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The G′G -expansion method for some nonlinear evolution equations
Applied Mathematics and Computation 217 (2010) 384–391
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The
G0 G
-expansion method for some nonlinear evolution equations
S. Kutluay, A. Esen *, O. Tasbozan Department of Mathematics, Inonu University, Faculty of Arts and Science, 44280 Malatya, Turkey
a r t i c l e
i n f o
a b s t r a c t 0 In this paper, the GG -expansion method is applied to the Liouville, sine–Gordon and new coupled MKdV equations to obtain their some generalized exact travelling wave solutions. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: 0 G G -expansion method Liouville equation Sine–Gordon equation Coupled MKdV equation
1. Introduction Mathematical modeling of many physical phenomena in various fields of physics and engineering generally leads to nonlinear ordinary or partial differential equations. It is known that investigating and constructing exact solutions of these equations are of great importance in applied mathematics. Therefore, in recent years, many effective methods such as sine–cosine method [1], tanh function method [2], variational iteration method [3], homotopy perturbation method [4–6], Exp-function method [7,8], F-expansion method [9,10], and others have been proposed for obtaining exact solutions to nonlinear partial differential equations. Some of these methods use transformations to reduce nonlinear equations into more simple equations, some others of methods use a trial function in an iterative scheme which converges rapidly to the exact solution and also there are still some other methods which consist of looking for the solution of the nonlinear evolution equations (NLEEs) considered as a polynomial in variable satisfying a subsidiary nonlinear ordinary differential equation. 0 Recently, Wang et al. [11] have proposed a method called the GG -expansion method to obtain exact travelling wave solutions of NLEEs arising in fluid dynamics, plasma, elastic media, optical fibers, etc. They successfully applied the method to find the travelling wave solutions involving parameters of the Korteweg–de Vries, the modified Korteweg–de Vries, the var 0 iant Boussinesq equations and the Hirota–Satsuma equations. The GG -expansion method has also successfully been applied G0 to various NLEEs 0 [12–16]. The key idea of the G -expansion method is to express the solution of NLLEs a finite series of the polynomial in GG satisfying the second order linear subsidiary ordinary differential equation G00 + kG0 + lG = 0. Before applying the method to the NLEEs considered in this paper, let us give the solution procedure of the method. 2. The basic idea behind the
G0 -expansion method G
To illustrate the basic idea behind this method, we consider the following nonlinear partial differential equation with only two independent variables x and t, and dependent variable u
Nðu; ut ; ux ; utt ; uxt ; uxx ; . . .Þ ¼ 0:
ð1Þ
Using the travelling wave transformation
u ¼ uðnÞ;
n ¼ x wt;
* Corresponding author. E-mail address: [email protected] (A. Esen). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.05.073
ð2Þ
S. Kutluay et al. / Applied Mathematics and Computation 217 (2010) 384–391
385
Eq. (1) reduces to an ordinary differential equation (ODE) in the form
N uðnÞ; wu0 ðnÞ; u0 ðnÞ; w2 u00 ðnÞ; wu00 ðnÞ; u00 ðnÞ; . . . ¼ 0:
ð3Þ
G0
The G -expansion method is based on the assumption that travelling wave solutions of Eq. (3) can be expressed by a poly 0 nomial in GG as
uðnÞ ¼
0 j n X G aj ; G j¼0
an – 0;
ð4Þ
with G = G(n) satisfying the second order linear ODE
G00 þ kG0 þ lG ¼ 0;
ð5Þ
where aj (j = 0, 1, 2,. . ., n), k and l are constants to be determined later. u(n) can be determined explicitly by using the following steps: Step 1. By considering the homogeneous balance between the highest nonlinear terms and the highest order derivatives of u(n) in Eq. (3), the positive integer n in Eq. (4) is determined. 0 Step 2. By substituting (4) with Eq. (5) into Eq. (3) and collecting all terms with the same power of GG together, the left hand side of Eq. (3) is converted into a polynomial. After setting each coefficient of this polynomial to zero, we obtain a set of algebraic equations in terms of aj (j = 0, 1, 2,. . ., n), w, k and l. Step 3. Solving the system of algebraic equations with the aid of symbolic computation and then substituting the results with the general solutions of Eq. (5) into Eq. (4) gives travelling wave solutions of Eq. (3). To illustrate the effectiveness and convenience of the method, we apply it to the Liouville, sine–Gordon and new coupled MKdV equations described in the next section. 3. Applications of the
G 0 -expansion method G
3.1. The Liouville equation We first consider the Liouville equation [17]
uxt þ eu ¼ 0:
ð6Þ
To look for travelling wave solutions of Eq. (6), we use the wave transformation n = x wt and change Eq. (6) into the form of an ODE
wu00 þ eu ¼ 0:
ð7Þ
Using the transformation
u ¼ lnv ;
ð8Þ
Eq. (7) reduces to nonlinear ODE in the form
wðvv 00 v 02 Þ þ v 3 ¼ 0;
ð9Þ
where the prime denotes differentiation with respect to n. We now suppose that the solution of Eq. (9) is given by Eq. (4). In 00 order to determine the value of integer n, we balance vv with v3, then we get 2n + 2 = 3n which yields n = 2. Therefore, the 0 solution of (9) can be expressed by a polynomial in GG as follows:
v ðnÞ ¼ a0 þ a1
0 0 2 G G þ a2 ; G G
a2 – 0:
ð10Þ
0 i Substituting Eq. (10) along with (5) into Eq. (9), and setting the coefficients of all powers of GG ði ¼ 0; 1; . . . ; 6Þ to zero, we obtain a system of nonlinear algebraic equations for a0, a1, a2 and w. Solving the resulting system with the help of Maple, we have the following set of solution:
a0 ¼ 2wl;
a1 ¼ 2wk;
a2 ¼ 2w;
w ¼ w;
ð11Þ
where k and l are arbitrary constants. Inserting Eq. (11) into (10), we obtain
0 0 2 G G : þ 2w G G
v ðnÞ ¼ 2wl þ 2wk
ð12Þ
Substituting the general solutions of Eq. (5) into Eq. (12) we have three types of travelling wave solutions of Eq. (9) as follows:
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S. Kutluay et al. / Applied Mathematics and Computation 217 (2010) 384–391
When k2 4l > 0
80 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 > > 2 2 1 1 < = c cosh 4 l sinh 4 l k n þ c k n 1 2 1 2 2 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1 : v 1 ðnÞ ¼ wðk2 4lÞ>B @ > 2 : c1 sinh 1 k2 4ln þ c2 cosh 1 k2 4ln ; 2 2
ð13Þ
When k2 4l < 0
80 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 > > 2 2 1 1 < = c sin l k cos l k 4 n þ c 4 n 1 2 1 2 2 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ 1 v 2 ðnÞ ¼ wðk2 4lÞ>B @ > 2 : c1 cos 1 4l k2 n þ c2 sin 1 4l k2 n ; 2 2
ð14Þ
and when k2 4l = 0
v 3 ðnÞ ¼
2wc22 ðc1 þ c2 nÞ2
ð15Þ
;
where n = x wt, c1 and c2 arbitrary constants. In particular, if we choose c2 – 0; c21 < c22 , then the solution (13) give the solitary wave solution
v 1 ðx; tÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 wðk2 4lÞsech ðx wtÞ k2 4l þ n0 ; 2 2
ð16Þ
1
where k2 4l > 0; n0 ¼ tanh ðcc12 Þ and the solution (14) give the travelling wave solution
v 2 ðx; tÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 wðk2 4lÞcsc2 ðx wtÞ 4l k2 þ n1 ; 2 2
ð17Þ
where k2 4l < 0; n1 ¼ tan1 ðcc12 Þ and recall that u(x, t) = lnv(x, t), hence we obtain the travelling wave solutions of the Liouville equation from Eqs. (16) and (17) as follows, respectively
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 ðx wtÞ k2 4l þ n0 ; u1 ðx; tÞ ¼ ln wðk2 4lÞsech 2 2
w < 0;
ð18Þ
1
where k2 4l > 0 and n0 ¼ tanh ðcc12 Þ,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ðx wtÞ 4l k2 þ n1 ; u2 ðx; tÞ ¼ ln wðk2 4lÞcsc2 2 2
w > 0;
ð19Þ
where k2 4l < 0; n1 ¼ tan1 ðcc12 Þ and the third travelling wave solution from Eq. (15) is as follows
"
u3 ðx; tÞ ¼ ln
#
2wc22
ðc1 þ c2 ðx wtÞÞ2
w > 0;
;
ð20Þ
where c1 and c2 are free parameters. Among these solutions, it is possible to obtain the solution given by Wazwaz [17] by taking n0 = 0 and some manipulation in Eq. (18). It should be noted that the other solutions derived here do not appear in Ref. [17]. 3.2. The sine–Gordon equation We secondly consider sine–Gordon equation [18–20]
utt uxx þ sin u ¼ 0: In order to apply the 2
G0 G
ð21Þ
method, we use the wave transformation n = x wt and change Eq. (21) into the form
00
ðw 1Þu þ sin u ¼ 0:
ð22Þ
We next use the transformation
v ¼ eiu ;
ð23Þ
so that
sin u ¼ which gives
v v 1 2i
;
cos u ¼
v þ v 1 2
;
ð24Þ
S. Kutluay et al. / Applied Mathematics and Computation 217 (2010) 384–391
u ¼ arccos
v þ v 1 2
:
387
ð25Þ
This transformation will change Eq. (22) into the ODE in the form
2ðw2 1Þðv 00 v ðv 0 Þ2 Þ þ v 3 v ¼ 0;
ð26Þ
where the prime denotes differentiation with respect to n. We now suppose that the solution of Eq. (26) is given by Eq. (4). In order to determine the value of integer n, we balance vv00 with v3, we get 2n + 2 = 3n which yields n = 2. Thus the solution of Eq. (26) is exactly the same with Eq. (10). 0 i Substituting Eq. (10) along with (5) into (26) and setting the coefficients of all powers of GG ði ¼ 0; 1; . . . ; 6Þ to zero, we obtain a system of nonlinear algebraic equations for a0, a1, a2 and w. Solving the resulting system with the help of Maple, we have the following sets of solutions:
a0 ¼
k2 k 2 4l
a1 ¼
;
4k k 2 4l
;
a2 ¼
4 k2 4l
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 w¼ 1 2 k 4l
;
ð27Þ
and
a0 ¼
k2 k 2 4l
a1 ¼
;
4k k2 4l
;
a2 ¼
4 k 2 4l
;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 : w¼ 1þ 2 k 4l
ð28Þ
Inserting Eqs. (27) and (28) into Eq. (10), we get
v 1 ðnÞ ¼
k2 2
k 4l
þ
0 0 2 G 4 G þ 2 k 4l G k 4l G 4k
2
ð29Þ
and
v 2 ðnÞ ¼
k2 k2 4l
0 0 2 G 4 G ; k 2 4l G k2 4l G 4k
ð30Þ
respectively. Using the general solutions of (5), we obtain travelling wave solutions of Eq. (26) as follows When k2 4l > 0
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 1 1 c cosh 4 l sinh k n þ c k2 4lnC 1 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ; v 1 ðx; tÞ ¼ B @ c1 sinh 12 k2 4ln þ c2 cosh 12 k2 4ln where n ¼ x
ð31Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k2 4 t and l
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 1 1 c cosh 4 l sinh k n þ c k2 4lnC 1 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ; v 2 ðx; tÞ ¼ B @ c1 sinh 12 k2 4ln þ c2 cosh 12 k2 4ln where n ¼ x
ð32Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ k2 4 t; c1 and c2 arbitrary constants. l
When k2 4l < 0
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 1 1 c sin l k cos 4 n þ c 4l k2 nC 1 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ; v 3 ðx; tÞ ¼ B @ c1 cos 12 4l k2 n þ c2 sin 12 4l k2 n where n ¼ x
ð33Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k2 4 t and l
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 1 1 Bc1 sin 2 4l k n þ c2 cos 2 4l k nC qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ; v 4 ðx; tÞ ¼ @ c1 cos 12 4l k2 n þ c2 sin 12 4l k2 n
ð34Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 where n ¼ x ð 1 þ k2 4 Þt; c1 and c2 arbitrary constants. In particular, if we choose c2 –0; c21 < c22 , then the solutions (31)– l (34) give the travelling wave solutions
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k2 4ln þ n0 ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 k2 4ln þ n0 ; v 2 ðx; tÞ ¼ tanh 2
v 1 ðx; tÞ ¼ tanh2
ð35Þ ð36Þ
388
S. Kutluay et al. / Applied Mathematics and Computation 217 (2010) 384–391 1
where n0 ¼ tanh ðcc12 Þ; k2 4l > 0 and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4l k2 n þ n1 ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4l k 2 n þ n 1 ; v 4 ðx; tÞ ¼ cot2 2
v 3 ðx; tÞ ¼ cot2
where n1 ¼ tan1
c1 c2
ð37Þ ð38Þ
. Using Eq. (25), we get the following travelling wave solutions of Eq. (21)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 2 1 2 1 ; tanh k2 4ln þ n0 þ coth k2 4ln þ n0 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 2 1 2 1 ; tanh k2 4ln þ n0 þ coth k2 4ln þ n0 u2 ðx; tÞ ¼ arccos 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 1 u3 ðx; tÞ ¼ arccos tan2 4l k2 n þ n1 þ cot2 4l k2 n þ n1 2 2 2
u1 ðx; tÞ ¼ arccos
ð39Þ ð40Þ ð41Þ
and
u4 ðx; tÞ ¼ arccos
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 1 ; tan2 4l k2 n þ n1 þ cot2 4l k2 n þ n1 2 2 2
ð42Þ
1
where n0 ¼ tanh ðcc12 Þ and n1 ¼ tan1 ðcc12 Þ. Comparing our results with Wazwaz’s results in Ref. [18], we can see that the results are the same, if we chose n0 and n1 zero, for appropriate parameters. 3.3. The new coupled MKdV equation We finally consider the new coupled MKdV equation [21]
1 uxxx 3u2 ux þ 3ðvv x Þx þ 3 uv 2 x ; 2 ¼ v xxx 3ðv ux Þx þ 6uv ux þ 3 u2 v 2 v x ;
ut ¼
vt
ð43Þ
where Eq. (43) becomes the MKdV equation for v = 0. In Ref. [20], Wu et al. take into account a 4 4 matrix spectral problem with three potentials and obtain a new hierarchy of nonlinear evolution equations. In Ref. [22], Cao et al. proposed some kinds of soliton solutions for Eq. (43). Substituting the wave transformation u(x, t) = u(n), v(x, t) = v(n), n = x wt into Eq. (43), integrating the resulting equation with respect to n yields the ODEs
1 00 u u3 þ 3vv 0 þ 3uv 2 ; 2 c2 wv ¼ v 00 3v u0 v 3 þ 3u2 v ;
c1 wu ¼
ð44Þ
where c1 and c2 are integration constants that are to be determined later. Considering the homogenous balance between u00 and uv2 and between v00 and u2v in Eq. (44), we get n + 2 = n + 2m and m + 2 = 2n + m which yield n = m = 1, then we suppose that
0 G ; G 0 G v ðnÞ ¼ b0 þ b1 : G uðnÞ ¼ a0 þ a1
ð45Þ
0 i Substituting Eq. (45) along with (5) into Eq. (44), collecting the coefficients of GG (i = 0, 1, 2, 3) for each equation and setting it to zero, we obtain a system of nonlinear algebraic equations. Solving the resulting system, we have the following sets of solutions: 2
3
c1 ¼ k2 b0 þ 6kb0 8b0 þ
1 lk 2b0 l; 2
1 lk þ 2b0 l; 2 1 2 1 2 k þ 2l 12kb0 þ 24b0 ; a0 ¼ k b0 ; w¼ 4 2 2
3
c2 ¼ k2 b0 6kb0 þ 8b0
and
ð46Þ a1 ¼
1 ; 2
b1 ¼
1 2
S. Kutluay et al. / Applied Mathematics and Computation 217 (2010) 384–391
1 lk þ 2b0 l; 2 1 2 3 c2 ¼ k2 b0 þ 6kb0 þ 8b0 þ lk þ 2b0 l; 2 1 2 1 2 k þ 2l þ 12kb0 þ 24b0 ; a0 ¼ k þ b0 ; w¼ 4 2 2
389
3
c1 ¼ k2 b0 þ 6kb0 þ 8b0 þ
ð47Þ a1 ¼
1 ; 2
1 b1 ¼ : 2
Inserting Eqs. (46) and (47) into (45), we get
1 G0 k þ b0 ; 2 G 2 1 G0 þ b0 v 1 ðnÞ ¼ 2 G
ð48Þ
1 G0 k þ þ b0 ; 2 G 2 1 G0 þ b0 ; v 2 ðnÞ ¼ 2 G
ð49Þ
u1 ðnÞ ¼
and
u2 ðnÞ ¼
respectively. Using the general solutions of (5), we obtain travelling wave solutions of Eq. (44) as follows: when k2 4l > 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k2 4l BA1 cosh 12 k2 4ln þ A2 sinh 12 k2 4lnC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ b0 ; u1 ðnÞ ¼ @ 4 4 A1 sinh 12 k2 4ln þ A2 cosh 12 k2 4ln 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 4l BA1 cosh 12 k2 4ln þ A2 sinh 12 k2 4lnC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ b0 ; v 1 ðnÞ ¼ @ 4 4 A1 sinh 12 k2 4ln þ A2 cosh 12 k2 4ln
ð50Þ
2 where n ¼ x 14 k2 þ 2l 12kb0 þ 24b0 t and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k2 4l BA1 cosh 12 k2 4ln þ A2 sinh 12 k2 4lnC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ þ b0 ; u2 ðnÞ ¼ @ 4 4 A1 sinh 12 k2 4ln þ A2 cosh 12 k2 4ln qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k2 4l BA1 cosh 12 k2 4ln þ A2 sinh 12 k2 4lnC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ þ b0 ; v 2 ðnÞ ¼ @ 4 4 A1 sinh 12 k2 4ln þ A2 cosh 12 k2 4ln
ð51Þ
2 where n ¼ x 14 k2 þ 2l þ 12kb0 þ 24b0 t, A1 and A2 arbitrary constants. When k2 4l < 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4l k2 BA1 sin 12 4l k2 n þ A2 cos 12 4l k2 nC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ b0 ; u3 ðnÞ ¼ @ 4 4 A1 cos 12 4l k2 n þ A2 sin 12 4l k2 n 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4l k2 BA1 sin 12 4l k2 n þ A2 cos 12 4l k2 nC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ b0 ; v 3 ðnÞ ¼ @ 4 4 A1 cos 12 4l k2 n þ A2 sin 12 4l k2 n
ð52Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4l k2 BA1 sin 12 4l k2 n þ A2 cos 12 4l k2 nC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ þ b0 ; u4 ðnÞ ¼ @ 4 4 A1 cos 12 4l k2 n þ A2 sin 12 4l k2 n 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4l k2 BA1 sin 12 4l k2 n þ A2 cos 12 4l k2 nC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ þ b0 ; v 4 ðnÞ ¼ @ 4 4 A1 cos 12 4l k2 n þ A2 sin 12 4l k2 n
ð53Þ
2 where n ¼ x 14 k2 þ 2l 12kb0 þ 24b0 t and
2 where n ¼ x 14 k2 þ 2l þ 12kb0 þ 24b0 t, A1 and A2 arbitrary constants.
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S. Kutluay et al. / Applied Mathematics and Computation 217 (2010) 384–391
When k2 4l = 0
1 2A2 k þ b0 ; k 4 2 A1 þ A2 n 1 2A2 v 5 ðnÞ ¼ k þ b0 ; 4 A1 þ A2 n
ð54Þ
1 2A2 k k þ þ b0 ; 4 2 A1 þ A2 n 1 2A2 v 6 ðnÞ ¼ k þ b0 ; 4 A1 þ A2 n
ð55Þ
u5 ðnÞ ¼
2 where n ¼ x 14 k2 þ 2l 12kb0 þ 24b0 t and
u6 ðnÞ ¼
2 where n ¼ x 14 k2 þ 2l þ 12kb0 þ 24b0 t; A1 and A2 arbitrary constants. In particular, if we choose A2 – 0; A21 < A22 , then the solutions (50)–(53) lead to the travelling wave solutions as follows: when k2 4l > 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k þ b0 ; k k2 4l tanh k2 4lðx wtÞ þ n0 4 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ b0 ; k2 4lðx wtÞ þ n0 v 1 ðx; tÞ ¼ k k2 4l tanh 4 2
ð56Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k þ þ b0 ; k k2 4l tanh k2 4lðx wtÞ þ n0 4 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ b0 ; k2 4lðx wtÞ þ n0 v 2 ðx; tÞ ¼ k k2 4l tanh 4 2
ð57Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k þ b0 ; k 4l k2 cot 4l k2 ðx wtÞ þ n1 4 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ b0 ; 4l k2 ðx wtÞ þ n1 v 3 ðx; tÞ ¼ k 4l k2 cot 4 2
ð58Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k þ þ b0 ; k 4l k2 cot 4l k2 ðx wtÞ þ n1 4 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ b0 ; 4l k2 ðx wtÞ þ n1 v 4 ðx; tÞ ¼ k 4l k2 cot 4 2
ð59Þ
u1 ðx; tÞ ¼
2 where w ¼ 14 k2 þ 2l 12kb0 þ 24b0 and
u2 ðx; tÞ ¼
2 1 A1 where w ¼ 14 k2 þ 2l þ 12kb0 þ 24b0 ; n0 ¼ tanh . A2 When k2 4l < 0
u3 ðx; tÞ ¼
2 where w ¼ 14 k2 þ 2l 12kb0 þ 24b0 and
u4 ðx; tÞ ¼
2 where w ¼ 14 k2 þ 2l þ 12kb0 þ 24b0 ; n1 ¼ tan1 AA12 . 4. Conclusion
0 In this paper, the GG -expansion method has been successfully used to obtain some exact travelling wave solutions for the Liouville, sine–Gordon and new coupled MKdV equations. It can be clearly seen that some solutions given in this paper are new solutions which have not been reported yet. The method can also be effectively used to construct new and numerous exact travelling wave solutions for some other nonlinear evolution equations arising in mathematical physics. Acknowledgment This study was supported by Inonu University Scientific Research Project (I.U.-BAP:2009/05, Malatya, TURKEY). References [1] C. Yan, A simple transformation for nonlinear waves, Phys. Lett. A 224 (1996) 77–84. [2] W. Malfliet, W. Hereman, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Phys. Scripta 54 (1996) 563–568. [3] 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. [4] J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. 6 (2005) 207–208. [5] J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals 26 (2005) 695–700.
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[6] J.H. He, New interpretation of homotopy perturbation method, Int. J. Modern Phys. B 20 (2006) 2561–2568. [7] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals 30 (2006) 700–708. [8] X.H. Wu, J.H. He, Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method, Comput. Math. Appl. 54 (2007) 966–986. [9] Y.B. Zhou, M.L. Wang, Y.M. Wang, Periodic wave solutions to a coupled KdV equations with variable coefficients, Phys. Lett. A 308 (2003) 31–36. [10] J.B. Liu, K.Q. Yang, The extended F-expansion method and exact solutions of nonlinear PDEs, Chaos Solitons Fractals 22 (2004) 111–121. 0 [11] M.L. Wang, X. Li, J. Zhang, The GG -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A 372 (2008) 417–423. 0 [12] M.L. Wang, J. Zhang, X. Li, Application of the GG -expansion to travelling wave solutions of the BroerKaup and the approximate long water wave equations, Appl. Math. Comput. 206 (2008) 321–326. 0 G G -expansion method for nonlinear evolution equations, Phys. Lett. A 372 (2008) 3400.
[13] A. Bekir, Application of the
[14] L.X. Li, W.L. Wang, The (G0 /G)-expansion method and travelling wave solutions for a higher-order nonlinear Schrdinger equation, Appl. Math. Comput. 208 (2009) 440–445. [15] S. Zhang, W. Wang, J. Tong, A generalized (G0 /G)-expansion method and its application to the (2+1)-dimensional BroerKaup equations, Appl. Math. Comput. 209 (2009) 399–404. [16] I. Aslan, T. Ozis, On the validity and reliability of the (G0 /G)-expansion method by using higher-order nonlinear equations, Appl. Math. Comput. 211 (2009) 531–536. [17] A.M. Wazwaz, The tanh method for travelling wave solutions to the ZhiberShabat equation and other related equations, Commun. Nonlinear Sci. Numer. Simul. 13 (2008) 584–592. [18] A.M. Wazwaz, The tanh method: exact solutions of the sine–Gordon and the sinh–Gordon equations, Appl. Math. Comput. 167 (2005) 1196–1210. [19] A.M. Wazwaz, Exact solutions for the generalized sine–Gordon and the generalized sinh–Gordon equations, Chaos Solitons Fractals 28 (2006) 127–135. [20] A.L. Fabian, R. Kohl, A. Biswas, Perturbation of topological solitons due to sine–Gordon equation and its type, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1227–1244. [21] Y. Wu, X. Geng, X. Hu, S. Zhu, A generalized HirotaSatsuma coupled Kortewegde Vries equation and Miura transformations, Phys. Lett. A 255 (1999) 259–264. [22] D.B. Cao, J.R. Yan, Y. Zhang, Exact solutions for a new coupled MKdV equations and a coupled KdV equations, Phys. Lett. A 297 (2002) 68–74.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
The
G0 G
-expansion method for some nonlinear evolution equations
S. Kutluay, A. Esen *, O. Tasbozan Department of Mathematics, Inonu University, Faculty of Arts and Science, 44280 Malatya, Turkey
a r t i c l e
i n f o
a b s t r a c t 0 In this paper, the GG -expansion method is applied to the Liouville, sine–Gordon and new coupled MKdV equations to obtain their some generalized exact travelling wave solutions. Ó 2010 Elsevier Inc. All rights reserved.
Keywords: 0 G G -expansion method Liouville equation Sine–Gordon equation Coupled MKdV equation
1. Introduction Mathematical modeling of many physical phenomena in various fields of physics and engineering generally leads to nonlinear ordinary or partial differential equations. It is known that investigating and constructing exact solutions of these equations are of great importance in applied mathematics. Therefore, in recent years, many effective methods such as sine–cosine method [1], tanh function method [2], variational iteration method [3], homotopy perturbation method [4–6], Exp-function method [7,8], F-expansion method [9,10], and others have been proposed for obtaining exact solutions to nonlinear partial differential equations. Some of these methods use transformations to reduce nonlinear equations into more simple equations, some others of methods use a trial function in an iterative scheme which converges rapidly to the exact solution and also there are still some other methods which consist of looking for the solution of the nonlinear evolution equations (NLEEs) considered as a polynomial in variable satisfying a subsidiary nonlinear ordinary differential equation. 0 Recently, Wang et al. [11] have proposed a method called the GG -expansion method to obtain exact travelling wave solutions of NLEEs arising in fluid dynamics, plasma, elastic media, optical fibers, etc. They successfully applied the method to find the travelling wave solutions involving parameters of the Korteweg–de Vries, the modified Korteweg–de Vries, the var 0 iant Boussinesq equations and the Hirota–Satsuma equations. The GG -expansion method has also successfully been applied G0 to various NLEEs 0 [12–16]. The key idea of the G -expansion method is to express the solution of NLLEs a finite series of the polynomial in GG satisfying the second order linear subsidiary ordinary differential equation G00 + kG0 + lG = 0. Before applying the method to the NLEEs considered in this paper, let us give the solution procedure of the method. 2. The basic idea behind the
G0 -expansion method G
To illustrate the basic idea behind this method, we consider the following nonlinear partial differential equation with only two independent variables x and t, and dependent variable u
Nðu; ut ; ux ; utt ; uxt ; uxx ; . . .Þ ¼ 0:
ð1Þ
Using the travelling wave transformation
u ¼ uðnÞ;
n ¼ x wt;
* Corresponding author. E-mail address: [email protected] (A. Esen). 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.05.073
ð2Þ
S. Kutluay et al. / Applied Mathematics and Computation 217 (2010) 384–391
385
Eq. (1) reduces to an ordinary differential equation (ODE) in the form
N uðnÞ; wu0 ðnÞ; u0 ðnÞ; w2 u00 ðnÞ; wu00 ðnÞ; u00 ðnÞ; . . . ¼ 0:
ð3Þ
G0
The G -expansion method is based on the assumption that travelling wave solutions of Eq. (3) can be expressed by a poly 0 nomial in GG as
uðnÞ ¼
0 j n X G aj ; G j¼0
an – 0;
ð4Þ
with G = G(n) satisfying the second order linear ODE
G00 þ kG0 þ lG ¼ 0;
ð5Þ
where aj (j = 0, 1, 2,. . ., n), k and l are constants to be determined later. u(n) can be determined explicitly by using the following steps: Step 1. By considering the homogeneous balance between the highest nonlinear terms and the highest order derivatives of u(n) in Eq. (3), the positive integer n in Eq. (4) is determined. 0 Step 2. By substituting (4) with Eq. (5) into Eq. (3) and collecting all terms with the same power of GG together, the left hand side of Eq. (3) is converted into a polynomial. After setting each coefficient of this polynomial to zero, we obtain a set of algebraic equations in terms of aj (j = 0, 1, 2,. . ., n), w, k and l. Step 3. Solving the system of algebraic equations with the aid of symbolic computation and then substituting the results with the general solutions of Eq. (5) into Eq. (4) gives travelling wave solutions of Eq. (3). To illustrate the effectiveness and convenience of the method, we apply it to the Liouville, sine–Gordon and new coupled MKdV equations described in the next section. 3. Applications of the
G 0 -expansion method G
3.1. The Liouville equation We first consider the Liouville equation [17]
uxt þ eu ¼ 0:
ð6Þ
To look for travelling wave solutions of Eq. (6), we use the wave transformation n = x wt and change Eq. (6) into the form of an ODE
wu00 þ eu ¼ 0:
ð7Þ
Using the transformation
u ¼ lnv ;
ð8Þ
Eq. (7) reduces to nonlinear ODE in the form
wðvv 00 v 02 Þ þ v 3 ¼ 0;
ð9Þ
where the prime denotes differentiation with respect to n. We now suppose that the solution of Eq. (9) is given by Eq. (4). In 00 order to determine the value of integer n, we balance vv with v3, then we get 2n + 2 = 3n which yields n = 2. Therefore, the 0 solution of (9) can be expressed by a polynomial in GG as follows:
v ðnÞ ¼ a0 þ a1
0 0 2 G G þ a2 ; G G
a2 – 0:
ð10Þ
0 i Substituting Eq. (10) along with (5) into Eq. (9), and setting the coefficients of all powers of GG ði ¼ 0; 1; . . . ; 6Þ to zero, we obtain a system of nonlinear algebraic equations for a0, a1, a2 and w. Solving the resulting system with the help of Maple, we have the following set of solution:
a0 ¼ 2wl;
a1 ¼ 2wk;
a2 ¼ 2w;
w ¼ w;
ð11Þ
where k and l are arbitrary constants. Inserting Eq. (11) into (10), we obtain
0 0 2 G G : þ 2w G G
v ðnÞ ¼ 2wl þ 2wk
ð12Þ
Substituting the general solutions of Eq. (5) into Eq. (12) we have three types of travelling wave solutions of Eq. (9) as follows:
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S. Kutluay et al. / Applied Mathematics and Computation 217 (2010) 384–391
When k2 4l > 0
80 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 > > 2 2 1 1 < = c cosh 4 l sinh 4 l k n þ c k n 1 2 1 2 2 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1 : v 1 ðnÞ ¼ wðk2 4lÞ>B @ > 2 : c1 sinh 1 k2 4ln þ c2 cosh 1 k2 4ln ; 2 2
ð13Þ
When k2 4l < 0
80 9 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 > > 2 2 1 1 < = c sin l k cos l k 4 n þ c 4 n 1 2 1 2 2 C qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ 1 v 2 ðnÞ ¼ wðk2 4lÞ>B @ > 2 : c1 cos 1 4l k2 n þ c2 sin 1 4l k2 n ; 2 2
ð14Þ
and when k2 4l = 0
v 3 ðnÞ ¼
2wc22 ðc1 þ c2 nÞ2
ð15Þ
;
where n = x wt, c1 and c2 arbitrary constants. In particular, if we choose c2 – 0; c21 < c22 , then the solution (13) give the solitary wave solution
v 1 ðx; tÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 wðk2 4lÞsech ðx wtÞ k2 4l þ n0 ; 2 2
ð16Þ
1
where k2 4l > 0; n0 ¼ tanh ðcc12 Þ and the solution (14) give the travelling wave solution
v 2 ðx; tÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 wðk2 4lÞcsc2 ðx wtÞ 4l k2 þ n1 ; 2 2
ð17Þ
where k2 4l < 0; n1 ¼ tan1 ðcc12 Þ and recall that u(x, t) = lnv(x, t), hence we obtain the travelling wave solutions of the Liouville equation from Eqs. (16) and (17) as follows, respectively
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 ðx wtÞ k2 4l þ n0 ; u1 ðx; tÞ ¼ ln wðk2 4lÞsech 2 2
w < 0;
ð18Þ
1
where k2 4l > 0 and n0 ¼ tanh ðcc12 Þ,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ðx wtÞ 4l k2 þ n1 ; u2 ðx; tÞ ¼ ln wðk2 4lÞcsc2 2 2
w > 0;
ð19Þ
where k2 4l < 0; n1 ¼ tan1 ðcc12 Þ and the third travelling wave solution from Eq. (15) is as follows
"
u3 ðx; tÞ ¼ ln
#
2wc22
ðc1 þ c2 ðx wtÞÞ2
w > 0;
;
ð20Þ
where c1 and c2 are free parameters. Among these solutions, it is possible to obtain the solution given by Wazwaz [17] by taking n0 = 0 and some manipulation in Eq. (18). It should be noted that the other solutions derived here do not appear in Ref. [17]. 3.2. The sine–Gordon equation We secondly consider sine–Gordon equation [18–20]
utt uxx þ sin u ¼ 0: In order to apply the 2
G0 G
ð21Þ
method, we use the wave transformation n = x wt and change Eq. (21) into the form
00
ðw 1Þu þ sin u ¼ 0:
ð22Þ
We next use the transformation
v ¼ eiu ;
ð23Þ
so that
sin u ¼ which gives
v v 1 2i
;
cos u ¼
v þ v 1 2
;
ð24Þ
S. Kutluay et al. / Applied Mathematics and Computation 217 (2010) 384–391
u ¼ arccos
v þ v 1 2
:
387
ð25Þ
This transformation will change Eq. (22) into the ODE in the form
2ðw2 1Þðv 00 v ðv 0 Þ2 Þ þ v 3 v ¼ 0;
ð26Þ
where the prime denotes differentiation with respect to n. We now suppose that the solution of Eq. (26) is given by Eq. (4). In order to determine the value of integer n, we balance vv00 with v3, we get 2n + 2 = 3n which yields n = 2. Thus the solution of Eq. (26) is exactly the same with Eq. (10). 0 i Substituting Eq. (10) along with (5) into (26) and setting the coefficients of all powers of GG ði ¼ 0; 1; . . . ; 6Þ to zero, we obtain a system of nonlinear algebraic equations for a0, a1, a2 and w. Solving the resulting system with the help of Maple, we have the following sets of solutions:
a0 ¼
k2 k 2 4l
a1 ¼
;
4k k 2 4l
;
a2 ¼
4 k2 4l
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 w¼ 1 2 k 4l
;
ð27Þ
and
a0 ¼
k2 k 2 4l
a1 ¼
;
4k k2 4l
;
a2 ¼
4 k 2 4l
;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 : w¼ 1þ 2 k 4l
ð28Þ
Inserting Eqs. (27) and (28) into Eq. (10), we get
v 1 ðnÞ ¼
k2 2
k 4l
þ
0 0 2 G 4 G þ 2 k 4l G k 4l G 4k
2
ð29Þ
and
v 2 ðnÞ ¼
k2 k2 4l
0 0 2 G 4 G ; k 2 4l G k2 4l G 4k
ð30Þ
respectively. Using the general solutions of (5), we obtain travelling wave solutions of Eq. (26) as follows When k2 4l > 0
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 1 1 c cosh 4 l sinh k n þ c k2 4lnC 1 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ; v 1 ðx; tÞ ¼ B @ c1 sinh 12 k2 4ln þ c2 cosh 12 k2 4ln where n ¼ x
ð31Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k2 4 t and l
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 1 1 c cosh 4 l sinh k n þ c k2 4lnC 1 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ; v 2 ðx; tÞ ¼ B @ c1 sinh 12 k2 4ln þ c2 cosh 12 k2 4ln where n ¼ x
ð32Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ k2 4 t; c1 and c2 arbitrary constants. l
When k2 4l < 0
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 1 1 c sin l k cos 4 n þ c 4l k2 nC 1 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ; v 3 ðx; tÞ ¼ B @ c1 cos 12 4l k2 n þ c2 sin 12 4l k2 n where n ¼ x
ð33Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k2 4 t and l
0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 2 2 1 1 Bc1 sin 2 4l k n þ c2 cos 2 4l k nC qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ; v 4 ðx; tÞ ¼ @ c1 cos 12 4l k2 n þ c2 sin 12 4l k2 n
ð34Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 where n ¼ x ð 1 þ k2 4 Þt; c1 and c2 arbitrary constants. In particular, if we choose c2 –0; c21 < c22 , then the solutions (31)– l (34) give the travelling wave solutions
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k2 4ln þ n0 ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 k2 4ln þ n0 ; v 2 ðx; tÞ ¼ tanh 2
v 1 ðx; tÞ ¼ tanh2
ð35Þ ð36Þ
388
S. Kutluay et al. / Applied Mathematics and Computation 217 (2010) 384–391 1
where n0 ¼ tanh ðcc12 Þ; k2 4l > 0 and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4l k2 n þ n1 ; 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4l k 2 n þ n 1 ; v 4 ðx; tÞ ¼ cot2 2
v 3 ðx; tÞ ¼ cot2
where n1 ¼ tan1
c1 c2
ð37Þ ð38Þ
. Using Eq. (25), we get the following travelling wave solutions of Eq. (21)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 2 1 2 1 ; tanh k2 4ln þ n0 þ coth k2 4ln þ n0 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 2 1 2 1 ; tanh k2 4ln þ n0 þ coth k2 4ln þ n0 u2 ðx; tÞ ¼ arccos 2 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 1 u3 ðx; tÞ ¼ arccos tan2 4l k2 n þ n1 þ cot2 4l k2 n þ n1 2 2 2
u1 ðx; tÞ ¼ arccos
ð39Þ ð40Þ ð41Þ
and
u4 ðx; tÞ ¼ arccos
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 1 ; tan2 4l k2 n þ n1 þ cot2 4l k2 n þ n1 2 2 2
ð42Þ
1
where n0 ¼ tanh ðcc12 Þ and n1 ¼ tan1 ðcc12 Þ. Comparing our results with Wazwaz’s results in Ref. [18], we can see that the results are the same, if we chose n0 and n1 zero, for appropriate parameters. 3.3. The new coupled MKdV equation We finally consider the new coupled MKdV equation [21]
1 uxxx 3u2 ux þ 3ðvv x Þx þ 3 uv 2 x ; 2 ¼ v xxx 3ðv ux Þx þ 6uv ux þ 3 u2 v 2 v x ;
ut ¼
vt
ð43Þ
where Eq. (43) becomes the MKdV equation for v = 0. In Ref. [20], Wu et al. take into account a 4 4 matrix spectral problem with three potentials and obtain a new hierarchy of nonlinear evolution equations. In Ref. [22], Cao et al. proposed some kinds of soliton solutions for Eq. (43). Substituting the wave transformation u(x, t) = u(n), v(x, t) = v(n), n = x wt into Eq. (43), integrating the resulting equation with respect to n yields the ODEs
1 00 u u3 þ 3vv 0 þ 3uv 2 ; 2 c2 wv ¼ v 00 3v u0 v 3 þ 3u2 v ;
c1 wu ¼
ð44Þ
where c1 and c2 are integration constants that are to be determined later. Considering the homogenous balance between u00 and uv2 and between v00 and u2v in Eq. (44), we get n + 2 = n + 2m and m + 2 = 2n + m which yield n = m = 1, then we suppose that
0 G ; G 0 G v ðnÞ ¼ b0 þ b1 : G uðnÞ ¼ a0 þ a1
ð45Þ
0 i Substituting Eq. (45) along with (5) into Eq. (44), collecting the coefficients of GG (i = 0, 1, 2, 3) for each equation and setting it to zero, we obtain a system of nonlinear algebraic equations. Solving the resulting system, we have the following sets of solutions: 2
3
c1 ¼ k2 b0 þ 6kb0 8b0 þ
1 lk 2b0 l; 2
1 lk þ 2b0 l; 2 1 2 1 2 k þ 2l 12kb0 þ 24b0 ; a0 ¼ k b0 ; w¼ 4 2 2
3
c2 ¼ k2 b0 6kb0 þ 8b0
and
ð46Þ a1 ¼
1 ; 2
b1 ¼
1 2
S. Kutluay et al. / Applied Mathematics and Computation 217 (2010) 384–391
1 lk þ 2b0 l; 2 1 2 3 c2 ¼ k2 b0 þ 6kb0 þ 8b0 þ lk þ 2b0 l; 2 1 2 1 2 k þ 2l þ 12kb0 þ 24b0 ; a0 ¼ k þ b0 ; w¼ 4 2 2
389
3
c1 ¼ k2 b0 þ 6kb0 þ 8b0 þ
ð47Þ a1 ¼
1 ; 2
1 b1 ¼ : 2
Inserting Eqs. (46) and (47) into (45), we get
1 G0 k þ b0 ; 2 G 2 1 G0 þ b0 v 1 ðnÞ ¼ 2 G
ð48Þ
1 G0 k þ þ b0 ; 2 G 2 1 G0 þ b0 ; v 2 ðnÞ ¼ 2 G
ð49Þ
u1 ðnÞ ¼
and
u2 ðnÞ ¼
respectively. Using the general solutions of (5), we obtain travelling wave solutions of Eq. (44) as follows: when k2 4l > 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k2 4l BA1 cosh 12 k2 4ln þ A2 sinh 12 k2 4lnC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ b0 ; u1 ðnÞ ¼ @ 4 4 A1 sinh 12 k2 4ln þ A2 cosh 12 k2 4ln 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 4l BA1 cosh 12 k2 4ln þ A2 sinh 12 k2 4lnC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ b0 ; v 1 ðnÞ ¼ @ 4 4 A1 sinh 12 k2 4ln þ A2 cosh 12 k2 4ln
ð50Þ
2 where n ¼ x 14 k2 þ 2l 12kb0 þ 24b0 t and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k2 4l BA1 cosh 12 k2 4ln þ A2 sinh 12 k2 4lnC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ þ b0 ; u2 ðnÞ ¼ @ 4 4 A1 sinh 12 k2 4ln þ A2 cosh 12 k2 4ln qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k2 4l BA1 cosh 12 k2 4ln þ A2 sinh 12 k2 4lnC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ þ b0 ; v 2 ðnÞ ¼ @ 4 4 A1 sinh 12 k2 4ln þ A2 cosh 12 k2 4ln
ð51Þ
2 where n ¼ x 14 k2 þ 2l þ 12kb0 þ 24b0 t, A1 and A2 arbitrary constants. When k2 4l < 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4l k2 BA1 sin 12 4l k2 n þ A2 cos 12 4l k2 nC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ b0 ; u3 ðnÞ ¼ @ 4 4 A1 cos 12 4l k2 n þ A2 sin 12 4l k2 n 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4l k2 BA1 sin 12 4l k2 n þ A2 cos 12 4l k2 nC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ b0 ; v 3 ðnÞ ¼ @ 4 4 A1 cos 12 4l k2 n þ A2 sin 12 4l k2 n
ð52Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 4l k2 BA1 sin 12 4l k2 n þ A2 cos 12 4l k2 nC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ þ b0 ; u4 ðnÞ ¼ @ 4 4 A1 cos 12 4l k2 n þ A2 sin 12 4l k2 n 0 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4l k2 BA1 sin 12 4l k2 n þ A2 cos 12 4l k2 nC k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A þ þ b0 ; v 4 ðnÞ ¼ @ 4 4 A1 cos 12 4l k2 n þ A2 sin 12 4l k2 n
ð53Þ
2 where n ¼ x 14 k2 þ 2l 12kb0 þ 24b0 t and
2 where n ¼ x 14 k2 þ 2l þ 12kb0 þ 24b0 t, A1 and A2 arbitrary constants.
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When k2 4l = 0
1 2A2 k þ b0 ; k 4 2 A1 þ A2 n 1 2A2 v 5 ðnÞ ¼ k þ b0 ; 4 A1 þ A2 n
ð54Þ
1 2A2 k k þ þ b0 ; 4 2 A1 þ A2 n 1 2A2 v 6 ðnÞ ¼ k þ b0 ; 4 A1 þ A2 n
ð55Þ
u5 ðnÞ ¼
2 where n ¼ x 14 k2 þ 2l 12kb0 þ 24b0 t and
u6 ðnÞ ¼
2 where n ¼ x 14 k2 þ 2l þ 12kb0 þ 24b0 t; A1 and A2 arbitrary constants. In particular, if we choose A2 – 0; A21 < A22 , then the solutions (50)–(53) lead to the travelling wave solutions as follows: when k2 4l > 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k þ b0 ; k k2 4l tanh k2 4lðx wtÞ þ n0 4 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ b0 ; k2 4lðx wtÞ þ n0 v 1 ðx; tÞ ¼ k k2 4l tanh 4 2
ð56Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k þ þ b0 ; k k2 4l tanh k2 4lðx wtÞ þ n0 4 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ b0 ; k2 4lðx wtÞ þ n0 v 2 ðx; tÞ ¼ k k2 4l tanh 4 2
ð57Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k þ b0 ; k 4l k2 cot 4l k2 ðx wtÞ þ n1 4 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ b0 ; 4l k2 ðx wtÞ þ n1 v 3 ðx; tÞ ¼ k 4l k2 cot 4 2
ð58Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k þ þ b0 ; k 4l k2 cot 4l k2 ðx wtÞ þ n1 4 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 þ b0 ; 4l k2 ðx wtÞ þ n1 v 4 ðx; tÞ ¼ k 4l k2 cot 4 2
ð59Þ
u1 ðx; tÞ ¼
2 where w ¼ 14 k2 þ 2l 12kb0 þ 24b0 and
u2 ðx; tÞ ¼
2 1 A1 where w ¼ 14 k2 þ 2l þ 12kb0 þ 24b0 ; n0 ¼ tanh . A2 When k2 4l < 0
u3 ðx; tÞ ¼
2 where w ¼ 14 k2 þ 2l 12kb0 þ 24b0 and
u4 ðx; tÞ ¼
2 where w ¼ 14 k2 þ 2l þ 12kb0 þ 24b0 ; n1 ¼ tan1 AA12 . 4. Conclusion
0 In this paper, the GG -expansion method has been successfully used to obtain some exact travelling wave solutions for the Liouville, sine–Gordon and new coupled MKdV equations. It can be clearly seen that some solutions given in this paper are new solutions which have not been reported yet. The method can also be effectively used to construct new and numerous exact travelling wave solutions for some other nonlinear evolution equations arising in mathematical physics. Acknowledgment This study was supported by Inonu University Scientific Research Project (I.U.-BAP:2009/05, Malatya, TURKEY). References [1] C. Yan, A simple transformation for nonlinear waves, Phys. Lett. A 224 (1996) 77–84. [2] W. Malfliet, W. Hereman, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Phys. Scripta 54 (1996) 563–568. [3] 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. [4] J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems, Int. J. Nonlinear Sci. 6 (2005) 207–208. [5] J.H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals 26 (2005) 695–700.
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