- Email: [email protected]
Some solutions of discrete sine-Gordon equation
Chaos, Solitons and Fractals 33 (2007) 1791–1795 www.elsevier.com/locate/chaos
Some solutions of discrete sine-Gordon equation Fuding Xie a
a,b,*
, Min Ji a, Hong Zhao
a
Department of Computer Science, Liaoning Normal University, Liaoning Dalian 116029, PR China Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing 100080, PR China
b
Accepted 7 March 2006
Communicated by Prof. M. Wadati
Abstract In this paper, a series of exact solutions of discrete sine-Gordon equation are obtained by the different transformations and symbolic computation. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction It is well known that the investigation of nonlinear differential–difference equations which describe many important phenomena and dynamical processes in many different fields, such as particle vibrations in lattices, currents in electrical networks, pulses in biological chains and so on, has played an important role in the study of modern physics. In recent years, remarkable progress has been achieved in the field of discrete integrable systems and associated difference equations [1–10]. The discrete sine-Gordon (DSG for short) equation occurs in many derivations of physical processes, and has consequently inspired many numerical simulations. The passage to the continuum sine-Gordon equation is not always a good approximation. In some applications it is the propagation of travelling waves that is of main interest [11] (and references therein). Hence it is need to develop methods to find the exact travelling wave solutions in systems where discreteness is left in the mathematical model. In general, it is difficult to find or give a method to solve DSG equation since the DSG equation possesses strong nonlinearity. Most recently, two different transformations are used to find exact travelling wave solutions of sine-Gordon equations in continuous case [12,13]. Inspired by their work and combined our previous ideas [14–16], in this paper, we develop two methods to solve DSG equation. As a result, a series of the exact travelling wave solutions for DSG equation have been obtained.
* Corresponding author. Address: Department of Computer Science, Liaoning Normal University, Liaoning Dalian 116029, PR China. E-mail address: [email protected] (F. Xie).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.018
1792
F. Xie et al. / Chaos, Solitons and Fractals 33 (2007) 1791–1795
The DSG equation we are interesting in is in the following form [1,17,18]: dunþ1 ðtÞ dun ðtÞ ¼ sin½unþ1 ðtÞ þ un ðtÞ; dt dt
ð1Þ
where un(t) = u(n,t) is the displacement of the nth particle from the equilibrium position. Eq. (1) may be thought as a discrete version of a nonlinear sine-Gordon equation in continuous case. For the investigation of Eq. (1), we recommend the reader to consult with Refs. [17,18].
2. Solutions to sine-Gordon equation For convenience, in what follows, we denote u(n, t), v(n, t), v(n + 1, t) and v(n 1, t) by un, vn, vn+1 and vn1, respectively. Looking for the travelling wave solutions of permanent profile in a moving reference frame with speed b, one can write un ¼ un ðtÞ ¼ uðn; tÞ ¼ uðnn Þ;
nn ¼ an bt;
where a and b are constants to be determined. We first make a transformation [13] pffiffiffiffiffiffiffi vn ¼ eiun ; i ¼ 1; or equivalently, 1 un ¼ ln vn ; i from which we have sinðun Þ ¼
vn v1 n ; 2i
that also gives un ¼ arccos
ð2Þ
ð3Þ
ð4Þ
cosðun Þ ¼
vn þ v1 n ; 2
vn þ v1 n . 2
ð5Þ
ð6Þ
Substituting the transformations introduced above into Eq. (1) gives 1 1 vnþ1 v1 vnþ1 þ v1 v_ nþ1 v_ n nþ1 vn þ vn nþ1 vn vn þ ; ¼ b vnþ1 vn 2 2 2 2 where the symbol Æ represents differentiation with respect to nj. Now, we suppose that Eq. (7) has the formal solution a1 ; vn ¼ a0 þ a2 þ /n
ð7Þ
ð8Þ
where /n = /(nn) satisfies the following Riccati equation d/n ¼ 1 þ d/2n . dt
ð9Þ
It is known that Riccati equation (9) possesses the solutions tanhðnn Þ; cothðnn Þ; d ¼ 1; /ðnn Þ ¼ tanðnn Þ; d ¼ 1:
ð10Þ
and the following rule is true /ðnn1 Þ ¼
/ðnn Þ þ f ðaÞ ; 1 þ l/ðnn Þf ðaÞ
where l = ±1 and tanhðaÞ; f ðaÞ ¼ tanðaÞ;
l ¼ 1; d ¼ 1; l ¼ 1; d ¼ 1:
ð11Þ
ð12Þ
F. Xie et al. / Chaos, Solitons and Fractals 33 (2007) 1791–1795
1793
We substitute Eq. (8) into Eq. (7) and employ the rule Eqs. (9) and (11); clearing the denominator and setting the coefficients of power terms in /(nn) to zero, gives a nonlinear algebraic system. Using the symbolic computation software Maple, we obtain When d = 1 and l = 1, we have Case Case Case Case Case Case
1. 2. 3. 4. 5. 6.
a2 = tanh(a) ± i sech(a), a1 = 2[±i sech2(a) tanh(a) sech(a)], a0 = [sech(a) ± i tanh(a)], b ¼ 12 cothðaÞ; a2 = tanh(a) ± i sech(a), a1 = 2[±i sech2(a) tanh(a) sech(a)], a0 = sech(a) ± i tanh(a), b ¼ 12 cothðaÞ; a2 = tanh(a) ± i sech(a), a1 = 2[sech2(a) + i tanh(a) sech(a)], a0 = i sech(a) ± tanh(a), b ¼ 12 cothðaÞ; a2 = tanh(a) ± i sech(a), a1 = 2[±sech2(a) + i tanh(a) sech(a)], a0 = i sech(a) tanh(a), b ¼ 12 cothðaÞ; a0 = a2 = 0, a1 = ±1, b ¼ 12 cothðaÞ; a0 = a2 = 0, a1 = ±i, b ¼ 12 cothðaÞ;
When d = 1 and l = 1, we also have Case 7. Case 8. Case 9. Case 10. Case 11. Case 12.
a2 = tan(a) ± sec(a), a1 = 2[±sec2(a) tan(a) sec(a)], a0 = sec(a) ± tan(a), b ¼ 12 cotðaÞ; a2 = tan(a) ± sec(a), a1 = 2[sec2(a) tan(a) sec(a)], a0 = sec(a) tan(a), b ¼ 12 cotðaÞ; a2 = tan(a) ± sec(a), a1 = 2i[sec2(a) + tan(a) sec(a)], a0 = i[sec(a) tan(a)], b ¼ 12 cotðaÞ; a2 = tan(a) ± sec(a), a1 = 2i[±sec2(a) tan(a) sec(a)], a0 = i[sec(a) ± tan(a)], b ¼ 12 cotðaÞ; a0 = a2 = 0, a1 = ±1, b ¼ 12 cotðaÞ; a0 = a2 = 0, a1 = ±i, b ¼ 12 cotðaÞ;
where a is left as a free parameter in all above cases. From Eqs. (8), (10) and above cases, we therefore obtain the solutions of Eq. (7) vðn; tÞ1 ¼ ½sechðaÞ i tanhðaÞ þ vðn; tÞ2 ¼ sechðaÞ i tanhðaÞ
2½i sech2 ðaÞ tanhðaÞ sechðaÞ ; tanhðaÞ i sechðaÞ þ tanh an 12 cothðaÞt
2½i sech2 ðaÞ tanhðaÞ sechðaÞ ; tanhðaÞ i sechðaÞ þ tanh an 12 cothðaÞt
vðn; tÞ3 ¼ i sechðaÞ tanhðaÞ
2½sech2 ðaÞ þ i tanhðaÞ sechðaÞ ; tanhðaÞ i sechðaÞ þ tanh an þ 12 cothðaÞt
2½sech2 ðaÞ þ i tanhðaÞ sechðaÞ ; tanhðaÞ i sechðaÞ þ tanh an þ 12 cothðaÞt 1 vðn; tÞ5 ¼ tanh an þ cothðaÞt ; 2 1 vðn; tÞ6 ¼ i tanh an cothðaÞt . 2 vðn; tÞ4 ¼ i sechðaÞ tanhðaÞ þ
Note: We can also use coth(nn) to replace tanh(nn) in v(n, t)k (k = 1, . . . , 6). vðn; tÞ7 ¼ secðaÞ tanðaÞ þ
2½sec2 ðaÞ tanðaÞ secðaÞ ; tanðaÞ secðaÞ þ tan an þ 12 cotðaÞt
vðn; tÞ8 ¼ secðaÞ tanðaÞ þ
2½sec2 ðaÞ tanðaÞ secðaÞ ; tanðaÞ secðaÞ þ tan an þ 12 cotðaÞt
vðn; tÞ9 ¼ i½secðaÞ tanðaÞ þ
2i½sec2 ðaÞ þ tanðaÞ secðaÞ ; tanðaÞ secðaÞ þ tan an 12 cotðaÞt
2i½sec2 ðaÞ tanðaÞ secðaÞ ; tanðaÞ secðaÞ þ tan an 12 cotðaÞt 1 vðn; tÞ11 ¼ cot an þ cotðaÞt ; 2 1 vðn; tÞ12 ¼ icot an cotðaÞt . 2 h 1 i vn þvn Recall that un ¼ arccos 2 from Eq. (6), we can easily get the solutions of the DSG equation (1). vðn; tÞ10 ¼ i½secðaÞ tanðaÞ þ
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F. Xie et al. / Chaos, Solitons and Fractals 33 (2007) 1791–1795
The second transformation is introduced in the form [12]
u n un ¼ 2arctanðvn Þ; or vn ¼ tan ; 2
ð13Þ
and then sinðun Þ ¼
2tanðun =2Þ 2vn ; ¼ 1 þ tan2 ðun =2Þ 1 þ v2n
cosðvn Þ ¼
1 v2n . 1 þ v2n
Combining Eqs. (13) and (14) with Eq. (1), the DSG Eq. (1) becomes, ! 2_vnþ1 ð1 v2nþ1 Þ 2_vn ð1 v2n Þ 2vnþ1 1 v2n 1 v2nþ1 2vn ¼ þ . b 2 2 2 2 1 þ v2nþ1 1 þ v2n 1 þ v2nþ1 1 þ v2n ð1 þ vnþ1 Þ ð1 þ vn Þ
ð14Þ
ð15Þ
In the travelling frame Eq. (2), we also assume that Eq. (15) has solutions as Eq. (8). Similarly, the coefficients in Eq. (8) can be determined as Case I. Case II. Case III. Case IV. Case V. Case VI.
a2 = ±1, a1 = 2[±cosh(a) + sinh(a)], a0 = cosh(a) ± sinh(a), b ¼ 12 cothðaÞ; a2 = ±1, a1 = 2[±cosh(a) + sinh(a)], a0 = cosh(a) sinh(a), b ¼ 12 cothðaÞ; a0 = a2 = 0, a1 = ±1, b ¼ 12 cothðaÞ; a2 = ±i, a1 = 2[ i cos(a) sin(a)], a0 = cos(a) i sin(a), b ¼ 12 cotðaÞ; a2 = ±i, a1 = 2[± i cos(a) + sin(a)], a0 = cos(a) ± i sin(a), b ¼ 12 cotðaÞ; a0 = a2 = 0, a1 = ±i, b ¼ 12 cotðaÞ.
Thus we can obtain nine families solutions to the DSG equation: ( ) 2½coshðaÞ þ sinhðaÞ ; uðn; tÞ1 ¼ 2 arctan coshðaÞ sinhðaÞ 1 þ tanhðan þ 12 cothðaÞtÞ ( ) 2½coshðaÞ þ sinhðaÞ ; uðn; tÞ2 ¼ 2 arctan coshðaÞ sinhðaÞ þ 1 þ tanhðan þ 12 cothðaÞtÞ 1 ; uðn; tÞ3 ¼ 2 arctan tanh an cothðaÞt 2 ( ) 2½coshðaÞ þ sinhðaÞ ; uðn; tÞ4 ¼ 2 arctan coshðaÞ sinhðaÞ 1 þ cothðan þ 12 cothðaÞtÞ ( ) 2½coshðaÞ þ sinhðaÞ ; uðn; tÞ5 ¼ 2 arctan coshðaÞ sinhðaÞ þ 1 þ cothðan þ 12 cothðaÞtÞ 1 uðn; tÞ6 ¼ 2 arctan coth an cothðaÞt ; 2 ( ) 2½icosðaÞ sinðaÞ ; uðn; tÞ7 ¼ 2 arctan cosðaÞ i sinðaÞ þ i þ tanðan 12 cotðaÞtÞ ( ) 2½icosðaÞ þ sinðaÞ uðn; tÞ8 ¼ 2 arctan cosðaÞ i sinðaÞ þ ; i þ tanðan 12 cotðaÞtÞ 1 1 ¼ 2i arctanh cot an þ cotðaÞt . uðn; tÞ9 ¼ 2 arctan i cot an þ cotðaÞt 2 2
3. Discussion and conclusions We should note that dHnþ1 ðtÞ dHn ðtÞ ¼ sin½Hnþ1 ðtÞ þ sin½Hn ðtÞ dt dt
ð16Þ
F. Xie et al. / Chaos, Solitons and Fractals 33 (2007) 1791–1795
1795
is an equivalent form of Eq. (1) under the transformation Hn ¼ unþ1 þ un .
ð17Þ
Therefore the solutions of Eq. (17) can obviously be obtained by the transformation (17). In this article, the discrete sine-Gordon equation is solved by two different transformations and the method based on Riccati equation expansion. Many solutions to the DSG equation have obtained. It reveals that different transformations lead to different form solutions for the DSG equation. Thus it is natural to ask whether or not there exists the other form solutions. The solve procedure also shows that it does not allow us to use the leading term analysis approach which is a valid method to investigate the nonlinear differential difference equation in polynomial style. Does there exist a relative systemic method to deal with this equation? It is worthwhile to further investigate these questions in the future.
Acknowledgements This work was supported by ‘‘973’’ Project (2004CB318000). This work was also supported by Doctor Start-up Foundation of Liaoning Province (20041066) and Science Research Plan of Liaoning Education Bureau (2004F099).
References [1] Ablowitz MJ, Clarkson PA. Soliton, nonlinear evolution equations and inverse scattering. New York: Cambridge University Press; 1991. [2] Toda M, Wadati M. J Phys Soc Jpn 1975;39:1204; Toda M. Theory of nonlinear lattices. Berlin, Heidelberg: Springer-Verlag; 1989. [3] Wadati M, Toda M. J Phys Soc Jpn 1975;39:1196; Wadati M. Prog Theor Phys Suppl 1976;59:36; Wadati M. J Phys Soc Jpn 1975;38:673; Wadati M, Watanabe M. Prog Theor Phys 1977;57:808. [4] Shabat AB, Yamilov RI. Phys Lett A 1997;227:15. ¨ , Hereman W. Comput Phys Commun 2004;162:203. [5] Baldwin D, Go¨ktas U [6] Levi D, Yamilov RI. J Math Phys 1997;38:6648. [7] Adler VE, Shabat AB, Yamilov RI. Theor Math Phys 2000;125:1603. [8] Cherdantsev IYu, Yamilov RI. Phys D 1995;87:140. [9] Hu XB, Ma WX. Phys Lett A 2002;293:161. [10] Ablowtiz MJ, Ladik JF. J Math Phys 1975;16:598. [11] Wattis J. Physica D 1995;82:333. [12] Liu SK, Fu Z, Liu SD. Phys Lett A 2006;351:59. [13] Wazwaz AM. Phys Lett A 2006;350:367. [14] Xie FD, Wang JQ. Chaos, Solitons & Fractals 2006;27:1067. [15] Xie FD. Commun Theor Phys (Beijing, China) 2005;44:293. [16] Xie FD, Ji M, Gong L. Commun Theor Phys (Beijing, China) 2006;45:36. [17] Pilloni L, Levi D. Phys Lett 1982;92A:5. [18] Orfanidis SJ. Phys Rev 1978;18D:3828.
Some solutions of discrete sine-Gordon equation Fuding Xie a
a,b,*
, Min Ji a, Hong Zhao
a
Department of Computer Science, Liaoning Normal University, Liaoning Dalian 116029, PR China Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing 100080, PR China
b
Accepted 7 March 2006
Communicated by Prof. M. Wadati
Abstract In this paper, a series of exact solutions of discrete sine-Gordon equation are obtained by the different transformations and symbolic computation. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction It is well known that the investigation of nonlinear differential–difference equations which describe many important phenomena and dynamical processes in many different fields, such as particle vibrations in lattices, currents in electrical networks, pulses in biological chains and so on, has played an important role in the study of modern physics. In recent years, remarkable progress has been achieved in the field of discrete integrable systems and associated difference equations [1–10]. The discrete sine-Gordon (DSG for short) equation occurs in many derivations of physical processes, and has consequently inspired many numerical simulations. The passage to the continuum sine-Gordon equation is not always a good approximation. In some applications it is the propagation of travelling waves that is of main interest [11] (and references therein). Hence it is need to develop methods to find the exact travelling wave solutions in systems where discreteness is left in the mathematical model. In general, it is difficult to find or give a method to solve DSG equation since the DSG equation possesses strong nonlinearity. Most recently, two different transformations are used to find exact travelling wave solutions of sine-Gordon equations in continuous case [12,13]. Inspired by their work and combined our previous ideas [14–16], in this paper, we develop two methods to solve DSG equation. As a result, a series of the exact travelling wave solutions for DSG equation have been obtained.
* Corresponding author. Address: Department of Computer Science, Liaoning Normal University, Liaoning Dalian 116029, PR China. E-mail address: [email protected] (F. Xie).
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.03.018
1792
F. Xie et al. / Chaos, Solitons and Fractals 33 (2007) 1791–1795
The DSG equation we are interesting in is in the following form [1,17,18]: dunþ1 ðtÞ dun ðtÞ ¼ sin½unþ1 ðtÞ þ un ðtÞ; dt dt
ð1Þ
where un(t) = u(n,t) is the displacement of the nth particle from the equilibrium position. Eq. (1) may be thought as a discrete version of a nonlinear sine-Gordon equation in continuous case. For the investigation of Eq. (1), we recommend the reader to consult with Refs. [17,18].
2. Solutions to sine-Gordon equation For convenience, in what follows, we denote u(n, t), v(n, t), v(n + 1, t) and v(n 1, t) by un, vn, vn+1 and vn1, respectively. Looking for the travelling wave solutions of permanent profile in a moving reference frame with speed b, one can write un ¼ un ðtÞ ¼ uðn; tÞ ¼ uðnn Þ;
nn ¼ an bt;
where a and b are constants to be determined. We first make a transformation [13] pffiffiffiffiffiffiffi vn ¼ eiun ; i ¼ 1; or equivalently, 1 un ¼ ln vn ; i from which we have sinðun Þ ¼
vn v1 n ; 2i
that also gives un ¼ arccos
ð2Þ
ð3Þ
ð4Þ
cosðun Þ ¼
vn þ v1 n ; 2
vn þ v1 n . 2
ð5Þ
ð6Þ
Substituting the transformations introduced above into Eq. (1) gives 1 1 vnþ1 v1 vnþ1 þ v1 v_ nþ1 v_ n nþ1 vn þ vn nþ1 vn vn þ ; ¼ b vnþ1 vn 2 2 2 2 where the symbol Æ represents differentiation with respect to nj. Now, we suppose that Eq. (7) has the formal solution a1 ; vn ¼ a0 þ a2 þ /n
ð7Þ
ð8Þ
where /n = /(nn) satisfies the following Riccati equation d/n ¼ 1 þ d/2n . dt
ð9Þ
It is known that Riccati equation (9) possesses the solutions tanhðnn Þ; cothðnn Þ; d ¼ 1; /ðnn Þ ¼ tanðnn Þ; d ¼ 1:
ð10Þ
and the following rule is true /ðnn1 Þ ¼
/ðnn Þ þ f ðaÞ ; 1 þ l/ðnn Þf ðaÞ
where l = ±1 and tanhðaÞ; f ðaÞ ¼ tanðaÞ;
l ¼ 1; d ¼ 1; l ¼ 1; d ¼ 1:
ð11Þ
ð12Þ
F. Xie et al. / Chaos, Solitons and Fractals 33 (2007) 1791–1795
1793
We substitute Eq. (8) into Eq. (7) and employ the rule Eqs. (9) and (11); clearing the denominator and setting the coefficients of power terms in /(nn) to zero, gives a nonlinear algebraic system. Using the symbolic computation software Maple, we obtain When d = 1 and l = 1, we have Case Case Case Case Case Case
1. 2. 3. 4. 5. 6.
a2 = tanh(a) ± i sech(a), a1 = 2[±i sech2(a) tanh(a) sech(a)], a0 = [sech(a) ± i tanh(a)], b ¼ 12 cothðaÞ; a2 = tanh(a) ± i sech(a), a1 = 2[±i sech2(a) tanh(a) sech(a)], a0 = sech(a) ± i tanh(a), b ¼ 12 cothðaÞ; a2 = tanh(a) ± i sech(a), a1 = 2[sech2(a) + i tanh(a) sech(a)], a0 = i sech(a) ± tanh(a), b ¼ 12 cothðaÞ; a2 = tanh(a) ± i sech(a), a1 = 2[±sech2(a) + i tanh(a) sech(a)], a0 = i sech(a) tanh(a), b ¼ 12 cothðaÞ; a0 = a2 = 0, a1 = ±1, b ¼ 12 cothðaÞ; a0 = a2 = 0, a1 = ±i, b ¼ 12 cothðaÞ;
When d = 1 and l = 1, we also have Case 7. Case 8. Case 9. Case 10. Case 11. Case 12.
a2 = tan(a) ± sec(a), a1 = 2[±sec2(a) tan(a) sec(a)], a0 = sec(a) ± tan(a), b ¼ 12 cotðaÞ; a2 = tan(a) ± sec(a), a1 = 2[sec2(a) tan(a) sec(a)], a0 = sec(a) tan(a), b ¼ 12 cotðaÞ; a2 = tan(a) ± sec(a), a1 = 2i[sec2(a) + tan(a) sec(a)], a0 = i[sec(a) tan(a)], b ¼ 12 cotðaÞ; a2 = tan(a) ± sec(a), a1 = 2i[±sec2(a) tan(a) sec(a)], a0 = i[sec(a) ± tan(a)], b ¼ 12 cotðaÞ; a0 = a2 = 0, a1 = ±1, b ¼ 12 cotðaÞ; a0 = a2 = 0, a1 = ±i, b ¼ 12 cotðaÞ;
where a is left as a free parameter in all above cases. From Eqs. (8), (10) and above cases, we therefore obtain the solutions of Eq. (7) vðn; tÞ1 ¼ ½sechðaÞ i tanhðaÞ þ vðn; tÞ2 ¼ sechðaÞ i tanhðaÞ
2½i sech2 ðaÞ tanhðaÞ sechðaÞ ; tanhðaÞ i sechðaÞ þ tanh an 12 cothðaÞt
2½i sech2 ðaÞ tanhðaÞ sechðaÞ ; tanhðaÞ i sechðaÞ þ tanh an 12 cothðaÞt
vðn; tÞ3 ¼ i sechðaÞ tanhðaÞ
2½sech2 ðaÞ þ i tanhðaÞ sechðaÞ ; tanhðaÞ i sechðaÞ þ tanh an þ 12 cothðaÞt
2½sech2 ðaÞ þ i tanhðaÞ sechðaÞ ; tanhðaÞ i sechðaÞ þ tanh an þ 12 cothðaÞt 1 vðn; tÞ5 ¼ tanh an þ cothðaÞt ; 2 1 vðn; tÞ6 ¼ i tanh an cothðaÞt . 2 vðn; tÞ4 ¼ i sechðaÞ tanhðaÞ þ
Note: We can also use coth(nn) to replace tanh(nn) in v(n, t)k (k = 1, . . . , 6). vðn; tÞ7 ¼ secðaÞ tanðaÞ þ
2½sec2 ðaÞ tanðaÞ secðaÞ ; tanðaÞ secðaÞ þ tan an þ 12 cotðaÞt
vðn; tÞ8 ¼ secðaÞ tanðaÞ þ
2½sec2 ðaÞ tanðaÞ secðaÞ ; tanðaÞ secðaÞ þ tan an þ 12 cotðaÞt
vðn; tÞ9 ¼ i½secðaÞ tanðaÞ þ
2i½sec2 ðaÞ þ tanðaÞ secðaÞ ; tanðaÞ secðaÞ þ tan an 12 cotðaÞt
2i½sec2 ðaÞ tanðaÞ secðaÞ ; tanðaÞ secðaÞ þ tan an 12 cotðaÞt 1 vðn; tÞ11 ¼ cot an þ cotðaÞt ; 2 1 vðn; tÞ12 ¼ icot an cotðaÞt . 2 h 1 i vn þvn Recall that un ¼ arccos 2 from Eq. (6), we can easily get the solutions of the DSG equation (1). vðn; tÞ10 ¼ i½secðaÞ tanðaÞ þ
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F. Xie et al. / Chaos, Solitons and Fractals 33 (2007) 1791–1795
The second transformation is introduced in the form [12]
u n un ¼ 2arctanðvn Þ; or vn ¼ tan ; 2
ð13Þ
and then sinðun Þ ¼
2tanðun =2Þ 2vn ; ¼ 1 þ tan2 ðun =2Þ 1 þ v2n
cosðvn Þ ¼
1 v2n . 1 þ v2n
Combining Eqs. (13) and (14) with Eq. (1), the DSG Eq. (1) becomes, ! 2_vnþ1 ð1 v2nþ1 Þ 2_vn ð1 v2n Þ 2vnþ1 1 v2n 1 v2nþ1 2vn ¼ þ . b 2 2 2 2 1 þ v2nþ1 1 þ v2n 1 þ v2nþ1 1 þ v2n ð1 þ vnþ1 Þ ð1 þ vn Þ
ð14Þ
ð15Þ
In the travelling frame Eq. (2), we also assume that Eq. (15) has solutions as Eq. (8). Similarly, the coefficients in Eq. (8) can be determined as Case I. Case II. Case III. Case IV. Case V. Case VI.
a2 = ±1, a1 = 2[±cosh(a) + sinh(a)], a0 = cosh(a) ± sinh(a), b ¼ 12 cothðaÞ; a2 = ±1, a1 = 2[±cosh(a) + sinh(a)], a0 = cosh(a) sinh(a), b ¼ 12 cothðaÞ; a0 = a2 = 0, a1 = ±1, b ¼ 12 cothðaÞ; a2 = ±i, a1 = 2[ i cos(a) sin(a)], a0 = cos(a) i sin(a), b ¼ 12 cotðaÞ; a2 = ±i, a1 = 2[± i cos(a) + sin(a)], a0 = cos(a) ± i sin(a), b ¼ 12 cotðaÞ; a0 = a2 = 0, a1 = ±i, b ¼ 12 cotðaÞ.
Thus we can obtain nine families solutions to the DSG equation: ( ) 2½coshðaÞ þ sinhðaÞ ; uðn; tÞ1 ¼ 2 arctan coshðaÞ sinhðaÞ 1 þ tanhðan þ 12 cothðaÞtÞ ( ) 2½coshðaÞ þ sinhðaÞ ; uðn; tÞ2 ¼ 2 arctan coshðaÞ sinhðaÞ þ 1 þ tanhðan þ 12 cothðaÞtÞ 1 ; uðn; tÞ3 ¼ 2 arctan tanh an cothðaÞt 2 ( ) 2½coshðaÞ þ sinhðaÞ ; uðn; tÞ4 ¼ 2 arctan coshðaÞ sinhðaÞ 1 þ cothðan þ 12 cothðaÞtÞ ( ) 2½coshðaÞ þ sinhðaÞ ; uðn; tÞ5 ¼ 2 arctan coshðaÞ sinhðaÞ þ 1 þ cothðan þ 12 cothðaÞtÞ 1 uðn; tÞ6 ¼ 2 arctan coth an cothðaÞt ; 2 ( ) 2½icosðaÞ sinðaÞ ; uðn; tÞ7 ¼ 2 arctan cosðaÞ i sinðaÞ þ i þ tanðan 12 cotðaÞtÞ ( ) 2½icosðaÞ þ sinðaÞ uðn; tÞ8 ¼ 2 arctan cosðaÞ i sinðaÞ þ ; i þ tanðan 12 cotðaÞtÞ 1 1 ¼ 2i arctanh cot an þ cotðaÞt . uðn; tÞ9 ¼ 2 arctan i cot an þ cotðaÞt 2 2
3. Discussion and conclusions We should note that dHnþ1 ðtÞ dHn ðtÞ ¼ sin½Hnþ1 ðtÞ þ sin½Hn ðtÞ dt dt
ð16Þ
F. Xie et al. / Chaos, Solitons and Fractals 33 (2007) 1791–1795
1795
is an equivalent form of Eq. (1) under the transformation Hn ¼ unþ1 þ un .
ð17Þ
Therefore the solutions of Eq. (17) can obviously be obtained by the transformation (17). In this article, the discrete sine-Gordon equation is solved by two different transformations and the method based on Riccati equation expansion. Many solutions to the DSG equation have obtained. It reveals that different transformations lead to different form solutions for the DSG equation. Thus it is natural to ask whether or not there exists the other form solutions. The solve procedure also shows that it does not allow us to use the leading term analysis approach which is a valid method to investigate the nonlinear differential difference equation in polynomial style. Does there exist a relative systemic method to deal with this equation? It is worthwhile to further investigate these questions in the future.
Acknowledgements This work was supported by ‘‘973’’ Project (2004CB318000). This work was also supported by Doctor Start-up Foundation of Liaoning Province (20041066) and Science Research Plan of Liaoning Education Bureau (2004F099).
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