Chaos, Solitons and Fractals 34 (2007) 940–946

www.elsevier.com/locate/chaos

Exact discrete soliton solutions of the nonlineardifferential-difference equations

Xiao-Fei Wu a,*, Hong-Liang Ge b, Zheng-Yi Ma a,c

a College of Information, Zhejiang Lishui University, Lishui 323000, PR Chinab Department of Applied Physics, China Jiliang University, Hangzhou 310018, PR China

c Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, PR China

Accepted 31 March 2006

Abstract

By using the extended hyperbolic function approach, we explore the discrete (2 + 1)-dimensional Toda lattice andthe discrete Hybrid lattice, and successfully obtain some new exact soliton solutions, including the discrete kink-typesoliton solution, bell-type soliton solution and some other types of soliton solutions.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Explicit excitations, especially the explicit solitary wave solutions, which are widely encountered in nature science, tothe model equations of physical systems are of fundamental importance in physical science and nonlinear science. In thestudy of systems modelling wave phenomena, one of the important things of investigation is the travelling wave exci-tation, i.e. a moving wave solution with a fixed velocity. For the continuous circumstance, it is well known that there areinfinite solutions for a nonlinear partial differential equation (PDE) and it is a difficult task to find an exact solution.However, in recent decades, many powerful approaches have been devised, such as inverse scattering method [1], Hir-ota’s bilinear method [2], Backlund transformation [3], homogeneous balance approach [4], Painleve truncation expan-sion [5], hyperbolic function approach [6], and so on [7–13].

Unlike difference equations which are fully discretized, differential-difference equations (DDEs) are semi-discretizedwith some (or all) of their spacial variables discretized while time is usually kept continuous. Since the work of Fermi,Pasta, and Ulam in the 1950s [14], DDEs have been the focus of many nonlinear studies. There is renewed interest inDDEs, which can be used to model such physical phenomena as particle vibrations in lattices, currents in electrical net-works, pulses in biological chains, etc. DDEs also play an important role in numerical simulations of nonlinear partialdifferential equations (PDEs), queuing problems, and discretizations in solid state, and quantum physics.

However, different from the considerably much work done on finding exact solutions to PDEs mentioned above, toour knowledge, little work has been done to investigate exact solutions for DDEs. For example, Tsuchida et al. [15,16]extended the inverse scattering method to study DDEs, Qian et al. [17] modified the multilinear variable separation

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.04.001

* Corresponding author.E-mail address: xfwu66@yahoo.com.cn (X.-F. Wu).

X.-F. Wu et al. / Chaos, Solitons and Fractals 34 (2007) 940–946 941

approach to solve a special DDE. More recently, Baldwin et al. [18] presented an algorithm to find exact travelling wavesolutions of DDEs in terms of tanh function and found kink-type solutions in many spatially discrete nonlinear modelssuch as the celebrated Toda lattice [12] and Ablowitz–Ladik lattice [19], etc. However, this method is unable to findsolutions other than polynomials in tanh. Here we expand this hyperbolic function approach. We not only obtainthe kink-type soliton solutions of DDEs, but also get the bell-type soliton solutions of DDEs. As our examples, weexplore the discrete (2 + 1)-dimensional Toda lattice and the discrete Hybrid lattice, and successfully derive somenew discrete kink-type soliton solution, bell-type soliton solution and other types of soliton solutions by using thisextended hyperbolic function approach.

2. The extended hyperbolic function approach to solve DDEs and its applications

2.1. The extended hyperbolic function approach

First, we recall the properties of hyperbolic functions,

cosh2n� sinh2n ¼ 1; sinh n= cosh n ¼ tanh n; 1= cosh n ¼ sechn; ðsinh nÞ0 ¼ cosh n;

ðcosh nÞ0 ¼ sinh n; sinhðn1 � n2Þ ¼ sinh n1 cosh n2 � cosh n1 sinh n2;

coshðn1 � n2Þ ¼ cosh n1 cosh n2 � sinh n1 sinh n2: ð1Þ

Given a system of M polynomial DDE,

Dðunþp1ðxÞ; . . . ; unþpk

ðxÞ; u0nþp1ðxÞ; . . . ; u0nþpk

ðxÞ; . . . ; uðrÞnþp1ðxÞ; . . . ; uðrÞnþpk

ðxÞÞ ¼ 0; ð2Þ

where the dependent variable u has M components ui, the continuous variable x has N components xi, the discrete var-iable n has Q components nj, the k shift vectors pi, and u(r)(x) denotes the collection of mixed derivative terms of order r.

The main steps of the extended hyperbolic function approach are outlined as follows:Step 1. When we seek the travelling wave solutions of Eq. (2), the first step is to introduce the wave transformation

nn ¼XQ

i¼1

dini þXN

j¼1

cjxj þ d; ð3Þ

where the coefficients di, cj and the phase d are constants to be determined later. In this way, Eq. (2) becomes

Dðunþp1ðnnÞ; . . . ; unþpk

ðnnÞ; u0nþp1ðnnÞ; . . . ; u0nþpk

ðnnÞ; . . . ; uðrÞnþp1ðnnÞ; . . . ; uðrÞnþpk

ðnnÞÞ ¼ 0: ð4Þ

Step 2. We propose that the travelling wave solutions of Eq. (2) or Eq. (4) is in the following frame

unðnnÞ ¼ a0 þXm

j¼1

f j�1ðnnÞðajf ðnnÞ þ bjgðnnÞÞ; ð5Þ

with

f ðnnÞ ¼sechnn

A tanh nn þ Bsechnn þ C; gðnnÞ ¼

tanh nn

A tanh nn þ Bsechnn þ C; ð6Þ

where a0, aj, bj (j = 1, . . . ,m), A, B and C are all constants to be determined and m is given according to the homoge-neous balance principle. Then, we can obtain

unþpsðnnÞ ¼ a0 þ

Xm

j¼1

f j�1ðnn þ usÞðajf ðnn þ usÞ þ bjgðnn þ usÞÞ

¼ a0 þXm

j¼1

sechðnn þ usÞA tanhðnn þ usÞ þ Bsechðnn þ usÞ þ C

� �j�1 ajsechðnn þ usÞA tanhðnn þ usÞ þ Bsechðnn þ usÞ þ C

�

þ bj tanhðnn þ usÞA tanhðnn þ usÞ þ Bsechðnn þ usÞ þ C

�; ð7Þ

with us satisfying

us ¼ ps1d1 þ ps2d2 þ � � � þ psQdQ: ð8Þ

942 X.-F. Wu et al. / Chaos, Solitons and Fractals 34 (2007) 940–946

Meanwhile, it is important to note that unþpsðnnÞ is a function of nn and not nnþps

.Step 3. Substituting the ansatzes (5)–(7) into Eq. (4), clearing the denominator and setting the coefficients of power

terms in coshrnn sinhsnn (r = 0,1, . . . , s = 0,1) to zero, a system of algebraic equations with a0, aj, bj (j = 1, . . . ,m), A, B

and C are obtained. Solving these equations by using of Wu’s method and symbolic computation software Maple, wecan obtain the corresponding undetermined coefficients.

Finally, substituting the obtained constants a0, aj, bj (j = 1, . . . ,m), A, B and C into ansatzes (5) with (6), a series ofexplicit and exact solitary wave solutions of the DDE (2) are constructed.

2.2. Soliton solutions of the (2 + 1)-dimensional Toda lattice

Let us consider the discrete nonlinear (2 + 1)-dimensional Toda lattice [20]

o2yn

oxot¼ expðyn�1 � ynÞ � expðyn � ynþ1Þ; ð9Þ

where yn(x, t) is the displacement from equilibrium of the nth unit mass under an exponentially decaying interactionforce between nearest neighbours. To write Eq. (9) as a polynomial DDE, setting

oun

ot¼ expðyn�1 � ynÞ � 1; ð10Þ

then Eq. (9) becomes

o2un

oxot¼ oun

otþ 1

� �ðun�1 � 2un þ unþ1Þ: ð11Þ

Let un = un(nn), nn = kn + c1x + c2t + d, then Eq. (11) becomes

c1c2u00n ¼ ðc2u0n þ 1Þðun�1 � 2un þ unþ1Þ: ð12Þ

We expand the solutions of Eq. (12) in the form of ansatzes Eqs. (5)–(7). In this case, ps = 1,�1,d1 = k. According tothe homogeneous balance principle, we may choose the solutions of Eq. (12) in the form

un ¼ a0 þa1sechnn

A tanh nn þ Bsechnn þ Cþ a2 tanh nn

A tanh nn þ Bsechnn þ C;

unþ1 ¼ a0 þa1sechðnn þ kÞ

A tanhðnn þ kÞ þ Bsechðnn þ kÞ þ Cþ a2 tanhðnn þ kÞ

A tanhðnn þ kÞ þ Bsechðnn þ kÞ þ C;

un�1 ¼ a0 þa1sechðnn � kÞ

A tanhðnn � kÞ þ Bsechðnn � kÞ þ Cþ a2 tanhðnn � kÞ

A tanhðnn � kÞ þ Bsechðnn � kÞ þ C:

ð13Þ

Substituting Eqs. (13) into Eq. (12), clearing the denominator and setting the coefficients of power terms like coshrnn-

sinhsnn (r = 0,1,2,3,4; s = 0,1) to zero, yield the following algebraic equations:

2a1B2C2 � 4a1A2B2 þ c1c2a1A4cosh2k � c1c2a2ABC2 � c1c2a1A2B2 � 4a2ABC2 þ 2c2a21ACB

þ 4c2a2a1C3 � 2c2a2a1B2C þ 2c1c2a1C2B2 þ c1c2a2AB3 þ 4a2AB3 þ c1c2a1A2C2 � 2a1A2C2 � 2c1c2a1C4

þ 2a1A4 cosh k � c1c2a2A3Bcosh2k � 2c1c2a1A2B2 cosh k � 2a2cosh2kB3Aþ 2c2a1a2 cosh kB2C

� 3c1c2a1A2C2cosh2k þ 2a2A3Bcosh2k � 4c2a2a1C3cosh2k þ 2c2a22 cosh kACB� 2c2a2

1ABC cosh k

þ 4a2ABC2cosh2k þ 2a1A2B2 cosh k þ 2c1c2a1C4cosh2k � 2c2a22cosh2kABC þ 2c1c2a2B3A cosh k

� 2a2B3A cosh k þ 2a1A2C2cosh2k þ 2a1B2A2cosh2k � 2a1A4cosh2k � 2a1B2C2cosh2k � 2a2A3B cosh k

� 2c2a2a1A2C cosh k þ c1c2a2ABC2cosh2k � 4c1c2a2AC2 cosh kBþ 2c2a1a2cosh2kA2C ¼ 0;

2a1A4 cosh k � 2a1C4 � 6a2AC2 cosh kB� c1c2a1A4 þ c1c2a2A3B� c1c2a1C4 þ 2a2A3Bþ 3c1c2a2ABC2

þ 6a2ABC2 � 6c1c2a1A2C2 � 2a1A4 � 2a2A3B cosh k þ 2a1C4 cosh k þ 12a1A2C2 cosh k � 12a1A2C2 ¼ 0;

6c2a2a1C2Bþ c1c2a1B3C � 2a2A3C � 4a2AC3 þ 2a1B3C þ 4a1C3Bþ 4a2AC3cosh2k � 2c1c2a1A2BC � c1c2a1BC3

þ 2c1c2a2AC3cosh2k � 4c2a1a2A2B cosh k � 6a2B2Ccosh2kA� 2c2a22cosh2kB2A� 2c2a1a2BC2cosh2k

þ 2c1c2a1A2C cosh kBþ 2c2a21A3 cosh k þ c1c2a1BC3cosh2k þ 4c2a2a1BA2cosh2k þ 4c1c2a1BC3 cosh k

X.-F. Wu et al. / Chaos, Solitons and Fractals 34 (2007) 940–946 943

þ 2c2a21AC2cosh2k � 2c1c2a2A3Ccosh2k � 2c2a2

2AC2cosh2k � 6a1A2BC cosh k þ 2c2a22AB2 cosh k

þ 2a2A3Ccosh2k þ 2a1A2CB� 4a1BC3cosh2k � 2c2a21A3cosh2k � 6c2a2

1AC2 þ 4c1c2a2ACB2

þ 2a2B2CA� 2a1B3C cosh k � 6c1c2a2AC3 þ 2c2a22AC2 � c1c2a1A2BCcosh2k þ 4a2AB2C cosh k

þ 4a1A2Bcosh2kC � 4c2a1a2 cosh kC2Bþ 4c2a21AC2 cosh k ¼ 0;

2c1c2a2C4cosh2k þ 2c2a1a2cosh2kABC � 4a1ABC2cosh2k � 2c2a21A2cosh2kC � c1c2a1A3Bcosh2k � 2a2B4

þ 4a1A3Bcosh2k � 4a2B2A2cosh2k � 2a2B2C2cosh2k � 2c1c2a2A2C2cosh2k � c1c2a2B2C2cosh2k

þ c1c2a1BC2Acosh2k þ 2c2a21C3cosh2k � 2c2a2

2C3cosh2k þ c1c2a2B2A2cosh2k � 2a1B3A cosh k

þ 2a1AB3 � c1c2a2B4 � 2c1c2a2C4 � c1c2a1BC2A� 2a1A3Bþ 2a2B2A2 þ 2c2a22C3 þ 2a2B4 cosh k

þ 3c1c2a2C2B2 � 2c2a21C3 þ 4a1AC2B� 2c2a2

2B2C þ 2c2a2a1ACBþ c1c2a1AB3 þ 2a2B2C2 � 2a2A2C2

þ 2a2A2C2cosh2k � 2a1A3B cosh k þ 2a2A2B2 cosh k � 4c2a1a2ABC cosh k � 2c1c2a1A3 cosh kB

þ 4c1c2a1AC2B cosh k þ 2c2a21A2C cosh k þ 2c1c2a2A2B2 cosh k þ 2c2a2

2 cosh kB2C ¼ 0;

3c1c2a2A2BC � 8a1C3A� 8a1A3C þ 2a2C3Bþ 6a2BCA2 � 2a2C3B cosh k � 4c1c2a1C3A� 4c1c2a1A3C

þ c1c2a2BC3 � 6a2A2BC cosh k þ 8a1A3C cosh k þ 8a1AC3 cosh k ¼ 0;

4a1C3Aþ 6a2A2BC cosh k � 8a1A3C cosh k � c1c2a2A2Bcosh2kC � 2c2a22 cosh kC2Bþ 4c1c2a2A2BC cosh k

� 2c2a21C2B� 2c1c2a1A3C þ 4a1A3C þ 6c1c2a1C3Aþ 2a2BCA2 þ 4a2C3Bþ 4a1ACB2 þ 4c2a2

2BC2

� 8c2a1a2AC2 þ 4c2a1a2AB2 � 2c2a21A2B� 2c2a2

2B3 þ c1c2a2B3C � c1c2a2BC3 þ 2c1c2a2A2BC

� 2a2B3C � 2c1c2a1ACB2 � 2c1c2a1AC3cosh2k þ 2c2a21BC2 cosh k � 2c1c2a2B3C cosh k þ c1c2a2BC3cosh2k

� 2c2a22BC2cosh2k � 4a1AC3cosh2k � 4a2BC3cosh2k � 4a1AB2C cosh k þ 2c2a2

1A2B cosh k þ 4c2a2a1AC2 cosh k

� 8a2A2BCcosh2k þ 4c1c2a2C3 cosh kBþ 4a1A3cosh2kC þ 2c2a22 cosh kB3 þ 2a2B3C cosh k

þ 4c2a1a2cosh2kAC2 þ 2c1c2a1A3Ccosh2k þ 4c1c2a1B2CA cosh k � 4c2a2a1B2A cosh k ¼ 0;

2a1A3B cosh k � 2c2a21C3 cosh k � 6a2A2C2cosh2k � 2a2A2B2 cosh k � 2a2B2C2 cosh k þ 6a1AC2B cosh k

þ 2c2a21C3 � c1c2a2C2B2 � 6a1AC2Bþ 2c2a2

2B2C þ 2a2C4 þ c1c2a1A3B� 8c2a2a1ACBþ 2a2B2C2 þ 6a2A2C2

þ 2c1c2a2C4 � c1c2a2B2A2 þ 6c2a21A2C þ 3c1c2a1BC2A� 2a1A3Bþ 2a2B2A2 � 2a2C4cosh2k þ 6c1c2a2A2C2

þ 8c2a1a2ABC cosh k � 2c1c2a1A3 cosh kB� 6c1c2a1AC2B cosh k þ 2c1c2a2B2C2 cosh k

þ 2c1c2a2A2B2 cosh k � 2c2a22 cosh kB2C � 6c2a2

1A2C cosh k ¼ 0;

2a2A3C � 4c2a2a1C2Bþ 6a2AC3 � 2a1C3B� 6a2AC3cosh2k þ 2c2a21A3 þ 3c1c2a1A2BC � 2a2A3Ccosh2k

� 4a2AB2C cosh k þ 4c2a1a2A2B cosh k þ 6a1A2BC cosh k � 2c2a22A cosh kB2 � 6c1c2a1A2BC cosh k

þ c1c2a1BC3 � 6a1A2CBþ 6c2a21AC2 � 2c1c2a2ACB2 � 4c2a2a1BA2 þ 2c2a2

2B2Aþ 6c1c2a2AC3

� 2c2a21A3 cosh k � 2c1c2a1C3 cosh kBþ 4c2a1a2 cosh kC2B� 6c2a2

1AC2 cosh k

þ 4c1c2a2AB2C cosh k þ 2a1C3B cosh k þ 2c1c2a2A3C þ 4a2B2CA ¼ 0;

c1c2a2ABC2cosh2k þ 8c1c2a2AC2 cosh kBþ 2c2a1a2cosh2kA2C � 2c2a22 cosh kACB� 2a1A2B2 cosh k

þ 2a1A2B2 þ 2a1B2C2 þ c1c2a1A4cosh2k � 2c1c2a2ABC2 � c1c2a1A2B2 þ 4a2ABC2 � 4c2a21ACB� 4c2a2a1C3

þ 6c1c2a1A2C2 � 4c2a1a2A2C þ 12a1A2C2 þ 3c1c2a1C4 � 2a1C4cosh2k þ 4c2a22BCAþ 2c2a2a1C3 cosh k

þ 6a2AC2 cosh kBþ 2c1c2a1B2C2 cosh k � 2a1B2C2 cosh k � 12a1A2C2 cosh k � 4a1A4 cosh k þ 2a1C4

� c1c2a2A3Bcosh2k þ 2c1c2a1A2B2 cosh k � 4c2a1a2 cosh kB2C � 2a2A3Bcosh2k þ 4a2A3B cosh k

þ 2a1A4cosh2k þ 2c2a2a1C3cosh2k þ 4c2a21ABC cosh k � 10a2ABC2cosh2k þ 2c2a2a1A2C cosh k

þ 4c2a2a1B2C þ c1c2a2A3B� 2a2A3Bþ 2a1A4 � c1c2a1C2B2 � c1c2a1A4 þ c1c2a2AB3 � 2a2AB3

� c1c2a1C4cosh2k � 2c2a22cosh2kABC � 2c1c2a2B3A cosh k þ 2a2B3A cosh k ¼ 0: ð14Þ

944 X.-F. Wu et al. / Chaos, Solitons and Fractals 34 (2007) 940–946

Solving the above system by using of Wu’s method and symbolic computation software Maple, we can find follow-ing meaningful solutions:

a0 ¼ a0; a1 ¼ 0; a2 ¼Csinh2k

c2

; c1 ¼sinh2k

c2

; c2 ¼ c2; A ¼ 0; B ¼ 0; C ¼ C;

where a0, c2, k and C are arbitrary constants;

a0 ¼ a0; a1 ¼ 0; a2 ¼ðC2 � A2Þsinh2k

c2C; c1 ¼

sinh2kc2

; c2 ¼ c2; A ¼ A; B ¼ 0; C ¼ C;

where a0, c2, k, A and C are arbitrary constants;

a0 ¼ a0; a1 ¼ �Iðcosh k � 1ÞC

c2

; a2 ¼ðcosh k � 1ÞC

c2

; c1 ¼2ðcosh k � 1Þ

c2

;

c2 ¼ c2; A ¼ 0; B ¼ 0; C ¼ C;

where I2 = �1, a0, c2, k and C are arbitrary constants; and

a0 ¼ a0; a1 ¼ð�BA� C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2 þ B2 � C2

pÞðcosh k � 1Þ

c2C; a2 ¼

ðC2 � A2Þðcosh k � 1Þc2C

;

c1 ¼2ðcosh k � 1Þ

c2

; c2 ¼ c2; A ¼ A; B ¼ B; C ¼ C;

where a0, c2, k, A, B and C are arbitrary constants and A2 + B2 P C2.Then, the closed form solutions of Eq. (11) are

un1 ¼ a0 þsinh2k

c2

tanh knþ sinh2kc2

xþ c2t þ d

� �; ð15Þ

un2 ¼ a0 þðC2 � A2Þsinh2k tanhðknþ sinh2k

c2xþ c2t þ dÞ

c2CðA tanhðknþ sinh2kc2

xþ c2t þ dÞ þ CÞ; ð16Þ

un3 ¼ a0 �Iðcosh k � 1Þ

c2

sech knþ 2ðcosh k � 1Þc2

xþ c2t þ d

� �

þ cosh k � 1

c2

tanh knþ 2ðcosh k � 1Þc2

xþ c2t þ d

� �; ð17Þ

un4 ¼ a0 þð�AB� C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2 þ B2 � C2

pÞðcosh k � 1Þsech knþ 2ðcosh k�1Þ

c2xþ c2t þ d

� �c2CðA tanh knþ 2ðcosh k�1Þ

c2xþ c2t þ d

� �þ Bsech knþ 2ðcosh k�1Þ

c2xþ c2t þ d

� �þ CÞ

þðC2 � A2Þðcosh k � 1Þ tanh knþ 2ðcosh k�1Þ

c2xþ c2t þ d

� �c2CðA tanh knþ 2ðcosh k�1Þ

c2xþ c2t þ d

� �þ Bsech knþ 2ðcosh k�1Þ

c2xþ c2t þ d

� �þ CÞ

: ð18Þ

The solution (15) is a kink-type soliton solution which is just as the same as (40) in Ref. [18]. To our knowledge, theother closed form soliton solutions to (2 + 1)-dimensional Toda lattice are novel.

2.3. Soliton solutions of the Hybrid lattice

The discrete nonlinear Hybrid lattice [18] reads

oun

ot¼ ð1þ aun þ bu2

nÞðun�1 � unþ1Þ; ð19Þ

where a and b are constants. Let un = un(nn), nn = kn + ct + d, then Eq. (19) becomes

cu0n ¼ ð1þ aun þ bu2nÞðun�1 � unþ1Þ: ð20Þ

Similarly, we can also expand the solutions of Eq. (20) in the form of ansatzes (5)–(7). In this case, ps = 1 � 1, d1 = k.According to the homogeneous balance principle, we may choose the solutions of Eq. (20) in the form

X.-F. Wu et al. / Chaos, Solitons and Fractals 34 (2007) 940–946 945

un ¼ a0 þa1sechnn

A tanh nn þ Bsechnn þ Cþ a2 tanh nn

A tanh nn þ Bsechnn þ C;

unþ1 ¼ a0 þa1sechðnn þ kÞ

A tanhðnn þ kÞ þ Bsechðnn þ kÞ þ Cþ a2 tanhðnn þ kÞ

A tanhðnn þ kÞ þ Bsechðnn þ kÞ þ C;

un�1 ¼ a0 þa1sechðnn � kÞ

A tanhðnn � kÞ þ Bsechðnn � kÞ þ Cþ a2 tanhðnn � kÞ

A tanhðnn � kÞ þ Bsechðnn � kÞ þ C:

ð21Þ

Substituting Eqs. (21) into Eq. (20), clearing the denominator and setting the coefficients of power terms like coshrnn-

sinhsnn (r = 0,1,2,3; s = 0,1) to zero, yield a system of algebraic equations. From there, we can find following meaning-ful solutions:

a0 ¼ �a

2b; a1 ¼ 0; a2 ¼ �

Cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � 4b

p2b

tanh k; c ¼ a2 � 4b2b

tanh k; A ¼ 0; B ¼ 0; C ¼ C;

where k and C are arbitrary constants and a2 > 4b;

a0 ¼ �a

2b; a1 ¼ �

Cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4b� a2

psinh k

2b; a2 ¼ 0; c ¼ �ð4b� a2Þ sinh k

2b; A ¼ 0; B ¼ 0; C ¼ C;

where k and C are arbitrary constants and a2 < 4b;

a0 ¼ a0; a1 ¼ a1; a2 ¼ �a1ðaþ 2ba0Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

a2 � 4b

scschk; c ¼ a2 � 4b

2bsinh k;

A ¼ �2a1b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

a2 � 4b

scschk; B ¼ 0; C ¼ 0;

where a0, a1 and k are arbitrary constants and a2 > 4b; and

a0 ¼ �a

2b; a1 ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2 � 4bÞðB2 � C2Þ

q2b

tanhk2; a2 ¼ �

Cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � 4b

p2b

tanhk2;

c ¼ a2 � 4bb

tanhk2; A ¼ 0; B ¼ B; C ¼ C;

where k, B and C are arbitrary constants, a2 > 4b and B2 > C2.Then, the closed form solutions of Eq. (19) are

un1 ¼ �a

2b�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � 4b

ptanh k

2btanhðknþ a2 � 4b

2btanh kt þ dÞ; ð22Þ

un2 ¼ �a

2b�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4b� a2

psinh

2bsechðkn� 4b� a2

2bsinh kt þ dÞ; ð23Þ

un3 ¼ �a

2b�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � 4b

psinh k

2bcschðknþ a2 � 4b

2bsinh kt þ dÞ; ð24Þ

un4 ¼ �a

2b�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiða2 � 4bÞðB2 � C2Þ

qtanh k

2sechðknþ a2�4b

b tanh k2t þ dÞ

2bðBsechðknþ a2�4bb tanh k

2t þ dÞ þ CÞ

�C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � 4b

ptanh k

2tanhðknþ a2�4b

b tanh k2t þ dÞ

2bðBsechðknþ a2�4bb tanh k

2t þ dÞ þ CÞ

: ð25Þ

The solution (22) is a kink-type soliton solution which is no other than the result for Hybrid lattice presented byBaldwin in Ref. [18]. The other solutions (23)–(25) are bell-type soliton solution, singular soliton solution and solitonsolution of new type respectively, which have not been found in previous literature.

3. Conclusion

In this paper, we have successfully used the extended hyperbolic function approach to construct exact solitary wavesolutions for the discrete (2 + 1)-dimensional Toda lattice and the discrete Hybrid lattice, obtained some new kink-type

946 X.-F. Wu et al. / Chaos, Solitons and Fractals 34 (2007) 940–946

soliton solution, bell-type soliton solution and abundant other types of soliton solutions. Not only has it obtained manykinds of solutions, but also the procedure is readily computable. Therefore, in view of the importance of the DDEs, theproposed method is worthy of further study and it may help us to find new and interesting solutions for a given non-linear discrete system.

Acknowledgements

The project was supported by the Scientific Research Fund of Zhejiang Provincial Education Department of Chinaunder the Grant No. 20051356 and the Natural Science Foundation of Zhejiang Lishui University of China under theGrant No. KZ05004.

References

[1] Cardener CS, Kruskal JM, Miura RM. Phys Rev Lett 1967;14:1095.[2] Hirota R. J Math Phys 1973;14:810.[3] Wadati M. Prog Theor Phys Suppl 1976;59:36.[4] Wang ML. Phys Lett A 1995;199:65.[5] Tian B, Gao YT. Phys Lett A 1995;209:297.[6] Fan EG, Zhang HQ. Phys Lett A 1998;246:403.[7] Wadati M, Toda M. J Phys Soc Jpn 1975;39:1196.[8] Toda M, Wadati M. J Phys Soc Jpn 1975;39:1204.[9] Wadati M, Sanuki K, Konno K. Prog Theor Phys 1975;53:419.

[10] Wadati M. J Phys Soc Jpn 1976;40:1517.[11] Wadati M, Watanabe M. Prog Theor Phys 1977;57:808.[12] Toda M. Theory of nonlinear lattices. Berlin Heidelberg: Springer-Verlag; 1988.[13] Tang XY, Qian XM, Ding W. Chaos, Solitons & Fractals 2005;23:1311.[14] Fermi E, Pasta J, Ulam S. Collected papers of Enrico Fermi. Chicago: University of Chicago Press; 1965.[15] Tsuchida T, Ujino H, Wadati M. J Math Phys 1998;39:4785.[16] Tsuchida T, Ujino H, Wadati M. J Phys A: Math Gen 1999;32:2239.[17] Qian XM, Lou SY, Hu XB. J Phys A: Math Gen 2003;37:2401.[18] Baldwin D, Goktas U, Hereman W. Comput Phys Commun 2004;162:203.[19] Ablowitz MJ, Ladik JF. Stud Appl Math 1976;55:213.[20] Kajiwara K, Satsuma J. J Math Phys 1991;32:506.