- Email: [email protected]
A sub-ODE method for generalized Gardner and BBM equation with nonlinear terms of any order
Applied Mathematics and Computation 217 (2010) 1404–1407
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A sub-ODE method for generalized Gardner and BBM equation with nonlinear terms of any order Dianchen Lu *, CuiLian Liu Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University Zhenjiang, Jiangsu 212013, PR China
a r t i c l e
i n f o
a b s t r a c t In this paper, a method with the aid of a sub-ODE and its solutions is used for constructing new periodic wave solutions for nonlinear Gardner equation and BBM equation with nonlinear terms of any order arising in mathematical physics. As a result, many exact traveling wave solutions are successfully obtained. The method in the paper is very direct and it can also be applied to other nonlinear evolution equations. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Sub-ODE Generalized Gardner equation Generalized BBM equation Traveling wave solutions
1. Introduction Looking for exact solutions to nonlinear partial differential equations (NLPDEs) has long been a major concern for both mathematicians and physicists. These solutions may well describe various phenomena in many fields, such as hydrodynamic, plasma physics, nonlinear optic, chemistry, and biology. Many powerful methods for obtaining exact solutions of NLPDEs have been presented, such as Riccati equation method [1], tanh-function method [2], homogeneous balance method [3], Exp-function method [4], Hirota bilinear method [5], and sine–cosine method [6]. In this paper, we consider Gardner and BBM equation with nonlinear terms of any order. Gardner equation is widely used in various branches of physics, such as plasma physics, fluid physics, and quantum field theory, and it also describes a variety of wave phenomena in plasma and solid state [7]. BBM equation represents the motion of nonlinear dispersive waves, which are often encountered in a number of important physical phenomena such as shallow water and ion acoustic plasmas, so it is very significant to investigate the exact solutions of Gardner equation and BBM equation. This paper is arranged as follows. In Section 2, we introduced the solutions of the subsidiary ODE. In Section 3, we obtain several families solutions of Gardner equation. In Section 4, we obtain several families solutions of BBM equation. In Section 5, some conclusions are given.
2. Special solutions of the sub-ODE Li and Wang [8] introduced a subsidiary ODE including an arbitrary positive power
ðF 0 ðnÞÞ2 ¼ AFðnÞ þ BF nþ2 ðnÞ þ CF 2nþ2 ðnÞ; 0
n > 0;
where A, B and C are constants, and ‘‘ ” denotes
d . dn
* Corresponding author. E-mail address: [email protected] (D. Lu). URL: http://[email protected] (D. Lu). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.05.049
ð2:1Þ
D. Lu, C. Liu / Applied Mathematics and Computation 217 (2010) 1404–1407
1405
Zhang et al. [9] gives the special solutions of Eq. (1), which are listed as follows. If A > 0, then Eq. (1) has the following hyperbolic function solutions:
8( )1n pffiffi > > 2Asechðn Aðnþn0 ÞÞ > pffiffiffiffiffiffiffiffiffiffiffiffi > ; B2 4AC > 0; > pffiffi > > B2 4AC sechðn Aðnþn0 ÞÞ > > > > )1n > >( pffiffi > > 2Acsch n Aðnþn0 ÞÞ > < pffiffiffiffiffiffiffiffiffiffiffiffi ð ; B2 4AC < 0; pffiffi 2 4ACB csch n A ðnþn Þ ð Þ 0 F ðnÞ ¼ > pffiffi 1n > > > ðn Aðnþn0 ÞÞ > A > 1 tanh ; B2 4AC ¼ 0; > B 2 > > > > > pffiffi 1n > > > ðn Aðnþn0 ÞÞ > : AB 1 coth ; B2 4AC ¼ 0: 2
ð2:2Þ
If A < 0, then Eq. (2.1) has the following trigonometric function solutions:
8( )1n pffiffi > > 2A sec ðn Aðnþn0 ÞÞ > > pffiffiffiffiffiffiffiffiffiffiffiffi > ; pffiffi > < B2 4AC sec ðn Aðnþn0 ÞÞ F ðnÞ ¼ ( )1n > pffiffi > > 2A csc 0ðn Aðnþn0 ÞÞ > > p ffiffiffiffiffiffiffiffiffiffiffiffi ; > pffiffi : 4ACB2 csc ðn Aðnþn0 ÞÞ
B2 4AC > 0; ð2:3Þ B2 4AC < 0:
If A = 0, then Eq. (2.1) has the following rational solutions:
FðnÞ ¼
1n
4B
; B – 0; 4C 1 F ðnÞ ¼ pffiffiffi ; B ¼ 0; C > 0: n C
ð2:4Þ
B2 n2 n2
ð2:5Þ
By using the solutions of Eq. (2.1), we will get the exact solutions of Gardner and BBM equation. 3. Exact traveling solutions of Gardner equation Considering the general Gardner equation with nonlinear terms of any order equation [8]:
ut þ ðp þ qun þ ru2n Þux þ uxxx ¼ 0;
n > 0;
ð3:1Þ
when n ¼ 1; q – 0; r – 0, Eq. (3.1) becomes the KdV–mKdV equation
ut þ ðp þ qu þ ru2 Þux þ uxxx ¼ 0;
ð3:2Þ
when n ¼ 1; q – 0; r ¼ 0, Eq. (3.1) becomes the KdV equation
ut þ ðp þ quÞux þ uxxx ¼ 0;
ð3:3Þ
when n ¼ 1; q ¼ 0; r – 0, Eq. (3.1) becomes the mKdV equation
ut þ ðp þ ru2 Þux þ uxxx ¼ 0:
ð3:4Þ
In the following, we shall construct exact traveling wave solutions of Eq. (3.1). Firstly Making the traveling wave transformation for Eq. (3.1):
uðx; tÞ ¼ uðnÞ;
n ¼ kðx ktÞ;
ð3:5Þ
where k and k are arbitrary constants to be determined later. Then, Eq. (3.1) reduces to a nonlinear ordinary differential equations (ODEs): 3
kku0 þ kðp þ qun þ ru2n Þu0 þ k u000 ¼ 0:
ð3:6Þ
Integrating the above equation twice and setting the integration constant to be zero, we obtain
ðu0 Þ2 ¼
kp 2
k
u
2q 2
k ðn þ 1Þðn þ 2Þ
unþ2
r 2
k ðn þ 1Þð2n þ 1Þ
u2nþ2 :
ð3:7Þ
2q r Let A ¼ kp ; B ¼ k2 ðnþ1Þðnþ2Þ ; C ¼ k2 ðnþ1Þð2nþ1Þ . Then Eq. (3.4) reduce to k2
ðu0 Þ2 ¼ Au þ Bunþ2 þ Cu2nþ2 : According to (2.1)–(2.4), we can obtain the solutions of Gardner equation with nonlinear terms of any order.
ð3:8Þ
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D. Lu, C. Liu / Applied Mathematics and Computation 217 (2010) 1404–1407
When A > 0, that is to say, k > p, the equation has the following hyperbolic function solutions (3.9)–(3.12):
pffiffiffi
91n = 2Asech n Aðkðx ktÞ þ n0 Þ
; u ðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : B2 4AC sech npffiffiffi Aðkðx ktÞ þ n0 Þ ; pffiffiffi
8 91n < = 2Acsch n Aðkðx ktÞ þ n0 Þ pffiffiffi
; u ðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 4AC B2 csch n Aðkðx ktÞ þ n Þ ; 0 8 <
B2 4AC > 0;
ð3:9Þ
B2 4AC < 0;
ð3:10Þ
(3.9)–(3.10) is bell-type solitary wave solution of Eq. (3.1) and when k ¼ 1, (3.9) is equivalent to the solution (28) given in [9], (3.10) is equivalent to the solution (29) given in [9].
1391n 8 2 0 pffiffiffi < A = n Aðkðx ktÞ þ n0 Þ A5 ; u ðx; tÞ ¼ 41 tanh @ : B ; 2
B2 4AC ¼ 0;
ð3:11Þ
1391n 8 2 0 pffiffiffi < A = n Aðkðx ktÞ þ n0 Þ A5 ; u ðx; tÞ ¼ 41 coth @ : B ; 2
B2 4AC ¼ 0;
ð3:12Þ
(3.11)–(3.12) is kink-type solitary wave solution of Eq. (3.1) and When k ¼ 1, (3.11) is equivalent to the solution (30) given in [9], (3.12) is equivalent to the solution (31) given in [9]. When A < 0, that is to say, k < p, the equation has the following trigonometric function solutions (3.13), (3.14):
pffiffiffi
91n = 2A sec n Aðkðx ktÞ þ n0 Þ pffiffiffi
; u ðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : B2 4AC sec n Aðkðx ktÞ þ n Þ ; 0 pffiffiffi
91n 8 = < 2A csc n Aðkðx ktÞ þ n0 Þ pffiffiffi
; u ðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 4AC B2 csc n Aðkðx ktÞ þ n Þ ; 0 8 <
B2 4AC > 0;
ð3:13Þ
B2 4AC < 0:
ð3:14Þ
(3.13) is Secant traveling wave solution of Eq. (3.1) and (3.14) is cosecant traveling wave solution of Eq. (3.1). When k ¼ 1, (3.13) is equivalent to the solution (32) given in [9], (3.14) is equivalent to the solution (33) given in [9]. When A = 0, that is to say, k ¼ p, the equation has the following rational solutions (3.15), (3.16):
(
)1n
4B
uðx; tÞ ¼
B2 n2 ½kðx ktÞ2 4C 1 B ¼ 0; u ðx; tÞ ¼ pffiffiffi ; n C
;
B – 0;
C > 0:
ð3:15Þ ð3:16Þ
When k ¼ 1, (3.15) is equivalent to the solution (34) given in [9], (3.16) is equivalent to the solution (3.13) given in [8]. In 2q r (3.9)–(3.16) where A ¼ kp ; B ¼ k2 ðnþ1Þðnþ2Þ ; C ¼ k2 ðnþ1Þð2nþ1Þ ; k; k are left as free parameters, a, b, and r, are arbitrary conk2 stants, n0 is an integration constant. Remark: the solutions (3.9)–(3.16) obtained in the paper is more extensive than [8,9] and our method is more brief and direct to find solutions to various solutions of nonlinear evolution equations.
4. Exact traveling solutions of BBM equation Considering the BBM equation with nonlinear terms of any order equation [10]: n
ut þ aux þ bu ux ruxxt ¼ 0;
n > 0;
ð4:1Þ
when n ¼ 1, Eq. (4.1) reduced to the BBM equation
ut þ aux þ buux ruxxt ¼ 0;
n > 0:
ð4:2Þ
Similarly, Eq. (4.1) can be reduced to the following equation:
ðu0 Þ2 ¼ Au þ Bunþ2 ; ka ; kk
2b krk2 ðnþ1Þðnþ2Þ ,
where A ¼ B¼ solutions of BBM equation.
ð4:3Þ k; k are left as free parameters. According to (2.1)–(2.4), where C = 0, we can obtain the
D. Lu, C. Liu / Applied Mathematics and Computation 217 (2010) 1404–1407
1407
When A > 0, that is to say, k > a, the equation has the following hyperbolic function solutions:
pffiffiffi
91n 8 < 2Asech n Aðkðx ktÞ þ n0 Þ =
u ðx; tÞ ¼ pffiffiffiffiffi : B2 sech npffiffiffi Aðkðx ktÞ þ n0 Þ ;
ð4:4Þ
(4.4) is similar to the solution (35) given in [11]. when A < 0, that is to say, k < a, the equation has the following trigonometric function solutions:
pffiffiffi
91n 8 < 2A sec n Aðkðx ktÞ þ n0 Þ =
u ðx; tÞ ¼ pffiffiffiffiffi : B2 sec npffiffiffi Aðkðx ktÞ þ n0 Þ ;
ð4:5Þ
when A = 0, that is to say, k ¼ a, the equation has the following rational solutions:
( uðx; tÞ ¼
4B
B2 n2 ½kðx ktÞ2
)1n ;
B – 0;
ð4:6Þ
2b where A ¼ ka ; B ¼ krk2 ðnþ1Þðnþ2Þ ; k; k are left as free parameters, a, b, r, are arbitrary constants, n0 is an integration constant. kk2
5. Conclusion In this paper, a method with the aid of a sub-ODE and its solutions is used for constructing new periodic wave solutions for nonlinear Gardner and BBM equation arising in mathematical physics, and many exact traveling wave solutions are obtained. This enriches the literature on the type of wave equations of Gardner and BBM equation. Of course, this method can also be applied to other nonlinear evolution equations. Acknowledgements The authors express their thanks to the referee for his suggestion. The work was supported by the National Nature Science Foundation of China (No. 10420130638). References [1] D.C. Lu, B.J. Hong, New exact solutions for the (2+1)-dimensional generalized Broer-Kaup system, Appl. Math. Comput. 199 (2008) 572–580. [2] E.J. Parkes, B.R. Duffy, An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Comput. Phys. Commun. 98 (1996) 288–296. [3] M.L. Wang, Y.B. Zhou, Z.B. Li, Applications of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996) 67–75. [4] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals 30 (3) (2006) 700–708. [5] A.M. Wazwaz, Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method, Appl. Math. Comput. 190 (1) (2007) 633–640. [6] A.M. Wazwaz, Nonlinear variants of KdV and KP equations with compactons, solitons and periodic solutions, Commun. Nonlinear Sci. Numer. Simul. 10 (4) (2005) 451–463. [7] Abdul-Majid Wazwaz, New solitons and kink solutions for the Gardner equation, Commun. Nonlinear Sci. Numer. Simul. 12 (2007) 1395–1404. [8] X.Z. Li, M.L. Wang, A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms, Phys. Lett. A 361 (2007) 115–118. [9] S. Zhang, W. Wang, J.L. Tong, The improved sub-ODE method for a generalized KdV–mKdV equation with nonlinear terms of any order, Phys. Lett. A 372 (2008) 3808–3813. [10] Y. Chen, B. Li, H.Q. Zhang, Exact solutions for two nonlinear wave equations with nonlinear terms of any order, Commun. Nonlinear Sci. Numer. Simul. 10 (2005) 133–138. [11] J. Nickel, Elliptic solutions to a generalized BBM equation, Phys. Lett. A 364 (2007) 221–226.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
A sub-ODE method for generalized Gardner and BBM equation with nonlinear terms of any order Dianchen Lu *, CuiLian Liu Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University Zhenjiang, Jiangsu 212013, PR China
a r t i c l e
i n f o
a b s t r a c t In this paper, a method with the aid of a sub-ODE and its solutions is used for constructing new periodic wave solutions for nonlinear Gardner equation and BBM equation with nonlinear terms of any order arising in mathematical physics. As a result, many exact traveling wave solutions are successfully obtained. The method in the paper is very direct and it can also be applied to other nonlinear evolution equations. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Sub-ODE Generalized Gardner equation Generalized BBM equation Traveling wave solutions
1. Introduction Looking for exact solutions to nonlinear partial differential equations (NLPDEs) has long been a major concern for both mathematicians and physicists. These solutions may well describe various phenomena in many fields, such as hydrodynamic, plasma physics, nonlinear optic, chemistry, and biology. Many powerful methods for obtaining exact solutions of NLPDEs have been presented, such as Riccati equation method [1], tanh-function method [2], homogeneous balance method [3], Exp-function method [4], Hirota bilinear method [5], and sine–cosine method [6]. In this paper, we consider Gardner and BBM equation with nonlinear terms of any order. Gardner equation is widely used in various branches of physics, such as plasma physics, fluid physics, and quantum field theory, and it also describes a variety of wave phenomena in plasma and solid state [7]. BBM equation represents the motion of nonlinear dispersive waves, which are often encountered in a number of important physical phenomena such as shallow water and ion acoustic plasmas, so it is very significant to investigate the exact solutions of Gardner equation and BBM equation. This paper is arranged as follows. In Section 2, we introduced the solutions of the subsidiary ODE. In Section 3, we obtain several families solutions of Gardner equation. In Section 4, we obtain several families solutions of BBM equation. In Section 5, some conclusions are given.
2. Special solutions of the sub-ODE Li and Wang [8] introduced a subsidiary ODE including an arbitrary positive power
ðF 0 ðnÞÞ2 ¼ AFðnÞ þ BF nþ2 ðnÞ þ CF 2nþ2 ðnÞ; 0
n > 0;
where A, B and C are constants, and ‘‘ ” denotes
d . dn
* Corresponding author. E-mail address: [email protected] (D. Lu). URL: http://[email protected] (D. Lu). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.05.049
ð2:1Þ
D. Lu, C. Liu / Applied Mathematics and Computation 217 (2010) 1404–1407
1405
Zhang et al. [9] gives the special solutions of Eq. (1), which are listed as follows. If A > 0, then Eq. (1) has the following hyperbolic function solutions:
8( )1n pffiffi > > 2Asechðn Aðnþn0 ÞÞ > pffiffiffiffiffiffiffiffiffiffiffiffi > ; B2 4AC > 0; > pffiffi > > B2 4AC sechðn Aðnþn0 ÞÞ > > > > )1n > >( pffiffi > > 2Acsch n Aðnþn0 ÞÞ > < pffiffiffiffiffiffiffiffiffiffiffiffi ð ; B2 4AC < 0; pffiffi 2 4ACB csch n A ðnþn Þ ð Þ 0 F ðnÞ ¼ > pffiffi 1n > > > ðn Aðnþn0 ÞÞ > A > 1 tanh ; B2 4AC ¼ 0; > B 2 > > > > > pffiffi 1n > > > ðn Aðnþn0 ÞÞ > : AB 1 coth ; B2 4AC ¼ 0: 2
ð2:2Þ
If A < 0, then Eq. (2.1) has the following trigonometric function solutions:
8( )1n pffiffi > > 2A sec ðn Aðnþn0 ÞÞ > > pffiffiffiffiffiffiffiffiffiffiffiffi > ; pffiffi > < B2 4AC sec ðn Aðnþn0 ÞÞ F ðnÞ ¼ ( )1n > pffiffi > > 2A csc 0ðn Aðnþn0 ÞÞ > > p ffiffiffiffiffiffiffiffiffiffiffiffi ; > pffiffi : 4ACB2 csc ðn Aðnþn0 ÞÞ
B2 4AC > 0; ð2:3Þ B2 4AC < 0:
If A = 0, then Eq. (2.1) has the following rational solutions:
FðnÞ ¼
1n
4B
; B – 0; 4C 1 F ðnÞ ¼ pffiffiffi ; B ¼ 0; C > 0: n C
ð2:4Þ
B2 n2 n2
ð2:5Þ
By using the solutions of Eq. (2.1), we will get the exact solutions of Gardner and BBM equation. 3. Exact traveling solutions of Gardner equation Considering the general Gardner equation with nonlinear terms of any order equation [8]:
ut þ ðp þ qun þ ru2n Þux þ uxxx ¼ 0;
n > 0;
ð3:1Þ
when n ¼ 1; q – 0; r – 0, Eq. (3.1) becomes the KdV–mKdV equation
ut þ ðp þ qu þ ru2 Þux þ uxxx ¼ 0;
ð3:2Þ
when n ¼ 1; q – 0; r ¼ 0, Eq. (3.1) becomes the KdV equation
ut þ ðp þ quÞux þ uxxx ¼ 0;
ð3:3Þ
when n ¼ 1; q ¼ 0; r – 0, Eq. (3.1) becomes the mKdV equation
ut þ ðp þ ru2 Þux þ uxxx ¼ 0:
ð3:4Þ
In the following, we shall construct exact traveling wave solutions of Eq. (3.1). Firstly Making the traveling wave transformation for Eq. (3.1):
uðx; tÞ ¼ uðnÞ;
n ¼ kðx ktÞ;
ð3:5Þ
where k and k are arbitrary constants to be determined later. Then, Eq. (3.1) reduces to a nonlinear ordinary differential equations (ODEs): 3
kku0 þ kðp þ qun þ ru2n Þu0 þ k u000 ¼ 0:
ð3:6Þ
Integrating the above equation twice and setting the integration constant to be zero, we obtain
ðu0 Þ2 ¼
kp 2
k
u
2q 2
k ðn þ 1Þðn þ 2Þ
unþ2
r 2
k ðn þ 1Þð2n þ 1Þ
u2nþ2 :
ð3:7Þ
2q r Let A ¼ kp ; B ¼ k2 ðnþ1Þðnþ2Þ ; C ¼ k2 ðnþ1Þð2nþ1Þ . Then Eq. (3.4) reduce to k2
ðu0 Þ2 ¼ Au þ Bunþ2 þ Cu2nþ2 : According to (2.1)–(2.4), we can obtain the solutions of Gardner equation with nonlinear terms of any order.
ð3:8Þ
1406
D. Lu, C. Liu / Applied Mathematics and Computation 217 (2010) 1404–1407
When A > 0, that is to say, k > p, the equation has the following hyperbolic function solutions (3.9)–(3.12):
pffiffiffi
91n = 2Asech n Aðkðx ktÞ þ n0 Þ
; u ðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : B2 4AC sech npffiffiffi Aðkðx ktÞ þ n0 Þ ; pffiffiffi
8 91n < = 2Acsch n Aðkðx ktÞ þ n0 Þ pffiffiffi
; u ðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 4AC B2 csch n Aðkðx ktÞ þ n Þ ; 0 8 <
B2 4AC > 0;
ð3:9Þ
B2 4AC < 0;
ð3:10Þ
(3.9)–(3.10) is bell-type solitary wave solution of Eq. (3.1) and when k ¼ 1, (3.9) is equivalent to the solution (28) given in [9], (3.10) is equivalent to the solution (29) given in [9].
1391n 8 2 0 pffiffiffi < A = n Aðkðx ktÞ þ n0 Þ A5 ; u ðx; tÞ ¼ 41 tanh @ : B ; 2
B2 4AC ¼ 0;
ð3:11Þ
1391n 8 2 0 pffiffiffi < A = n Aðkðx ktÞ þ n0 Þ A5 ; u ðx; tÞ ¼ 41 coth @ : B ; 2
B2 4AC ¼ 0;
ð3:12Þ
(3.11)–(3.12) is kink-type solitary wave solution of Eq. (3.1) and When k ¼ 1, (3.11) is equivalent to the solution (30) given in [9], (3.12) is equivalent to the solution (31) given in [9]. When A < 0, that is to say, k < p, the equation has the following trigonometric function solutions (3.13), (3.14):
pffiffiffi
91n = 2A sec n Aðkðx ktÞ þ n0 Þ pffiffiffi
; u ðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : B2 4AC sec n Aðkðx ktÞ þ n Þ ; 0 pffiffiffi
91n 8 = < 2A csc n Aðkðx ktÞ þ n0 Þ pffiffiffi
; u ðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 4AC B2 csc n Aðkðx ktÞ þ n Þ ; 0 8 <
B2 4AC > 0;
ð3:13Þ
B2 4AC < 0:
ð3:14Þ
(3.13) is Secant traveling wave solution of Eq. (3.1) and (3.14) is cosecant traveling wave solution of Eq. (3.1). When k ¼ 1, (3.13) is equivalent to the solution (32) given in [9], (3.14) is equivalent to the solution (33) given in [9]. When A = 0, that is to say, k ¼ p, the equation has the following rational solutions (3.15), (3.16):
(
)1n
4B
uðx; tÞ ¼
B2 n2 ½kðx ktÞ2 4C 1 B ¼ 0; u ðx; tÞ ¼ pffiffiffi ; n C
;
B – 0;
C > 0:
ð3:15Þ ð3:16Þ
When k ¼ 1, (3.15) is equivalent to the solution (34) given in [9], (3.16) is equivalent to the solution (3.13) given in [8]. In 2q r (3.9)–(3.16) where A ¼ kp ; B ¼ k2 ðnþ1Þðnþ2Þ ; C ¼ k2 ðnþ1Þð2nþ1Þ ; k; k are left as free parameters, a, b, and r, are arbitrary conk2 stants, n0 is an integration constant. Remark: the solutions (3.9)–(3.16) obtained in the paper is more extensive than [8,9] and our method is more brief and direct to find solutions to various solutions of nonlinear evolution equations.
4. Exact traveling solutions of BBM equation Considering the BBM equation with nonlinear terms of any order equation [10]: n
ut þ aux þ bu ux ruxxt ¼ 0;
n > 0;
ð4:1Þ
when n ¼ 1, Eq. (4.1) reduced to the BBM equation
ut þ aux þ buux ruxxt ¼ 0;
n > 0:
ð4:2Þ
Similarly, Eq. (4.1) can be reduced to the following equation:
ðu0 Þ2 ¼ Au þ Bunþ2 ; ka ; kk
2b krk2 ðnþ1Þðnþ2Þ ,
where A ¼ B¼ solutions of BBM equation.
ð4:3Þ k; k are left as free parameters. According to (2.1)–(2.4), where C = 0, we can obtain the
D. Lu, C. Liu / Applied Mathematics and Computation 217 (2010) 1404–1407
1407
When A > 0, that is to say, k > a, the equation has the following hyperbolic function solutions:
pffiffiffi
91n 8 < 2Asech n Aðkðx ktÞ þ n0 Þ =
u ðx; tÞ ¼ pffiffiffiffiffi : B2 sech npffiffiffi Aðkðx ktÞ þ n0 Þ ;
ð4:4Þ
(4.4) is similar to the solution (35) given in [11]. when A < 0, that is to say, k < a, the equation has the following trigonometric function solutions:
pffiffiffi
91n 8 < 2A sec n Aðkðx ktÞ þ n0 Þ =
u ðx; tÞ ¼ pffiffiffiffiffi : B2 sec npffiffiffi Aðkðx ktÞ þ n0 Þ ;
ð4:5Þ
when A = 0, that is to say, k ¼ a, the equation has the following rational solutions:
( uðx; tÞ ¼
4B
B2 n2 ½kðx ktÞ2
)1n ;
B – 0;
ð4:6Þ
2b where A ¼ ka ; B ¼ krk2 ðnþ1Þðnþ2Þ ; k; k are left as free parameters, a, b, r, are arbitrary constants, n0 is an integration constant. kk2
5. Conclusion In this paper, a method with the aid of a sub-ODE and its solutions is used for constructing new periodic wave solutions for nonlinear Gardner and BBM equation arising in mathematical physics, and many exact traveling wave solutions are obtained. This enriches the literature on the type of wave equations of Gardner and BBM equation. Of course, this method can also be applied to other nonlinear evolution equations. Acknowledgements The authors express their thanks to the referee for his suggestion. The work was supported by the National Nature Science Foundation of China (No. 10420130638). References [1] D.C. Lu, B.J. Hong, New exact solutions for the (2+1)-dimensional generalized Broer-Kaup system, Appl. Math. Comput. 199 (2008) 572–580. [2] E.J. Parkes, B.R. Duffy, An automated tanh-function method for finding solitary wave solutions to nonlinear evolution equations, Comput. Phys. Commun. 98 (1996) 288–296. [3] M.L. Wang, Y.B. Zhou, Z.B. Li, Applications of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996) 67–75. [4] J.H. He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals 30 (3) (2006) 700–708. [5] A.M. Wazwaz, Multiple-soliton solutions for the KP equation by Hirota’s bilinear method and by the tanh–coth method, Appl. Math. Comput. 190 (1) (2007) 633–640. [6] A.M. Wazwaz, Nonlinear variants of KdV and KP equations with compactons, solitons and periodic solutions, Commun. Nonlinear Sci. Numer. Simul. 10 (4) (2005) 451–463. [7] Abdul-Majid Wazwaz, New solitons and kink solutions for the Gardner equation, Commun. Nonlinear Sci. Numer. Simul. 12 (2007) 1395–1404. [8] X.Z. Li, M.L. Wang, A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms, Phys. Lett. A 361 (2007) 115–118. [9] S. Zhang, W. Wang, J.L. Tong, The improved sub-ODE method for a generalized KdV–mKdV equation with nonlinear terms of any order, Phys. Lett. A 372 (2008) 3808–3813. [10] Y. Chen, B. Li, H.Q. Zhang, Exact solutions for two nonlinear wave equations with nonlinear terms of any order, Commun. Nonlinear Sci. Numer. Simul. 10 (2005) 133–138. [11] J. Nickel, Elliptic solutions to a generalized BBM equation, Phys. Lett. A 364 (2007) 221–226.