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Travelling wave solutions for a class of generalized KdV equation
Applied Mathematics and Computation 215 (2009) 2768–2774
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Travelling wave solutions for a class of generalized KdV equation q Shengqiang Tang *, Jianxian Zheng, Wentao Huang School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, PR China
a r t i c l e
i n f o
a b s t r a c t In this work, the K ðl; pÞ equation is investigated. The sine–cosine method, the tanh method and the extended tanh method are efficiently used for analytic study of this equation. New solitary patterns solutions and compactons solutions are formally derived. The proposed schemes are reliable and manageable. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Compactons Solitary patterns solutions Sine–cosine method The extended tanh method
1. Introduction In 1993, Cooper et al. [1] considered the following generalized KdV equation:
K ðl; pÞ :
ut ¼ ux ul2 þ a½2uxxx up þ 4pup1 ux uxx þ pðp 1Þup2 ðux Þ3 ;
ð1Þ
þ
where p; l P 2; l; p 2 Z . This equation was derived from the Lagrangian
# Z " 1 ðwx Þl p 2 Lðl pÞ ¼ ww þ aðwx Þ ðwxx Þ dx; 2 x t lðl 1Þ
ð2Þ
where uðx; tÞ was defined by uðx; tÞ ¼ wx ðx; tÞ: The authors of [1] investigated the Hamiltonian structure and integrability properties for this class of KdV equations. The authors of [2] investigated the bifurcation behavior of the travelling wave solutions of the corresponding travelling wave equations in its parameter space. It is well known that searching for explicit solutions for nonlinear evolution equation, by using different methods, is the goal for many researchers. Many powerful methods, such as Bäcklund transformation, inverse scattering method, Hirota bilinear forms, pseudo spectral method, the tanh–sech method [3–8], the sine–cosine method [9], and many other techniques were successfully used to investigate these types of equations. Practically, there is no unified method that can be used to handle all types of nonlinear problems. The sine–cosine method and the extended tanh method are direct and effective algebraic method for finding exact solutions of nonlinear diffusion equations and Wazwaz developed the methods of sine–cosine [10,11, and the references therein] and the extended tanh [6–8, and the references therein]. The sine–cosine and the extended tanh algorithms, that provides a systematic framework for many nonlinear dispersive equations, will be employed to back up our analysis to determine compactons and solitary patterns travelling waves solutions. Let uðx; tÞ ¼ /ðx ctÞ ¼ /ðnÞ, where c is the wave speed. Then (1) becomes to
c/0 ¼
1 ð/l1 Þ0 þ a½2ð/p /00 Þ0 þ pð/p1 ð/0 Þ2 Þ0 ; l1
ð3Þ
q This research was supported by Science foundation of the Education Department of Guangxi Province (D2008007) and Program for Excellent Talents in Guangxi Higher Education Institutions. * Corresponding author. E-mail address: [email protected] (S. Tang).
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.09.019
S. Tang et al. / Applied Mathematics and Computation 215 (2009) 2768–2774
2769
where ‘‘0 ” is the derivative with respect to n. Integrating (3) once and setting integration constant as 0, we have
2a/p /00 þ ap/p1 ð/0 Þ2 þ
1 /l1 þ c/ ¼ 0: l1
ð4Þ
The paper is organized as follows: In Section 2, the sine–cosine method and the tanh method are briefly discussed. In Section 3, represents exact analytical solutions of (1) by using the sine–cosine method. In Section 4, represents exact analytical solutions of (1) by using the tanh method and the extended tanh method. In the last section, we conclude the paper and give some discussions. 2. Analysis of the two methods The sine–cosine method, the tanh method and the extended tanh method have been applied for a wide variety of nonlinear problems. The main features of the two methods will be reviewed briefly. For both methods, we first use the wave variable n ¼ x ct to carry a PDE in two independent variables
Pðu; ut ; ux ; uxx ; uxxx ; . . .Þ ¼ 0
ð5Þ
into an ODE
Q ðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0:
ð6Þ
Eq. (6) is then integrated as long as all terms contain derivatives where integration constants are considered zeros. 2.1. The sine–cosine method The sine–cosine method admits the use of the solution in the form [5, and references therein].
uðx; tÞ ¼
k cosb ðlnÞ; jlnj < p2 ; 0;
otherwise
ð7Þ
or in the form
( uðx; tÞ ¼
b
k sin ðlnÞ; jlnj < p2 ; 0;
otherwise;
ð8Þ
where k; l and b are parameters that will be determined. We substitute (7) or (8) into the reduced ordinary differential equation obtained above in (6), balance the terms of the cosine functions when (7) is used, or balance the terms of the sine functions when (8) is used, and solving the resulting system of algebraic equations by using the computerized symbolic calculations to obtain all possible value of the parameters k; l and b. 2.2. The tanh method and the extended tanh method The standard tanh method introduced in [3–5] where the tanh is used as a new variable, since all derivatives of a tanh are represented by a tanh itself. We use a new independent variable
Y ¼ tanhðlnÞ;
ð9Þ
that leads to the change of derivatives:
d d ¼ lð1 Y 2 Þ ; dn dY 2
d
dn2
¼l
2
! 2 d 2 d ð1 Y Þ 2Y þ ð1 Y Þ 2 : dY dY 2
ð10Þ
We then apply the following finite expansion:
uðlnÞ ¼ SðYÞ ¼
M X
ak Y k
ð11Þ
k¼0
and
uðlnÞ ¼ SðYÞ ¼
M X k¼0
ak Y k þ
M X
ak Y k ;
ð12Þ
k¼1
where M is a positive integer that will be determined to derive a closed form analytic solution. However, if M is not an integer, a transformation formula is usually used. Substituting (9) and (10) into the simplified ODE (6) results in an equation in
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S. Tang et al. / Applied Mathematics and Computation 215 (2009) 2768–2774
powers of Y. To determine the parameter M, we usually balance the linear terms of highest order in the resulting equation with the highest order nonlinear terms. With M determined, we collect all coefficients of powers of Y in the resulting equation where these coefficients have to vanish. This will give a system of algebraic equations involving the parameters ak ðk ¼ 0; . . . ; MÞ; l and c. Having determined these parameters, knowing that M is a positive integer in most cases, and using (11) or (12) we obtain an analytic solution uðx; tÞ in a closed form. 3. Using the sine–cosine method Substituting (7) into (4) yields
akpþ1 l2 b½bðp þ 2Þ 2 cosbðpþ1Þ2 ðlnÞ akpþ1 l2 b2 ðp þ 2Þ cosbðpþ1Þ ðlnÞ þ
1 l1 k cosbðl1Þ þck cosb ðlnÞ ¼ 0: l1
ð13Þ
It is clear that Eq. (13) is satisfied if the following system of algebraic equations holds:
bðp þ 1Þ 2 ¼ b;
bðp þ 1Þ ¼ bðl 1Þ;
akpþ1 l2 b½bðp þ 2Þ 2 ¼ ck; akpþ1 l2 b2 ðp þ 2Þ ¼
1 l1 k : l1
ð14Þ
Solving this system yields
p ¼ l 2;
b¼
2 ; p
p
1
1p
l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; k ¼ cðp þ 1Þðp þ 2Þ ; 2 2 aðp þ 1Þðp þ 2Þ
ð15Þ
that can also be obtained by using the sine ansatz (8). Consequently, for a > 0 we obtain a family of compactons solutions
8 1p > < pðxctÞ ; jlnj < p2 ; 12 cðp þ 1Þðp þ 2Þ cos2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; tÞ ¼ 2 aðpþ1Þðpþ2Þ > : 0; otherwise
ð16Þ
8 1p > < 2 pðxctÞ ; jlnj < p2 ; 12 cðp þ 1Þðp þ 2Þ sin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; tÞ ¼ 2 aðpþ1Þðpþ2Þ > : 0; otherwise:
ð17Þ
and
However, for a < 0 we obtain solitary patterns solutions
( uðx; tÞ ¼
" #)1p 1 pðx ctÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cðp þ 1Þðp þ 2Þcosh 2 2 aðp þ 1Þðp þ 2Þ
ð18Þ
" #)1p 1 pðx ctÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : cðp þ 1Þðp þ 2Þsinh 2 2 aðp þ 1Þðp þ 2Þ
ð19Þ
and
( uðx; tÞ ¼
4. Using the tanh method and the extended tanh method The K ðl; pÞ equation: the following two cases will be investigated: Case I l ¼ p þ 2; p ¼ p > 2. Case II l ¼ 2ðp þ 1Þ; p ¼ p. 4.1. Case I. l ¼ p þ 2; p ¼ p > 2 In this case, Eq. (4) becomes
2a/p /00 þ ap/p1 ð/0 Þ2 þ
1 /pþ1 þ c/ ¼ 0: pþ1
ð20Þ
Balancing / with /p /00 we find
M ¼ pM þ 4 þ M 2;
ð21Þ
so that
2 M¼ : p
ð22Þ
S. Tang et al. / Applied Mathematics and Computation 215 (2009) 2768–2774
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To get a closed form analytic solution, the parameter M should be an integer. A transformation formula 1
/ ¼ v p
ð23Þ
should be used to achieve our goal. This in turn transforms (20) to
að3p þ 2Þ p2 Balancing
2 1 ðv 0 Þ2 avv 00 þ v 2 þ cv 3 ¼ 0: p pþ1
ð24Þ
vv 00 and v 3 gives M ¼ 2. The tanh method allows us to use the substitution
v ðx; tÞ ¼ SðYÞ ¼ a0 þ a1 Y þ a2 Y 2 :
ð25Þ
Substituting (25) into (24), collecting the coefficients of each power of Y, and solve the resulting system of algebraic equations to find the following sets of solutions:
2½4al2 ð6p þ 1Þðp þ 1Þ þ p2 ; a1 ¼ 0; 3cp2 ðp þ 1Þðp þ 2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4al2 ðp 2Þ p 15p2 þ 24p þ 8 ; l¼ ; a2 ¼ 3cp2 4 aðp þ 1Þð3p3 12p2 9p 1Þ a0 ¼
ð26Þ
1
where c is selected as a free parameter. Noting that / ¼ v p , for a > 0; p P 5 or a < 0; 2 6 p 6 4, we find a family of solitary patterns solutions
(
" sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1p p 15p2 þ 24p þ 8 a0 þ a2 tanh ðx ctÞ 4 aðp þ 1Þð3p3 12p2 9p 1Þ 2
uðx; tÞ ¼
ð27Þ
and
( 2
uðx; tÞ ¼
a0 þ a2 coth
" sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1p p 15p2 þ 24p þ 8 ðx ctÞ : 4 aðp þ 1Þð3p3 12p2 9p 1Þ
ð28Þ
However, for a < 0; p P 5 or a > 0; 2 6 p 6 4, we obtain a family of compactons solutions given by
(
" sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1p p ð15p2 þ 24p þ 8Þ ðx ctÞ a0 a2 tan 4 aðp þ 1Þð3p3 12p2 9p 1Þ 2
uðx; tÞ ¼
ð29Þ
and
( uðx; tÞ ¼
" sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1p p ð15p2 þ 24p þ 8Þ a0 a2 cot : ðx ctÞ 4 aðp þ 1Þð3p3 12p2 9p 1Þ 2
ð30Þ
We next apply the extended tanh method give in (12), where we substitute
uðnÞ ¼ a0 þ a1 Y þ a2 Y 2 þ b1 Y 1 þ b2 Y 2
ð31Þ
into (24), and proceed as before to find the following sets for a0 ; a1 ; a2 ; b1 ; b2 and
l
8al ð2p þ 1Þðp þ 2Þ þ p ; 3cp2 ðp þ 2Þ 8al2 ; b1 ¼ 0; a1 ¼ 0; a2 ¼ b2 ¼ cp2 2
2
a0 ¼
where
ð32Þ
l2 satisfies the following equation cpa2 ð6a2 7a0 Þ þ
1 ða0 þ 2a22 þ 8a0 a2 Þ þ ca0 ða20 þ 6a22 Þ þ 12ca2 ða20 þ a22 Þ ¼ 0: pþ1
The following three cases will be investigated: Case (I) l2 > 0. Case (II) l2 < 0. Case (III) l2 ¼ a1 þ ia2 , where a1 ; a2 are two real number. In view (32) we find a family of solitary patterns solutions for Case (I): l2 > 0.
n h io1p 2 2 uðx; tÞ ¼ a0 þ a2 tanh lðx ctÞ þ coth lðx ctÞ : In view (32) we find a family of solitary patterns solutions for Case (II): given by
ð33Þ
l2 < 0, we obtain a family of compactons solutions
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S. Tang et al. / Applied Mathematics and Computation 215 (2009) 2768–2774
uðx; tÞ ¼
1p qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi a0 a2 tan2 l2 ðx ctÞ þ cot2 l2 ðx ctÞ :
In view (32) we find a family of solitary patterns solutions for Case (III):
ð34Þ
l2 ¼ a1 þ ia2 . Denote that two second roots of l2 :
l1 ¼ a3 þ ia4 ; l2 ¼ a5 þ ia6 ; we obtain the complex solutions:
(
uðx; tÞ ¼
2 cos a4 ðx ctÞ sinh a3 ðx ctÞ þ i sin a4 ðx ctÞ cosh a3 ðx ctÞ cos a4 ðx ctÞ cosh a3 ðx ctÞ þ i sin a4 ðx ctÞ sinh a3 ðx ctÞ 2 )1p cos a4 ðx ctÞ cosh a3 ðx ctÞ þ i sin a4 ðx ctÞ sinh a3 ðx ctÞ þ cos a4 ðx ctÞ sinh a3 ðx ctÞ þ i sin a4 ðx ctÞ cosh a3 ðx ctÞ a0 þ a2
ð35Þ
and
(
uðx; tÞ ¼
2 cos a6 ðx ctÞ sinh a5 ðx ctÞ þ i sin a6 ðx ctÞ cosh a5 ðx ctÞ cos a6 ðx ctÞ cosh a5 ðx ctÞ þ i sin a6 ðx ctÞ sinh a5 ðx ctÞ 2 )1p cos a6 ðx ctÞ cosh a5 ðx ctÞ þ i sin a6 ðx ctÞ sinh a5 ðx ctÞ þ : cos a6 ðx ctÞ sinh a5 ðx ctÞ þ i sin a6 ðx ctÞ cosh a5 ðx ctÞ a0 þ a2
ð36Þ
4.2. Case II l ¼ 2ðp þ 1Þ; p ¼ p In this case, Eq. (4) becomes
2a/p /00 þ ap/p1 ð/0 Þ2 þ
1 /2pþ1 þ c/ ¼ 0: 2p þ 1
ð37Þ
Balancing / with /p /00 we find
M ¼ pM þ 4 þ M 2;
ð38Þ
so that
2 M¼ : p
ð39Þ
To get a closed form analytic solution, the parameter M should be an integer. A transformation formula 1
/ ¼ v p
ð40Þ
should be used to achieve our goal. This in turn transforms (37) to
að3p þ 2Þ p2 Balancing
vv 00
2 1 ðv 0 Þ2 avv 00 þ v þ cv 3 ¼ 0: p 2p þ 1
and
ð41Þ
v 3 gives M ¼ 2. The tanh method allows us to use the substitution
v ðx; tÞ ¼ SðYÞ ¼ a0 þ a1 Y þ a2 Y 2 :
ð42Þ
Substituting (42) into (41), collecting the coefficients of each power of Y, and solve the resulting system of algebraic equations to find the following sets of solutions:
a0 ¼
8al2 ð2p þ 1Þ
a1 ¼ 0;
a2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½8al2 ð2p þ 1Þ2 cð3p þ 5Þð5p þ 2Þð2p þ 1Þ
8al2 ; cp2
l
ð3p þ 2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 8½3cp2 ðp þ 2Þð2p þ 1Þ D ¼ ; 9cap2 ðp þ 2Þ2
; ð43Þ
where D ¼ 9c2 p4 ð2p þ 1Þ2 ðp þ 2Þ2 þ cð9p2 4Þð2p þ 1Þð6p3 3p2 þ 4p þ 4Þ, c is selected as a free parameter. Noting that 1 / ¼ v p , we find a family of solitary patterns solutions for a < 0; c > 0
( uðx; tÞ ¼ and
2
a0 þ a2 tanh
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ þ D 9cap2 ðp þ 2Þ2
ð44Þ
S. Tang et al. / Applied Mathematics and Computation 215 (2009) 2768–2774
( 2
uðx; tÞ ¼
a0 þ a2 coth
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ þ D 9cap2 ðp þ 2Þ2
2773
ð45Þ
:
For a > 0; c > 0, we also find a family of solitary patterns solutions
( 2
uðx; tÞ ¼
a0 þ a2 tanh
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ D
ð46Þ
9cap2 ðp þ 2Þ2
and
( 2
uðx; tÞ ¼
a0 þ a2 coth
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ D 9cap2 ðp þ 2Þ2
ð47Þ
:
However, for a > 0; c > 0, we obtain a family of compactons solutions given by
( uðx; tÞ ¼
a0 a2 tan
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 2 ðp þ 2Þð2p þ 1Þ þ 8½3cp D 2
ð48Þ
9cap2 ðp þ 2Þ2
and
( 2
uðx; tÞ ¼
a0 a2 cot
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ þ D 9cap2 ðp þ 2Þ2
ð49Þ
:
For a < 0; c > 0, we also obtain a family of compactons solutions given by
( uðx; tÞ ¼
a0 a2 tan
2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ D
ð50Þ
9cap2 ðp þ 2Þ2
and
( uðx; tÞ ¼
2
a0 a2 cot
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ D 9cap2 ðp þ 2Þ2
ð51Þ
:
We next apply the extended tanh method give in (12), where we substitute
uðnÞ ¼ a0 þ a1 Y þ a2 Y 2 þ b1 Y 1 þ b2 Y 2 ;
ð52Þ
into (24), and proceed as before to find the following sets for a0 ; a1 ; a2 ; b1 ; b2 and
a0 ¼
l4 ¼
16al2 ; 3cp2
a1 ¼ b1 ¼ 0;
a2 ¼ b2 ¼
45cp4
a2 ð2p þ 1Þð9216p þ 11264Þ
l
8al2 ; cp2 ð53Þ
:
The following two cases will be investigated: Case (I) l4 > 0. Case (II) l4 < 0. In view (53) we find a family of solitary patterns solutions for case (I):
n h io1p 2 2 uðx; tÞ ¼ a0 þ a2 tanh lðx ctÞ þ coth lðx ctÞ : Denote that four fourth roots of
l1;2
¼ i
ð54Þ
l4 < 0:
14 45cp4 ; a2 ð2p þ 1Þð9216p þ 11264Þ
lk ¼ ak þ ibk ; k ¼ 3; 4;
where ak ; bk are real number. In view (53) we find a family of compactons solutions for
l1;2 given by
1 uðx; tÞ ¼ a0 a2 tan2 l1 ðx ctÞ þ cot2 l1 ðx ctÞ p : However, for
ð55Þ
lk ðk ¼ 3; 4Þ, we obtain the complex solutions:
n h io1p 2 2 uðx; tÞ ¼ a0 þ a2 tanh lk ðx ctÞ þ coth lk ðx ctÞ ;
k ¼ 3; 4:
ð56Þ
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S. Tang et al. / Applied Mathematics and Computation 215 (2009) 2768–2774
5. Discussion The sine–cosine method, the tanh method and the extended tanh method were used to investigate the K ðl; pÞ equation. The study revealed compactons solutions and solitary patterns solutions for some examined variants. The study emphasized the fact that the two methods are reliable in handling nonlinear problems. The obtained results clearly demonstrate the efficiency of the two methods used in this work. Moreover, the methods are capable of greatly minimizing the size of computational work compared to other existing techniques. The two methods worked successfully in handling nonlinear dispersive equations. This emphasizes the fact that the two methods are applicable to a wide variety of nonlinear problems. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Travelling wave solutions for a class of generalized KdV equation q Shengqiang Tang *, Jianxian Zheng, Wentao Huang School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, PR China
a r t i c l e
i n f o
a b s t r a c t In this work, the K ðl; pÞ equation is investigated. The sine–cosine method, the tanh method and the extended tanh method are efficiently used for analytic study of this equation. New solitary patterns solutions and compactons solutions are formally derived. The proposed schemes are reliable and manageable. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Compactons Solitary patterns solutions Sine–cosine method The extended tanh method
1. Introduction In 1993, Cooper et al. [1] considered the following generalized KdV equation:
K ðl; pÞ :
ut ¼ ux ul2 þ a½2uxxx up þ 4pup1 ux uxx þ pðp 1Þup2 ðux Þ3 ;
ð1Þ
þ
where p; l P 2; l; p 2 Z . This equation was derived from the Lagrangian
# Z " 1 ðwx Þl p 2 Lðl pÞ ¼ ww þ aðwx Þ ðwxx Þ dx; 2 x t lðl 1Þ
ð2Þ
where uðx; tÞ was defined by uðx; tÞ ¼ wx ðx; tÞ: The authors of [1] investigated the Hamiltonian structure and integrability properties for this class of KdV equations. The authors of [2] investigated the bifurcation behavior of the travelling wave solutions of the corresponding travelling wave equations in its parameter space. It is well known that searching for explicit solutions for nonlinear evolution equation, by using different methods, is the goal for many researchers. Many powerful methods, such as Bäcklund transformation, inverse scattering method, Hirota bilinear forms, pseudo spectral method, the tanh–sech method [3–8], the sine–cosine method [9], and many other techniques were successfully used to investigate these types of equations. Practically, there is no unified method that can be used to handle all types of nonlinear problems. The sine–cosine method and the extended tanh method are direct and effective algebraic method for finding exact solutions of nonlinear diffusion equations and Wazwaz developed the methods of sine–cosine [10,11, and the references therein] and the extended tanh [6–8, and the references therein]. The sine–cosine and the extended tanh algorithms, that provides a systematic framework for many nonlinear dispersive equations, will be employed to back up our analysis to determine compactons and solitary patterns travelling waves solutions. Let uðx; tÞ ¼ /ðx ctÞ ¼ /ðnÞ, where c is the wave speed. Then (1) becomes to
c/0 ¼
1 ð/l1 Þ0 þ a½2ð/p /00 Þ0 þ pð/p1 ð/0 Þ2 Þ0 ; l1
ð3Þ
q This research was supported by Science foundation of the Education Department of Guangxi Province (D2008007) and Program for Excellent Talents in Guangxi Higher Education Institutions. * Corresponding author. E-mail address: [email protected] (S. Tang).
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.09.019
S. Tang et al. / Applied Mathematics and Computation 215 (2009) 2768–2774
2769
where ‘‘0 ” is the derivative with respect to n. Integrating (3) once and setting integration constant as 0, we have
2a/p /00 þ ap/p1 ð/0 Þ2 þ
1 /l1 þ c/ ¼ 0: l1
ð4Þ
The paper is organized as follows: In Section 2, the sine–cosine method and the tanh method are briefly discussed. In Section 3, represents exact analytical solutions of (1) by using the sine–cosine method. In Section 4, represents exact analytical solutions of (1) by using the tanh method and the extended tanh method. In the last section, we conclude the paper and give some discussions. 2. Analysis of the two methods The sine–cosine method, the tanh method and the extended tanh method have been applied for a wide variety of nonlinear problems. The main features of the two methods will be reviewed briefly. For both methods, we first use the wave variable n ¼ x ct to carry a PDE in two independent variables
Pðu; ut ; ux ; uxx ; uxxx ; . . .Þ ¼ 0
ð5Þ
into an ODE
Q ðu; u0 ; u00 ; u000 ; . . .Þ ¼ 0:
ð6Þ
Eq. (6) is then integrated as long as all terms contain derivatives where integration constants are considered zeros. 2.1. The sine–cosine method The sine–cosine method admits the use of the solution in the form [5, and references therein].
uðx; tÞ ¼
k cosb ðlnÞ; jlnj < p2 ; 0;
otherwise
ð7Þ
or in the form
( uðx; tÞ ¼
b
k sin ðlnÞ; jlnj < p2 ; 0;
otherwise;
ð8Þ
where k; l and b are parameters that will be determined. We substitute (7) or (8) into the reduced ordinary differential equation obtained above in (6), balance the terms of the cosine functions when (7) is used, or balance the terms of the sine functions when (8) is used, and solving the resulting system of algebraic equations by using the computerized symbolic calculations to obtain all possible value of the parameters k; l and b. 2.2. The tanh method and the extended tanh method The standard tanh method introduced in [3–5] where the tanh is used as a new variable, since all derivatives of a tanh are represented by a tanh itself. We use a new independent variable
Y ¼ tanhðlnÞ;
ð9Þ
that leads to the change of derivatives:
d d ¼ lð1 Y 2 Þ ; dn dY 2
d
dn2
¼l
2
! 2 d 2 d ð1 Y Þ 2Y þ ð1 Y Þ 2 : dY dY 2
ð10Þ
We then apply the following finite expansion:
uðlnÞ ¼ SðYÞ ¼
M X
ak Y k
ð11Þ
k¼0
and
uðlnÞ ¼ SðYÞ ¼
M X k¼0
ak Y k þ
M X
ak Y k ;
ð12Þ
k¼1
where M is a positive integer that will be determined to derive a closed form analytic solution. However, if M is not an integer, a transformation formula is usually used. Substituting (9) and (10) into the simplified ODE (6) results in an equation in
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S. Tang et al. / Applied Mathematics and Computation 215 (2009) 2768–2774
powers of Y. To determine the parameter M, we usually balance the linear terms of highest order in the resulting equation with the highest order nonlinear terms. With M determined, we collect all coefficients of powers of Y in the resulting equation where these coefficients have to vanish. This will give a system of algebraic equations involving the parameters ak ðk ¼ 0; . . . ; MÞ; l and c. Having determined these parameters, knowing that M is a positive integer in most cases, and using (11) or (12) we obtain an analytic solution uðx; tÞ in a closed form. 3. Using the sine–cosine method Substituting (7) into (4) yields
akpþ1 l2 b½bðp þ 2Þ 2 cosbðpþ1Þ2 ðlnÞ akpþ1 l2 b2 ðp þ 2Þ cosbðpþ1Þ ðlnÞ þ
1 l1 k cosbðl1Þ þck cosb ðlnÞ ¼ 0: l1
ð13Þ
It is clear that Eq. (13) is satisfied if the following system of algebraic equations holds:
bðp þ 1Þ 2 ¼ b;
bðp þ 1Þ ¼ bðl 1Þ;
akpþ1 l2 b½bðp þ 2Þ 2 ¼ ck; akpþ1 l2 b2 ðp þ 2Þ ¼
1 l1 k : l1
ð14Þ
Solving this system yields
p ¼ l 2;
b¼
2 ; p
p
1
1p
l ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; k ¼ cðp þ 1Þðp þ 2Þ ; 2 2 aðp þ 1Þðp þ 2Þ
ð15Þ
that can also be obtained by using the sine ansatz (8). Consequently, for a > 0 we obtain a family of compactons solutions
8 1p > < pðxctÞ ; jlnj < p2 ; 12 cðp þ 1Þðp þ 2Þ cos2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; tÞ ¼ 2 aðpþ1Þðpþ2Þ > : 0; otherwise
ð16Þ
8 1p > < 2 pðxctÞ ; jlnj < p2 ; 12 cðp þ 1Þðp þ 2Þ sin pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uðx; tÞ ¼ 2 aðpþ1Þðpþ2Þ > : 0; otherwise:
ð17Þ
and
However, for a < 0 we obtain solitary patterns solutions
( uðx; tÞ ¼
" #)1p 1 pðx ctÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cðp þ 1Þðp þ 2Þcosh 2 2 aðp þ 1Þðp þ 2Þ
ð18Þ
" #)1p 1 pðx ctÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : cðp þ 1Þðp þ 2Þsinh 2 2 aðp þ 1Þðp þ 2Þ
ð19Þ
and
( uðx; tÞ ¼
4. Using the tanh method and the extended tanh method The K ðl; pÞ equation: the following two cases will be investigated: Case I l ¼ p þ 2; p ¼ p > 2. Case II l ¼ 2ðp þ 1Þ; p ¼ p. 4.1. Case I. l ¼ p þ 2; p ¼ p > 2 In this case, Eq. (4) becomes
2a/p /00 þ ap/p1 ð/0 Þ2 þ
1 /pþ1 þ c/ ¼ 0: pþ1
ð20Þ
Balancing / with /p /00 we find
M ¼ pM þ 4 þ M 2;
ð21Þ
so that
2 M¼ : p
ð22Þ
S. Tang et al. / Applied Mathematics and Computation 215 (2009) 2768–2774
2771
To get a closed form analytic solution, the parameter M should be an integer. A transformation formula 1
/ ¼ v p
ð23Þ
should be used to achieve our goal. This in turn transforms (20) to
að3p þ 2Þ p2 Balancing
2 1 ðv 0 Þ2 avv 00 þ v 2 þ cv 3 ¼ 0: p pþ1
ð24Þ
vv 00 and v 3 gives M ¼ 2. The tanh method allows us to use the substitution
v ðx; tÞ ¼ SðYÞ ¼ a0 þ a1 Y þ a2 Y 2 :
ð25Þ
Substituting (25) into (24), collecting the coefficients of each power of Y, and solve the resulting system of algebraic equations to find the following sets of solutions:
2½4al2 ð6p þ 1Þðp þ 1Þ þ p2 ; a1 ¼ 0; 3cp2 ðp þ 1Þðp þ 2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4al2 ðp 2Þ p 15p2 þ 24p þ 8 ; l¼ ; a2 ¼ 3cp2 4 aðp þ 1Þð3p3 12p2 9p 1Þ a0 ¼
ð26Þ
1
where c is selected as a free parameter. Noting that / ¼ v p , for a > 0; p P 5 or a < 0; 2 6 p 6 4, we find a family of solitary patterns solutions
(
" sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1p p 15p2 þ 24p þ 8 a0 þ a2 tanh ðx ctÞ 4 aðp þ 1Þð3p3 12p2 9p 1Þ 2
uðx; tÞ ¼
ð27Þ
and
( 2
uðx; tÞ ¼
a0 þ a2 coth
" sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1p p 15p2 þ 24p þ 8 ðx ctÞ : 4 aðp þ 1Þð3p3 12p2 9p 1Þ
ð28Þ
However, for a < 0; p P 5 or a > 0; 2 6 p 6 4, we obtain a family of compactons solutions given by
(
" sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1p p ð15p2 þ 24p þ 8Þ ðx ctÞ a0 a2 tan 4 aðp þ 1Þð3p3 12p2 9p 1Þ 2
uðx; tÞ ¼
ð29Þ
and
( uðx; tÞ ¼
" sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #)1p p ð15p2 þ 24p þ 8Þ a0 a2 cot : ðx ctÞ 4 aðp þ 1Þð3p3 12p2 9p 1Þ 2
ð30Þ
We next apply the extended tanh method give in (12), where we substitute
uðnÞ ¼ a0 þ a1 Y þ a2 Y 2 þ b1 Y 1 þ b2 Y 2
ð31Þ
into (24), and proceed as before to find the following sets for a0 ; a1 ; a2 ; b1 ; b2 and
l
8al ð2p þ 1Þðp þ 2Þ þ p ; 3cp2 ðp þ 2Þ 8al2 ; b1 ¼ 0; a1 ¼ 0; a2 ¼ b2 ¼ cp2 2
2
a0 ¼
where
ð32Þ
l2 satisfies the following equation cpa2 ð6a2 7a0 Þ þ
1 ða0 þ 2a22 þ 8a0 a2 Þ þ ca0 ða20 þ 6a22 Þ þ 12ca2 ða20 þ a22 Þ ¼ 0: pþ1
The following three cases will be investigated: Case (I) l2 > 0. Case (II) l2 < 0. Case (III) l2 ¼ a1 þ ia2 , where a1 ; a2 are two real number. In view (32) we find a family of solitary patterns solutions for Case (I): l2 > 0.
n h io1p 2 2 uðx; tÞ ¼ a0 þ a2 tanh lðx ctÞ þ coth lðx ctÞ : In view (32) we find a family of solitary patterns solutions for Case (II): given by
ð33Þ
l2 < 0, we obtain a family of compactons solutions
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S. Tang et al. / Applied Mathematics and Computation 215 (2009) 2768–2774
uðx; tÞ ¼
1p qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi a0 a2 tan2 l2 ðx ctÞ þ cot2 l2 ðx ctÞ :
In view (32) we find a family of solitary patterns solutions for Case (III):
ð34Þ
l2 ¼ a1 þ ia2 . Denote that two second roots of l2 :
l1 ¼ a3 þ ia4 ; l2 ¼ a5 þ ia6 ; we obtain the complex solutions:
(
uðx; tÞ ¼
2 cos a4 ðx ctÞ sinh a3 ðx ctÞ þ i sin a4 ðx ctÞ cosh a3 ðx ctÞ cos a4 ðx ctÞ cosh a3 ðx ctÞ þ i sin a4 ðx ctÞ sinh a3 ðx ctÞ 2 )1p cos a4 ðx ctÞ cosh a3 ðx ctÞ þ i sin a4 ðx ctÞ sinh a3 ðx ctÞ þ cos a4 ðx ctÞ sinh a3 ðx ctÞ þ i sin a4 ðx ctÞ cosh a3 ðx ctÞ a0 þ a2
ð35Þ
and
(
uðx; tÞ ¼
2 cos a6 ðx ctÞ sinh a5 ðx ctÞ þ i sin a6 ðx ctÞ cosh a5 ðx ctÞ cos a6 ðx ctÞ cosh a5 ðx ctÞ þ i sin a6 ðx ctÞ sinh a5 ðx ctÞ 2 )1p cos a6 ðx ctÞ cosh a5 ðx ctÞ þ i sin a6 ðx ctÞ sinh a5 ðx ctÞ þ : cos a6 ðx ctÞ sinh a5 ðx ctÞ þ i sin a6 ðx ctÞ cosh a5 ðx ctÞ a0 þ a2
ð36Þ
4.2. Case II l ¼ 2ðp þ 1Þ; p ¼ p In this case, Eq. (4) becomes
2a/p /00 þ ap/p1 ð/0 Þ2 þ
1 /2pþ1 þ c/ ¼ 0: 2p þ 1
ð37Þ
Balancing / with /p /00 we find
M ¼ pM þ 4 þ M 2;
ð38Þ
so that
2 M¼ : p
ð39Þ
To get a closed form analytic solution, the parameter M should be an integer. A transformation formula 1
/ ¼ v p
ð40Þ
should be used to achieve our goal. This in turn transforms (37) to
að3p þ 2Þ p2 Balancing
vv 00
2 1 ðv 0 Þ2 avv 00 þ v þ cv 3 ¼ 0: p 2p þ 1
and
ð41Þ
v 3 gives M ¼ 2. The tanh method allows us to use the substitution
v ðx; tÞ ¼ SðYÞ ¼ a0 þ a1 Y þ a2 Y 2 :
ð42Þ
Substituting (42) into (41), collecting the coefficients of each power of Y, and solve the resulting system of algebraic equations to find the following sets of solutions:
a0 ¼
8al2 ð2p þ 1Þ
a1 ¼ 0;
a2 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½8al2 ð2p þ 1Þ2 cð3p þ 5Þð5p þ 2Þð2p þ 1Þ
8al2 ; cp2
l
ð3p þ 2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi 8½3cp2 ðp þ 2Þð2p þ 1Þ D ¼ ; 9cap2 ðp þ 2Þ2
; ð43Þ
where D ¼ 9c2 p4 ð2p þ 1Þ2 ðp þ 2Þ2 þ cð9p2 4Þð2p þ 1Þð6p3 3p2 þ 4p þ 4Þ, c is selected as a free parameter. Noting that 1 / ¼ v p , we find a family of solitary patterns solutions for a < 0; c > 0
( uðx; tÞ ¼ and
2
a0 þ a2 tanh
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ þ D 9cap2 ðp þ 2Þ2
ð44Þ
S. Tang et al. / Applied Mathematics and Computation 215 (2009) 2768–2774
( 2
uðx; tÞ ¼
a0 þ a2 coth
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ þ D 9cap2 ðp þ 2Þ2
2773
ð45Þ
:
For a > 0; c > 0, we also find a family of solitary patterns solutions
( 2
uðx; tÞ ¼
a0 þ a2 tanh
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ D
ð46Þ
9cap2 ðp þ 2Þ2
and
( 2
uðx; tÞ ¼
a0 þ a2 coth
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ D 9cap2 ðp þ 2Þ2
ð47Þ
:
However, for a > 0; c > 0, we obtain a family of compactons solutions given by
( uðx; tÞ ¼
a0 a2 tan
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 2 ðp þ 2Þð2p þ 1Þ þ 8½3cp D 2
ð48Þ
9cap2 ðp þ 2Þ2
and
( 2
uðx; tÞ ¼
a0 a2 cot
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ þ D 9cap2 ðp þ 2Þ2
ð49Þ
:
For a < 0; c > 0, we also obtain a family of compactons solutions given by
( uðx; tÞ ¼
a0 a2 tan
2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ D
ð50Þ
9cap2 ðp þ 2Þ2
and
( uðx; tÞ ¼
2
a0 a2 cot
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi )1p 8½3cp2 ðp þ 2Þð2p þ 1Þ D 9cap2 ðp þ 2Þ2
ð51Þ
:
We next apply the extended tanh method give in (12), where we substitute
uðnÞ ¼ a0 þ a1 Y þ a2 Y 2 þ b1 Y 1 þ b2 Y 2 ;
ð52Þ
into (24), and proceed as before to find the following sets for a0 ; a1 ; a2 ; b1 ; b2 and
a0 ¼
l4 ¼
16al2 ; 3cp2
a1 ¼ b1 ¼ 0;
a2 ¼ b2 ¼
45cp4
a2 ð2p þ 1Þð9216p þ 11264Þ
l
8al2 ; cp2 ð53Þ
:
The following two cases will be investigated: Case (I) l4 > 0. Case (II) l4 < 0. In view (53) we find a family of solitary patterns solutions for case (I):
n h io1p 2 2 uðx; tÞ ¼ a0 þ a2 tanh lðx ctÞ þ coth lðx ctÞ : Denote that four fourth roots of
l1;2
¼ i
ð54Þ
l4 < 0:
14 45cp4 ; a2 ð2p þ 1Þð9216p þ 11264Þ
lk ¼ ak þ ibk ; k ¼ 3; 4;
where ak ; bk are real number. In view (53) we find a family of compactons solutions for
l1;2 given by
1 uðx; tÞ ¼ a0 a2 tan2 l1 ðx ctÞ þ cot2 l1 ðx ctÞ p : However, for
ð55Þ
lk ðk ¼ 3; 4Þ, we obtain the complex solutions:
n h io1p 2 2 uðx; tÞ ¼ a0 þ a2 tanh lk ðx ctÞ þ coth lk ðx ctÞ ;
k ¼ 3; 4:
ð56Þ
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S. Tang et al. / Applied Mathematics and Computation 215 (2009) 2768–2774
5. Discussion The sine–cosine method, the tanh method and the extended tanh method were used to investigate the K ðl; pÞ equation. The study revealed compactons solutions and solitary patterns solutions for some examined variants. The study emphasized the fact that the two methods are reliable in handling nonlinear problems. The obtained results clearly demonstrate the efficiency of the two methods used in this work. Moreover, the methods are capable of greatly minimizing the size of computational work compared to other existing techniques. The two methods worked successfully in handling nonlinear dispersive equations. This emphasizes the fact that the two methods are applicable to a wide variety of nonlinear problems. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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