From the SelectedWorks of Ji-Huan He
2008
Exp-function method to solve the nonlinear
dispersive K(m,n) equations
Xin-Wei Zhou
Yi-Xin Wen
Ji-Huan He, Donghua University
Available at: http://works.bepress.com/ji_huan_he/39/
©Freund Publishing House Ltd. International Journal of Nonlinear Sciences and Numerical Simulation,9(3),301-306, 2008
Exp-function method to solve the nonlinear dispersive K(m,n)
equations
Xin-Wei Zhou1 2, Yi-Xin Wen2, Ji-Huan He1
，
1
Modern Textile Institute, Donghua University, 1882 Yan-an Xilu Road, Shanghai 200051,
P.R. China
2
Department of Mathematics, Kunming College, 2 Kun-shi Road, Yunnan, 650031, P.R.
China
Abstract
Some new exact solutions are obtained for the nonlinear dispersive K(m, n) equations using the expfunction method. The results show that the method is straightforward and concise and its applications are
promising.
Keywords: Exp-function method; Nonlinear dispersive K(m,n) equations; Higher order nonlinearity..
general solution procedure, we consider the
following general partial differential equation：
1 Introduction
In this paper we consider the following
nonlinear dispersive K(m,n) equation:
ut + a ( u m ) + ( u n )
x
xxx
=0
(1)
first proposed by Rosenau and Hyman [1]. For
certain values of m and n , the K(m,n) equation
has solitary waves which are compactly
supported. It was shown in Refs.[1,2] that Eq.(1)
presents compactly supported solutions with
nonsmooth fronts. A large number of methods
were suggested recently to study the nonlinear
equations, such as the variational iteration
method [3-8], the homotopy perturbation
method [9-13], the variational method [14] and
the parameter-expansion method[15], a complete
review on recently developed analytical methods
is available in Refs[16,17]. In this paper, we will
apply the Exp-function method[18-20] to
K(s+1,1) equation in the form [21].
ut + au s u x + uxxx = 0
(2)
F ( u , ut , u x , utt , u xx ,") = 0 .
(3)
We aim at its wave solution, so we
introduce a complex variable, η , defined as
η = kx + ωt .
(4)
We, therefore, can convert the partial differential
equation, Eq.(3), into an ordinary differential
equation :
G ( u , ωu′, ku′, ω 2u′′, k 2u ′′,") = 0 .
(5)
The Exp-function method is to search for
its solution in the form
∑
u (η ) =
∑
d
j =− c
q
a j exp ( jη )
b exp ( iη )
i =− q i
(6)
where c, d , p and q are positive integers
unknown to be further determined , a j and bi
are unknown constants.
2 Exp-function method
3 Solution Procedure
The Exp-function method[18-20] is widely
used to search for generalized solitary solutions
and periodic solutions[22-26]. To illustrate the
Using the transformation (4), Eq. (2) becomes
ωu′ + kau s u′ + k 3u′′′ = 0
(7)
302
X.W. Zhou, Y.-X. Wen, J.-H. He,Exp-function method to solve the nonlinear dispersive K(m,n) equations
Integrating (7), and setting the constant of
integration to be zero, we obtain
ωu +
aks s +1 3
u + k u′′ = 0
s +1
(8)
where the prime denotes the differential with
respect to η .
Making the transformation
u=v ,
1/ s
(9)
A1 − B1
A − B1
, a0 ≠ 0 , a−1 = 1
,
6sab−1
6 sa
A +B
b0 = 1 1 ， b−1 ≠ 0 ,
saa0
a1 =
k ≠ 0 ，ω =
−k ( A1 − B1 )
;
6s ( s + 1) b−1
(14)
where
A1 = 4( s + 1)k 2b−1 and B1 = A12 − 6 s 4 a 2 a02b−1 .
Eq.(8) becomes
Cases 3 and 4:
s 2 ( s + 1)ωv 2 + aks 2 v3
+ k (1 − s
3
2
) ( v′ )
2
+ k s ( s + 1)vv′′ = 0 (10)
3
We assume that the solution of Eq.(10) can be
expressed in the form
v (η ) =
a1 exp (η ) + a0 + a−1 exp ( −η )
(11)
exp (η ) + b0 + b−1 exp ( −η )
( s + 1)k 2 A2
( s + 1) k
a1 =
, a0 = ±
15saB2
s
a−1 =
1
{C0 + C1 exp(η ) + " + C8 exp(8η )} = 0 (12)
A
Equating the coefficients of exp(nη ) in Eq.(12)
to be zero yields a set of algebraic equations:
3 b−1 (19 B2 − 192b−1 )
, b−1 = b−1 ,
3
5
2
B
b0 = ±2
( 2)
4kb−21C1
;
15sB23
where
A2 = 215b−1 + 145b−21 ,
B2 = 23b−1 + 145b−21 ,
+5b−1 ( 7579k 2 + 16629 s ) .
C4 = 0, C5 = 0, C6 = 0, C7 = 0, C8 = 0
Solving the above algebraic system with the help
of symbolic computation system, we obtain the
following results
Cases 5 and 6
Case 1:
a1 =
A1 + B1
A + B1
, a0 ≠ 0 , a−1 = 1
,
6sab−1
6sa
A −B
b0 = 1 1 , b−1 ≠ 0 ,
saa0
a1 =
Case 2:
−k ( A1 + B1 )
;
6 s ( s + 1) b−1
(15)
C1 = 145b−21 ( 2641k 2 + 6495s )
C0 = 0, C1 = 0, C2 = 0, C3 = 0,
k ≠ 0, ω =
B2
,
15a 2
( s + 1)b−1k 2 A2
,
15saB2
ω=k−
Substituting Eq. (11) into Eq. (10), we have
2
( s + 1)k 2 A3
( s + 1) k
, a0 = ±
15saB3
s
a−1 =
2
B3
,
15a 2
( s + 1)k 2b−1 A3
,
15saB3
b0 = ±2
(13)
ω=k−
3 b−1 (19 B3 − 192b−1 )
,
3
5
B 2
( 3)
4kb−21C2
15sB33
(16)
ISSN: 1565-1339 International Journal of Nonlinear Sciences and Numerical Simulation, 9(3), 301-306, 2008
where
Cases 9 and 10:
A3 = 215b−1 − 145b ,
2
−1
a1 =
B3 = 23b−1 − 145b−21 ,
C2 = 145b−21 ( 2641k 2 + 6495s )
−5b−1 ( 7579k 2 + 16629 s ) .
Cases 7 and 8
a1 =
303
5a − 29a 2
0
ω=−
(5
b =±
5a + 29a 2
0
)
B5b−1
16 3
, b−1 = b−1 ,
aA5 k 3
ω=−
.
30s
(18)
where
A4C3
ACb
Bb
, a0 = ± 4 −1 C3 , a−1 = 4 3 −1 ,
30
30
15
(5
b =±
A5C3
AC b
Bb
, a0 = ± 5 −1 C3 , a−1 = 5 3 −1 ,
30
30
15
)
B4b−1
16 3
, b−1 = b−1 ,
aA4 k 3
.
30s
(17)
25a + 145a 2
23a − 145a 2
, B4 =
,
A4 =
a2
a3
25a − 145a 2
23a + 145a 2
,
,
A5 =
B
=
5
a2
a3
k 2 ( s + 1)
C3 =
.
s
For each case, we obtain the corresponding
solutions as follows:
⎛ a ( A + B )e
+ 6 saa b + a0b−1 ( A1 + B1 ) e
u1 ( x, t ) = ⎜ 0 1 1 kx +ω t
⎜ 6saa b e 1 + 6b ( A − B ) + 6saa b 2 e−( kx +ω1t )
0 −1
1
1
0 −1
−1
⎝
kx +ω1t
−( kx +ω1t )
2
0 −1
1
s
⎞
⎟ ,
⎟
⎠
(19)
1
⎛ a0 ( A1 − B1 ) ekx +ω2 t + 6 saa02b−1 + a0b−1 ( A1 − B1 ) e−( kx +ω2t ) ⎞ s
u 2 ( x, t ) = ⎜
⎟ ,
⎜ 6saa b ekx +ω2 t + 6b ( A + B ) + 6saa b 2 e−( kx +ω2t ) ⎟
0
1
1
1
1
0
1
−
−
−
⎝
⎠
2
where A1 = 4( s + 1)k b−1 , B1 =
A12 − 6 s 4 a 2 a02b−1 , ω1 =
(20)
−k ( A1 + B1 )
−k ( A1 − B1 )
, and ω2 =
.
6s ( s + 1) b−1
6s ( s + 1) b−1
1
1
1
⎛
⎞s
−( kx +ω1t )
kx +ω1t
2
2 2
2
2
2
2
ak
A
B
e
k
B
15
a
ab
k
A
B
±
+
5( s + 1)
−1
2( 2)
2
2( 2) e
⎜
⎟ , (21)
u3,4 ( x, t ) =
3
3
⎜⎜ 15sa 2
⎟
5 ( B2 ) 2 ekx +ω1t ± 2 3b−1 (19 B2 − 192b−1 ) + 5b−1 ( B2 ) 2 e−( kx +ω1t ) ⎟⎠
⎝
1
s
⎛
⎞
−( kx +ω2t )
kx +ω t
2
2
2
2
2
5( s + 1) ak A3 ( B3 ) e 2 ± k aB3 15a + ab−1k A3 ( B3 ) e
⎟ , (22)
u5,6 ( x, t ) = ⎜
3
3
⎜⎜ 15sa 2
⎟
5 ( B3 ) 2 ekx +ω2t ± 2 3b−1 (19 B3 − 192b−1 ) + 5b−1 ( B3 ) 2 e−( kx +ω2t ) ⎟⎠
⎝
1
2
1
2
where A2 = 215b−1 + 145b−21 , B2 = 23b−1 + 145b−21 , A3 = 215b−1 − 145b−21 ,
4b−21kC1
4b−21kC2
, ω2 = k −
,
B3 = 23b−1 − 145b , ω1 = k −
15sB23
15sB33
2
−1
304
X.W. Zhou, Y.-X. Wen, J.-H. He,Exp-function method to solve the nonlinear dispersive K(m,n) equations
C1 = 145b−21 ( 2641k 2 + 6495s ) + 5b−1 ( 7579k 2 + 16629 s ) ,
C2 = 145b−21 ( 2641k 2 + 6495s ) − 5b−1 ( 7579k 2 + 16629s ) .
1
⎛
⎞s
−( kx +ω1t )
kx +ω1t
8
3
A
C
e
48
5
B
b
C
8
3
A
C
b
e
±
+
⎟ ,
4 3
4 −1 3
4 3 −1
u7,8 ( x, t ) = ⎜⎜
− ( kx +ω1t ) ⎟
kx +ω1t
2
± 15 5 5a − 29a
B4b−1 + 240 3b−1e
⎜ 240 3e
⎟
⎝
⎠
)
(
(23)
1
⎛
⎞s
8 3 A5C3ekx +ω2t ± 48 5B5b−1 C3 + 8 3 A5C3b−1e −( kx +ω2t ) ⎟
⎜
u9,10 ( x, t ) = ⎜
⎟ ,
kx +ω t
2
B5b−1 + 240b−1e −( kx +ω2t ) ⎟
⎜ 240e 2 ± 15 5 5a + 29a
⎝
⎠
(
)
(24)
25a + 145a 2
23a − 145a 2
25a − 145a 2
23a + 145a 2
,
,
,
,
=
=
=
B
A
B
4
5
5
a2
a3
a2
a3
k 2 ( s + 1)
aA5 k 3
aA4 k 3
ω1 = −
, ω2 = −
and C3 =
.
s
30 s
30s
where A4 =
When k is an imaginary number, the obtained solitary solutions can be converted into periodic
solutions [18]. Setting k = iK and ω = iΩ , we obtain the following transformations:
ekx +ωt = cos[ Kx + Ωt ] + i sin[ Kx + Ωt ] ,
(25)
e−( kx +ωt ) = cos[ Kx + Ωt ] − i sin[ Kx + Ωt ] ,.
(26)
and
Substituting (25) and (26) into Eqs. (19)-(24), respectively, leads to the following periodic
solutions:
1
⎛ a0 (δ1 + δ 2 ) cos[ Kx + Ω1t ] + 3saa02 ⎞ s
u11 ( x, t ) = ⎜⎜
⎟⎟ ,
⎝ 6 saa0 cos[ Kx + Ω1t ] + 3 (δ1 + δ 2 ) ⎠
(27)
1
⎛ a0 (δ1 − δ 2 ) cos[ Kx + Ω 2t ] + 3saa02 ⎞ s
u12 ( x, t ) = ⎜⎜
⎟⎟ ,
⎝ 6 saa0 cos[ Kx + Ω 2t ] + 3 (δ1 − δ 2 ) ⎠
− K ( δ1 + δ 2 )
− K ( δ1 − δ 2 )
2
where δ1 = 4k ( s + 1) ， δ 2 = δ12 − 6a 2 a02 s 4 , Ω1 =
and Ω 2 =
.
6s ( s + 1)
6 s ( s + 1)
(28)
1
1
s
⎛
2
2 2 2 ⎞
2 cos[ Kx + Ω t ] ± 5 3 ( s + 1) a k δ
2
5(
s
1)
ak
δ
δ
+
(
)
3
4
3
4
⎟ ,
u13,14 ( x, t ) = ⎜
3
⎜⎜
⎟⎟
2
2
⎝ 30 5sa (δ 4 ) 2 cos[ Kx + Ω3t ] ± 30 3sa (19δ 4 − 192 ) ⎠
(29)
ISSN: 1565-1339 International Journal of Nonlinear Sciences and Numerical Simulation, 9(3), 301-306, 2008
305
1
1
s
⎛
2
2 2 2 ⎞
2 cos[ Kx + Ω t ] ± 5 3 ( s + 1) a k δ
2
5(
s
1)
ak
δ
δ
+
(
)
5
6
4
6 ⎟
⎜
,
u15,16 ( x, t ) =
3
⎜⎜
⎟⎟
2
2
2
⎝ 30 5sa (δ 6 ) cos[ Kx + Ω 4t ] ± 30 3sa (19δ 6 − 192 ) ⎠
where δ 3 = 215 + 145 , δ 4 = 23 + 145 , δ 5 = 215 − 145 , δ 6 = 23 − 145 , Ω3 = K −
Ω4 = K −
4 Kα 2
, α1 = 145 ( 2641k 2 + 6495s ) + 5 ( 7579k 2 + 16629s ) and
15sδ 63
(30)
4 K α1
,
15sδ 43
α 2 = 145 ( 2641k 2 + 6495s ) − 5 ( 7579k 2 + 16629s ) .
1
⎛
⎞s
16
3
δ
δ
cos[
Kx
t
]
48
5
δ
δ
+
Ω
±
⎟ ,
7 11
5
8 11
u17,18 ( x, t ) = ⎜⎜
⎟
2
δ8 ⎟
⎜ 480 3 cos[ Kx + Ω5t ] ± 15 5 5a − 29a
⎝
⎠
(
)
(31)
1
⎛
⎞s
16 3δ 9δ11 cos[ Kx + Ω 6t ] ± 48 5δ10 δ11
⎜
⎟ ,
u19,20 ( x, t ) = ⎜
⎟
2
δ10 ⎟
⎜ 480 3 cos[ Kx + Ω 6t ] ± 15 5 5a + 29a
⎝
⎠
(
)
(32)
where
25a + 145a 2
23a − 145a 2
25a − 145a 2
,
,
=
=
δ
δ
8
9
a2
a3
a2
k 2 ( s + 1)
aδ K 3
aδ K 3
δ11 =
, Ω5 = 7 and Ω6 = 9 .
s
30s
30s
δ7 =
4. Conclusion
References
The exp-function method itself is of utter
simplicity. Using the method we can obtain a
series of exact solutions with some free parameters
which can be determined according to the
boundary/initial conditions.
[1]
Acknowlegements
The work is supported by the Foundation of
Yunnan Province (07C10208) and the Program for
New Century Excellent Talents in University
under grand No. NCET-05-0417.
, δ10 =
23a + 145a 2
a3
,
P. Rosenau, J.M. Hyman, Compactons:
solitons with finite wavelengths. Phys Rev
Lett., 70(1993), 564–567.
[2] P. Rosenau, Compact and noncompact
dispersive structure. Phys. Lett. A. 275(2000),
193–203.
[3] M. Inc, Numerical simulation of KdV and
mKdV equations with initial conditions by
the variational iteration method, Chaos
Soliton. Fract., 34(2007),1075-1081
[4] M. Moghimi, F.S.A. Hejazi, Variational
iteration method for solving generalized
Burger-Fisher and Burger equations, Chaos
Soliton. Fract., 33(2007)1756-1761
306
X.W. Zhou, Y.-X. Wen, J.-H. He,Exp-function method to solve the nonlinear dispersive K(m,n) equations
[5] E. Yusufoglu, Variational iteration method
for construction of some compact and
noncompact structures of Klein-Gordon
equations , Int. J. Nonlinear Sci., 8(2007)
153-158
[6] N. Bildik, A. Konuralp, Two-dimensional
differential transform method, Adomian's
decomposition method, and variational
iteration method for partial differential
equations, Int. J. Comput. Math., 83(2006),
973-987
[7] J.H. He, X.H. Wu, Construction of solitary
solution and compacton-like solution by
variational iteration method. Chaos, Solitons
& Fractals, 29(2006), 108–113.
[8] A.M.Wazwaz, The variational iteration
method for rational solutions for KdV, K(2,2),
Burgers, and cubic Boussinesq equations, J.
Comput. Appl. Math. 207 (2007) 18 – 23.
[9] H. Tari, D.D. Ganji, M. Rostamian,
Approximate solutions of K (2,2), KdV and
modified KdV equations by variational
iteration method, homotopy perturbation
method and homotopy analysis method, Int. J.
Nonlinear Sci., 8(2007), 203-210
[10] T. Ozis, A. Yildirim, Traveling wave solution
of Korteweg-de Vries equation using He's
homotopy perturbation method , Int. J.
Nonlinear Sci., 8(2007) 239-242
[11] J.H. He, Application of homotopy
perturbation method to nonlinear wave
equations,
Chaos
Soliton.
Fract.,
26(2005),695-700
[12] J.H. He, Homotopy perturbation method for
bifurcation of nonlinear problems, Int. J.
Nonlinear Sci., 6(2005) 207-208
[13] M. Gorji, D.D. Ganji, S. Soleimani, New
application of He's homotopy perturbation
method, Int. J. Nonlinear Sci., 8(2007)319328.
[14] L. Xu, Variational approach to solitons of
nonlinear dispersive K(m,n) equations, Chaos,
Solitons & Fractals 37 (2008) 137–143.
[15] L. Xu, He's parameter-expanding methods for
strongly nonlinear oscillators. J. Comput.
Appl. Math. 207 (1) (2007) 148-154.
[16] J.H. He, Some asymptotic methods for
strongly nonlinear equations. Int. J. Modern
Phys. B 2006;20(10):1141–1199.
[17] J.H. He,
Non-perturbative methods for
strongly
nonlinear
problems.
Berlin:
dissertation. de-Verlag im Internet GmbH,
2006.
[18] A.M. Wazwaz, New sets of solitary wave
solutions to the KdV, mKdV,and the
generalized KdV equations, Communications
in Nonlinear Science and Numerical
Simulation 13 (2008) 331–339.
[19] J.H. He, X.H. Wu, Exp-function method for
nonlinear wave equations, Chaos Solitons &
Fractals, 30(2006)700-708.
[20] X.H. Wu, J.H. He, Solitary solutions, periodic
solutions and compacton-like solutions using
the Exp-function method, Computers &
Mathematics with Applications,54(2007)966986.
[21] J.H. He, M.A.
Abdou, New periodic
solutions for nonlinear evolution equations
using Exp-function method, Chaos Solitons &
Fractals, 34(2007):1421-1429.
[22] S.D. Zhu, Exp-function method for the
Hybrid-Lattice system, Int. J. Nonlinear Sci.,
8(2007)461-464.
[23] S.D. Zhu,Exp-function method for the
discrete mKdV lattice, Int. J. Nonlinear Sci. ,
8(2007) 465-468.
[24] Xin-Wei Zhou, Exp-Function Method for
Solving Huxley Equation, Mathematical
Problems in Engineering, vol. 2008, Article
ID 538489
[25] X.W. Zhou, Exp-function method for solving
Fisher’s Equation, Journal. of Physics:
Conference Series , 96(2008): 012063.
[26] A, Bekir, A.Boz Exact solutions for a class of
nonlinear partial differential equations using
exp-function method, Int. J. Nonlinear Sci. ,
8(2007)505-512.