Discrete (2+1) -dimensional Toda lattice equation via Exp-function method

Physics Letters A 372 (2008) 654–657 www.elsevier.com/locate/pla

Discrete (2 + 1)-dimensional Toda lattice equation via Exp-function method Shun-dong Zhu Department of Science, Zhejiang Lishui University, Lishui 323000, PR China Received 24 May 2007; received in revised form 30 May 2007; accepted 25 July 2007 Available online 23 August 2007 Communicated by A.R. Bishop

Abstract In this Letter, we utilize the Exp-function method to construct two families of new generalized soliton solutions for the discrete (2 + 1)dimensional Toda lattice equation. Our solutions naturally include those in open literature as special cases. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving typical discrete nonlinear evolution equation in physics. © 2007 Elsevier B.V. All rights reserved. PACS: 03.40.Kf; 42.65.Tg Keywords: Discrete (2 + 1)-dimensional Toda lattice equation; Exp-function method; Generalized soliton solution

1. Introduction Seeking exact solutions of nonlinear evolution equations is of important significance in mathematical physics. In the past decades, several powerful methods have been developed to construct many types of exact solutions of nonlinear partial differential equations (PDEs), such as inverse scattering theory [1], Bäcklund transformation [2], the tanh function method [3,4], homogeneous balance method [5,6], multilinear variable separation approach [7,8], Jacobian elliptic function method [9,10], and homotopy perturbation method [11,12], variational iteration method [13], a heuristic review on recently developed analytical method is available in Refs. [14,15]. In recent years, the direct search for exact solutions of PDEs has become more and more attractive partly due to the availability of computer symbolic systems like Maple or Mathematica, which allows us to perform the complicated and tedious algebraic calculations on computer. In particular, one of the most effective direct methods to construct exact solutions of PDEs is the Exp-function method [16–20]. Since the work of Fermi, Pasta and Ulam in the 1950s [21], the investigation of exact solutions of the nonlinear differential– difference 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. One also encounters 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. However, different from the considerable works done on finding exact solutions to PDEs just mentioned above, to our knowledge, only a few works have investigated exact solutions for DDEs. Recently, using the Exp-function method, which was first presented by He [16], is proposed to seek solitary solutions, periodic solutions and compacton-like solutions of nonlinear differential equations. In this Letter, we further extend the Exp-function method for the nonlinear differential–difference equations (DDEs).

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2. The Exp-function method for DDEs The Exp-function method can be adapted to solve nonlinear polynomial DDEs. Given a system of DDEs ∂un = F (. . . , un−1 , un , un+1 , . . .) = 0, (1) ∂t where un = u(n, x, t), and F is assumed to be a polynomial with constant coefficients. The equations are continuous in time, and discretized in the (single) space variable. There are no restrictions on the level of shifts or the degree of nonlinearity. According to the Exp-function method [22,23]. Let us begin with the ansatz for DDEs as follows: f an exp(nξ ) , un (x, t, n) = un (ξ ) = qn=−e (2) m=−p bm exp(mξ ) f an exp(n(ξ − id)) un−i (x, t, n) = un−i (ξ ) = qn=−e (3) , m=−p bm exp(m(ξ − id)) f an exp(n(ξ + id)) un+i (x, t, n) = un+i (ξ ) = qn=−e (4) , b m=−p m exp(m(ξ + id)) where ξ = dn + c1 x + c2 t + ξ0 , i is a given integral number. The coefficients d, c1 , c2 and the phase ξ0 are constants to be determined, the integral number e, f , p, q is given according to the homogeneous balance principle, an and bm are unknown constants. Substituting Eqs. (2)–(4) into Eq. (1), clearing the denominator and setting the coefficients of power terms in exp(j ξ ), j = 1, 2, . . . , to zero, a system of nonlinear algebraic equations with exp(d) is obtained. From these equations, we can obtain the corresponding undetermined coefficients. Finally, a series of explicit exact solutions of the DDEs (1) is constructed. 3. Exact traveling wave solutions of the discrete (2 + 1)-dimensional Toda lattice equation Let us consider the discrete (2 + 1)-dimensional Toda lattice equation which reads [24] ∂ 2 yn = exp(yn−1 − yn ) − exp(yn − yn+1 ), (5) ∂x∂t where yn (x, t) is the displacement from equilibrium of the nth unit mass under an exponentially decaying interaction force between nearest neighbours. To write (5) as a polynomial DDE, we set ∂un = exp(yn−1 − yn ) − 1. ∂t Then (5) becomes   ∂ 2 un ∂un = + 1 (un−1 − 2un + un+1 ). ∂x∂t ∂t

(6)

(7)

Let un = un (ξn ), ξn = dn + c1 x + c2 t + ξ0 , then Eq. (7) becomes   c1 c2 un = c2 un + 1 (un−1 − 2un + un+1 ).

(8)

According to the homogeneous balance principle, which leads to the result e = p, f = q. For simplicity, we set e = p = 1 and f = q = 1, so Eqs. (2)–(4) reduce to a1 exp(ξn ) + a0 + a−1 exp(−ξn ) , b1 exp(ξn ) + b0 + exp(−ξn ) a1 exp(ξn − d) + a0 + a−1 exp(−ξn + d) , un−1 = b1 exp(ξn − d) + b0 + exp(−ξn + d) a1 exp(ξn + d) + a0 + a−1 exp(−ξn − d) . un+1 = b1 exp(ξn + d) + b0 + exp(−ξn − d) un =

(9) (10) (11)

Substituting Eqs. (9)–(11) into Eq. (8), and by the help of Maple, clearing the denominator and setting the coefficients of power terms like exp(j ξn ), j = 1, 2, . . . , to zero yields a system of algebraic equations, we obtain the following exact solutions: Case 1: a0 = 0,

b0 = 0,

c1 =

(exp(2d) − 1)2 , 4c2 exp(2d)

a−1 =

a1 (exp(2d) − 1)2 − , b1 2c2 exp(2d)

(12)

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S.-d. Zhu / Physics Letters A 372 (2008) 654–657

where a1 , b1 and c2 are arbitrary constants. According to (9) and (12), we can obtain un =

(exp(2d)−1)2 (exp(2d)−1)2 (exp(2d)−1)2 a1 4c2 exp(2d) x + c2 t + ξ0 ) + ( b1 − 2c2 exp(2d) ) exp(−nd − 4c2 exp(2d) x − c2 t 2 (exp(2d)−1)2 b1 exp(nd + (exp(2d)−1) 4c2 exp(2d) x + c2 t + ξ0 ) + exp(−nd − 4c2 exp(2d) x − c2 t − ξ0 )

a1 exp(nd +

− ξ0 )

(13)

,

which are new travelling wave solutions. Case 2: c1 =

(exp(d) − 1)2 , c2 exp(d)

a1 = b0 (a0 − a−1 b0 ) −

b1 =

b02 [c2 (a−1 b0 − a0 ) + 12 b0 (exp(d) − 1)2 ]2 , − 4 (exp(d) − 1)4

a02 c2 exp(d) a−1 c2 (a0 − a−1 b0 )[(b0 + c2 ) exp(2d) + b0 ] + , (exp(d) − 1)2 (exp(d) − 1)4

(14)

where a0 , b0 , a−1 and c2 are arbitrary constants. According to (9) and (14), we can obtain a1 exp(nd +

un = (

b02 4

−

(exp(d)−1)2 c2 exp(d) x

[c2 (a−1 b0 −a0 )+ 12 b0 (exp(d)−1)2 ]2 (exp(d)−1)4

+ c2 t + ξ0 ) + a0 + a−1 exp(−nd −

) exp(nd +

(exp(d)−1)2 c2 exp(d) x

(exp(d)−1)2 c2 exp(d) x

− c 2 t − ξ0 )

+ c2 t + ξ0 ) + b0 + exp(−nd −

(exp(d)−1)2 c2 exp(d) x

, − c2 t − ξ0 ) (15)

which are new travelling wave solutions. To compare our result, Eqs. (13) and (15), with that in open literature, we write down Zhu’s solutions [25], which read un1 = A0 + un2,2

sinh2 (d) tanh(nd +

sinh2 (d) c2 x

+ c 2 t + ξ0 )

(16)

,   I (cosh(d) − 1) 2(cosh(d) − 1) = A0 ± sec h nd + x + c2 t + ξ0 c2 c2 +

c2

(cosh(d) − 1) tanh(nd +

un3 = A0 + un4 = A0 +

2(cosh(d)−1) x c2

+ c2 t + ξ0 )

c2

(17)

,

2(cosh(d)−1) x + c2 t + ξ0 ) c2 , 2(cosh(d)−1) x + c2 t + ξ0 )) c2

(18)

2(cosh(d)−1) x + c2 t + ξ0 ) c2 , 2(cosh(d)−1) x + c2 t + ξ0 ) − 1) c2

(19)

(cosh(d) − 1) sinh(nd + c2 (1 + cosh(nd +

(cosh(d) − 1) sinh(nd + c2 (cosh(nd +

where A0 and c2 are arbitrary constants, √ un5,5 = A0 ±

r 2 − 1B1

r + cosh(nd + 2B1 x +

cosh(d)−1 t B1

+ ξ0 )

+

cosh(d)−1 t + ξ0 ) B1 , cosh(d)−1 + t + ξ0 ) B1

B1 sinh(nd + 2B1 x + r + cosh(nd + 2B1 x

(20)

where A0 and B1 are arbitrary constants, it requires that r 2 − 1 > 0. 2 If we choose b1 = 1, a1 = 2A0 + sinhc2(d) , our solution, Eq. (13), turns out to be Zhu’s kink-type solitary solutions [25] as expressed in Eqs. (16). , a−1 = A0 + (cosh(d)−1) , our solution, Eq. (15), turns out to be Zhu’s travelling We choose b0 = 0, b1 = 1, a0 = ± 2I (cosh(d)−1) c2 c2 wave solutions [25] as expressed in Eqs. (17). We choose b0 = 2, b1 = 1, a0 = 2A0 , a−1 = A0 − (cosh(d)−1) , our solution, Eq. (15), turns out to be Zhu’s travelling wave c2 solutions [25] as expressed in Eqs. (18). , our solution, Eq. (15), turns out to be Zhu’s travelling wave solutions [25] If b0 = −2, b1 = 1, a0 = 2A0 , a−1 = A0 − (cosh(d)−1) c2 as expressed in Eqs. (19). √ , our solution, Eq. (15), is Zhu’s travelling wave If b0 = 2r, b1 = 1, a0 = 2A0 r ± 2 r 2 − 1B1 , a−1 = A0 − B1 , c2 = (cosh(d)−1) B1 solutions [25] as expressed in Eqs. (20). So the suggested Exp-function method can obtain easily the generalized soliton solutions, kink-type solitary solutions and travelling wave solutions for the discrete (2 + 1)-dimensional Toda lattice equation.

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4. Conclusion In this Letter, we have utilized the Exp-function method to study the discrete (2 + 1)-dimensional Toda lattice equation. As a result, some new explicit exact travelling wave solutions of the discrete (2+1)-dimensional Toda lattice equation have been obtained which include generalized soliton solutions, kink-type solitary solutions and travelling wave solutions, and successfully cover the previously known exact solutions in Ref. [25]. So we think the Exp-function method will become a promising and powerful new method for a lot of DDEs. Acknowledgements The work is supported by the National Natural Science Foundation of China under the Grant No. 10172056. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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