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Exp-function method for constructing explicit and exact solutions of a lattice equation
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Applied Mathematics and Computation 199 (2008) 242–249 www.elsevier.com/locate/amc
Exp-function method for constructing explicit and exact solutions of a lattice equation Sheng Zhang Department of Mathematics, Bohai University, Jinzhou 121000, PR China
Abstract In this paper, the Exp-function method is use to obtain explicit and exact solutions of a lattice equation. As a result, some new generalized solitonary solutions with parameters are obtained. Taking full advantages of the generalized solitonary solutions, some known solitary wave solutions reported in open literature are derived as special cases. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful new method for discrete nonlinear evolution equations in mathematical physics. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Exp-function method; Generalized solitonary solutions; Nonlinear evolution equations; Lattice equation
1. Introduction Seeking exact solutions of nonlinear evolution equations (NLEEs) is of important significance in mathematical physics and becomes one of the most exciting and extremely active areas of research investigation. In the past several decades, many effective methods for obtaining exact solutions of NLEEs have been presented, such as inverse scattering method [1], Hirota’s bilinear method [2], Backlund transformation [3], Painleve´ expansion [4], sine–cosine method [5], homogenous balance method [6], homotopy perturbation method [7–9], variational method [10–14], Adomian decomposition method [15], tanh-function method [16–21], algebraic method [22–25], Jacobi elliptic function expansion method [26–29], F-expansion method [30–37], auxiliary equation method [38–41] and so on. Recently, He and Wu [42] proposed the Exp-function method to obtain exact solutions of NLEEs. The basic idea of the Exp-function method was proposed in He’s monograph [43]. The solution procedure of this method, by the help of Matlab or Mathematica, is of utter simplicity and this method has been successfully applied to many kinds of NLEEs [44–57]. The Exp-function method leads to not only generalized solitonary solutions but also periodic solutions. Taking full advantages of the generalized solitonary solutions, we can recover some known solutions gained by the most existing methods such as Adomian decomposition method,
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S. Zhang / Applied Mathematics and Computation 199 (2008) 242–249
243
tanh-function method, algebraic method, Jacobi elliptic function expansion method, F-expansion method, auxiliary equation method and others. Since the work of Fermi et al. in the 1950s [58], the investigation of exact solutions of nonlinear differentialdifference equations (DDEs) has played a crucial role in the modelling of many phenomena in different fields which include condensed matter physics, biophysics and mechanical engineering. We can also encounter such systems in numerical simulation of soliton dynamics in high energy physics where they arise as approximations of continuum models. Unlike difference equations which are fully discretized, DDEs are semi-discretized with some (or all) of their spacial variables discretized while time is usually kept continuous [54]. The present paper is motivated by the desire to extend the Exp-function method to a lattice equation, which reads dun ¼ ða þ bun þ cu2n Þðun1 unþ1 Þ; dt
ð1Þ
where un = u(n, t), n 2 Z, which contains Hybrid lattice equation, mKdV lattice equation and modified Volterra lattice equation as special cases [59]. 2. Exact solutions of the lattice equation Using the transformation U n ðgn Þ ¼ un ðtÞ;
gn ¼ n d þ ct þ g0 ;
where d and c are constants to be determined later, and g0 is the phase, then Eq. (1) becomes cU 0n ¼ a þ bU n þ cU 2n ðU n1 U nþ1 Þ: According to the Exp-function method, we suppose Pg N ¼f aN expðN gn Þ U n ¼ Pq ; M¼p bM expðMgn Þ
ð2Þ
ð3Þ
ð4Þ
where f, g, p and q are positive integers which are unknown to be further determined, aN and bM are unknown constants. Eq. (8) can be re-written in an alternative form [44] as follows Un ¼
af expðf gn Þ þ þ ag expðggn Þ ; expðpgn Þ þ þ bq expðqgn Þ
ð5Þ
then we have af expðfdÞ expðf gn Þ þ þ ag expðgdÞ expðdgn Þ ; expðpdÞ expðpgn Þ þ þ bq expðqdÞ expðqgn Þ af expðfdÞ expðf gn Þ þ þ ag expðgdÞ expðdgn Þ : ¼ expðpdÞ expðpgn Þ þ þ bq expðqdÞ expðqgn Þ
U n1 ¼
ð6Þ
U nþ1
ð7Þ
In order to determine values of f and p, we balance the linear term of highest order in Eq. (3) with the highest order nonlinear term [42]. By simple calculation, we have U 0n ¼
c1 exp½ðp þ f Þgn þ c2 expð2pgn Þ þ
ð8Þ
and U 2n U n1 ¼
c3 expð3f gn Þ þ c3 exp½ð3f pÞgn þ ¼ ; c4 expð3pgn Þ þ c4 expð2pgn Þ þ
ð9Þ
where ci are determined coefficients only for simplicity. Balancing highest order of Exp-function in Eqs. (8) and (9), we have f þ p ¼ 3f p;
ð10Þ
244
S. Zhang / Applied Mathematics and Computation 199 (2008) 242–249
which leads to the result p ¼ f:
ð11Þ
Similarly to determine values of g and q, we balance the linear term of lowest order in Eq. (3) þ d 1 exp½ðg þ qÞgn þ d 2 expð2qgn Þ
U 0n ¼
ð12Þ
and U 2n U n1 ¼
þ d 3 expð3ggn Þ þ d 3 exp½ð3g qÞgn ¼ ; þ d 4 expð3qgn Þ þ d 4 expð2qgn Þ
ð13Þ
where di are determined coefficients only for simplicity. Balancing lowest order of Exp-function in Eqs. (12) and (13), we have ðg þ qÞ ¼ ð3g qÞ;
ð14Þ
which leads to the result g ¼ q:
ð15Þ
We can freely choose the values of f and g, but the final solution does not strongly depend upon the choice of values of f and g [42,44]. For simplicity, we set p = f = 1 and g = q = 1, then Eqs. (5)–(7) become Un ¼ U n1 U nþ1
a1 expðgn Þ þ a0 þ a1 expðgn Þ ; expðgn Þ þ b0 þ b1 expðgn Þ a1 expðdÞ expðgn Þ þ a0 þ a1 expðdÞ expðgn Þ ; ¼ expðdÞ expðgn Þ þ b0 þ b1 expðdÞ expðgn Þ a1 expðdÞ expðgn Þ þ a0 þ a1 expðdÞ expðgn Þ : ¼ expðdÞ expðgn Þ þ b0 þ b1 expðdÞ expðgn Þ
ð16Þ ð17Þ ð18Þ
Substituting Eqs. (16)–(18) into Eq. (3), and using Mathematica, equating to zero the coefficients of all powers of exp(jgn) (j = 0, ±1, ±2, . . .) yields a set of algebraic equations for a1, a0, a1, b0, b1 and c. Solving the system of algebraic equations by use of Mathematica, we obtain the following results: Case 1 2
a1 ¼ 0; b1 ¼
a0 ¼
b0 a½1 þ expðdÞ ; b expðdÞ
a1 ¼ 0;
b20 fac½1 þ expð2dÞ þ expðdÞðb2 2acÞg b2 ½1 þ expðdÞ
2
b0 ¼ b0 ; c¼
;
ð19Þ
a½1 expð2dÞ : expðdÞ
ð20Þ
Case 2 a1 ¼
b½1 þ expðdÞ ½1 þ expðdÞ b0
2c½1 þ expðdÞ
ð21Þ
;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b½1 þ expðdÞ ½1 þ expðdÞ b2 4ac
a0 ¼ b0 ¼ b 0 ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac
;
2c½1 þ expðdÞ b1 ¼ 0;
c¼
½1 þ expðdÞðb2 4acÞ : c½1 þ expðdÞ
a1 ¼ 0;
ð22Þ ð23Þ
Case 3 a1 ¼
b½1 þ expðdÞ ½1 þ expðdÞ 2c½1 þ expðdÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac
;
ð24Þ
S. Zhang / Applied Mathematics and Computation 199 (2008) 242–249
b1 a0 ¼ 0;
a1 ¼
b0 ¼ 0;
b1 ¼ b1 ;
245
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b½1 þ expð2dÞ ½1 þ expð2dÞ b2 4ac
c¼
ð25Þ
;
2c½1 þ expð2dÞ ½1 þ expð2dÞðb2 4acÞ : 2c½1 þ expð2dÞ
ð26Þ
Case 4 a1 ¼
b½1 þ expðdÞ ½1 þ expðdÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac
; a0 ¼ a0 ; 2c½1 þ expðdÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a0 c½1 þ expðdÞ b½1 þ expðdÞ ½1 þ expðdÞ b2 4ac
a1 ¼
b1 ¼
a20 c2 ½1 þ expðdÞ
2
2
½1 þ expðdÞ ðb2 4acÞ
ð28Þ
;
2c½1 þ expðdÞ2
b0 ¼ 0;
ð27Þ
c¼
;
½1 þ expðdÞðb2 4acÞ : c½1 þ expðdÞ
ð29Þ
Case 5 a1 ¼
b ; 2c
b0 ¼ 0;
a0 ¼ a0 ;
b1 ¼
a1 ¼
2a20 bc expð2dÞ 2
½1 þ expð2dÞ ðb2 4acÞ
4a20 c2 expð2dÞ ½1 þ expð2dÞ2 ðb2 4acÞ
;
c¼
ð30Þ
;
½1 þ expð2dÞðb2 4acÞ : 4c expðdÞ
ð31Þ
Case 6 a1 ¼ a1 ; a1 ¼ b1 ¼
a0 ¼
b0 fa½1 þ expðdÞ2 þ a1 b½1 expðdÞ þ expð2dÞ þ a21 c½1 þ expð2dÞg ; expðdÞðb þ 2a1 cÞ
ð32Þ
a1 b20 fc½a þ a1 ðb þ a1 cÞ þ c expð2dÞ½a þ a1 ðb þ a1 cÞ þ expðdÞ½b2 þ 2a1 bc þ 2cða þ a21 cÞg 2
½1 þ expðdÞ ðb þ 2a1 cÞ
2
b20 fc½a þ a1 ðb þ a1 cÞ þ c expð2dÞ½a þ a1 ðb þ a1 cÞ þ expðdÞ½b2 þ 2a1 bc þ 2cða þ a21 cÞg
b0 ¼ b 0 ;
2
½1 þ expðdÞ ðb þ 2a1 cÞ c¼
2
½1 þ expð2dÞ½a þ a1 ðb þ a1 cÞ : expðdÞ
;
;
ð33Þ ð34Þ ð35Þ
Substituting Cases 1–6 into Eq. (16) respectively, and using Eq. (2), we obtain the following generalized solitonary solutions of Eq. (1): b0 a½1þexpðdÞ2 b expðdÞ
un ¼ expðgn Þ þ b0 þ
b20 fac½1þexpð2dÞþexpðdÞðb2 2acÞg 2
b ½1þexpðdÞ
t þ g0 . where gn ¼ n d þ a½1expð2dÞ expðdÞ pffiffiffiffiffiffiffiffiffiffi ffi 2 un ¼
b½1þexpðdÞ½1þexpðdÞ 2c½1þexpðdÞ
b 4ac
2
expðgn Þ
b0 b½1þexpðdÞ½1þexpðdÞ
expðgn Þ þ b0 2
4acÞ where gn ¼ n d þ ½1þexpðdÞðb t þ g0 . c½1þexpðdÞ
ð36Þ
; expðgn Þ
2c½1þexpðdÞ
pffiffiffiffiffiffiffiffiffiffi ffi 2 b 4ac
;
ð37Þ
246
S. Zhang / Applied Mathematics and Computation 199 (2008) 242–249
un ¼
b½1þexpðdÞ½1þexpðdÞ 2c½1þexpðdÞ
pffiffiffiffiffiffiffiffiffiffi ffi 2 b 4ac
expðgn Þ
b1 b½1þexpð2dÞ½1þexpð2dÞ
pffiffiffiffiffiffiffiffiffiffi ffi 2 b 4ac
2c½1þexpð2dÞ
expðgn Þ
expðgn Þ þ b1 expðgn Þ
ð38Þ
;
2
4acÞ where gn ¼ n d þ ½1þexpð2dÞðb t þ g0 . 2c½1þexpð2dÞ
un ¼
b½1þexpðdÞ½1þexpðdÞ 2c½1þexpðdÞ
pffiffiffiffiffiffiffiffiffiffi ffi 2 b 4ac
expðgn Þ þ a0 þ
a20 c½1þexpðdÞ b½1þexpðdÞ½1þexpðdÞ
pffiffiffiffiffiffiffiffiffiffi ffi 2 b 4ac
2c½1þexpðdÞ2
expðgn Þ
a20 c2 ½1þexpðdÞ2 2 2
expðgn Þ ½1þexpðdÞ
ðb
;
ð39Þ
expðgn Þ 4acÞ
2
4acÞ t þ g0 . where gn ¼ n d þ ½1þexpðdÞðb c½1þexpðdÞ 2a2 bc expð2dÞ
un ¼
0 2cb expðgn Þ þ a0 þ ½1þexpð2dÞ expðgn Þ 2 2 ðb 4acÞ
4a2 c2 expð2dÞ
ð40Þ
;
0 expðgn Þ expðgn Þ ½1þexpð2dÞ 2 2 ðb 4acÞ
2
4acÞ t þ g0 . where gn ¼ n d þ ½1þexpð2dÞðb 4c expðdÞ
un ¼
a1 expðgn Þ þ expðgn Þ þ b0 þ
where gn ¼ n d
b0 fa½1þexpðdÞ2 þa1 b½1expðdÞþexpð2dÞþa21 c½1þexpð2dÞg expðdÞðbþ2a1 cÞ
b20 fc½aþa1 ðbþa1 cÞþc
þ a1 expðgn Þ
2
expð2dÞ½aþa1 ðbþa1 cÞþexpðdÞ½b þ2a1 bcþ2cðaþa21 cÞg ½1þexpðdÞ2 ðbþ2a1 cÞ2
½1þexpð2dÞðaþa1 ðbþa1 cÞÞ t expðdÞ
;
ð41Þ
expðgn Þ
þ g0 , a1 is determined in Eq. (33).
To our knowledge, above obtained generalized solitonary solutions are new, they have not been reported in literature yet. Furthermore, if we set the parameters in above generalized solitonary solutions as special values, some known solitary wave solutions can be recovered. For example, setting b0 = 1 in Eq. (37) we obtain b un ¼ 2c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac 2c
g d tanh tanh n ; 2 2
2 tanh d2 t þ g0 . where gn ¼ n d þ b 4ac c Setting b0 = 1 in Eq. (37) we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g b2 4ac b d un ¼ tanh coth n 2c 2 2 2c or
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g g b2 4ac d d n tanh tanh tanh coth n ; 2 2 4 4 4c 4c d b2 4ac where gn ¼ n d þ c tanh 2 t þ g0 . pffiffiffiffiffiffiffiffiffiffiffi b2 4ac½1þexpð2dÞ Setting a0 ¼ in Eq. (39) we have c½1þexpðdÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g g b2 4ac b2 4ac b d d n tanh tanh un ¼ coth csch n ; 2c 2 2 2 2 2c 2c d b2 4ac where gn ¼ n d þ c tanh 2 t þ g0 . b un ¼ 2c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac
ð42Þ
ð43Þ
ð44Þ
ð45Þ
S. Zhang / Applied Mathematics and Computation 199 (2008) 242–249
Setting a0 ¼
i
pffiffiffiffiffiffiffiffiffiffi ffi 2
b 4ac½1þexpð2dÞ c½1þexpðdÞ
247
in Eq. (39) we have
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g b2 4ac b d g d tanh tanh tanh n i sech n ; un ¼ 2c 2 2 2c 2c 2 2 d b2 4ac where gn ¼ n d þ c tanh 2 t þ g0 . pffiffiffiffiffiffiffiffiffiffiffi b2 4ac½1þexpð2dÞ Setting a0 ¼ in Eq. (40) we get 2c expðdÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g g b2 4ac b2 4ac b sinhðdÞ tanh n sinhðdÞ coth n ; un ¼ 2c 2 2 4c 4c
ð46Þ
ð47Þ
or un ¼
b 2c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac 2c
sinhðdÞcschðgn Þ;
ð48Þ
2
where gn ¼ n d þ b 4ac sinhðdÞt þ g0 . 2c pffiffiffiffiffiffiffiffiffiffi ffi i b2 4ac½1þexpð2dÞ Setting a0 ¼ in Eq. (40) we get 2c expðdÞ b un ¼ i 2c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac 2c
sinhðdÞsechðgn Þ;
ð49Þ
2
where gn ¼ n d þ b 4ac sinhðdÞt þ g0 . 2c It is easy to see that above solutions (42)–(49) contain all the solutions obtained from Cases 1–8 in [59]. From Cases 1–6 obtained in this paper we can also derive some other solitary wave solutions, we omit them here for simplicity. 3. Conclusion In this paper, the Exp-function method has been successfully used to obtain explicit and exact solutions of a lattice equation. As a result, some new generalized solitonary solutions with parameters are obtained. To our knowledge, these solutions have not been reported. It may be important to explain some physical phenomena. By setting the parameters as special values, some known solitary wave solutions reported in open literature are derived. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful new method for some discrete nonlinear evolution equations in mathematical physics. Acknowledgements I would like to express my sincere thanks to referee for the valuable advice. This work was supported by the Natural Science Foundation of Educational Committee of Liaoning Province of China under Grant No. 2006022. References [1] M.J. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York, 1991. [2] R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971) 1192–1194. [3] M.R. Miurs, Backlund Transformation, Springer, Berlin, 1978. [4] J. Weiss, M. Tabor, G. Carnevale, The Painleve´ property for partial differential equations, J. Math. Phys. 24 (1983) 522–526. [5] C.T. Yan, A simple transformation for nonlinear waves, Phys. Lett. A 224 (1996) 77–84. [6] M.L. Wang, Exact solutions for a compound KdV–Burgers equations, Phys. Lett. A 213 (1996) 279–287.
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Applied Mathematics and Computation 199 (2008) 242–249 www.elsevier.com/locate/amc
Exp-function method for constructing explicit and exact solutions of a lattice equation Sheng Zhang Department of Mathematics, Bohai University, Jinzhou 121000, PR China
Abstract In this paper, the Exp-function method is use to obtain explicit and exact solutions of a lattice equation. As a result, some new generalized solitonary solutions with parameters are obtained. Taking full advantages of the generalized solitonary solutions, some known solitary wave solutions reported in open literature are derived as special cases. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful new method for discrete nonlinear evolution equations in mathematical physics. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Exp-function method; Generalized solitonary solutions; Nonlinear evolution equations; Lattice equation
1. Introduction Seeking exact solutions of nonlinear evolution equations (NLEEs) is of important significance in mathematical physics and becomes one of the most exciting and extremely active areas of research investigation. In the past several decades, many effective methods for obtaining exact solutions of NLEEs have been presented, such as inverse scattering method [1], Hirota’s bilinear method [2], Backlund transformation [3], Painleve´ expansion [4], sine–cosine method [5], homogenous balance method [6], homotopy perturbation method [7–9], variational method [10–14], Adomian decomposition method [15], tanh-function method [16–21], algebraic method [22–25], Jacobi elliptic function expansion method [26–29], F-expansion method [30–37], auxiliary equation method [38–41] and so on. Recently, He and Wu [42] proposed the Exp-function method to obtain exact solutions of NLEEs. The basic idea of the Exp-function method was proposed in He’s monograph [43]. The solution procedure of this method, by the help of Matlab or Mathematica, is of utter simplicity and this method has been successfully applied to many kinds of NLEEs [44–57]. The Exp-function method leads to not only generalized solitonary solutions but also periodic solutions. Taking full advantages of the generalized solitonary solutions, we can recover some known solutions gained by the most existing methods such as Adomian decomposition method,
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.09.051
S. Zhang / Applied Mathematics and Computation 199 (2008) 242–249
243
tanh-function method, algebraic method, Jacobi elliptic function expansion method, F-expansion method, auxiliary equation method and others. Since the work of Fermi et al. in the 1950s [58], the investigation of exact solutions of nonlinear differentialdifference equations (DDEs) has played a crucial role in the modelling of many phenomena in different fields which include condensed matter physics, biophysics and mechanical engineering. We can also encounter such systems in numerical simulation of soliton dynamics in high energy physics where they arise as approximations of continuum models. Unlike difference equations which are fully discretized, DDEs are semi-discretized with some (or all) of their spacial variables discretized while time is usually kept continuous [54]. The present paper is motivated by the desire to extend the Exp-function method to a lattice equation, which reads dun ¼ ða þ bun þ cu2n Þðun1 unþ1 Þ; dt
ð1Þ
where un = u(n, t), n 2 Z, which contains Hybrid lattice equation, mKdV lattice equation and modified Volterra lattice equation as special cases [59]. 2. Exact solutions of the lattice equation Using the transformation U n ðgn Þ ¼ un ðtÞ;
gn ¼ n d þ ct þ g0 ;
where d and c are constants to be determined later, and g0 is the phase, then Eq. (1) becomes cU 0n ¼ a þ bU n þ cU 2n ðU n1 U nþ1 Þ: According to the Exp-function method, we suppose Pg N ¼f aN expðN gn Þ U n ¼ Pq ; M¼p bM expðMgn Þ
ð2Þ
ð3Þ
ð4Þ
where f, g, p and q are positive integers which are unknown to be further determined, aN and bM are unknown constants. Eq. (8) can be re-written in an alternative form [44] as follows Un ¼
af expðf gn Þ þ þ ag expðggn Þ ; expðpgn Þ þ þ bq expðqgn Þ
ð5Þ
then we have af expðfdÞ expðf gn Þ þ þ ag expðgdÞ expðdgn Þ ; expðpdÞ expðpgn Þ þ þ bq expðqdÞ expðqgn Þ af expðfdÞ expðf gn Þ þ þ ag expðgdÞ expðdgn Þ : ¼ expðpdÞ expðpgn Þ þ þ bq expðqdÞ expðqgn Þ
U n1 ¼
ð6Þ
U nþ1
ð7Þ
In order to determine values of f and p, we balance the linear term of highest order in Eq. (3) with the highest order nonlinear term [42]. By simple calculation, we have U 0n ¼
c1 exp½ðp þ f Þgn þ c2 expð2pgn Þ þ
ð8Þ
and U 2n U n1 ¼
c3 expð3f gn Þ þ c3 exp½ð3f pÞgn þ ¼ ; c4 expð3pgn Þ þ c4 expð2pgn Þ þ
ð9Þ
where ci are determined coefficients only for simplicity. Balancing highest order of Exp-function in Eqs. (8) and (9), we have f þ p ¼ 3f p;
ð10Þ
244
S. Zhang / Applied Mathematics and Computation 199 (2008) 242–249
which leads to the result p ¼ f:
ð11Þ
Similarly to determine values of g and q, we balance the linear term of lowest order in Eq. (3) þ d 1 exp½ðg þ qÞgn þ d 2 expð2qgn Þ
U 0n ¼
ð12Þ
and U 2n U n1 ¼
þ d 3 expð3ggn Þ þ d 3 exp½ð3g qÞgn ¼ ; þ d 4 expð3qgn Þ þ d 4 expð2qgn Þ
ð13Þ
where di are determined coefficients only for simplicity. Balancing lowest order of Exp-function in Eqs. (12) and (13), we have ðg þ qÞ ¼ ð3g qÞ;
ð14Þ
which leads to the result g ¼ q:
ð15Þ
We can freely choose the values of f and g, but the final solution does not strongly depend upon the choice of values of f and g [42,44]. For simplicity, we set p = f = 1 and g = q = 1, then Eqs. (5)–(7) become Un ¼ U n1 U nþ1
a1 expðgn Þ þ a0 þ a1 expðgn Þ ; expðgn Þ þ b0 þ b1 expðgn Þ a1 expðdÞ expðgn Þ þ a0 þ a1 expðdÞ expðgn Þ ; ¼ expðdÞ expðgn Þ þ b0 þ b1 expðdÞ expðgn Þ a1 expðdÞ expðgn Þ þ a0 þ a1 expðdÞ expðgn Þ : ¼ expðdÞ expðgn Þ þ b0 þ b1 expðdÞ expðgn Þ
ð16Þ ð17Þ ð18Þ
Substituting Eqs. (16)–(18) into Eq. (3), and using Mathematica, equating to zero the coefficients of all powers of exp(jgn) (j = 0, ±1, ±2, . . .) yields a set of algebraic equations for a1, a0, a1, b0, b1 and c. Solving the system of algebraic equations by use of Mathematica, we obtain the following results: Case 1 2
a1 ¼ 0; b1 ¼
a0 ¼
b0 a½1 þ expðdÞ ; b expðdÞ
a1 ¼ 0;
b20 fac½1 þ expð2dÞ þ expðdÞðb2 2acÞg b2 ½1 þ expðdÞ
2
b0 ¼ b0 ; c¼
;
ð19Þ
a½1 expð2dÞ : expðdÞ
ð20Þ
Case 2 a1 ¼
b½1 þ expðdÞ ½1 þ expðdÞ b0
2c½1 þ expðdÞ
ð21Þ
;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b½1 þ expðdÞ ½1 þ expðdÞ b2 4ac
a0 ¼ b0 ¼ b 0 ;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac
;
2c½1 þ expðdÞ b1 ¼ 0;
c¼
½1 þ expðdÞðb2 4acÞ : c½1 þ expðdÞ
a1 ¼ 0;
ð22Þ ð23Þ
Case 3 a1 ¼
b½1 þ expðdÞ ½1 þ expðdÞ 2c½1 þ expðdÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac
;
ð24Þ
S. Zhang / Applied Mathematics and Computation 199 (2008) 242–249
b1 a0 ¼ 0;
a1 ¼
b0 ¼ 0;
b1 ¼ b1 ;
245
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b½1 þ expð2dÞ ½1 þ expð2dÞ b2 4ac
c¼
ð25Þ
;
2c½1 þ expð2dÞ ½1 þ expð2dÞðb2 4acÞ : 2c½1 þ expð2dÞ
ð26Þ
Case 4 a1 ¼
b½1 þ expðdÞ ½1 þ expðdÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac
; a0 ¼ a0 ; 2c½1 þ expðdÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a0 c½1 þ expðdÞ b½1 þ expðdÞ ½1 þ expðdÞ b2 4ac
a1 ¼
b1 ¼
a20 c2 ½1 þ expðdÞ
2
2
½1 þ expðdÞ ðb2 4acÞ
ð28Þ
;
2c½1 þ expðdÞ2
b0 ¼ 0;
ð27Þ
c¼
;
½1 þ expðdÞðb2 4acÞ : c½1 þ expðdÞ
ð29Þ
Case 5 a1 ¼
b ; 2c
b0 ¼ 0;
a0 ¼ a0 ;
b1 ¼
a1 ¼
2a20 bc expð2dÞ 2
½1 þ expð2dÞ ðb2 4acÞ
4a20 c2 expð2dÞ ½1 þ expð2dÞ2 ðb2 4acÞ
;
c¼
ð30Þ
;
½1 þ expð2dÞðb2 4acÞ : 4c expðdÞ
ð31Þ
Case 6 a1 ¼ a1 ; a1 ¼ b1 ¼
a0 ¼
b0 fa½1 þ expðdÞ2 þ a1 b½1 expðdÞ þ expð2dÞ þ a21 c½1 þ expð2dÞg ; expðdÞðb þ 2a1 cÞ
ð32Þ
a1 b20 fc½a þ a1 ðb þ a1 cÞ þ c expð2dÞ½a þ a1 ðb þ a1 cÞ þ expðdÞ½b2 þ 2a1 bc þ 2cða þ a21 cÞg 2
½1 þ expðdÞ ðb þ 2a1 cÞ
2
b20 fc½a þ a1 ðb þ a1 cÞ þ c expð2dÞ½a þ a1 ðb þ a1 cÞ þ expðdÞ½b2 þ 2a1 bc þ 2cða þ a21 cÞg
b0 ¼ b 0 ;
2
½1 þ expðdÞ ðb þ 2a1 cÞ c¼
2
½1 þ expð2dÞ½a þ a1 ðb þ a1 cÞ : expðdÞ
;
;
ð33Þ ð34Þ ð35Þ
Substituting Cases 1–6 into Eq. (16) respectively, and using Eq. (2), we obtain the following generalized solitonary solutions of Eq. (1): b0 a½1þexpðdÞ2 b expðdÞ
un ¼ expðgn Þ þ b0 þ
b20 fac½1þexpð2dÞþexpðdÞðb2 2acÞg 2
b ½1þexpðdÞ
t þ g0 . where gn ¼ n d þ a½1expð2dÞ expðdÞ pffiffiffiffiffiffiffiffiffiffi ffi 2 un ¼
b½1þexpðdÞ½1þexpðdÞ 2c½1þexpðdÞ
b 4ac
2
expðgn Þ
b0 b½1þexpðdÞ½1þexpðdÞ
expðgn Þ þ b0 2
4acÞ where gn ¼ n d þ ½1þexpðdÞðb t þ g0 . c½1þexpðdÞ
ð36Þ
; expðgn Þ
2c½1þexpðdÞ
pffiffiffiffiffiffiffiffiffiffi ffi 2 b 4ac
;
ð37Þ
246
S. Zhang / Applied Mathematics and Computation 199 (2008) 242–249
un ¼
b½1þexpðdÞ½1þexpðdÞ 2c½1þexpðdÞ
pffiffiffiffiffiffiffiffiffiffi ffi 2 b 4ac
expðgn Þ
b1 b½1þexpð2dÞ½1þexpð2dÞ
pffiffiffiffiffiffiffiffiffiffi ffi 2 b 4ac
2c½1þexpð2dÞ
expðgn Þ
expðgn Þ þ b1 expðgn Þ
ð38Þ
;
2
4acÞ where gn ¼ n d þ ½1þexpð2dÞðb t þ g0 . 2c½1þexpð2dÞ
un ¼
b½1þexpðdÞ½1þexpðdÞ 2c½1þexpðdÞ
pffiffiffiffiffiffiffiffiffiffi ffi 2 b 4ac
expðgn Þ þ a0 þ
a20 c½1þexpðdÞ b½1þexpðdÞ½1þexpðdÞ
pffiffiffiffiffiffiffiffiffiffi ffi 2 b 4ac
2c½1þexpðdÞ2
expðgn Þ
a20 c2 ½1þexpðdÞ2 2 2
expðgn Þ ½1þexpðdÞ
ðb
;
ð39Þ
expðgn Þ 4acÞ
2
4acÞ t þ g0 . where gn ¼ n d þ ½1þexpðdÞðb c½1þexpðdÞ 2a2 bc expð2dÞ
un ¼
0 2cb expðgn Þ þ a0 þ ½1þexpð2dÞ expðgn Þ 2 2 ðb 4acÞ
4a2 c2 expð2dÞ
ð40Þ
;
0 expðgn Þ expðgn Þ ½1þexpð2dÞ 2 2 ðb 4acÞ
2
4acÞ t þ g0 . where gn ¼ n d þ ½1þexpð2dÞðb 4c expðdÞ
un ¼
a1 expðgn Þ þ expðgn Þ þ b0 þ
where gn ¼ n d
b0 fa½1þexpðdÞ2 þa1 b½1expðdÞþexpð2dÞþa21 c½1þexpð2dÞg expðdÞðbþ2a1 cÞ
b20 fc½aþa1 ðbþa1 cÞþc
þ a1 expðgn Þ
2
expð2dÞ½aþa1 ðbþa1 cÞþexpðdÞ½b þ2a1 bcþ2cðaþa21 cÞg ½1þexpðdÞ2 ðbþ2a1 cÞ2
½1þexpð2dÞðaþa1 ðbþa1 cÞÞ t expðdÞ
;
ð41Þ
expðgn Þ
þ g0 , a1 is determined in Eq. (33).
To our knowledge, above obtained generalized solitonary solutions are new, they have not been reported in literature yet. Furthermore, if we set the parameters in above generalized solitonary solutions as special values, some known solitary wave solutions can be recovered. For example, setting b0 = 1 in Eq. (37) we obtain b un ¼ 2c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac 2c
g d tanh tanh n ; 2 2
2 tanh d2 t þ g0 . where gn ¼ n d þ b 4ac c Setting b0 = 1 in Eq. (37) we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g b2 4ac b d un ¼ tanh coth n 2c 2 2 2c or
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g g b2 4ac d d n tanh tanh tanh coth n ; 2 2 4 4 4c 4c d b2 4ac where gn ¼ n d þ c tanh 2 t þ g0 . pffiffiffiffiffiffiffiffiffiffiffi b2 4ac½1þexpð2dÞ Setting a0 ¼ in Eq. (39) we have c½1þexpðdÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g g b2 4ac b2 4ac b d d n tanh tanh un ¼ coth csch n ; 2c 2 2 2 2 2c 2c d b2 4ac where gn ¼ n d þ c tanh 2 t þ g0 . b un ¼ 2c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac
ð42Þ
ð43Þ
ð44Þ
ð45Þ
S. Zhang / Applied Mathematics and Computation 199 (2008) 242–249
Setting a0 ¼
i
pffiffiffiffiffiffiffiffiffiffi ffi 2
b 4ac½1þexpð2dÞ c½1þexpðdÞ
247
in Eq. (39) we have
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g b2 4ac b d g d tanh tanh tanh n i sech n ; un ¼ 2c 2 2 2c 2c 2 2 d b2 4ac where gn ¼ n d þ c tanh 2 t þ g0 . pffiffiffiffiffiffiffiffiffiffiffi b2 4ac½1þexpð2dÞ Setting a0 ¼ in Eq. (40) we get 2c expðdÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g g b2 4ac b2 4ac b sinhðdÞ tanh n sinhðdÞ coth n ; un ¼ 2c 2 2 4c 4c
ð46Þ
ð47Þ
or un ¼
b 2c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac 2c
sinhðdÞcschðgn Þ;
ð48Þ
2
where gn ¼ n d þ b 4ac sinhðdÞt þ g0 . 2c pffiffiffiffiffiffiffiffiffiffi ffi i b2 4ac½1þexpð2dÞ Setting a0 ¼ in Eq. (40) we get 2c expðdÞ b un ¼ i 2c
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 4ac 2c
sinhðdÞsechðgn Þ;
ð49Þ
2
where gn ¼ n d þ b 4ac sinhðdÞt þ g0 . 2c It is easy to see that above solutions (42)–(49) contain all the solutions obtained from Cases 1–8 in [59]. From Cases 1–6 obtained in this paper we can also derive some other solitary wave solutions, we omit them here for simplicity. 3. Conclusion In this paper, the Exp-function method has been successfully used to obtain explicit and exact solutions of a lattice equation. As a result, some new generalized solitonary solutions with parameters are obtained. To our knowledge, these solutions have not been reported. It may be important to explain some physical phenomena. By setting the parameters as special values, some known solitary wave solutions reported in open literature are derived. It is shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful new method for some discrete nonlinear evolution equations in mathematical physics. Acknowledgements I would like to express my sincere thanks to referee for the valuable advice. This work was supported by the Natural Science Foundation of Educational Committee of Liaoning Province of China under Grant No. 2006022. References [1] M.J. Ablowitz, P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, New York, 1991. [2] R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971) 1192–1194. [3] M.R. Miurs, Backlund Transformation, Springer, Berlin, 1978. [4] J. Weiss, M. Tabor, G. Carnevale, The Painleve´ property for partial differential equations, J. Math. Phys. 24 (1983) 522–526. [5] C.T. Yan, A simple transformation for nonlinear waves, Phys. Lett. A 224 (1996) 77–84. [6] M.L. Wang, Exact solutions for a compound KdV–Burgers equations, Phys. Lett. A 213 (1996) 279–287.
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