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Some new and general solutions to the compound KdV–Burgers system with nonlinear terms of any order
Applied Mathematics and Computation 217 (2010) 1652–1657
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Some new and general solutions to the compound KdV–Burgers system with nonlinear terms of any order Jing Wang Department of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454000, PR China
a r t i c l e
i n f o
Keywords: Riccati equation Compound KdV–Burgers equation with nonlinear terms of any order Triangular periodic solution Exp function solution Soliton-like solutions
a b s t r a c t In this letter, a new Riccati equation expansion method is presented for constructing exact travelling-wave solutions of nonlinear partial differential equations. The main idea of this method is to take full advantage of the solutions of the Riccati equation to construct exact travelling-wave solutions of nonlinear partial differential equations. As a result, some more generalized solutions, which contain triangular periodic solutions, exp function solutions and the soliton-like solutions, are obtained. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction In recent years, the nonlinear evolution equations are widely used to describe many important phenomena and dynamic processes in physics, mechanics, chemistry, biology, etc. Much work has been focused on constructing the exact solutions of nonlinear evolution equations [1–19]. Due to the availability of computer symbolic system like Mathematica or Maple, many powerful methods have been presented to obtain the exact solutions of nonlinear evolution equations, such as the hyperbolic tangent function expansion method [1–3], Riccati equation expansion method [4–6], the generalized Riccati equation method [7,8], auxiliary ordinary differential equation method [9], modified extended tanh-function method [10,11], the Fans unified algebraic method [12], the extended Fans sub-equation method [13] and so on. Most of these methods are constructed by use of the solutions of some simple and solvable nonlinear ordinary differential equations. In this paper, by use of the Riccati equation, we propose a new algebraic method to construct some new exact solutions of the compound KdV–Burgers system with nonlinear terms of any order. The presented method can also be applied to other nonlinear evolution equations with nonlinear terms of any order. The rest of paper is arranged as follows. In Section 2, we briefly describe our method – the new Riccati equation rational expansion method. In Section 3, we apply the new method to compound KdV–Burgers equation with nonlinear terms of any order and obtain some generalized solutions. Finally, in Section 4, some conclusions are given. 2. Summary of the generalized Riccati equation rational expansion method In the following we would like to outline the main steps of our method: Step 1. For a given nonlinear evolution system with some physical fields ui ðx; y; tÞ in three variables x; y; t,
rðui ; uit ; uix ; uiy ; uitt ; uixt ; uiyt ; uixx ; uiyy ; uixy ; . . .Þ ¼ 0;
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.09.020
ð1Þ
J. Wang / Applied Mathematics and Computation 217 (2010) 1652–1657
1653
by using the wave transformation
ui ðx; y; tÞ ¼ U i ðnÞ;
n ¼ kðmx þ ny ktÞ;
ð2Þ
where k; m; n and k are constants to be determined later. Then the nonlinear partial differential (1) is reduced to a nonlinear ordinary differential equation (ODE):
MðU i ; U 0i ; U 00i ; . . .Þ ¼ 0; where ‘‘0” =
ð3Þ
d . dðnÞ
Step 2. We introduce a new ansätz in terms of finite rational formal expansion in the following forms:
U i ¼ ai0 þ
mi X
aij wj ðnÞ
j¼1
ð1 þ lwðnÞÞj
ð4Þ
;
the new variable w ¼ wðnÞ satisfying
w0 ðq0 þ q2 w2 Þ ¼
dw ðq0 þ q2 w2 Þ ¼ 0; dn
ð5Þ
where ai0 ; aij ði ¼ 1; 2; . . . ; j ¼ 1; 2; . . . ; mi Þ and l are constants to be determined later. Step 3. The parameter mi can be found by balancing the highest order derivative term and the nonlinear terms in (1) or (3): (i) If mi is a positive integer then Step 4; q (ii) If mi ¼ qp, we make the transformation UðnÞ ¼ /p ðnÞ and then return to Step 1; (iii) If mi is a negative integer, we make the transformation UðnÞ ¼ /mi ðnÞ and then return to Step 1. Step 4. Substitute (4) into (3) along with (5) and then set all coefficients of wi ðnÞ ði ¼ 1; 2; . . .Þ of the resulting system’s numerator to be zero to get an over-determined system of nonlinear algebraic equations with respect to k; k; l; ai0 ; aij ði ¼ 1; 2; . . . ; j ¼ 1; 2; ::; mi Þ and l. Step 5. Solving the over-determined system of nonlinear algebraic equations by use of Maple, we would end up with the explicit expressions for k; k; m; n; ai0 ; aij ði ¼ 1; 2; . . . ; j ¼ 1; 2; . . . ; mi Þ and l. Step 6. With the aid of Maple, we obtain the general solutions of (5) which are now listed as following: (1) when q0 q2 > 0
wðnÞ ¼
pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi q0 q2 q0 q2 tanð q0 q2 n þ c1 Þ ¼ cotð q0 q2 n þ c2 Þ; q2 q2
ð6Þ
(2) when q0 q2 < 0
wðnÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q0 q2 q0 q2 tanhð q0 q2 n þ c3 Þ ¼ cothð q0 q2 n þ c4 Þ; q2 q2
ð7Þ
(3) when q0 ¼ 0 and q2 – 0
wðnÞ ¼
1 ; q2 n þ c
ð8Þ
where n ¼ kðmx þ ny ktÞ and c; c1 ; c2 ; c3 ; c4 are arbitrary constants. Remark 1. As is well known, there exist the following relationship: 2
cosð2nÞ ¼ 2 cos2 ðnÞ 1 ¼ 1 2 sin ðnÞ; 2
2
sinð2nÞ ¼ 2 sinðnÞ cosðnÞ:
coshð2nÞ ¼ 2cosh ðnÞ 1 ¼ 2sinh ðnÞ þ 1;
sinhð2nÞ ¼ 2 sinhðnÞ coshðnÞ;
so (6) can be listed as:
wðnÞ ¼
pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi q0 q2 q0 q2 ðcscð 4q0 q2 n þ c5 Þ cotð 4q0 q2 n þ c5 ÞÞ ¼ ðsecð 4q0 q2 n þ c6 Þ tanð 4q0 q2 n þ c6 ÞÞ; q2 q2
ð9Þ
while (7) can also be listed as:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q0 q2 ðcothð 4q0 q2 n þ c7 Þ cschð 4q0 q2 n þ c7 ÞÞ q2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q0 q2 ðtanhð 4q0 q2 n þ c8 Þ isechð 4q0 q2 n þ c8 ÞÞ; ¼ q2
wðnÞ ¼
where n ¼ kðmx þ ny ktÞ and c5 ; c6 ; c7 ; c8 are arbitrary constants.
ð10Þ
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J. Wang / Applied Mathematics and Computation 217 (2010) 1652–1657
Remark 2. By use of the Euler formula, (6) and (7) can also be listed as exp functions: pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2in q0 q2 q0 q2 ð1 ice Þ pffiffiffiffiffiffiffi ; wðnÞ ¼ q2 ð1 þ ce2in q0 q2 Þ
ð11Þ
while (7) can also be listed as:
wðnÞ ¼
pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q0 q2 1 þ ce2 q0 q2 n pffiffiffiffiffiffiffiffiffiffi : 1 ce2 q0 q2 n q2
ð12Þ
Remark 3. If we set the parameters q0 and q2 in (5) to different values, the ansätz in the tanh-function method [5], extended tanh-function method [6], modified extended tanh-function method [10], generalized hyperbolic-function method [14], the Riccati equation rational expansion method [7] and the generalized Riccati equation rational expansion [8] can all be recovered. That is to say, the ansätz proposed here, is more generalized.
3. Applications to the nonlinear evolution equations with nonlinear terms of any order In the following, we use the improved method to construct some new and more general rational formal solutions for the compound KdV–Burgers-type equation:
ut þ aup ux þ bu2p ux þ uxx þ duxxx ¼ 0;
ð13Þ
which have been studied by many authors [15–19]. The equation with p P 1 is a model for long-wave propagation in nonlinear media with dispersion and dissipation [15–19]. In Ref. [19], Zhang et al. obtained kink-profile solitary-wave solutions for it. The stability of travelling-wave solutions of (13) with b ¼ 0; 6 0 and p P 1 are studied by Pego et al. [16]. Firstly we take the form of the required solution as follows:
uðx; tÞ ¼ UðnÞ;
n ¼ kðmx ktÞ:
ð14Þ
We change (13) to the form 00
2
kU 0 þ amU p U 0 þ bmU 2p U 0 þ m2 kU þ dm3 k U 000 ¼ 0;
ð15Þ
d where ‘‘0” = dn .
By balancing the highest order partial derivative term and the nonlinear term in (15), we get the value of m; m ¼ 1p. Therefore we make the following transformation: 1
UðnÞ ¼ /p ðnÞ:
ð16Þ
Then substituting (16) into (15) reads
p2 /2 ðk/0 þ am//0 þ bm/2 /0 Þ þ m2 kp/ðð1 pÞð/0 Þ2 þ p//00 Þþ 2
dm3 k ðð1 pÞð1 2pÞð/0 Þ3 þ 3pð1 pÞ//0 /00 þ p2 /2 /000 Þ ¼ 0:
ð17Þ
According to the proposed method, we expand the solution of (17) in the form
/ðnÞ ¼ a0 þ
a1 wðnÞ ; 1 þ lwðnÞ
ð18Þ
where wðnÞ satisfies (5). With the aid of Maple, substituting (18) along with (5) into (17), yields a set of algebraic equations for wi ðnÞ ði ¼ 0; 1; . . .Þ. Setting the coefficients of these terms wi ðnÞ to zero yields a set of over-determined algebraic equations with respect to a0 ; a1 ; l; k; k; a and b (because the over-determined algebraic equations are so many, they are omit). Then we get the following results: Case 1:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðlq0 q0 q2 Þa0 a1 ¼ ; l ¼ l; k ¼ k; q0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi kmq0 q2 þ l2 q0 ðp þ p2 2dmkð5p2 3p 2Þ q0 q2 Þ a¼ ; ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p a0 p2 ðlq0 q0 q2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dm2 k ðp þ 1Þð2p þ 1Þ l2 q0 þ q2 q0 2m2 kðp q0 q2 þ 2dmkq0 q2 Þ : ; k¼ b¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 p2 a0 p ðl q0 2l q0 q2 þ q2 Þ a0 ¼ a0 ;
ð19Þ
J. Wang / Applied Mathematics and Computation 217 (2010) 1652–1657
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Case 2:
a0 ¼ a0 ;
a1 ¼
ðlq0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q0 q2 Þa0 ; q0
l ¼ l; k ¼
2 mðp 1Þ
; dðp 2Þ2 2 l2 q0 þ q2 2 ð4p þ 1Þðp 1Þ p ; b¼ k¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2dm q0 q2 ðp 2Þ 4da20 q2 ðp 2Þ2 q2 2l q0 q2 l2 q0 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi l q þ q q0 l3 q2 q0 3l2 q0 q2 3q2 l q2 q0 þ q22 ðp þ 1Þ2 p a¼ 2 0 2 : pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2q0 q2 6l2q0 q2 4q2 l q2 q0 þ l4 q20 4q0 l3 q2 q0 þ q22 ð2 þ pÞ2 da0
ð20Þ
From (14), (16), (18), (6)–(12), we obtain the following solutions for (13): Family 1: According to (19), when q0 q2 > 0 (a)
From (6) and (9) we obtain the triangular periodic solutions of the compound KdV–Burgers equation:
1 pffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 i tanð q0 q2 kðmx þ ktÞ þ c1 Þ p ; pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 tanð q0 q2 kðmx þ ktÞ þ c1 Þ 1 pffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 i cotð q0 q2 kðmx þ ktÞ þ c2 Þ p ; u2 ¼ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 cotð q0 q2 kðmx þ ktÞ þ c2 Þ !1p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 iðsecð 4q0 q2 kðmx þ ktÞ þ c3 Þ tanð 4q0 q2 kðmx þ ktÞ þ c3 ÞÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffiffiffiffiffi ffi ; u3 ¼ pffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 ðsecð 4q0 q2 kðmx þ ktÞ þ c3 Þ tanð 4q0 q2 kðmx þ ktÞ þ c3 ÞÞ !1p pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 iðcscð 4q0 q2 kðmx þ ktÞ þ c4 Þ cotð 4q0 q2 kðmx þ ktÞ þ c4 ÞÞ ffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffi ffi p p u4 ¼ : pffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 ðcscð 4q0 q2 kðmx þ ktÞ þ c4 Þ cotð 4q0 q2 kðmx þ ktÞ þ c4 ÞÞ
u1 ¼
(b)
ð21Þ ð22Þ
ð23Þ
ð24Þ
From (11) we obtain the exp function solutions of the compound KdV–Burgers equation:
0
pffiffiffiffiffiffi 11p 2i q0 q2 kðmxþktÞ a0 q2 a0 q2 i 1ic5 e2ipffiffiffiffiffiffi q0 q2 kðmxþktÞ B C 1þc5 e A ; u5 ¼ @ q0 q2 kðmxþktÞ pffiffiffiffiffiffiffiffiffiffi1ic5 e2ipffiffiffiffiffiffi p ffiffiffiffiffiffi q2 þ l q0 q2 2i q0 q2 kðmxþktÞ
ð25Þ
1þc5 e
pffiffiffiffiffiffiffiffiffiffi 2 2m kðp qp02q2 þ2dmkq0 q2 Þ ;
where k ¼ mined by (19).
l; k; m and c1 ; c2 ; c3 ; c4 ; c5 are arbitrary constants and a; b are deter-
a0 ;
Family 2: According to (19), when q0 q2 < 0 (a)
From (7) and (10) we obtain the soliton-like solutions of the compound KdV–Burgers equation:
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 tanhð q0 q2 kðmx þ ktÞ þ c6 Þ p ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 tanhð q0 q2 kðmx þ ktÞ þ c6 Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 cothð q0 q2 kðmx þ ktÞ þ c7 Þ p ; u7 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 cothð q0 q2 kðmx þ ktÞ þ c7 Þ !1p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 ðtanhð 4q0 q2 kðmx þ ktÞ þ c8 Þ isechð 4q0 q2 kðmx þ ktÞ þ c8 ÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u8 ¼ ; pffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 ðtanhð 4q0 q2 kðmx þ ktÞ þ c8 Þ isechð 4q0 q2 kðmx þ ktÞ þ c8 ÞÞ !1p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 ðcothð 4q0 q2 kðmx þ ktÞ þ c9 Þ cschð 4q0 q2 kðmx þ ktÞ þ c9 ÞÞ p p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : u9 ¼ pffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 ðcothð 4q0 q2 kðmx þ ktÞ þ c9 Þ cschð 4q0 q2 kðmx þ ktÞ þ c9 ÞÞ
u6 ¼
(b)
ð26Þ ð27Þ
ð28Þ
ð29Þ
From (12), we obtain the exp function solutions of the compound KdV–Burgers equation:
0 B u10 ¼ @
a0 q2 a0 q2
pffiffiffiffiffiffi 0 q2 kðmxþktÞ 1þc10 e2pqffiffiffiffiffiffi
11p
1c10 e2 q0 q2 kðmxþktÞ C A ; q0 q2 kðmxþktÞ pffiffiffiffiffiffiffiffiffiffi1þc10 e2pffiffiffiffiffiffi pffiffiffiffiffiffi q2 þ l q0 q2 2 q0 q2 kðmxþktÞ
ð30Þ
1c10 e
pffiffiffiffiffiffiffiffiffiffi 2 2m kðp qp02q2 þ2dmkq0 q2 Þ ;
where k ¼ mined by (19).
a0 ;
l; k; m and c6 ; c7 ; c8 ; c9 ; c10 are arbitrary constants and a; b are deter-
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J. Wang / Applied Mathematics and Computation 217 (2010) 1652–1657
Family 3: According to (20), when q0 q2 > 0 (a)
From (6) and (9), we obtain the triangular periodic solutions of the compound KdV–Burgers equation:
u11
11p 2 ðp1Þ i p a0 q2 a0 q2 i tan 2dðp2Þ x d ðp2Þ þ c11 2 t A ; ¼@ pffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ i p x d ðp2Þ t þ c q2 þ l q0 q2 tan 2dðp2Þ 11 2
ð31Þ
u12
11p 2 ðp1Þ i p a0 q2 a0 q2 i cot 2dðp2Þ x d ðp2Þ þ c12 2 t A ; ¼@ pffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ i p x d ðp2Þ þ c12 q2 þ l q0 q2 cot 2dðp2Þ 2 t
ð32Þ
u13
11p 2 ðp1Þ 2 ðp1Þ i p i p a0 q2 a0 q2 i sec dðp2Þ x d ðp2Þ þ c13 tan dðp2Þ x d ðp2Þ þ c13 2 t 2 t A ; ¼@ pffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ 2 ðp1Þ i p i p x d ðp2Þ þ c13 tan dðp2Þ x d ðp2Þ þ c13 q2 þ l q0 q2 sec dðp2Þ 2 t 2 t
ð33Þ
u14
11p 2 ðp1Þ 2 ðp1Þ i p i p a0 q2 a0 q2 i csc dðp2Þ x d ðp2Þ þ c14 cot dðp2Þ x d ðp2Þ þ c14 2 t 2 t A : ¼@ pffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ 2 ðp1Þ i p i p x d ðp2Þ þ c14 cot dðp2Þ x d ðp2Þ þ c14 q2 þ l q0 q2 csc dðp2Þ 2 t 2 t
ð34Þ
0
0
0
0
(b)
From (11), we obtain the exp function solutions of the compound KdV–Burgers equation:
0
u15
0
p dðp2Þ
2 ðp1Þ t x d ðp2Þ2
1 11p
1ic15 e B a q a q iB C AC 0 2 @ C B 0 2 2 ðp1Þ t p C B x dðp2Þ C B d ðp2Þ2 1þc15 e B 0 1C ¼B C; 2 ðp1Þ p C B x t B pffiffiffiffiffiffiffiffiffiffiB1ic15 edðp2Þ d ðp2Þ2 CC A @q2 þ l q0 q2 @ A 2 1þc15 e
where a0 ;
p dðp2Þ
x
ð35Þ
ðp1Þ t d ðp2Þ2
l; m and c11 ; c12 ; c13 ; c14 ; c15 are arbitrary constants and a; b are determined by (20).
Family 4: According to (20), when q0 q2 < 0 (a)
From (7) and (10), we obtain the soliton-like solutions of the compound KdV–Burgers equation:
u16
11p 2 ðp1Þ p a0 q2 a0 q2 tanh 2dðp2Þ x d ðp2Þ þ c16 2 t A ; ¼@ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ p x d ðp2Þ þ c16 q2 þ l q0 q2 tanh 2dðp2Þ 2 t
ð36Þ
u17
11p 2 ðp1Þ p a0 q2 a0 q2 coth 2dðp2Þ x d ðp2Þ þ c17 2 t A ; ¼@ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ p x d ðp2Þ þ c17 q2 þ l q0 q2 coth 2dðp2Þ 2 t
ð37Þ
u18
11p 2 ðp1Þ 2 ðp1Þ p p a0 q2 a0 q2 tanh dðp2Þ x d ðp2Þ þ c18 isech dðp2Þ x d ðp2Þ þ c18 2 t 2 t A ; ¼@ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ 2 ðp1Þ p p x dðp2Þ þ c18 isech dðp2Þ x d ðp2Þ þ c18 q2 þ l q0 q2 tanh dðp2Þ 2 t 2 t
ð38Þ
u19
11p 2 ðp1Þ 2 ðp1Þ p p a0 q2 a0 q2 coth dðp2Þ x d ðp2Þ þ c19 csch dðp2Þ x d ðp2Þ þ c19 2 t 2 t A : ¼@ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ 2 ðp1Þ p p x d ðp2Þ þ c19 csch dðp2Þ x d ðp2Þ þ c19 q2 þ l q0 q2 coth dðp2Þ 2 t 2 t
ð39Þ
0
0
0
0
(b)
From (12), we obtain the exp function solutions of the compound KdV–Burgers equation:
0
u20
0
p dðp2Þ
x
2 ðp1Þ t
1 11p
1þc20 e Ba q a q B C AC 0 2@ C B 0 2 2 ðp1Þ t p C B x 2 dðp2Þ C B d ðp2Þ 1c e 20 C; 0 1 ¼B C B 2 ðp1Þ p C B x t dðp2Þ 2 d ðp2Þ pffiffiffiffiffiffiffiffiffiffiB1þc20 e B C AC A @q2 þ l q0 q2 @
1c20 e
where a0 ;
p dðp2Þ
d ðp2Þ2
2 ðp1Þ t x d ðp2Þ2
l; m and c16 ; c17 ; c18 ; c19 ; c20 are arbitrary constants and a; b are determined by (20).
ð40Þ
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4. Summary and conclusions In summary, based on the new Riccati equation rational expansion method, many generalized exact solutions of the compound KdV–Burgers equation with nonlinear terms of any order have been derived. The method can also easily to be extended to treat other nonlinear evolution equations and is sufficient to seek more solitary-wave solutions and other formal solutions of nonlinear evolution equations. In addition, this method is also computerizable, which allows us to perform complicated and tedious symbolic algebraic calculation on a computer. Acknowledgements The work is supported by The Youth Funds of Henan Polytechnic University under the Grant Q2006-10-646172, The Natural Science Foundation of Henan Province of China under the Grant 0611055500, The Youth Mainstay teacher Funds of Henan Polytechnic University of China under the Grant 649071. References [1] E.J. Parkes, B.R. Duffy, An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Commun. 98 (1996) 288–300; E.J. Parkes, B.R. Duffy, Travelling solitary wave solutions to a compound KdV–Burgers equation, Phys. Lett. A 229 (1997) 217–220. [2] W. Hereman, Exact solitary wave solutions of coupled nonlinear evolution equations using MACSYMA, Comput. Phys. Commun. 65 (1991) 143–150. [3] Z.B. Li, Y.P. Liu, RATH: a Maple package for finding travelling solitary wave solutions to nonlinear evolution equations, Comput. Phys. Commun. 148 (2002) 256–266. [4] Z.Y. Yan, New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water, Phys. Lett. A 285 (2001) 355–362. [5] A.H. Khater, W. Malfliet, D.K. Callebaut, E.S. Kamel, The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction– diffusion equations, Chaos Solitons Fract. 14 (2002) 513–522. [6] E.G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212–218. [7] Q. Wang, Y. Chen, H.Q. Zhang, A new Riccati equation rational expansion method and its application to (2 + 1)-dimensional Burgers equation, Chaos Solitons Fract. 25 (2005) 1019–1028. [8] J. Wang, X.L. Zhang, L.N. Song, H.Q. Zhang, A new general algebraic method with symbolic computation and its application to two nonlinear differential equations with nonlinear terms of any order, Appl. Math. Comput. 182 (2006) 1330–1340. [9] D. Siren, J. Sun, Auxiliary equation method for solving nonlinear partial differential equations, Phys. Lett. A 309 (2003) 387–396. [10] S.A. Elwakil, S.K. El-labany, M.A. Zahran, R. Sabry, Modified extended tanh-function method for solving nonlinear partial differential equations, Phys. Lett. A 299 (2002) 179–188. [11] E. Yomba, Construction of new soliton-like solutions of the (2 + 1) dimensional dispersive long wave equation, Chaos Solitons Fract. 20 (2004) 1135– 1139. [12] E.G. Fan, Y.C. Hon, A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves, Chaos Solitons Fract. 15 (2003) 559–566. [13] E. Yomba, The extended Fan’s sub-equation method and its application to KdV–MKdV, BKK and variant Boussinesq equations, Phys. Lett. A 336 (2005) 463–476. [14] Y.T. Gao, B. Tian, Generalized hyperbolic-function method with computerized symbolic computation to construct the solitonic solutions to nonlinear equations of mathematical physics, Comput. Phys. Commun. 133 (2001) 158–164. [15] M. Wadati, Wave propagation in nonlinear lattice. I, J. Phys. Soc. Jpn. 38 (1975) 673–680. [16] R.L. Pego, P. Smereka, M.I. Weinstein, Oscillatory instability of traveling waves for a KdV–Burgers equation, Physica D 67 (1993) 45–65. [17] B. Dey, Domain wall solutions of KdV like equations with higher order nonlinearity, J. Phys. A 19 (1986) L9–L12. [18] M.W. Coffey, On series expansions giving closed-form solutions of Korteweg–DeVries-like equations, J. SIAM Appl. Math. 50 (1990) 1580–1592. [19] W.G. Zhang, Q.S. Chang, B.G. Jiang, Explicit exact solitary-wave solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order, Chaos Solitons Fract. 13 (2002) 311–319.
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Some new and general solutions to the compound KdV–Burgers system with nonlinear terms of any order Jing Wang Department of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo 454000, PR China
a r t i c l e
i n f o
Keywords: Riccati equation Compound KdV–Burgers equation with nonlinear terms of any order Triangular periodic solution Exp function solution Soliton-like solutions
a b s t r a c t In this letter, a new Riccati equation expansion method is presented for constructing exact travelling-wave solutions of nonlinear partial differential equations. The main idea of this method is to take full advantage of the solutions of the Riccati equation to construct exact travelling-wave solutions of nonlinear partial differential equations. As a result, some more generalized solutions, which contain triangular periodic solutions, exp function solutions and the soliton-like solutions, are obtained. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction In recent years, the nonlinear evolution equations are widely used to describe many important phenomena and dynamic processes in physics, mechanics, chemistry, biology, etc. Much work has been focused on constructing the exact solutions of nonlinear evolution equations [1–19]. Due to the availability of computer symbolic system like Mathematica or Maple, many powerful methods have been presented to obtain the exact solutions of nonlinear evolution equations, such as the hyperbolic tangent function expansion method [1–3], Riccati equation expansion method [4–6], the generalized Riccati equation method [7,8], auxiliary ordinary differential equation method [9], modified extended tanh-function method [10,11], the Fans unified algebraic method [12], the extended Fans sub-equation method [13] and so on. Most of these methods are constructed by use of the solutions of some simple and solvable nonlinear ordinary differential equations. In this paper, by use of the Riccati equation, we propose a new algebraic method to construct some new exact solutions of the compound KdV–Burgers system with nonlinear terms of any order. The presented method can also be applied to other nonlinear evolution equations with nonlinear terms of any order. The rest of paper is arranged as follows. In Section 2, we briefly describe our method – the new Riccati equation rational expansion method. In Section 3, we apply the new method to compound KdV–Burgers equation with nonlinear terms of any order and obtain some generalized solutions. Finally, in Section 4, some conclusions are given. 2. Summary of the generalized Riccati equation rational expansion method In the following we would like to outline the main steps of our method: Step 1. For a given nonlinear evolution system with some physical fields ui ðx; y; tÞ in three variables x; y; t,
rðui ; uit ; uix ; uiy ; uitt ; uixt ; uiyt ; uixx ; uiyy ; uixy ; . . .Þ ¼ 0;
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.09.020
ð1Þ
J. Wang / Applied Mathematics and Computation 217 (2010) 1652–1657
1653
by using the wave transformation
ui ðx; y; tÞ ¼ U i ðnÞ;
n ¼ kðmx þ ny ktÞ;
ð2Þ
where k; m; n and k are constants to be determined later. Then the nonlinear partial differential (1) is reduced to a nonlinear ordinary differential equation (ODE):
MðU i ; U 0i ; U 00i ; . . .Þ ¼ 0; where ‘‘0” =
ð3Þ
d . dðnÞ
Step 2. We introduce a new ansätz in terms of finite rational formal expansion in the following forms:
U i ¼ ai0 þ
mi X
aij wj ðnÞ
j¼1
ð1 þ lwðnÞÞj
ð4Þ
;
the new variable w ¼ wðnÞ satisfying
w0 ðq0 þ q2 w2 Þ ¼
dw ðq0 þ q2 w2 Þ ¼ 0; dn
ð5Þ
where ai0 ; aij ði ¼ 1; 2; . . . ; j ¼ 1; 2; . . . ; mi Þ and l are constants to be determined later. Step 3. The parameter mi can be found by balancing the highest order derivative term and the nonlinear terms in (1) or (3): (i) If mi is a positive integer then Step 4; q (ii) If mi ¼ qp, we make the transformation UðnÞ ¼ /p ðnÞ and then return to Step 1; (iii) If mi is a negative integer, we make the transformation UðnÞ ¼ /mi ðnÞ and then return to Step 1. Step 4. Substitute (4) into (3) along with (5) and then set all coefficients of wi ðnÞ ði ¼ 1; 2; . . .Þ of the resulting system’s numerator to be zero to get an over-determined system of nonlinear algebraic equations with respect to k; k; l; ai0 ; aij ði ¼ 1; 2; . . . ; j ¼ 1; 2; ::; mi Þ and l. Step 5. Solving the over-determined system of nonlinear algebraic equations by use of Maple, we would end up with the explicit expressions for k; k; m; n; ai0 ; aij ði ¼ 1; 2; . . . ; j ¼ 1; 2; . . . ; mi Þ and l. Step 6. With the aid of Maple, we obtain the general solutions of (5) which are now listed as following: (1) when q0 q2 > 0
wðnÞ ¼
pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi q0 q2 q0 q2 tanð q0 q2 n þ c1 Þ ¼ cotð q0 q2 n þ c2 Þ; q2 q2
ð6Þ
(2) when q0 q2 < 0
wðnÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q0 q2 q0 q2 tanhð q0 q2 n þ c3 Þ ¼ cothð q0 q2 n þ c4 Þ; q2 q2
ð7Þ
(3) when q0 ¼ 0 and q2 – 0
wðnÞ ¼
1 ; q2 n þ c
ð8Þ
where n ¼ kðmx þ ny ktÞ and c; c1 ; c2 ; c3 ; c4 are arbitrary constants. Remark 1. As is well known, there exist the following relationship: 2
cosð2nÞ ¼ 2 cos2 ðnÞ 1 ¼ 1 2 sin ðnÞ; 2
2
sinð2nÞ ¼ 2 sinðnÞ cosðnÞ:
coshð2nÞ ¼ 2cosh ðnÞ 1 ¼ 2sinh ðnÞ þ 1;
sinhð2nÞ ¼ 2 sinhðnÞ coshðnÞ;
so (6) can be listed as:
wðnÞ ¼
pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi q0 q2 q0 q2 ðcscð 4q0 q2 n þ c5 Þ cotð 4q0 q2 n þ c5 ÞÞ ¼ ðsecð 4q0 q2 n þ c6 Þ tanð 4q0 q2 n þ c6 ÞÞ; q2 q2
ð9Þ
while (7) can also be listed as:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q0 q2 ðcothð 4q0 q2 n þ c7 Þ cschð 4q0 q2 n þ c7 ÞÞ q2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q0 q2 ðtanhð 4q0 q2 n þ c8 Þ isechð 4q0 q2 n þ c8 ÞÞ; ¼ q2
wðnÞ ¼
where n ¼ kðmx þ ny ktÞ and c5 ; c6 ; c7 ; c8 are arbitrary constants.
ð10Þ
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J. Wang / Applied Mathematics and Computation 217 (2010) 1652–1657
Remark 2. By use of the Euler formula, (6) and (7) can also be listed as exp functions: pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2in q0 q2 q0 q2 ð1 ice Þ pffiffiffiffiffiffiffi ; wðnÞ ¼ q2 ð1 þ ce2in q0 q2 Þ
ð11Þ
while (7) can also be listed as:
wðnÞ ¼
pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q0 q2 1 þ ce2 q0 q2 n pffiffiffiffiffiffiffiffiffiffi : 1 ce2 q0 q2 n q2
ð12Þ
Remark 3. If we set the parameters q0 and q2 in (5) to different values, the ansätz in the tanh-function method [5], extended tanh-function method [6], modified extended tanh-function method [10], generalized hyperbolic-function method [14], the Riccati equation rational expansion method [7] and the generalized Riccati equation rational expansion [8] can all be recovered. That is to say, the ansätz proposed here, is more generalized.
3. Applications to the nonlinear evolution equations with nonlinear terms of any order In the following, we use the improved method to construct some new and more general rational formal solutions for the compound KdV–Burgers-type equation:
ut þ aup ux þ bu2p ux þ uxx þ duxxx ¼ 0;
ð13Þ
which have been studied by many authors [15–19]. The equation with p P 1 is a model for long-wave propagation in nonlinear media with dispersion and dissipation [15–19]. In Ref. [19], Zhang et al. obtained kink-profile solitary-wave solutions for it. The stability of travelling-wave solutions of (13) with b ¼ 0; 6 0 and p P 1 are studied by Pego et al. [16]. Firstly we take the form of the required solution as follows:
uðx; tÞ ¼ UðnÞ;
n ¼ kðmx ktÞ:
ð14Þ
We change (13) to the form 00
2
kU 0 þ amU p U 0 þ bmU 2p U 0 þ m2 kU þ dm3 k U 000 ¼ 0;
ð15Þ
d where ‘‘0” = dn .
By balancing the highest order partial derivative term and the nonlinear term in (15), we get the value of m; m ¼ 1p. Therefore we make the following transformation: 1
UðnÞ ¼ /p ðnÞ:
ð16Þ
Then substituting (16) into (15) reads
p2 /2 ðk/0 þ am//0 þ bm/2 /0 Þ þ m2 kp/ðð1 pÞð/0 Þ2 þ p//00 Þþ 2
dm3 k ðð1 pÞð1 2pÞð/0 Þ3 þ 3pð1 pÞ//0 /00 þ p2 /2 /000 Þ ¼ 0:
ð17Þ
According to the proposed method, we expand the solution of (17) in the form
/ðnÞ ¼ a0 þ
a1 wðnÞ ; 1 þ lwðnÞ
ð18Þ
where wðnÞ satisfies (5). With the aid of Maple, substituting (18) along with (5) into (17), yields a set of algebraic equations for wi ðnÞ ði ¼ 0; 1; . . .Þ. Setting the coefficients of these terms wi ðnÞ to zero yields a set of over-determined algebraic equations with respect to a0 ; a1 ; l; k; k; a and b (because the over-determined algebraic equations are so many, they are omit). Then we get the following results: Case 1:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðlq0 q0 q2 Þa0 a1 ¼ ; l ¼ l; k ¼ k; q0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi kmq0 q2 þ l2 q0 ðp þ p2 2dmkð5p2 3p 2Þ q0 q2 Þ a¼ ; ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p a0 p2 ðlq0 q0 q2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dm2 k ðp þ 1Þð2p þ 1Þ l2 q0 þ q2 q0 2m2 kðp q0 q2 þ 2dmkq0 q2 Þ : ; k¼ b¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 p2 a0 p ðl q0 2l q0 q2 þ q2 Þ a0 ¼ a0 ;
ð19Þ
J. Wang / Applied Mathematics and Computation 217 (2010) 1652–1657
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Case 2:
a0 ¼ a0 ;
a1 ¼
ðlq0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q0 q2 Þa0 ; q0
l ¼ l; k ¼
2 mðp 1Þ
; dðp 2Þ2 2 l2 q0 þ q2 2 ð4p þ 1Þðp 1Þ p ; b¼ k¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2dm q0 q2 ðp 2Þ 4da20 q2 ðp 2Þ2 q2 2l q0 q2 l2 q0 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi l q þ q q0 l3 q2 q0 3l2 q0 q2 3q2 l q2 q0 þ q22 ðp þ 1Þ2 p a¼ 2 0 2 : pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2q0 q2 6l2q0 q2 4q2 l q2 q0 þ l4 q20 4q0 l3 q2 q0 þ q22 ð2 þ pÞ2 da0
ð20Þ
From (14), (16), (18), (6)–(12), we obtain the following solutions for (13): Family 1: According to (19), when q0 q2 > 0 (a)
From (6) and (9) we obtain the triangular periodic solutions of the compound KdV–Burgers equation:
1 pffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 i tanð q0 q2 kðmx þ ktÞ þ c1 Þ p ; pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 tanð q0 q2 kðmx þ ktÞ þ c1 Þ 1 pffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 i cotð q0 q2 kðmx þ ktÞ þ c2 Þ p ; u2 ¼ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 cotð q0 q2 kðmx þ ktÞ þ c2 Þ !1p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 iðsecð 4q0 q2 kðmx þ ktÞ þ c3 Þ tanð 4q0 q2 kðmx þ ktÞ þ c3 ÞÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p ffiffiffiffiffiffiffiffiffiffiffiffi ffi ; u3 ¼ pffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 ðsecð 4q0 q2 kðmx þ ktÞ þ c3 Þ tanð 4q0 q2 kðmx þ ktÞ þ c3 ÞÞ !1p pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 iðcscð 4q0 q2 kðmx þ ktÞ þ c4 Þ cotð 4q0 q2 kðmx þ ktÞ þ c4 ÞÞ ffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffi ffi p p u4 ¼ : pffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 ðcscð 4q0 q2 kðmx þ ktÞ þ c4 Þ cotð 4q0 q2 kðmx þ ktÞ þ c4 ÞÞ
u1 ¼
(b)
ð21Þ ð22Þ
ð23Þ
ð24Þ
From (11) we obtain the exp function solutions of the compound KdV–Burgers equation:
0
pffiffiffiffiffiffi 11p 2i q0 q2 kðmxþktÞ a0 q2 a0 q2 i 1ic5 e2ipffiffiffiffiffiffi q0 q2 kðmxþktÞ B C 1þc5 e A ; u5 ¼ @ q0 q2 kðmxþktÞ pffiffiffiffiffiffiffiffiffiffi1ic5 e2ipffiffiffiffiffiffi p ffiffiffiffiffiffi q2 þ l q0 q2 2i q0 q2 kðmxþktÞ
ð25Þ
1þc5 e
pffiffiffiffiffiffiffiffiffiffi 2 2m kðp qp02q2 þ2dmkq0 q2 Þ ;
where k ¼ mined by (19).
l; k; m and c1 ; c2 ; c3 ; c4 ; c5 are arbitrary constants and a; b are deter-
a0 ;
Family 2: According to (19), when q0 q2 < 0 (a)
From (7) and (10) we obtain the soliton-like solutions of the compound KdV–Burgers equation:
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 tanhð q0 q2 kðmx þ ktÞ þ c6 Þ p ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 tanhð q0 q2 kðmx þ ktÞ þ c6 Þ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 cothð q0 q2 kðmx þ ktÞ þ c7 Þ p ; u7 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 cothð q0 q2 kðmx þ ktÞ þ c7 Þ !1p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 ðtanhð 4q0 q2 kðmx þ ktÞ þ c8 Þ isechð 4q0 q2 kðmx þ ktÞ þ c8 ÞÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u8 ¼ ; pffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 ðtanhð 4q0 q2 kðmx þ ktÞ þ c8 Þ isechð 4q0 q2 kðmx þ ktÞ þ c8 ÞÞ !1p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a0 q2 a0 q2 ðcothð 4q0 q2 kðmx þ ktÞ þ c9 Þ cschð 4q0 q2 kðmx þ ktÞ þ c9 ÞÞ p p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : u9 ¼ pffiffiffiffiffiffiffiffiffiffi q2 þ l q0 q2 ðcothð 4q0 q2 kðmx þ ktÞ þ c9 Þ cschð 4q0 q2 kðmx þ ktÞ þ c9 ÞÞ
u6 ¼
(b)
ð26Þ ð27Þ
ð28Þ
ð29Þ
From (12), we obtain the exp function solutions of the compound KdV–Burgers equation:
0 B u10 ¼ @
a0 q2 a0 q2
pffiffiffiffiffiffi 0 q2 kðmxþktÞ 1þc10 e2pqffiffiffiffiffiffi
11p
1c10 e2 q0 q2 kðmxþktÞ C A ; q0 q2 kðmxþktÞ pffiffiffiffiffiffiffiffiffiffi1þc10 e2pffiffiffiffiffiffi pffiffiffiffiffiffi q2 þ l q0 q2 2 q0 q2 kðmxþktÞ
ð30Þ
1c10 e
pffiffiffiffiffiffiffiffiffiffi 2 2m kðp qp02q2 þ2dmkq0 q2 Þ ;
where k ¼ mined by (19).
a0 ;
l; k; m and c6 ; c7 ; c8 ; c9 ; c10 are arbitrary constants and a; b are deter-
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J. Wang / Applied Mathematics and Computation 217 (2010) 1652–1657
Family 3: According to (20), when q0 q2 > 0 (a)
From (6) and (9), we obtain the triangular periodic solutions of the compound KdV–Burgers equation:
u11
11p 2 ðp1Þ i p a0 q2 a0 q2 i tan 2dðp2Þ x d ðp2Þ þ c11 2 t A ; ¼@ pffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ i p x d ðp2Þ t þ c q2 þ l q0 q2 tan 2dðp2Þ 11 2
ð31Þ
u12
11p 2 ðp1Þ i p a0 q2 a0 q2 i cot 2dðp2Þ x d ðp2Þ þ c12 2 t A ; ¼@ pffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ i p x d ðp2Þ þ c12 q2 þ l q0 q2 cot 2dðp2Þ 2 t
ð32Þ
u13
11p 2 ðp1Þ 2 ðp1Þ i p i p a0 q2 a0 q2 i sec dðp2Þ x d ðp2Þ þ c13 tan dðp2Þ x d ðp2Þ þ c13 2 t 2 t A ; ¼@ pffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ 2 ðp1Þ i p i p x d ðp2Þ þ c13 tan dðp2Þ x d ðp2Þ þ c13 q2 þ l q0 q2 sec dðp2Þ 2 t 2 t
ð33Þ
u14
11p 2 ðp1Þ 2 ðp1Þ i p i p a0 q2 a0 q2 i csc dðp2Þ x d ðp2Þ þ c14 cot dðp2Þ x d ðp2Þ þ c14 2 t 2 t A : ¼@ pffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ 2 ðp1Þ i p i p x d ðp2Þ þ c14 cot dðp2Þ x d ðp2Þ þ c14 q2 þ l q0 q2 csc dðp2Þ 2 t 2 t
ð34Þ
0
0
0
0
(b)
From (11), we obtain the exp function solutions of the compound KdV–Burgers equation:
0
u15
0
p dðp2Þ
2 ðp1Þ t x d ðp2Þ2
1 11p
1ic15 e B a q a q iB C AC 0 2 @ C B 0 2 2 ðp1Þ t p C B x dðp2Þ C B d ðp2Þ2 1þc15 e B 0 1C ¼B C; 2 ðp1Þ p C B x t B pffiffiffiffiffiffiffiffiffiffiB1ic15 edðp2Þ d ðp2Þ2 CC A @q2 þ l q0 q2 @ A 2 1þc15 e
where a0 ;
p dðp2Þ
x
ð35Þ
ðp1Þ t d ðp2Þ2
l; m and c11 ; c12 ; c13 ; c14 ; c15 are arbitrary constants and a; b are determined by (20).
Family 4: According to (20), when q0 q2 < 0 (a)
From (7) and (10), we obtain the soliton-like solutions of the compound KdV–Burgers equation:
u16
11p 2 ðp1Þ p a0 q2 a0 q2 tanh 2dðp2Þ x d ðp2Þ þ c16 2 t A ; ¼@ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ p x d ðp2Þ þ c16 q2 þ l q0 q2 tanh 2dðp2Þ 2 t
ð36Þ
u17
11p 2 ðp1Þ p a0 q2 a0 q2 coth 2dðp2Þ x d ðp2Þ þ c17 2 t A ; ¼@ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ p x d ðp2Þ þ c17 q2 þ l q0 q2 coth 2dðp2Þ 2 t
ð37Þ
u18
11p 2 ðp1Þ 2 ðp1Þ p p a0 q2 a0 q2 tanh dðp2Þ x d ðp2Þ þ c18 isech dðp2Þ x d ðp2Þ þ c18 2 t 2 t A ; ¼@ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ 2 ðp1Þ p p x dðp2Þ þ c18 isech dðp2Þ x d ðp2Þ þ c18 q2 þ l q0 q2 tanh dðp2Þ 2 t 2 t
ð38Þ
u19
11p 2 ðp1Þ 2 ðp1Þ p p a0 q2 a0 q2 coth dðp2Þ x d ðp2Þ þ c19 csch dðp2Þ x d ðp2Þ þ c19 2 t 2 t A : ¼@ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðp1Þ 2 ðp1Þ p p x d ðp2Þ þ c19 csch dðp2Þ x d ðp2Þ þ c19 q2 þ l q0 q2 coth dðp2Þ 2 t 2 t
ð39Þ
0
0
0
0
(b)
From (12), we obtain the exp function solutions of the compound KdV–Burgers equation:
0
u20
0
p dðp2Þ
x
2 ðp1Þ t
1 11p
1þc20 e Ba q a q B C AC 0 2@ C B 0 2 2 ðp1Þ t p C B x 2 dðp2Þ C B d ðp2Þ 1c e 20 C; 0 1 ¼B C B 2 ðp1Þ p C B x t dðp2Þ 2 d ðp2Þ pffiffiffiffiffiffiffiffiffiffiB1þc20 e B C AC A @q2 þ l q0 q2 @
1c20 e
where a0 ;
p dðp2Þ
d ðp2Þ2
2 ðp1Þ t x d ðp2Þ2
l; m and c16 ; c17 ; c18 ; c19 ; c20 are arbitrary constants and a; b are determined by (20).
ð40Þ
J. Wang / Applied Mathematics and Computation 217 (2010) 1652–1657
1657
4. Summary and conclusions In summary, based on the new Riccati equation rational expansion method, many generalized exact solutions of the compound KdV–Burgers equation with nonlinear terms of any order have been derived. The method can also easily to be extended to treat other nonlinear evolution equations and is sufficient to seek more solitary-wave solutions and other formal solutions of nonlinear evolution equations. In addition, this method is also computerizable, which allows us to perform complicated and tedious symbolic algebraic calculation on a computer. Acknowledgements The work is supported by The Youth Funds of Henan Polytechnic University under the Grant Q2006-10-646172, The Natural Science Foundation of Henan Province of China under the Grant 0611055500, The Youth Mainstay teacher Funds of Henan Polytechnic University of China under the Grant 649071. References [1] E.J. Parkes, B.R. Duffy, An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations, Comput. Phys. Commun. 98 (1996) 288–300; E.J. Parkes, B.R. Duffy, Travelling solitary wave solutions to a compound KdV–Burgers equation, Phys. Lett. A 229 (1997) 217–220. [2] W. Hereman, Exact solitary wave solutions of coupled nonlinear evolution equations using MACSYMA, Comput. Phys. Commun. 65 (1991) 143–150. [3] Z.B. Li, Y.P. Liu, RATH: a Maple package for finding travelling solitary wave solutions to nonlinear evolution equations, Comput. Phys. Commun. 148 (2002) 256–266. [4] Z.Y. Yan, New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water, Phys. Lett. A 285 (2001) 355–362. [5] A.H. Khater, W. Malfliet, D.K. Callebaut, E.S. Kamel, The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction– diffusion equations, Chaos Solitons Fract. 14 (2002) 513–522. [6] E.G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212–218. [7] Q. Wang, Y. Chen, H.Q. Zhang, A new Riccati equation rational expansion method and its application to (2 + 1)-dimensional Burgers equation, Chaos Solitons Fract. 25 (2005) 1019–1028. [8] J. Wang, X.L. Zhang, L.N. Song, H.Q. Zhang, A new general algebraic method with symbolic computation and its application to two nonlinear differential equations with nonlinear terms of any order, Appl. Math. Comput. 182 (2006) 1330–1340. [9] D. Siren, J. Sun, Auxiliary equation method for solving nonlinear partial differential equations, Phys. Lett. A 309 (2003) 387–396. [10] S.A. Elwakil, S.K. El-labany, M.A. Zahran, R. Sabry, Modified extended tanh-function method for solving nonlinear partial differential equations, Phys. Lett. A 299 (2002) 179–188. [11] E. Yomba, Construction of new soliton-like solutions of the (2 + 1) dimensional dispersive long wave equation, Chaos Solitons Fract. 20 (2004) 1135– 1139. [12] E.G. Fan, Y.C. Hon, A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves, Chaos Solitons Fract. 15 (2003) 559–566. [13] E. Yomba, The extended Fan’s sub-equation method and its application to KdV–MKdV, BKK and variant Boussinesq equations, Phys. Lett. A 336 (2005) 463–476. [14] Y.T. Gao, B. Tian, Generalized hyperbolic-function method with computerized symbolic computation to construct the solitonic solutions to nonlinear equations of mathematical physics, Comput. Phys. Commun. 133 (2001) 158–164. [15] M. Wadati, Wave propagation in nonlinear lattice. I, J. Phys. Soc. Jpn. 38 (1975) 673–680. [16] R.L. Pego, P. Smereka, M.I. Weinstein, Oscillatory instability of traveling waves for a KdV–Burgers equation, Physica D 67 (1993) 45–65. [17] B. Dey, Domain wall solutions of KdV like equations with higher order nonlinearity, J. Phys. A 19 (1986) L9–L12. [18] M.W. Coffey, On series expansions giving closed-form solutions of Korteweg–DeVries-like equations, J. SIAM Appl. Math. 50 (1990) 1580–1592. [19] W.G. Zhang, Q.S. Chang, B.G. Jiang, Explicit exact solitary-wave solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order, Chaos Solitons Fract. 13 (2002) 311–319.