2016 International Conference on Mathematical, Computational and Statistical Sciences and Engineering (MCSSE 2016) ISBN: 978-1-60595-396-0
Exact Solutions of Three Nonlinear Systems via Exp-Function Method
Sheng ZHANG
1,*and Jian WANG
1 1School of Mathematics and Physics, Bohai University, Jinzhou, 121013, China *Corresponding author
Keywords: Exact solution, Fokas equation, Burger’s equation, BBM equation.
Abstract. In this paper, the exp-function method is used to exactly solve three nonlinear partial differential systems, i.e., the (4+1)-dimensional Fokas equation, the variable-coefficient Burger’s equation and the variable-coefficient Benjamin-Bona-Mahony (BBM) equation. As a result, three exact solutions are obtained. It is shown that the exp-function method is a useful mathematical tool for solving some other high-dimensional nonlinear partial differential equations (PDEs) and variable-coefficient nonlinear PDEs.
Introduction
In 2006, He and Wu [1] proposed a simple and direct method for solving nonlinear PDEs. The existing researches show that the exp-function method and its improvements are available for many nonlinear PDEs, such as those in [2-9]. This is due to the exp-function method possesses a more general ansätz [10]:
exp( ) , exp( )
g n n f q
m m p
a n
u
b m
1,
s
i x i
k x wt
(1)which is supposed for a given nonlinear PDE with independent variables t x x, ,1 2, ,xs and
dependent variable u:
1 2 1 2 1 1 2 2
( ) 0.
m s s s
t x x x x t x t x t tt x x x x x x
F u u u u u u u u u u u u (2)
where an and bm are undetermined constants, the integers ,f p, g and q can be determined by balancing the highest order linear term with the highest order nonlinear term in Eq. (2). In general, the final solution does not strongly depend on the choices of values of f, p, g, q and usually
1
f p g q is the simplest choice. The present paper is motivated by the desire to extend the
exp-function method to the (4+1)-dimensional equation [11]:
1 2 2 1 1 2 1 2
2
3 3 2
1
1 3 3
( ) ( ) ,
4 x x x x 2 x x 2 y y
u
u u u
t x
(3)
the variable-coefficient Burger’s equation [12]:
( ) ( ) 0,
t x xx
u t uu t u (4) and the variable-coefficient BBM equation [13]:
( ) ( ) 0,
t x xxt
Exact Solutions
Firstly, we consider the (4+1)-dimensional Eq. (3). Taking the following travelling wave transformation:
( ),
uu ax1bx2cy1dy2et, (6) where a, ,b c, d and e are undetermined constants, we convert Eq. (3) into an ordinary differential equation (ODE):
3 (4) 3 (4) 2
4eaua bu ab u 12abu 12abu6cdu0. (7) Supposing Eq. (7) has a solution of the form:
1 0 11
1 0 11
exp( ) exp( )
( ) ,
exp( ) exp( )
a a a
u
b b b
(8)
where a1, a0, a11, b1, b0 and b11 are constants determined later. Substituting Eq. (8) into Eq. (7) and then equating each coefficient of the same order power of exp(j)(j1, 2, ,9) to zero yields a system of nonlinear algebraic equations for a1, a0, a1, b1, b0, b1, a, ,b c, d and e:
2 2 3 3 3 3 3 3 4 3 4
11 0 11 0 11 11 11 0 11 11 0 11 0 11 0 11
12aa bb b 12aa a bb a a bb b aa b b b a a bb aa b b
3 4 3 4
11 0 11 0 11 11 0 11 0 11
6a b b cd 6a b cd 4aa b b e 4aa b e 0,
(9)
2 2 4 3 2 2 3 2 2 2 2
11 0 11 1 11 11 0 11 11 0 11 11 1 11
12aa bb b 16aa b e11a a bb b 11aa b b b 48aa bb b
3 3 3 3 3 3 3 3 3
1 11 11 0 0 11 0 0 11 11 1 11 11 1 11
48aa a bb 11a a bb b 11aa b b b 16a a bb b 16aa b b b
3 4 3 4 2 2 3 3 4
1 11 1 11 11 0 11 0 0 11 11 1 11 1 11
16a a bb 16aa b b 6a b b cd 6a b b cd 24a b b cd 24a b cd
2 2 3 3 2 2 3
11 0 11 0 0 11 11 1 11 0 11 0 11 0 11
4aa b b e 4aa b b e 16aa b b e 36aa a bb b 24aa bb 0,
(10)
3 3 2 3 3 3 3 2
0 1 11 0 11 0 11 11 0 11 11 0 11 11 0 1 11
76a a bb b 36aa a bb b 11a a bb b 11aa b b b 24aa bb b b
3 2 2 3 2 2 3 2 2 2
11 0 1 11 1 11 0 11 0 0 11 0 0 11 0 11 1 11
77a a bb b b 24aa a bb b 11a a bb b 11aa b b b 156aa a bb b
2 2 3 2 3 2 3 3
0 0 11 11 0 1 11 0 1 11 11 0 1 11 1 0 11
12aa bb b 77aa b b b b 108aa a bb 42a b b b cd aa b b b
2 3 3 3 3 2 2 3 3 3
11 0 0 1 11 11 0 11 0 0 11 1 0 11 1 0 11
24aa bb 76aa b b b 6a b b cd 6a b b cd a a bb b 66a b b cd
3 3 2 2 2 3 3
0 1 11 11 0 11 0 0 11 11 0 1 11 1 0 11 0 1 11
24a b b cd 4aa b b e 4aa b b e 28aa b b b e 44aa b b e 16aa b b e 0,
(11)
3 3 4 3 4 2 2 2 2 2
0 11 0 11 0 11 0 11 0 1 0 0 11 1 11 0 11
12aa a bb a a bb aa b b 108aa bb b 12aa bb b 24aa a bb b
3 3 3 3 3 2 3 2
0 0 11 0 0 11168 0 11 0 1 11 58 11 0 1 11 58 11 0 1 11
a a bb b aa b b b aa a bb b b a a bb b b aa b b b b
2 2 2 3 2 2 3 2 2 2 2
11 1 11 0 1 0 11 1 0 11 1 0 11 0 1 11
48aa bb b 156aa a bb b 11a a bb b 11aa b b b 72aa bb b
2 2 3 2 3 2 3 2 2 3 2 2
11 1 11 0 0 1 11 0 0 1 11 11 1 11 11 1 11
16aa b b e 47a a bb b b 47aa b b b b 176a a bb b 176aa b b b
2 3 2 3 3 4 3 2
1 11 1 1 11 1 1 11 11 0 0 0 11 11 0 1 11
96aa bb 16aa b b e 176aa b b b 6a b cd 6a b b cd 12a b b b cd
2 2 2 2 2 3 4 3
1 0 11 0 0 1 11 11 1 11 1 1 11 11 0 0 0 11
66a b b cd 78a b b b cd 24a b b cd 24a b b cd 4aa b e 4aa b b e
2 2 2 2 2 3 3
11 0 1 11 1 0 11 0 0 1 11 1 11 1 11 1 1 11
8aa b b b e 44aa b b e 52aa b b b e 144aa a bb b 176a a bb b 0,
(12)
3 2 2 3 3 2 2 2
0 0 1 11 0 11 1 11 1 0 11 11 0 1 11 1 0 11
10aa b b b b 60aa a bb b 20aa b b e 115aa b b b b 180aa bb b
2 3 3 2 3 2 2 3 2 2
0 1 1 11 11 0 1 1 0 1 11 0 1 11 0 1 11
60aa a bb b 20aa b b e 115aa b b b b 230a a bb b 230aa b b b
3 3 2 2 2
11 0 1 1 0 11 0 0 1 11 11 0 1 11 1 0 1 11
30a b b cd 30a b b cd 60a b b b cd 30a b b b cd 30a b b b cd
2 2 3 2 3 2 2 2
0 1 11 1 0 1 11 11 0 1 11 0 0 1 11 11 0 1 11
60a b b cd 115a a bb b b 115a a bb b b 40aa b b b e 20aa b b b e
2 2 2
1 0 1 11 0 1 11
20aa b b b e 40aa b b e 0,
(13)
3 3 4 3 4 2 2 2 3 3
0 1 0 1 0 1 0 0 0 1 1 11 0 1 0 0 1
12aa a bb a a bb aa b b 12aa bb b 24aa a bb b a a bb b
3 3 2 3 2 2 3 2 2 3 2 2
0 0 1 156 0 11 0 1 11 11 0 1 11 11 0 1 11 11 0 1
aa b b b aa a bb b a a bb b a a bb b aa b b b
2 3 2 2 3 2 3 2
11 1 1 0 11 0 1 0 1 11 1 0 1 11 1 0 1 11
96aa bb 108aa bb b 168aa a bb b b 58a a bb b b 58aa b b b b
2 2 2 3 2 3 2 4
0 1 11 1 11 11 0 0 1 11 0 0 1 11 1 0
72aa bb b 144aa a bb bb 47a a bb b b 47aa b b b b 4aa b e
3 3 2 2 3 2 2 3 2 2 4
11 1 11 1 1 11 1 1 11 1 1 11 1 0
176aa b b b 48aa bb b 176a a bb b 176aa b b b 6a b cd
3 2 2 2 2 3
0 0 1 11 0 1 1 0 1 11 0 0 1 11 11 1 11
6a b b cd 66a b b cd 12a b b b cd 78a b b b cd 24a b b cd
2 2 3 2 2 2 2
1 1 11 0 0 1 11 0 1 1 0 1 11 0 0 1 11
24a b b cd 4aa b b e 44aa b b e 8aa b b b e 52aa b b b e
3 2 2 3 3
11 1 11 1 1 11 11 1 11
16aa b b e 16aa b b e 176a a bb b 0,
(14)
2 3 2 3 3 3 3 2 2 3
1 0 0 1 0 1 1 0 1 1 0 1 0 0 1 0 1 11
24aa bb 36aa a bb b 11a a bb b 11aa b b b 12aa bb b 16aa b b e
2 3 2 2 3 2 2 3 3 3
1 11 0 1 0 0 1 0 0 1 0 11 1 11 0 1
24aa a bb b 11a a bb b 11aa b b b 108aa a bb a a bb b
3 3 2 2 3 2 3 2
11 0 1 24 1 0 1 11 156 0 1 1 11 77 1 0 1 11 77 1 0 1 11
aa b b b aa bb b b aa a bb b a a bb b b aa b b b b
3 3 3 3 3 3 3 2 2
0 1 11 0 1 11 0 1 11 1 0 1 0 0 1
76a a bb b 76aa b b b 76aa b b b 6a b b cd 6a b b cd
3 2 3 3 2 2 3
11 0 1 1 0 1 11 0 1 1 1 0 1 0 0 1 11 0 1
66a b b cd 42a b b b cd 24a b b cd 4aa b b e 4aa b b e 44aa b b e
2 1 0 1 11
28aa b b b e,
(15)
2 2 2 3 2 2 3 2 2 2 3
1 0 1 0 1 0 1 1 0 1 1 0 1 0 1
12aa bb b 36aa a bb b 11a a bb b 11aa b b b 24aa bb
3 3 3 3 3 3 4 3 4
1 11 1 0 0 1 0 0 1 11 1 11 1
48aa a bb 11a a bb b 11aa b b b 16a a bb 16aa b b
2 2 3 3 3 3 2 2 3
1 1 11 1 1 11 1 1 11 1 0 1 0 0 1
48aa bb b 16a a bb b 16aa b b b 6a b b cd 6a b b cd
4 3 2 2 3 4 3
11 1 1 1 11 1 0 1 0 0 1 11 1 1 1 11
24a b cd 24a b b cd 4aa b b e 4aa b b e 16aa b e 16aa b b e 0,
(16)
2 2 3 3 3 3 3 3 4
1 0 1 0 1 1 1 0 1 1 0 1 0 1
12aa bb b 12aa a bb a a bb b aa b b b a a bb
3 4 3 4 3 4
0 1 6 1 0 1 6 0 1 4 1 0 1 4 0 1 0.
aa b b a b b cd a b cd aa b b e aa b e
(17)
Solving the set of algebraic Eqs. (9)-(17) yields:
3 3
1 1
( 6 4 )
, 12
b a b ab cd ae a
ab
11 3 3
11
( 6 4 )
12
b a b ab cd ae a
ab
, (18)
3 3
0 0
(5 5 6 4 )
, 12
b a b ab cd ae a
ab
02
1 11
, 4
b b
b
(19)
And then we obtain an exact solution of Eq. (3):
3 3 3 3
1 11 0
2 0
0 11 11
( 6 4 ) (5 5 6 4 )
,
12 ( )
4
a b ab cd ae b e b e b a b ab cd ae u
b
ab e b b e b
（ ）
(20)
where ax1bx2cy1dy2et, a, ,b c, d and e are arbitrary constants.
Secondly, we consider the variable-coefficient Burger’s Eq. (4). We use the transformation:
( ),
where ( )k t and ( )p t are undetermined functions, then Eq. (4) is converted into an ODE:
2
[ ( )k t x p t u( )] k t( ) ( ) t uuk t( ) ( ) t u0. (22) We suppose that Eq. (22) has a solution of the form:
1 0 11
1 0 11
exp( ) exp( )
( ) ,
exp( ) exp( )
a a a
u
b b b
(23)
where a1, a0, a11, b1, b0 and b11 are constants to be determined. Substituting Eq. (23) into Eq. (22) and then equating each coefficient of the same order power of xexp(j)(j1, 2,3, 4,5) and exp(j)
(j1, 2,3, 4,5） to zero yields a system of nonlinear differential equations for a1, a0, a1, b1, b0,
1,
b ( )k t and ( ).p t Solving the set of nonlinear differential equations, we have
2
0 11 0 0 1 11
11 2
0 0 1 11
( 2 )
,
( )
a b a b a b
a
a b a b
1 0 0 1 11
1 2
0
( )
,
a a b a b
b
a
( )k t k, (24)
3 2
0 0 0 1 11 2 0 0 1 11
( 2 3 )
( ) ( )d ,
2( )
k a b a a b
p t t t
a b a b
2 0 1 11
2 0 0 1 11
( )
( ) ,
2( )
a a b t
t
a b a b k
(25)
and hence obtain an exact solution of Eq. (4):
2
0 11 0 0 1 11
1 0 2
0 0 1 11
1 0 0 1 11
0 11 2
0
( 2 )
( )
,
( )
a b a b a b
a e a e
a b a b u
a a b a b
e b b e a 3 2
0 0 0 1 11 2 0 0 1 11
( 2 3 )
( )d ,
2( )
k a b a a b
kx t t
a b a b
(26)where a0, a1, b0 and b11 are arbitrary constants.
Finally, we solve the variable-coefficient BBM Eq. (5). Using the transformation (21) yields an ODE:
2
[ ( )k t x p t u( )] ( ) ( )t k t uu( )t k t k t x( )[ ( ) p t u( )] 0. (27) We suppose that Eq. (27) has a solution of the form (23). Substituting Eq. (23) into Eq. (27) and then equating each coefficient of the same order power of xexp(j)(j1, 2,3, 4,5) and exp(j)
(j1, 2,3, 4,5） to zero yields a system of nonlinear differential equations for a1, a0, a1, b1, b0,
1,
b ( )k t and ( ).p t Solving the set of nonlinear differential equations, we have
2 2
0 0
1 2 2
11
[ ( )] , 4 [1 5 ( )]
a a k t
a
a k t
2 2
11 0 0
1 2 2 2
11
[ ( )] , 4 [1 5 ( )]
b a a k t
b
a k t
2 0 11 0 2 11
[1 ( )] , [ 1 5 ( )]
a b k t
b
a k t
( )k t k, (28)
( )t b,
11
2 11
( )
( ) d ,
[1 ( )]
a k t
p t t
b k t
(29) and hence obtain an exact solution of Eq. (5):2 2 0 0
0 11 2 2
11
2 2 2
11 0 0 0 11
11
2 2 2 2
11 11
( )
4 (1 5 )
,
( ) (1 )
4 (1 5 ) ( 1 5 )
a a k b
e a a e
a k b
u
b a a k b a b k b
e b e
a k b a k b
11 2 11 ( ) d , [1 ( )]
a k t
kx t
b k t
(30)Summary
The (4+1)-dimensional Fokas Eq. (3), the variable-coefficient Burger’s Eq. (4) and the variable-coefficient Benjamin-Bona-Mahony (BBM) Eq. (5) have been solved by the exp-function method. As a result, three exact solutions (20), (26) and (30) with free parameters are obtained. This paper shows that the exp-function method is effective for some other high-dimensional nonlinear PDEs and variable-coefficient nonlinear PDEs.
Acknowledgement
This work was supported by the Natural Science Foundation of China (11547005), the PhD Start-up Fund of Liaoning Province of China (20141137), the Liaoning BaiQianWan Talents Program (2013921055) and the Natural Science Foundation of Liaoning Province of China (L2012404).
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