. Feng, Math. Manus. 84 Ž1994. 361. w7x X. Feng, Phys. Lett. A 213 Ž1996. 167. w8x R. Hirota, J. Satsuma, Phys. Lett. A 85 Ž1981. 407. w9x R. Hirota, J. Satsuma, J. Phys. Soc. Japan 51 Ž1982. 3390. w10x B.Q. Lu, Phys. Lett. A 189 Ž1994. 25. w11x B.Q. Lu et al., Phys. Lett. A 180 Ž1993. 61. w12x J.D. Murray, Mathematical biology ŽSpringer, New York, 1989.. w13x P.L. Olver, Hamiltonian and non-Hamiltonian models for water waves, in Lecture Notes in Physics, No.195, pp.273290, Springer-Verlag, New York, 1984. w14x M.L. Wang et al., Phys. Lett. A 216 Ž1996. 67. w15x M.X. Wang et al., J. Math. Anal. Appl. 182 Ž1994. 705. w16x M.X. Wang, Nonlinear equations of parabolic type ŽScience Press, Beijing, 1994.win Chinesex. w17x G. Wilson, Phys. Lett. A 89 Ž1982. 332.