Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 756896 10 pageshttpdxdoiorg1011552013756896

Research ArticleExact Solutions for Nonlinear Differential Difference Equationsin Mathematical Physics

Khaled A Gepreel12 Taher A Nofal23 and Fawziah M Alotaibi2

1 Mathematics Department Faculty of Science Zagazig University Egypt2Mathematics Department Faculty of Science Taif University Saudi Arabia3Mathematics Department Faculty of Science Minia University Egypt

Correspondence should be addressed to Khaled A Gepreel kagepreelyahoocom

Received 24 July 2012 Revised 13 October 2012 Accepted 20 November 2012

Academic Editor Patricia J Y Wong

Copyright copy 2013 Khaled A Gepreel et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Wemodified the truncated expansionmethod to construct the exact solutions for somenonlinear differential difference equations inmathematical physics via the general lattice equation the discrete nonlinear Schrodinger with a saturable nonlinearity the quinticdiscrete nonlinear Schrodinger equation and the relativistic Toda lattice system Also we put a rational solitary wave functionmethod to find the rational solitary wave solutions for some nonlinear differential difference equationsThe proposed methods aremore effective and powerful to obtain the exact solutions for nonlinear difference differential equations

1 Introduction

Nonlinear differential difference equations (NDDEs) play acrucial role in many branches of applied physical sciencessuch as condensed matter physics biophysics atomic chainsmolecular crystals and discretization in solid-state and quan-tum physics They also play an important role in numeri-cal simulation of soliton dynamics in high-energy physicsbecause of their rich structures Therefore researchers haveshown a wide interest in studying NDDEs since the originalwork of Fermi et al [1] in the 1950s Contrary to differ-ence equations that are being fully discretized NDDEs aresemidiscretized with some (or all) of their space variablesbeing discretized while time is usually kept continuous Asfar as we could verify little work has been done to search forexact solutions of NDDEs Hence it would make sense to domore research on solving NDDEs

The study of discrete nonlinear system governed bydifferential difference equations (DDEs) has drawn muchattention in recent years particularly from the point of viewof complete integrabilityThere is a vast body of work on non-linear DDEs including investigation of integrability criteriathe computation of densities Backlund transformation and

recursion operator [2ndash17] Xie [18] and Zayed et al [19] haveput the rational solitary wave solutions for nonlinear partialdifferential equations

In the recent years there have been a lot of papersdevoted to obtain the solitary wave or periodic solutionsfor a variety of nonlinear differential difference equationsby using the symbolic computations Among these methodsLiu [20] used the exponential function rational expansionmethod to some NDDEs Zhang et al [21] have modifiedthe (1198661015840119866) expansion method form solving the nonlinearpartial differential equations to solve the nonlinear differ-ential difference equations Aslan [22 23] has applied the(1198661015840119866) expansion method for solving the discrete nonlinear

Schrodinger equations with a saturable nonlinearity discreteBurgers equation and the relativistic Toda lattice systemMore recently Gepreel et al [24ndash26] have used the modifiedrational Jacobi elliptic functions method to construct sometypes of Jacobi elliptic solutions of the lattice equation thediscrete nonlinear Schrodinger equation with a saturablenonlinearity and the quintic discrete nonlinear Schrodingerequation

In this paper we use a modified truncated expansionmethod to construct the exact solutions of the following

2 Abstract and Applied Analysis

nonlinear difference differential equations in mathematicalphysics

(i) the general lattice equation [21 25 27]

119906119899119905

= (120572 + 120573119906119899+ 1205741199062

119899) (119906119899+1minus 119906119899minus1) (1)

(ii) the discrete nonlinear Schrodinger equation with asaturable nonlinearity [23 25]

119894120597120595119899

120597119905+ (120595119899+1+ 120595119899minus1minus 2120595119899) +

120578100381610038161003816100381612059511989910038161003816100381610038162

1 + 120583100381610038161003816100381612059511989910038161003816100381610038162120595119899= 0 (2)

(iii) the quintic discrete nonlinear Schrodinger equation[26 28]

119894120597120595119899

120597119905+ 120572 (120595

119899+1minus 2120595119899+ 120595119899minus1) + 120573

100381610038161003816100381612059511989910038161003816100381610038162

120595119899

+ 120574100381610038161003816100381612059511989910038161003816100381610038162

(120595119899+1+ 120595119899minus1) + 120575

100381610038161003816100381612059511989910038161003816100381610038164

(120595119899+1+ 120595119899minus1) = 0

(3)

(iv) the relativistic Toda lattice system [21]

119906119899119905

minus (1 + 120572119906119899) (119907119899minus 119907119899minus1) = 0

119907119899119905

minus 119907119899(119906119899+1minus 119906119899+ 120572119907119899+1minus 120572119907119899minus1) = 0

(4)

where 120572 120573 120574 120575 120583 and 120578 are arbitrary constants Also weput a rational solitary wave method to the nonlinear differ-ential difference equations We use the proposed method tofind the rational solitary wave solutions for the nonlineardifferential difference equations via the discrete nonlinearSchrodinger equation with a saturable nonlinearity thequintic discrete nonlinear Schrodinger equation and therelativistic Toda lattice equation

2 Description of the Modified TruncatedExpansion Method to Nonlinear DDEs

In this section we would like to outline the algorithm forusing the modified truncated expansion method to solve thenonlinear DDEs Consider a given system of119872 polynomialnonlinear DDEs

Δ (119906119899+1199011

(119883) 119906119899+119901119896

(119883) 1199061015840

119899+1199011

(119883) 1199061015840

119899+119901119896

(119883)

119906(119903)

119899+1199011

(119883) 119906(119903)

119899+119901119896

(119883)) = 0

(5)

where the dependent variable 119906 has 119872 components 119906119894 the

continuous variable 119909 has 119873 components 119909119895 the discrete

variable 119899 has 119876 components 119899119894 the 119896 shift vectors 119875

119904isin 119885119876

and 119906(119903) denotes the collection of mixed derivative terms oforder 119903

The main steps of the algorithm for the modified trun-cated expansion method to solve NDDEs are outlined asfollows

Step 1 We seek the traveling wave transformation in thefollowing form

119906119899+119901119904

(119883) = 119880 (120585119899+119901119904

) 120585119899=

119876

sum119894=1

119889119894119899119894+

119898

sum119895=1

119888119895119909119895+ 1205850

119904 = 1 2 119896

(6)

where the coefficients 119889119894(119894 = 1 119876) 119888

119895(119895 = 1 119873) and

the phase 1205850are constants The transformations (6) lead to

write (5) into the following form

Δ (119880119899+1199011

(120585119899) 119880

119899+119901119896

(120585119899) 1198801015840

119899+1199011

(120585119899)

1198801015840

119899+119901119896

(120585119899) 119880

(119903)

119899+1199011

(120585119899) 119880

(119903)

119899+119901119896

(120585119899)) = 0

(7)

Step 2 We suppose the following series expansion as asolution of (7)

119880(120585119899) =

119896

sum119894=0

119886119894(120593 (120585119899))119894

(8)

where 119886119894= (0 1 119870) are arbitrary constants to be

determined later 120593(120585119899) has the following form

120593 (120585119899) =

1

119860 + 119890120585119899 (9)

and 119860 is a nonzero arbitrary constantFurther using the properties of expansion functions the

iterative relations can be written in the following form

119880119899+119901119904

(120585119899)

=

119870

sum119894=0

119886119894(

120593 (120585119899) 120593 (120590119904)

119860120593 (120585119899) 120593 (120590119904) (1 + 119860)minus119860 [120593 (120590

119904) + 120593 (120585

119899)]+1

)

119894

119880119899minus119901119904

(120585119899)

=

119870

sum119894=0

119886119894(

120593 (120585119899) minus 120593 (120590

119904) 120593 (120585119899) 119860

119860120593 (120585119899) + 120593 (120590

119904) minus 119860120593 (120590

119904) 120593 (120585119899) (1 + 119860)

)

119894

(10)

where

120585119899+119901119904

= 120585119899+ 120590119904 120590119904= 11990111990411198891+ 11990111990421198892+ sdot sdot sdot + 119901

119904119876119889119876 (11)

Step 3 Determine the degree119870 of (8) by balancing the high-est order nonlinear term(s) and the highest order derivativeof 119880(120585

119899) in (7)

Step 4 Substituting (8)ndash(10) and given the value of 119870 deter-mined in (7) Collecting all terms with the same power of120593(120585119899) the left-hand side of (7) is converted into polynomial

in 120593(120585119899) Setting each coefficient of this polynomial to be

zero we will derive a set of algebraic equations for 119860 119886119894=

(0 1 119870)

Step 5 Solving the overdetermined system of nonlinearalgebraic equations by using Mathematica or Maple we endup with explicit expressions of 119860 119886

119894

Abstract and Applied Analysis 3

Step 6 Using the results obtained in above steps we canfinally obtain exact solutions of (5)

3 Applications of the Modified TruncatedExpansion Method

In this section we apply the proposed modified truncatedexpansion method to construct the exact solutions for thenonlinear DDEs via the lattice equation the discrete non-linear Schrodinger equation with a saturable nonlinearitythe quintic discrete nonlinear Schrodinger equation and therelativistic Toda lattice system which are very important inthe mathematical physics and have been paid attention bymany researchers

31 Example 1 The General Lattice Equation In this subsec-tion we use the modified truncated expansion method tofind the exact solutions of the general lattice equation Thetraveling wave variable (6) permits us converting (1) into thefollowing form

11986211198801015840(120585119899) minus [120572 + 120573119880 (120585

119899) + 120574119880

2(120585119899)]

times [119880 (120585119899+ 119889) minus 119880 (120585

119899minus 119889)] = 0

(12)

where (1015840) = 119889119889120585119899 Considering the homogeneous balance

between the highest order derivatives and nonlinear term in(12) we get 119870 = 1 So we look for the solution of (12) in thefollowing form

119880 (120585119899) = 1198860+ 1198861120593 (120585119899) (13)

where 1198860and 119886

1are arbitrary constants to be determined

later and 120593(120585119899) satisfies (9) and (10) We substitute (13) (9)

and (10) into (12) and collect all terms with the same powerin [120593(120585

119899)]119894

(119894 = 0 1 2 ) Setting each coefficient of thispolynomial to zero we derive a set of algebraic equationsfor 1198860 1198861 119889 and 119862

1 Solving the set of algebraic equations by

using Maple or Mathematica we have the following results

1198860= minus

1

2

120573119890119889 + 120573 plusmn radicminus (119890119889 minus 1)2

(minus1205732+ 4120572120574)

(119890119889 + 1) 120574

1198861= plusmn

119860radicminus (119890119889 minus 1)2

(minus1205732+ 4120572120574)

(119890119889 + 1) 120574

1198621=(119890119889 minus 1) (4120572120574 minus 1205732)

(119890119889 + 1) 120574

(14)

where 120572 119860 119889 120573 and 120574 are arbitrary constants In this casethe exact wave solution takes the following form

119880 (120585119899) = minus

1

2

120573119890119889 + 120573 plusmn radicminus (119890119889 minus 1)2

(minus1205732+ 4120572120574)

(119890119889 + 1) 120574

plusmn119860radicminus (119890119889 minus 1)

2

(minus1205732+ 4120572120574)

(119890119889 + 1) 120574 (119860 + 119890120585119899)

(15)

where 120585119899= ((119890119889 minus 1)(4120572120574 minus 1205732)(119890119889 + 1)120574)119905 + 119899119889 + 120585

0

32 Example 2 The Discrete Nonlinear Schrodinger Equationwith a Saturable Nonlinearity If we take the transformation

120595119899= 119884 (120585

119899) 119890minus119894(120590119905+120588)

120585119899= 119889119899 + 120573 (16)

where 119889 120590 120573 and 120588 are arbitrary constants to be determinedlater

The transformation (16) leads to write (2) into the follow-ing form

(120590 minus 2) 119884 (120585119899) + 119884 (120585

119899+ 119889) + 119884 (120585

119899minus 119889) +

1205781198843 (120585119899)

1 + 1205831198842 (120585119899)= 0

(17)

We suppose the solution of (17) takes the form

119884 (120585119899) = 1198860+ 1198861120593 (120585119899) (18)

where 1198860and 1198861are arbitrary constants to be determined later

and 120593(120585119899) satisfies (9) and (10)

We substitute (18) (9) and (10) into (17) and collect allterms with the same power in [120593(120585

119899)]119894

(119894 = 0 1 2 )Setting each coefficient of this polynomial to zero we derivea set of algebraic equations for 119886

0 1198861 119889 and 119862

1 Solving the

set of algebraic equations by usingMaple orMathematica wehave the following results

1198860= plusmn

(119890120572 minus 1)

radicminus120583 (119890120572 + 1) 119886

1= plusmn

2119860 (1 minus 119890120572)

radicminus120583 (119890120572 + 1)

120590 =2(119890120572 minus 1)

2

(119890120572 + 1)2 120578 =

8119890120572120583

(119890120572 + 1)2

(19)

where 120572 120583 119860 and 119889 are arbitrary constants In this case theexact wave solution of (2) takes the following form

120595119899= (plusmn

(119890120572 minus 1)

radicminus120583 (119890120572 + 1)plusmn

2119860 (1 minus 119890120572)

radicminus120583 (119890120572 + 1)119860 + 119890120585119899)

times 119890minus119894((2(119890

120572minus1)2(119890120572+1)2)119905+120588)

(20)

where 120585119899= 119889119899 + 120573

33 Example 3 The Quintic Discrete Nonlinear SchrodingerEquation In this subsection we study the quintic discretenonlinear Schrodinger equation (3) by using the modifiedtruncated expansion method

If we take the transformation

120595119899= 119884119899119890119894120596119905 (21)

where 120596 is arbitrary constant to be determined laterThe transformation (21) leads to write (3) into the follow-

ing form

119884119899+1+ 119884119899minus1

=(2120572 minus 120596)119884

119899minus 1205731198843119899

120572 + 1205741198842119899+ 1205751198844119899

(22)

If we suppose

119884119899= 119880 (120585

119899) 120585

119899= 119889119899 + 119896 (23)

4 Abstract and Applied Analysis

Equation (23) leads to write (22) into the following form

119880 (120585119899+ 119889) + 119880 (120585

119899minus 119889) =

(2120572 minus 120596)119880 (120585119899) minus 1205731198803 (120585

119899)

120572 + 1205741198802 (120585119899) + 1205751198804 (120585

119899)

(24)

We suppose that the solution of (24) takes the form

119880 (120585119899) = 1198860+ 1198861120593 (120585119899) (25)

where 1198860and 119886

1are arbitrary constants to be determined

later and 120593(120585119899) satisfies (9) and (10) We substitute (25) (9)

and (10) into (24) and collect all terms with the same powerin [120593(120585

119899)]119894 (119894 = 0 1 2 ) Setting each coefficient of this

polynomial to zero we derive a set of algebraic equations for1198860 1198861 119889 and 119862

1 Solving the set of algebraic equations by

using Maple or Mathematica we have the following results

1198860= plusmn

1

4

radic2radic120575119890119889120573 (119890119889 minus 1)

120575119890119889

1198861= plusmn

1

2

radic2radic120575119890119889120573 (119890119889 minus 1)119860

120575119890119889

120596 = minus120573

32

[8120574119890119889(119890119889 minus 1)2

+ 120573((119890119889)2

minus 1)2

]

120575(119890119889)2

120572 = minus120573

64

(119890119889 + 1)

2

(8120574119890119889 + 120573(119890119889 + 1)2

)

120575(119890119889)2

(26)

where 120573119860 119889 120574 and 120575 are arbitrary constants In this case thesolitary wave solution takes the following form

119880 (120585119899) = plusmn

1

4

radic2radic120575119890119889120573 (119890119889 minus 1)

120575119890119889plusmn1

2

radic2radic120575119890119889120573 (119890119889 minus 1)119860

120575119890119889 (119860 + 119890120585119899)

(27)

Consequently the exact solution of (3) is given by

120595119899= (plusmn

1

4

radic2radic120575119890119889120573 (119890119889 minus 1)

120575119890119889plusmn1

2

radic2radic120575119890119889120573 (119890119889 minus 1)119860

120575119890119889 (119860 + 119890120585119899))

times 119890119894(minus(12057332)([8120574119890

119889(119890119889minus1)

2

+120573((119890119889)

2

minus1)

2

]120575(119890119889)

2

))119905

(28)

where 120585119899= 119899119889 + 119896

34 Example 4 The Relativistic Toda Lattice System In thissubsection we use themodified truncated expansionmethodto study the relativistic Toda lattice system (4) The travelingwave variables 119906

119899= 119880(120585

119899) 119907119899= 119881(120585

119899) and 120585

119899= 1198621119905 + 119899119889+ 120585

0

permit us to reduce (4) to the following nonlinear differencedifferential equations

11986211198801015840(120585119899) minus (1 + 120572119880 (120585

119899))

times (119881 (120585119899) minus 119881 (120585

119899minus 119889)) = 0

11986211198811015840(120585119899) minus 119881 (120585

119899) (119880 (120585

119899+ 119889) minus 119880 (120585

119899)

+120572119881 (120585119899+ 119889) minus 120572119881 (120585

119899minus 119889)) = 0

(29)

Considering the homogeneous balance between the highestorder derivatives and nonlinear terms in (30) we get 119896 = 1So we look for the solutions of (30) in the form

119880(120585119899) = 1198860+ 1198861120593 (120585119899) 119881 (120585

119899) = 1198870+ 1198871120593 (120585119899) (30)

where 1198860 1198870 1198861 and 119887

1are arbitrary constants to be deter-

mined later and 120593(120585119899) satisfies (9) and (10) We substitute

(31) (9) and (10) into (30) and collect all terms with the samepower in [120593(120585

119899)]119894 (119894 = 0 1 2 ) Setting each coefficient of

this polynomial to zero we derive a set of algebraic equationsfor 1198860 1198861 119889 and 119862

1 Solving the set of algebraic equations by

using Maple or Mathematica we have the following results

1198860= minus

(119890119889 minus 1) + 1205721198621

120572 (119890119889 minus 1) 119886

1= minus1198621119860

1198870=

1198621

120572 (119890119889 minus 1) 119887

1=1198621119860

120572

(31)

where 119860 119889 1198621 and 120572 are arbitrary constants In this case the

solitary wave solutions take the following form

119880 (120585119899) = minus

119890119889 minus 1 + 1205721198621

120572 (119890119889 minus 1)+minus1198621119860

119860 + 119890120585119899

119881 (120585119899) =

1198621

120572 (119890119889 minus 1)+

1198621119860

120572 (119860 + 119890120585119899)

(32)

where 120585119899= 1198621119905 + 119899119889 + 120585

0

4 Description of the Rational Solitary WaveFunctions Method

In this section we would like to outline the algorithm forusing the rational solitary wave functions method to solvenonlinear DDEs Consider a given system of119872 polynomialNDDEs

Δ (119906119899+1199011

(119883) 119906119899+119901119896

(119883) 1199061015840

119899+1199011

(119883) 1199061015840

119899+119901119896

(119883)

119906(119903)

119899+1199011

(119883) 119906(119903)

119899+119901119896

(119883)) = 0

(33)

where the dependent variable 119906 has 119872 components 119906119894 the

continuous variable 119909 has 119873 components 119909119895 the discrete

variable 119899 has 119876 components 119899119894 the 119896 shift vectors 119875

119904isin 119885119876

Abstract and Applied Analysis 5

and 119906(119903) denotes the collection of mixed derivative terms oforder 119903

The main steps of the algorithm for the rational solitarywave functions method to solve nonlinear DDEs are outlinedas follows

Step 1 We suppose the wave transformation in the followingform

119906119899+119901119904

(119883) = 119880 (120585119899+119901119904

) 120585119899=

119876

sum119894=1

119889119894119899119894+

119898

sum119895=1

119888119895119909119895+ 1205850

119904 = 1 2 119896

(34)

where the coefficients 119889119894(119894 = 1 119876) 119888

119895(119895 = 1 119873) and

the phase 1205850are constants The transformations (34) lead to

write (33) into the following form

Δ (119880119899+1199011

(120585119899) 119880

119899+119901119896

(120585119899) 1198801015840

119899+1199011

(120585119899)

1198801015840

119899+119901119896

(120585119899) 119880

(119903)

119899+1199011

(120585119899) 119880

(119903)

119899+119901119896

(120585119899)) = 0

(35)

Step 2 We suppose the rational solitary wave series expan-sion solutions of (35) in the following form

119880 (120585119899) =

119873

sum119894=0

119886119894[119892 (120585119899)]119894

+

119873

sum119895=1

119887119895[119892 (120585119899)]119895minus1

119891 (120585119899) (36)

with

119891 (120585119899) =

1

119860 tanh (120585119899) + 119861 sec ℎ (120585

119899)

119892 (120585119899) =

sec ℎ (120585119899)

119860 tanh (120585119899) + 119861 sec ℎ (120585

119899)

(37)

which satisfy

1198911015840(120585119899) = minus119860119892

2(120585119899) +

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) +

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) plusmn [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(38)

where 119886119894 119887119895 119860 and 119861 are arbitrary constants to be deter-

mined later and

120590119904= 1199011199041

1198891+ 1199011199042

1198892+ sdot sdot sdot + 119901

119904119876119889119876 (39)

Also we can assume that

119891 (120585119899) =

1

119860 tan (120585119899) + 119861 sec (120585

119899)

119892 (120585119899) =

sec (120585119899)

119860 tan (120585119899) + 119861 sec (120585

119899)

(40)

which satisfy

1198911015840(120585119899) = minus119860119892

2(120585119899) minus

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) minus

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) ∓ [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(41)

Equations (38) and (40) can be written into unified form

1198911015840(120585119899) = minus119860119892

2(120585119899) + 120588

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) + 120588

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) plusmn 120588 [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(42)

where 120588 = plusmn1

6 Abstract and Applied Analysis

Step 3 Determine the degree 119873 of (35) by balancing thehighest order nonlinear term(s) and the highest order deriva-tives of 119880(120585

119899) in (35)

Step 4 Substituting (36)ndash(41) and given the value of 119873determined in Step 3 into (35) and collecting all terms withthe same degree of 119891(120585

119899) and 119892(120585

119899) together the left-hand

side of (35) is converted into polynomial in 119891(120585119899) and 119892(120585

119899)

Then setting each coefficient 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) of this polynomial to zero we derive a set ofalgebraic equations for 119886

119894 119887119895 119862119894 119860 119861

Step 5 Solving the overdetermined system of nonlinearalgebraic equations by using Maple or Mathematica soft-ware package we end up with explicit expressions for119886119894 119887119895 119862119894 119860 and 119861

Step 6 Substituting 119886119894 119887119895 119862119894 119860 and 119861 into (36) along with

(37) and (39) we can finally obtain the rational solitary wavesolutions for nonlinear difference differential equations (33)

5 Applications

In this section we apply the proposed rational solitary wavefunctions method to construct the rational solitary wavesolutions for some nonlinear DDEs via the discrete non-linear Schrodinger equation with a saturable nonlinearitythe quintic discrete nonlinear Schrodinger equation and therelativistic Toda lattice system which are very important inthe mathematical physics and modern physics

51 Example 1 The Discrete Nonlinear Schrodinger Equationwith a Saturable Nonlinearity We suppose that the solutionof (17) takes the form

119880 (120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (43)

where 1198860 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (43)(41) into (17) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585

119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) to zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198861= plusmn119860radic

minus2

120578

sinh (1198892)cosh2 (1198892)

1198862= plusmnradic

minus2 (1198602 + 1198612)

120578

sinh (1198892)cosh2 (1198892)

120583 =1

4(1 + cosh (119889)) 120578

120590 =2 (cosh (119889) minus 1)1 + cos (119889)

1198860= 0

(44)

where 119860 119861 and 120578 are arbitrary constants and 120578 lt 0 In thiscase the rational hyperbolic solitary wave solution of (17)takes the following form

119880(120585119899) = plusmn 119860radic

minus2

120578

sinh (1198892)cosh2 (1198892) [119860 tanh (120585

119899) + 119861 sech (120585

119899)]

plusmn radicminus2 (1198602 + 1198612)

120578

sinh (1198892) sech (120585119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(45)

Consequently the rational hyperbolic solitary wave solu-tion of (2) has the following form

120595119899=[[

[

= plusmn119860radicminus2

120578

sinh (1198892)cosh2 (1198892) [119860 tanh (120585

119899) + 119861 sech (120585

119899)]

plusmnradicminus2 (1198602 + 1198612)

120578

sinh (1198892) sech (120585119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

]]

]

times 119890minus119894((2(cosh(119889)minus1)(1+cos(119889)))119905+120588)

(46)

where 120585119899= 119899119889 + 120573

Case 2 (120588 = minus1)

1198861= plusmn119860radic

minus2

120578

sin (1198892)cos2 (1198892)

1198862= plusmnradic

minus2 (1198602 minus 1198612)

120578

sin (1198892)cos2 (1198892)

120583 =1

4(cos (119889) + 1) 120578

120590 =2 (minus1 + 2 cos (119889))

cos (119889) + 1 119886

0= 0

(47)

where 119860 119861 and 120578 are arbitrary constants and 120578 lt 0 In thiscase the rational trigonometric solitary wave solution of (17)takes the following form

119880(120585119899) = plusmn 119860radic

minus2

120578

sin (1198892)cos2 (1198892) [119860 tan (120585

119899) + 119861 sec (120585

119899)]

plusmn radicminus2 (1198602 minus 1198612)

120578

timessin (1198892) sec (120585

119899)

cos2 (1198892) [119860 tan (120585119899) + 119861 sec (120585

119899)]

(48)

Abstract and Applied Analysis 7

Consequently the rational trigonometric solitary wave solu-tion of (2) has the following form

120595119899= [plusmn119860radic

minus2

120578

sin (1198892)cos2 (1198892) [119860 tan (120585

119899) + 119861 sec (120585

119899)]

plusmn radicminus2 (1198602 minus 1198612)

120578

timessin (1198892) sec (120585

119899)

cos2 (1198892) [119860 tan (120585119899) + 119861 sec (120585

119899)]]

times 119890minus119894((2(minus1+2 cos(119889))(cos(119889)+1))119905+120588)

(49)

where 120585119899= 119899119889 + 120573

52 Example 2 The Quintic Discrete Nonlinear SchrodingerEquation In this subsection we study the quintic discretenonlinear Schrodinger equation (3) by using the rationalsolitary wave functions method

We suppose that the solution of (24) takes the form

119880 (120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (50)

where 1198860 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (50) and(41) into (24) collect all terms with the same power in119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 = 0 1 2 ) and setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) to zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198861= plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

1198862= plusmn

1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) )

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

120572 =minus1

16

[4120574 (cos ℎ (119889) + 1)+120573 (1+2 cosh (119889) + cosh2 (119889))]cosh (119889) minus 1

120596 =minus1

8

120573 [4120574 (cos ℎ (119889) minus 1) + 120573 (cosh2 (119889) minus 1)]120575

1198860= 0

(51)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational hyperbolic solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

)

(52)

Consequently the rational hyperbolic solitary wave solutionof (3) has the following form

120595119899= [plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

) ]

times 119890119894((minus18)(120573[4120574(cos ℎ(119889)minus1)+120573(cosh2(119889)minus1)]120575))119905

(53)

where 120585119899= 119899119889 + 119896

Case 2 (120588 = minus1)

1198861= plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575

1198862= plusmn

1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))120575

8 Abstract and Applied Analysis

1198860= 0

120572 =minus1

16

120573 [(cos (119889) + 1) (4120574 + 2120573) minus 120573 sin2 (119889)]120575

120596 =minus1

8

120573 [4120574 (cos (119889) minus 1) minus 120573 sin2 (119889)]120575

(54)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational trigonometric solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

(55)

Consequently the rational trigonometric solitary wave solu-tion of (3) has the following form

120595119899= ( plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

)

times119890119894((minus18)(120573[4120574(cos(119889)minus1)minus120573 sin2(119889)]120575))119905

(56)

where 120585119899= 119899119889 + 119896

53 Example 3 The Relativistic Toda Lattice System In thissubsection we study the relativistic Toda lattice system (4)by using the rational solitary wave functions method

If we take the transformation

119907119899= minus

1

120572119906119899minus1

1205722 (57)

the transformation (57) reduced the relativistic Toda latticesystem (4) into the following difference differential equation

119906119899119905

= (119906119899+1

120572) (119906119899minus1minus 119906119899) (58)

According to the main steps of rational solitary wave func-tions method we seek traveling wave solutions of (58) in thefollowing form

119906119899(119905) = 119880 (120585

119899) 120585

119899= 1198621119905 + 119899119889 + 120585

0 (59)

where 119889 1198621 and 120585

0are constants The transformation (59)

permits us converting (58) into the following form

11986211198801015840(120585119899) = (119880 (120585

119899) +

1

120572) [119880 (120585

119899minus 119889) minus 119880 (120585

119899)] (60)

Considering the homogeneous balance between the highestorder derivatives and nonlinear terms in (60) we get119873 = 1Thus the solution of (60) has the following form

119880(120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (61)

where 1198860 1198870 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (61) and(41) into (60) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585119899) (119894 = 0 1 119895 =

0 1 2 ) to be zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198860=

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

1198861=

plusmn1198601198862

radic1198602 + 1198612 119862

1=

plusmn21198862

radic1198602 + 1198612

(62)

where 119860 119861 1198862 and 120572 are arbitrary constants In this case

the rational hyperbolic solitary wave solution of (58) has thefollowing form

119880 (120585119899) =

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

plusmn1198601198862

radic1198602 + 1198612 [119860 tanh (120585119899) + 119861 sech (120585

119899)]

+1198862sech (120585

119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(63)

where

120585119899= plusmn

21198862

radic1198602 + 1198612119905 + 119899119889 + 120585

0 (64)

Case 2 (120588 = minus1)

1198860=

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

1198861=

plusmn1198601198862

radic1198602 minus 1198612 119862

1=

plusmn21198862

radic1198602 minus 1198612

(65)

In this case the rational trigonometric solitary wave solutionof (58) has the following form

119880(120585119899) =

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

plusmn1198601198862

radic1198602 minus 1198612 [119860 tan (120585119899) + 119861 sec (120585

119899)]

+1198862sec (120585119899)

[119860 tan (120585119899) + 119861 sec (120585

119899)]

(66)

where

120585119899= plusmn

21198862

radic1198602 minus 1198612119905 + 119899119889 + 120585

0 (67)

Abstract and Applied Analysis 9

6 Discussion

Whenwe compare between the results which obtained in thispaper and other exact solutions we get the following

The solutions obtained in the modified truncated expan-sion functionsmethod are equivalent to the solution obtainedby the exp-functions method but the modified truncatedexpansion is simple and allowed us to solvemore complicatednonlinear difference differential equations such as the dis-crete nonlinear Schrodinger equation with a saturable non-linearity the quintic discrete nonlinear Schrodinger equa-tion and the relativistic Toda lattice system For example thesolution (15) equivalent is to the solution (40) in [20]

In the special case when 119861 = 0 in the rational solitarywave function method we get this method which is equiv-alent to the tanh-function method which discussed in [29]The rational solitary wave functionmethod is extended to thenew rational formal solution which is discussed by [30] when119887119895= 0 in (36)

Remarks These methods which are discussed in this paperallowed us to obtain some new rational solitary wave solu-tions for some complicated nonlinear differential differenceequations

These methods prefer to another methods to convertthe complicated rational methods into a direct nonrationalmethod

7 Conclusions

In this paper we use the modified truncated expansionmethod to obtain the exact solutions for some nonlineardifferential difference equations in the mathematical physicsAlso we calculate the rational solitary wave solutions for thenonlinear differential difference equations As a result manynew and more rational solitary wave solutions are obtainedfrom the hyperbolic function solutions and trigonometricfunction

Acknowledgment

The authors wish to thank the referees for their suggestionsand very useful comments

References

[1] E Fermi J Pasta and S Ulam Collected Papers of Enrico FermiII University of Chicago Press Chicago Ill USA 1965

[2] W P Su J R Schrieffer and A J Heeger ldquoSolitons inpolyacetylenerdquo Physical Review Letters vol 42 no 25 pp 1698ndash1701 1979

[3] A S Davydov ldquoThe theory of contraction of proteins undertheir excitationrdquo Journal ofTheoretical Biology vol 38 no 3 pp559ndash569 1973

[4] P Marquie J M Bilbault and M Remoissenet ldquoObservationof nonlinear localized modes in an electrical latticerdquo PhysicalReview E vol 51 no 6 pp 6127ndash6133 1995

[5] M Toda Theory of Nonlinear Lattices Springer Berlin Ger-many 1989

[6] M Wadati ldquoTransformation theories for nonlinear discretesystemsrdquo Progress ofTheoretical Physics vol 59 pp 36ndash63 1976

[7] YOhta andRHirota ldquoA discrete KdV equation and its Casoratideterminant solutionrdquo Journal of the Physical Society of Japanvol 60 p 2095 1991

[8] M J Ablowitz and J F Ladik ldquoNonlinear differential-differenceequationsrdquo Journal of Mathematical Physics vol 16 no 3 pp598ndash603 1974

[9] X B Hu and W X Ma ldquoApplication of Hirotarsquos bilinearformalism to the Toeplitz latticemdashsome special soliton-likesolutionsrdquo Physics Letters Section A vol 293 no 3-4 pp 161ndash165 2002

[10] D Baldwin U Goktas and W Hereman ldquoSymbolic computa-tion of hyperbolic tangent solutions for nonlinear differential-difference equationsrdquo Computer Physics Communications vol162 no 3 pp 203ndash217 2004

[11] S K Liu Z T Fu Z GWang and S D Liu ldquoPeriodic solutionsfor a class of nonlinear differential-difference equationsrdquo Com-munications in Theoretical Physics vol 49 no 5 pp 1155ndash11582008

[12] C Qiong and L Bin ldquoApplications of jacobi elliptic functionexpansion method for nonlinear differential-difference equa-tionsrdquo Communications inTheoretical Physics vol 43 no 3 pp385ndash388 2005

[13] F Xie M Ji and H Zhao ldquoSome solutions of discrete sine-Gordon equationrdquo Chaos Solitons and Fractals vol 33 no 5pp 1791ndash1795 2007

[14] S D Zhu ldquoExp-function method for the Hybrid-Lattice sys-temrdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 no 3 pp 461ndash464 2007

[15] I Aslan ldquoA discrete generalization of the extended simplestequation methodrdquo Communications in Nonlinear Science andNumerical Simulation vol 15 no 8 pp 1967ndash1973 2010

[16] P Yang Y Chen and Z B Li ldquoADM-Pade technique for thenonlinear lattice equationsrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 362ndash375 2009

[17] SD Zhu YMChu and S LQiu ldquoThehomotopy perturbationmethod for discontinued problems arising in nanotechnologyrdquoComputers andMathematics with Applications vol 58 no 11-12pp 2398ndash2401 2009

[18] F Xie ldquoNew travellingwave solutions of the generalized coupledHirota-Satsuma KdV systemrdquo Chaos Solitons and Fractals vol20 no 5 pp 1005ndash1012 2004

[19] E M E Zayed A M Abourabia K A Gepreel and MM El Horbaty ldquoOn the rational solitary wave solutions forthe nonlinear Hirota-Satsuma coupled KdV systemrdquoApplicableAnalysis vol 85 pp 751ndash768 2006

[20] C S Liu ldquoExponential function rational expansion method fornonlinear differential-difference equationsrdquoChaos Solitons andFractals vol 40 no 2 pp 708ndash716 2009

[21] S Zhang L Dong J M Ba and Y N Sun ldquoThe (frac(1198661015840119866))-expansion method for nonlinear differential-difference equa-tionsrdquo Physics Letters Section A vol 373 no 10 pp 905ndash9102009

[22] I Aslan ldquoThe Discrete (1198661015840119866)-expansion method applied tothe differential-difference Burgers equation and the relativisticToda lattice systemrdquo Numerical Methods for Partial DifferentialEquations vol 28 pp 127ndash137 2012

[23] I Aslan ldquoExact and explicit solutions to the discrete nonlinearSchrodinger equation with asaturable nonlinearityrdquo PhysicsLetters A vol 375 pp 4214ndash4217 2011

10 Abstract and Applied Analysis

[24] K A Gepreel ldquoRational Jacobi Elliptic solutions for nonlineardifference differential equationsrdquo Nonlinear Science Letters Avol 2 pp 151ndash158 2011

[25] K A Gepreel and A R Shehata ldquoRational Jacobi ellipticsolutions for nonlinear differential-difference lattice equationsrdquoApplied Mathematics Letters vol 25 pp 1173ndash1178 2012

[26] K A Gepreel T A Nofal and A A AlThobaiti ldquoThe modi-fied rational Jacobi elliptic solutions for nonlinear differentialdifference equationsrdquo Journal of Applied Mathematics vol 2012Article ID 427479 30 pages 2012

[27] G CWu and T C Xia ldquoA newmethod for constructing solitonsolutions to differential-difference equationwith symbolic com-putationrdquo Chaos Solitons and Fractals vol 39 no 5 pp 2245ndash2248 2009

[28] J Zhang Z Liu S Li and M Wang ldquoSolitary waves andstable analysis for the quintic discrete nonlinear Schrodingerequationrdquo Physica Scripta vol 86 pp 015401ndash015409 2012

[29] F Xie and J Wang ldquoA new method for solving nonlineardifferential-difference equationrdquo Chaos Solitons and Fractalsvol 27 no 4 pp 1067ndash1071 2006

[30] Q Wang and Y Yu ldquoNew rational formal solutions for (1 + 1)-dimensional Toda equation and another Toda equationrdquoChaosSolitons and Fractals vol 29 no 4 pp 904ndash915 2006

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

nonlinear difference differential equations in mathematicalphysics

(i) the general lattice equation [21 25 27]

119906119899119905

= (120572 + 120573119906119899+ 1205741199062

119899) (119906119899+1minus 119906119899minus1) (1)

(ii) the discrete nonlinear Schrodinger equation with asaturable nonlinearity [23 25]

119894120597120595119899

120597119905+ (120595119899+1+ 120595119899minus1minus 2120595119899) +

120578100381610038161003816100381612059511989910038161003816100381610038162

1 + 120583100381610038161003816100381612059511989910038161003816100381610038162120595119899= 0 (2)

(iii) the quintic discrete nonlinear Schrodinger equation[26 28]

119894120597120595119899

120597119905+ 120572 (120595

119899+1minus 2120595119899+ 120595119899minus1) + 120573

100381610038161003816100381612059511989910038161003816100381610038162

120595119899

+ 120574100381610038161003816100381612059511989910038161003816100381610038162

(120595119899+1+ 120595119899minus1) + 120575

100381610038161003816100381612059511989910038161003816100381610038164

(120595119899+1+ 120595119899minus1) = 0

(3)

(iv) the relativistic Toda lattice system [21]

119906119899119905

minus (1 + 120572119906119899) (119907119899minus 119907119899minus1) = 0

119907119899119905

minus 119907119899(119906119899+1minus 119906119899+ 120572119907119899+1minus 120572119907119899minus1) = 0

(4)

where 120572 120573 120574 120575 120583 and 120578 are arbitrary constants Also weput a rational solitary wave method to the nonlinear differ-ential difference equations We use the proposed method tofind the rational solitary wave solutions for the nonlineardifferential difference equations via the discrete nonlinearSchrodinger equation with a saturable nonlinearity thequintic discrete nonlinear Schrodinger equation and therelativistic Toda lattice equation

2 Description of the Modified TruncatedExpansion Method to Nonlinear DDEs

In this section we would like to outline the algorithm forusing the modified truncated expansion method to solve thenonlinear DDEs Consider a given system of119872 polynomialnonlinear DDEs

Δ (119906119899+1199011

(119883) 119906119899+119901119896

(119883) 1199061015840

119899+1199011

(119883) 1199061015840

119899+119901119896

(119883)

119906(119903)

119899+1199011

(119883) 119906(119903)

119899+119901119896

(119883)) = 0

(5)

where the dependent variable 119906 has 119872 components 119906119894 the

continuous variable 119909 has 119873 components 119909119895 the discrete

variable 119899 has 119876 components 119899119894 the 119896 shift vectors 119875

119904isin 119885119876

and 119906(119903) denotes the collection of mixed derivative terms oforder 119903

The main steps of the algorithm for the modified trun-cated expansion method to solve NDDEs are outlined asfollows

Step 1 We seek the traveling wave transformation in thefollowing form

119906119899+119901119904

(119883) = 119880 (120585119899+119901119904

) 120585119899=

119876

sum119894=1

119889119894119899119894+

119898

sum119895=1

119888119895119909119895+ 1205850

119904 = 1 2 119896

(6)

where the coefficients 119889119894(119894 = 1 119876) 119888

119895(119895 = 1 119873) and

the phase 1205850are constants The transformations (6) lead to

write (5) into the following form

Δ (119880119899+1199011

(120585119899) 119880

119899+119901119896

(120585119899) 1198801015840

119899+1199011

(120585119899)

1198801015840

119899+119901119896

(120585119899) 119880

(119903)

119899+1199011

(120585119899) 119880

(119903)

119899+119901119896

(120585119899)) = 0

(7)

Step 2 We suppose the following series expansion as asolution of (7)

119880(120585119899) =

119896

sum119894=0

119886119894(120593 (120585119899))119894

(8)

where 119886119894= (0 1 119870) are arbitrary constants to be

determined later 120593(120585119899) has the following form

120593 (120585119899) =

1

119860 + 119890120585119899 (9)

and 119860 is a nonzero arbitrary constantFurther using the properties of expansion functions the

iterative relations can be written in the following form

119880119899+119901119904

(120585119899)

=

119870

sum119894=0

119886119894(

120593 (120585119899) 120593 (120590119904)

119860120593 (120585119899) 120593 (120590119904) (1 + 119860)minus119860 [120593 (120590

119904) + 120593 (120585

119899)]+1

)

119894

119880119899minus119901119904

(120585119899)

=

119870

sum119894=0

119886119894(

120593 (120585119899) minus 120593 (120590

119904) 120593 (120585119899) 119860

119860120593 (120585119899) + 120593 (120590

119904) minus 119860120593 (120590

119904) 120593 (120585119899) (1 + 119860)

)

119894

(10)

where

120585119899+119901119904

= 120585119899+ 120590119904 120590119904= 11990111990411198891+ 11990111990421198892+ sdot sdot sdot + 119901

119904119876119889119876 (11)

Step 3 Determine the degree119870 of (8) by balancing the high-est order nonlinear term(s) and the highest order derivativeof 119880(120585

119899) in (7)

Step 4 Substituting (8)ndash(10) and given the value of 119870 deter-mined in (7) Collecting all terms with the same power of120593(120585119899) the left-hand side of (7) is converted into polynomial

in 120593(120585119899) Setting each coefficient of this polynomial to be

zero we will derive a set of algebraic equations for 119860 119886119894=

(0 1 119870)

Step 5 Solving the overdetermined system of nonlinearalgebraic equations by using Mathematica or Maple we endup with explicit expressions of 119860 119886

119894

Abstract and Applied Analysis 3

Step 6 Using the results obtained in above steps we canfinally obtain exact solutions of (5)

3 Applications of the Modified TruncatedExpansion Method

In this section we apply the proposed modified truncatedexpansion method to construct the exact solutions for thenonlinear DDEs via the lattice equation the discrete non-linear Schrodinger equation with a saturable nonlinearitythe quintic discrete nonlinear Schrodinger equation and therelativistic Toda lattice system which are very important inthe mathematical physics and have been paid attention bymany researchers

31 Example 1 The General Lattice Equation In this subsec-tion we use the modified truncated expansion method tofind the exact solutions of the general lattice equation Thetraveling wave variable (6) permits us converting (1) into thefollowing form

11986211198801015840(120585119899) minus [120572 + 120573119880 (120585

119899) + 120574119880

2(120585119899)]

times [119880 (120585119899+ 119889) minus 119880 (120585

119899minus 119889)] = 0

(12)

where (1015840) = 119889119889120585119899 Considering the homogeneous balance

between the highest order derivatives and nonlinear term in(12) we get 119870 = 1 So we look for the solution of (12) in thefollowing form

119880 (120585119899) = 1198860+ 1198861120593 (120585119899) (13)

where 1198860and 119886

1are arbitrary constants to be determined

later and 120593(120585119899) satisfies (9) and (10) We substitute (13) (9)

and (10) into (12) and collect all terms with the same powerin [120593(120585

119899)]119894

(119894 = 0 1 2 ) Setting each coefficient of thispolynomial to zero we derive a set of algebraic equationsfor 1198860 1198861 119889 and 119862

1 Solving the set of algebraic equations by

using Maple or Mathematica we have the following results

1198860= minus

1

2

120573119890119889 + 120573 plusmn radicminus (119890119889 minus 1)2

(minus1205732+ 4120572120574)

(119890119889 + 1) 120574

1198861= plusmn

119860radicminus (119890119889 minus 1)2

(minus1205732+ 4120572120574)

(119890119889 + 1) 120574

1198621=(119890119889 minus 1) (4120572120574 minus 1205732)

(119890119889 + 1) 120574

(14)

where 120572 119860 119889 120573 and 120574 are arbitrary constants In this casethe exact wave solution takes the following form

119880 (120585119899) = minus

1

2

120573119890119889 + 120573 plusmn radicminus (119890119889 minus 1)2

(minus1205732+ 4120572120574)

(119890119889 + 1) 120574

plusmn119860radicminus (119890119889 minus 1)

2

(minus1205732+ 4120572120574)

(119890119889 + 1) 120574 (119860 + 119890120585119899)

(15)

where 120585119899= ((119890119889 minus 1)(4120572120574 minus 1205732)(119890119889 + 1)120574)119905 + 119899119889 + 120585

0

32 Example 2 The Discrete Nonlinear Schrodinger Equationwith a Saturable Nonlinearity If we take the transformation

120595119899= 119884 (120585

119899) 119890minus119894(120590119905+120588)

120585119899= 119889119899 + 120573 (16)

where 119889 120590 120573 and 120588 are arbitrary constants to be determinedlater

The transformation (16) leads to write (2) into the follow-ing form

(120590 minus 2) 119884 (120585119899) + 119884 (120585

119899+ 119889) + 119884 (120585

119899minus 119889) +

1205781198843 (120585119899)

1 + 1205831198842 (120585119899)= 0

(17)

We suppose the solution of (17) takes the form

119884 (120585119899) = 1198860+ 1198861120593 (120585119899) (18)

where 1198860and 1198861are arbitrary constants to be determined later

and 120593(120585119899) satisfies (9) and (10)

We substitute (18) (9) and (10) into (17) and collect allterms with the same power in [120593(120585

119899)]119894

(119894 = 0 1 2 )Setting each coefficient of this polynomial to zero we derivea set of algebraic equations for 119886

0 1198861 119889 and 119862

1 Solving the

set of algebraic equations by usingMaple orMathematica wehave the following results

1198860= plusmn

(119890120572 minus 1)

radicminus120583 (119890120572 + 1) 119886

1= plusmn

2119860 (1 minus 119890120572)

radicminus120583 (119890120572 + 1)

120590 =2(119890120572 minus 1)

2

(119890120572 + 1)2 120578 =

8119890120572120583

(119890120572 + 1)2

(19)

where 120572 120583 119860 and 119889 are arbitrary constants In this case theexact wave solution of (2) takes the following form

120595119899= (plusmn

(119890120572 minus 1)

radicminus120583 (119890120572 + 1)plusmn

2119860 (1 minus 119890120572)

radicminus120583 (119890120572 + 1)119860 + 119890120585119899)

times 119890minus119894((2(119890

120572minus1)2(119890120572+1)2)119905+120588)

(20)

where 120585119899= 119889119899 + 120573

33 Example 3 The Quintic Discrete Nonlinear SchrodingerEquation In this subsection we study the quintic discretenonlinear Schrodinger equation (3) by using the modifiedtruncated expansion method

If we take the transformation

120595119899= 119884119899119890119894120596119905 (21)

where 120596 is arbitrary constant to be determined laterThe transformation (21) leads to write (3) into the follow-

ing form

119884119899+1+ 119884119899minus1

=(2120572 minus 120596)119884

119899minus 1205731198843119899

120572 + 1205741198842119899+ 1205751198844119899

(22)

If we suppose

119884119899= 119880 (120585

119899) 120585

119899= 119889119899 + 119896 (23)

4 Abstract and Applied Analysis

Equation (23) leads to write (22) into the following form

119880 (120585119899+ 119889) + 119880 (120585

119899minus 119889) =

(2120572 minus 120596)119880 (120585119899) minus 1205731198803 (120585

119899)

120572 + 1205741198802 (120585119899) + 1205751198804 (120585

119899)

(24)

We suppose that the solution of (24) takes the form

119880 (120585119899) = 1198860+ 1198861120593 (120585119899) (25)

where 1198860and 119886

1are arbitrary constants to be determined

later and 120593(120585119899) satisfies (9) and (10) We substitute (25) (9)

and (10) into (24) and collect all terms with the same powerin [120593(120585

119899)]119894 (119894 = 0 1 2 ) Setting each coefficient of this

polynomial to zero we derive a set of algebraic equations for1198860 1198861 119889 and 119862

1 Solving the set of algebraic equations by

using Maple or Mathematica we have the following results

1198860= plusmn

1

4

radic2radic120575119890119889120573 (119890119889 minus 1)

120575119890119889

1198861= plusmn

1

2

radic2radic120575119890119889120573 (119890119889 minus 1)119860

120575119890119889

120596 = minus120573

32

[8120574119890119889(119890119889 minus 1)2

+ 120573((119890119889)2

minus 1)2

]

120575(119890119889)2

120572 = minus120573

64

(119890119889 + 1)

2

(8120574119890119889 + 120573(119890119889 + 1)2

)

120575(119890119889)2

(26)

where 120573119860 119889 120574 and 120575 are arbitrary constants In this case thesolitary wave solution takes the following form

119880 (120585119899) = plusmn

1

4

radic2radic120575119890119889120573 (119890119889 minus 1)

120575119890119889plusmn1

2

radic2radic120575119890119889120573 (119890119889 minus 1)119860

120575119890119889 (119860 + 119890120585119899)

(27)

Consequently the exact solution of (3) is given by

120595119899= (plusmn

1

4

radic2radic120575119890119889120573 (119890119889 minus 1)

120575119890119889plusmn1

2

radic2radic120575119890119889120573 (119890119889 minus 1)119860

120575119890119889 (119860 + 119890120585119899))

times 119890119894(minus(12057332)([8120574119890

119889(119890119889minus1)

2

+120573((119890119889)

2

minus1)

2

]120575(119890119889)

2

))119905

(28)

where 120585119899= 119899119889 + 119896

34 Example 4 The Relativistic Toda Lattice System In thissubsection we use themodified truncated expansionmethodto study the relativistic Toda lattice system (4) The travelingwave variables 119906

119899= 119880(120585

119899) 119907119899= 119881(120585

119899) and 120585

119899= 1198621119905 + 119899119889+ 120585

0

permit us to reduce (4) to the following nonlinear differencedifferential equations

11986211198801015840(120585119899) minus (1 + 120572119880 (120585

119899))

times (119881 (120585119899) minus 119881 (120585

119899minus 119889)) = 0

11986211198811015840(120585119899) minus 119881 (120585

119899) (119880 (120585

119899+ 119889) minus 119880 (120585

119899)

+120572119881 (120585119899+ 119889) minus 120572119881 (120585

119899minus 119889)) = 0

(29)

Considering the homogeneous balance between the highestorder derivatives and nonlinear terms in (30) we get 119896 = 1So we look for the solutions of (30) in the form

119880(120585119899) = 1198860+ 1198861120593 (120585119899) 119881 (120585

119899) = 1198870+ 1198871120593 (120585119899) (30)

where 1198860 1198870 1198861 and 119887

1are arbitrary constants to be deter-

mined later and 120593(120585119899) satisfies (9) and (10) We substitute

(31) (9) and (10) into (30) and collect all terms with the samepower in [120593(120585

119899)]119894 (119894 = 0 1 2 ) Setting each coefficient of

this polynomial to zero we derive a set of algebraic equationsfor 1198860 1198861 119889 and 119862

1 Solving the set of algebraic equations by

using Maple or Mathematica we have the following results

1198860= minus

(119890119889 minus 1) + 1205721198621

120572 (119890119889 minus 1) 119886

1= minus1198621119860

1198870=

1198621

120572 (119890119889 minus 1) 119887

1=1198621119860

120572

(31)

where 119860 119889 1198621 and 120572 are arbitrary constants In this case the

solitary wave solutions take the following form

119880 (120585119899) = minus

119890119889 minus 1 + 1205721198621

120572 (119890119889 minus 1)+minus1198621119860

119860 + 119890120585119899

119881 (120585119899) =

1198621

120572 (119890119889 minus 1)+

1198621119860

120572 (119860 + 119890120585119899)

(32)

where 120585119899= 1198621119905 + 119899119889 + 120585

0

4 Description of the Rational Solitary WaveFunctions Method

In this section we would like to outline the algorithm forusing the rational solitary wave functions method to solvenonlinear DDEs Consider a given system of119872 polynomialNDDEs

Δ (119906119899+1199011

(119883) 119906119899+119901119896

(119883) 1199061015840

119899+1199011

(119883) 1199061015840

119899+119901119896

(119883)

119906(119903)

119899+1199011

(119883) 119906(119903)

119899+119901119896

(119883)) = 0

(33)

where the dependent variable 119906 has 119872 components 119906119894 the

continuous variable 119909 has 119873 components 119909119895 the discrete

variable 119899 has 119876 components 119899119894 the 119896 shift vectors 119875

119904isin 119885119876

Abstract and Applied Analysis 5

and 119906(119903) denotes the collection of mixed derivative terms oforder 119903

The main steps of the algorithm for the rational solitarywave functions method to solve nonlinear DDEs are outlinedas follows

Step 1 We suppose the wave transformation in the followingform

119906119899+119901119904

(119883) = 119880 (120585119899+119901119904

) 120585119899=

119876

sum119894=1

119889119894119899119894+

119898

sum119895=1

119888119895119909119895+ 1205850

119904 = 1 2 119896

(34)

where the coefficients 119889119894(119894 = 1 119876) 119888

119895(119895 = 1 119873) and

the phase 1205850are constants The transformations (34) lead to

write (33) into the following form

Δ (119880119899+1199011

(120585119899) 119880

119899+119901119896

(120585119899) 1198801015840

119899+1199011

(120585119899)

1198801015840

119899+119901119896

(120585119899) 119880

(119903)

119899+1199011

(120585119899) 119880

(119903)

119899+119901119896

(120585119899)) = 0

(35)

Step 2 We suppose the rational solitary wave series expan-sion solutions of (35) in the following form

119880 (120585119899) =

119873

sum119894=0

119886119894[119892 (120585119899)]119894

+

119873

sum119895=1

119887119895[119892 (120585119899)]119895minus1

119891 (120585119899) (36)

with

119891 (120585119899) =

1

119860 tanh (120585119899) + 119861 sec ℎ (120585

119899)

119892 (120585119899) =

sec ℎ (120585119899)

119860 tanh (120585119899) + 119861 sec ℎ (120585

119899)

(37)

which satisfy

1198911015840(120585119899) = minus119860119892

2(120585119899) +

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) +

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) plusmn [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(38)

where 119886119894 119887119895 119860 and 119861 are arbitrary constants to be deter-

mined later and

120590119904= 1199011199041

1198891+ 1199011199042

1198892+ sdot sdot sdot + 119901

119904119876119889119876 (39)

Also we can assume that

119891 (120585119899) =

1

119860 tan (120585119899) + 119861 sec (120585

119899)

119892 (120585119899) =

sec (120585119899)

119860 tan (120585119899) + 119861 sec (120585

119899)

(40)

which satisfy

1198911015840(120585119899) = minus119860119892

2(120585119899) minus

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) minus

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) ∓ [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(41)

Equations (38) and (40) can be written into unified form

1198911015840(120585119899) = minus119860119892

2(120585119899) + 120588

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) + 120588

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) plusmn 120588 [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(42)

where 120588 = plusmn1

6 Abstract and Applied Analysis

Step 3 Determine the degree 119873 of (35) by balancing thehighest order nonlinear term(s) and the highest order deriva-tives of 119880(120585

119899) in (35)

Step 4 Substituting (36)ndash(41) and given the value of 119873determined in Step 3 into (35) and collecting all terms withthe same degree of 119891(120585

119899) and 119892(120585

119899) together the left-hand

side of (35) is converted into polynomial in 119891(120585119899) and 119892(120585

119899)

Then setting each coefficient 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) of this polynomial to zero we derive a set ofalgebraic equations for 119886

119894 119887119895 119862119894 119860 119861

Step 5 Solving the overdetermined system of nonlinearalgebraic equations by using Maple or Mathematica soft-ware package we end up with explicit expressions for119886119894 119887119895 119862119894 119860 and 119861

Step 6 Substituting 119886119894 119887119895 119862119894 119860 and 119861 into (36) along with

(37) and (39) we can finally obtain the rational solitary wavesolutions for nonlinear difference differential equations (33)

5 Applications

In this section we apply the proposed rational solitary wavefunctions method to construct the rational solitary wavesolutions for some nonlinear DDEs via the discrete non-linear Schrodinger equation with a saturable nonlinearitythe quintic discrete nonlinear Schrodinger equation and therelativistic Toda lattice system which are very important inthe mathematical physics and modern physics

51 Example 1 The Discrete Nonlinear Schrodinger Equationwith a Saturable Nonlinearity We suppose that the solutionof (17) takes the form

119880 (120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (43)

where 1198860 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (43)(41) into (17) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585

119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) to zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198861= plusmn119860radic

minus2

120578

sinh (1198892)cosh2 (1198892)

1198862= plusmnradic

minus2 (1198602 + 1198612)

120578

sinh (1198892)cosh2 (1198892)

120583 =1

4(1 + cosh (119889)) 120578

120590 =2 (cosh (119889) minus 1)1 + cos (119889)

1198860= 0

(44)

where 119860 119861 and 120578 are arbitrary constants and 120578 lt 0 In thiscase the rational hyperbolic solitary wave solution of (17)takes the following form

119880(120585119899) = plusmn 119860radic

minus2

120578

sinh (1198892)cosh2 (1198892) [119860 tanh (120585

119899) + 119861 sech (120585

119899)]

plusmn radicminus2 (1198602 + 1198612)

120578

sinh (1198892) sech (120585119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(45)

Consequently the rational hyperbolic solitary wave solu-tion of (2) has the following form

120595119899=[[

[

= plusmn119860radicminus2

120578

sinh (1198892)cosh2 (1198892) [119860 tanh (120585

119899) + 119861 sech (120585

119899)]

plusmnradicminus2 (1198602 + 1198612)

120578

sinh (1198892) sech (120585119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

]]

]

times 119890minus119894((2(cosh(119889)minus1)(1+cos(119889)))119905+120588)

(46)

where 120585119899= 119899119889 + 120573

Case 2 (120588 = minus1)

1198861= plusmn119860radic

minus2

120578

sin (1198892)cos2 (1198892)

1198862= plusmnradic

minus2 (1198602 minus 1198612)

120578

sin (1198892)cos2 (1198892)

120583 =1

4(cos (119889) + 1) 120578

120590 =2 (minus1 + 2 cos (119889))

cos (119889) + 1 119886

0= 0

(47)

where 119860 119861 and 120578 are arbitrary constants and 120578 lt 0 In thiscase the rational trigonometric solitary wave solution of (17)takes the following form

119880(120585119899) = plusmn 119860radic

minus2

120578

sin (1198892)cos2 (1198892) [119860 tan (120585

119899) + 119861 sec (120585

119899)]

plusmn radicminus2 (1198602 minus 1198612)

120578

timessin (1198892) sec (120585

119899)

cos2 (1198892) [119860 tan (120585119899) + 119861 sec (120585

119899)]

(48)

Abstract and Applied Analysis 7

Consequently the rational trigonometric solitary wave solu-tion of (2) has the following form

120595119899= [plusmn119860radic

minus2

120578

sin (1198892)cos2 (1198892) [119860 tan (120585

119899) + 119861 sec (120585

119899)]

plusmn radicminus2 (1198602 minus 1198612)

120578

timessin (1198892) sec (120585

119899)

cos2 (1198892) [119860 tan (120585119899) + 119861 sec (120585

119899)]]

times 119890minus119894((2(minus1+2 cos(119889))(cos(119889)+1))119905+120588)

(49)

where 120585119899= 119899119889 + 120573

52 Example 2 The Quintic Discrete Nonlinear SchrodingerEquation In this subsection we study the quintic discretenonlinear Schrodinger equation (3) by using the rationalsolitary wave functions method

We suppose that the solution of (24) takes the form

119880 (120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (50)

where 1198860 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (50) and(41) into (24) collect all terms with the same power in119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 = 0 1 2 ) and setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) to zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198861= plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

1198862= plusmn

1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) )

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

120572 =minus1

16

[4120574 (cos ℎ (119889) + 1)+120573 (1+2 cosh (119889) + cosh2 (119889))]cosh (119889) minus 1

120596 =minus1

8

120573 [4120574 (cos ℎ (119889) minus 1) + 120573 (cosh2 (119889) minus 1)]120575

1198860= 0

(51)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational hyperbolic solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

)

(52)

Consequently the rational hyperbolic solitary wave solutionof (3) has the following form

120595119899= [plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

) ]

times 119890119894((minus18)(120573[4120574(cos ℎ(119889)minus1)+120573(cosh2(119889)minus1)]120575))119905

(53)

where 120585119899= 119899119889 + 119896

Case 2 (120588 = minus1)

1198861= plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575

1198862= plusmn

1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))120575

8 Abstract and Applied Analysis

1198860= 0

120572 =minus1

16

120573 [(cos (119889) + 1) (4120574 + 2120573) minus 120573 sin2 (119889)]120575

120596 =minus1

8

120573 [4120574 (cos (119889) minus 1) minus 120573 sin2 (119889)]120575

(54)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational trigonometric solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

(55)

Consequently the rational trigonometric solitary wave solu-tion of (3) has the following form

120595119899= ( plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

)

times119890119894((minus18)(120573[4120574(cos(119889)minus1)minus120573 sin2(119889)]120575))119905

(56)

where 120585119899= 119899119889 + 119896

53 Example 3 The Relativistic Toda Lattice System In thissubsection we study the relativistic Toda lattice system (4)by using the rational solitary wave functions method

If we take the transformation

119907119899= minus

1

120572119906119899minus1

1205722 (57)

the transformation (57) reduced the relativistic Toda latticesystem (4) into the following difference differential equation

119906119899119905

= (119906119899+1

120572) (119906119899minus1minus 119906119899) (58)

According to the main steps of rational solitary wave func-tions method we seek traveling wave solutions of (58) in thefollowing form

119906119899(119905) = 119880 (120585

119899) 120585

119899= 1198621119905 + 119899119889 + 120585

0 (59)

where 119889 1198621 and 120585

0are constants The transformation (59)

permits us converting (58) into the following form

11986211198801015840(120585119899) = (119880 (120585

119899) +

1

120572) [119880 (120585

119899minus 119889) minus 119880 (120585

119899)] (60)

Considering the homogeneous balance between the highestorder derivatives and nonlinear terms in (60) we get119873 = 1Thus the solution of (60) has the following form

119880(120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (61)

where 1198860 1198870 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (61) and(41) into (60) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585119899) (119894 = 0 1 119895 =

0 1 2 ) to be zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198860=

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

1198861=

plusmn1198601198862

radic1198602 + 1198612 119862

1=

plusmn21198862

radic1198602 + 1198612

(62)

where 119860 119861 1198862 and 120572 are arbitrary constants In this case

the rational hyperbolic solitary wave solution of (58) has thefollowing form

119880 (120585119899) =

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

plusmn1198601198862

radic1198602 + 1198612 [119860 tanh (120585119899) + 119861 sech (120585

119899)]

+1198862sech (120585

119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(63)

where

120585119899= plusmn

21198862

radic1198602 + 1198612119905 + 119899119889 + 120585

0 (64)

Case 2 (120588 = minus1)

1198860=

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

1198861=

plusmn1198601198862

radic1198602 minus 1198612 119862

1=

plusmn21198862

radic1198602 minus 1198612

(65)

In this case the rational trigonometric solitary wave solutionof (58) has the following form

119880(120585119899) =

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

plusmn1198601198862

radic1198602 minus 1198612 [119860 tan (120585119899) + 119861 sec (120585

119899)]

+1198862sec (120585119899)

[119860 tan (120585119899) + 119861 sec (120585

119899)]

(66)

where

120585119899= plusmn

21198862

radic1198602 minus 1198612119905 + 119899119889 + 120585

0 (67)

Abstract and Applied Analysis 9

6 Discussion

Whenwe compare between the results which obtained in thispaper and other exact solutions we get the following

The solutions obtained in the modified truncated expan-sion functionsmethod are equivalent to the solution obtainedby the exp-functions method but the modified truncatedexpansion is simple and allowed us to solvemore complicatednonlinear difference differential equations such as the dis-crete nonlinear Schrodinger equation with a saturable non-linearity the quintic discrete nonlinear Schrodinger equa-tion and the relativistic Toda lattice system For example thesolution (15) equivalent is to the solution (40) in [20]

In the special case when 119861 = 0 in the rational solitarywave function method we get this method which is equiv-alent to the tanh-function method which discussed in [29]The rational solitary wave functionmethod is extended to thenew rational formal solution which is discussed by [30] when119887119895= 0 in (36)

Remarks These methods which are discussed in this paperallowed us to obtain some new rational solitary wave solu-tions for some complicated nonlinear differential differenceequations

These methods prefer to another methods to convertthe complicated rational methods into a direct nonrationalmethod

7 Conclusions

In this paper we use the modified truncated expansionmethod to obtain the exact solutions for some nonlineardifferential difference equations in the mathematical physicsAlso we calculate the rational solitary wave solutions for thenonlinear differential difference equations As a result manynew and more rational solitary wave solutions are obtainedfrom the hyperbolic function solutions and trigonometricfunction

Acknowledgment

The authors wish to thank the referees for their suggestionsand very useful comments

References

[1] E Fermi J Pasta and S Ulam Collected Papers of Enrico FermiII University of Chicago Press Chicago Ill USA 1965

[2] W P Su J R Schrieffer and A J Heeger ldquoSolitons inpolyacetylenerdquo Physical Review Letters vol 42 no 25 pp 1698ndash1701 1979

[3] A S Davydov ldquoThe theory of contraction of proteins undertheir excitationrdquo Journal ofTheoretical Biology vol 38 no 3 pp559ndash569 1973

[4] P Marquie J M Bilbault and M Remoissenet ldquoObservationof nonlinear localized modes in an electrical latticerdquo PhysicalReview E vol 51 no 6 pp 6127ndash6133 1995

[5] M Toda Theory of Nonlinear Lattices Springer Berlin Ger-many 1989

[6] M Wadati ldquoTransformation theories for nonlinear discretesystemsrdquo Progress ofTheoretical Physics vol 59 pp 36ndash63 1976

[7] YOhta andRHirota ldquoA discrete KdV equation and its Casoratideterminant solutionrdquo Journal of the Physical Society of Japanvol 60 p 2095 1991

[8] M J Ablowitz and J F Ladik ldquoNonlinear differential-differenceequationsrdquo Journal of Mathematical Physics vol 16 no 3 pp598ndash603 1974

[9] X B Hu and W X Ma ldquoApplication of Hirotarsquos bilinearformalism to the Toeplitz latticemdashsome special soliton-likesolutionsrdquo Physics Letters Section A vol 293 no 3-4 pp 161ndash165 2002

[10] D Baldwin U Goktas and W Hereman ldquoSymbolic computa-tion of hyperbolic tangent solutions for nonlinear differential-difference equationsrdquo Computer Physics Communications vol162 no 3 pp 203ndash217 2004

[11] S K Liu Z T Fu Z GWang and S D Liu ldquoPeriodic solutionsfor a class of nonlinear differential-difference equationsrdquo Com-munications in Theoretical Physics vol 49 no 5 pp 1155ndash11582008

[12] C Qiong and L Bin ldquoApplications of jacobi elliptic functionexpansion method for nonlinear differential-difference equa-tionsrdquo Communications inTheoretical Physics vol 43 no 3 pp385ndash388 2005

[13] F Xie M Ji and H Zhao ldquoSome solutions of discrete sine-Gordon equationrdquo Chaos Solitons and Fractals vol 33 no 5pp 1791ndash1795 2007

[14] S D Zhu ldquoExp-function method for the Hybrid-Lattice sys-temrdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 no 3 pp 461ndash464 2007

[15] I Aslan ldquoA discrete generalization of the extended simplestequation methodrdquo Communications in Nonlinear Science andNumerical Simulation vol 15 no 8 pp 1967ndash1973 2010

[16] P Yang Y Chen and Z B Li ldquoADM-Pade technique for thenonlinear lattice equationsrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 362ndash375 2009

[17] SD Zhu YMChu and S LQiu ldquoThehomotopy perturbationmethod for discontinued problems arising in nanotechnologyrdquoComputers andMathematics with Applications vol 58 no 11-12pp 2398ndash2401 2009

[18] F Xie ldquoNew travellingwave solutions of the generalized coupledHirota-Satsuma KdV systemrdquo Chaos Solitons and Fractals vol20 no 5 pp 1005ndash1012 2004

[19] E M E Zayed A M Abourabia K A Gepreel and MM El Horbaty ldquoOn the rational solitary wave solutions forthe nonlinear Hirota-Satsuma coupled KdV systemrdquoApplicableAnalysis vol 85 pp 751ndash768 2006

[20] C S Liu ldquoExponential function rational expansion method fornonlinear differential-difference equationsrdquoChaos Solitons andFractals vol 40 no 2 pp 708ndash716 2009

[21] S Zhang L Dong J M Ba and Y N Sun ldquoThe (frac(1198661015840119866))-expansion method for nonlinear differential-difference equa-tionsrdquo Physics Letters Section A vol 373 no 10 pp 905ndash9102009

[22] I Aslan ldquoThe Discrete (1198661015840119866)-expansion method applied tothe differential-difference Burgers equation and the relativisticToda lattice systemrdquo Numerical Methods for Partial DifferentialEquations vol 28 pp 127ndash137 2012

[23] I Aslan ldquoExact and explicit solutions to the discrete nonlinearSchrodinger equation with asaturable nonlinearityrdquo PhysicsLetters A vol 375 pp 4214ndash4217 2011

10 Abstract and Applied Analysis

[24] K A Gepreel ldquoRational Jacobi Elliptic solutions for nonlineardifference differential equationsrdquo Nonlinear Science Letters Avol 2 pp 151ndash158 2011

[25] K A Gepreel and A R Shehata ldquoRational Jacobi ellipticsolutions for nonlinear differential-difference lattice equationsrdquoApplied Mathematics Letters vol 25 pp 1173ndash1178 2012

[26] K A Gepreel T A Nofal and A A AlThobaiti ldquoThe modi-fied rational Jacobi elliptic solutions for nonlinear differentialdifference equationsrdquo Journal of Applied Mathematics vol 2012Article ID 427479 30 pages 2012

[27] G CWu and T C Xia ldquoA newmethod for constructing solitonsolutions to differential-difference equationwith symbolic com-putationrdquo Chaos Solitons and Fractals vol 39 no 5 pp 2245ndash2248 2009

[28] J Zhang Z Liu S Li and M Wang ldquoSolitary waves andstable analysis for the quintic discrete nonlinear Schrodingerequationrdquo Physica Scripta vol 86 pp 015401ndash015409 2012

[29] F Xie and J Wang ldquoA new method for solving nonlineardifferential-difference equationrdquo Chaos Solitons and Fractalsvol 27 no 4 pp 1067ndash1071 2006

[30] Q Wang and Y Yu ldquoNew rational formal solutions for (1 + 1)-dimensional Toda equation and another Toda equationrdquoChaosSolitons and Fractals vol 29 no 4 pp 904ndash915 2006

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Step 6 Using the results obtained in above steps we canfinally obtain exact solutions of (5)

3 Applications of the Modified TruncatedExpansion Method

In this section we apply the proposed modified truncatedexpansion method to construct the exact solutions for thenonlinear DDEs via the lattice equation the discrete non-linear Schrodinger equation with a saturable nonlinearitythe quintic discrete nonlinear Schrodinger equation and therelativistic Toda lattice system which are very important inthe mathematical physics and have been paid attention bymany researchers

31 Example 1 The General Lattice Equation In this subsec-tion we use the modified truncated expansion method tofind the exact solutions of the general lattice equation Thetraveling wave variable (6) permits us converting (1) into thefollowing form

11986211198801015840(120585119899) minus [120572 + 120573119880 (120585

119899) + 120574119880

2(120585119899)]

times [119880 (120585119899+ 119889) minus 119880 (120585

119899minus 119889)] = 0

(12)

where (1015840) = 119889119889120585119899 Considering the homogeneous balance

between the highest order derivatives and nonlinear term in(12) we get 119870 = 1 So we look for the solution of (12) in thefollowing form

119880 (120585119899) = 1198860+ 1198861120593 (120585119899) (13)

where 1198860and 119886

1are arbitrary constants to be determined

later and 120593(120585119899) satisfies (9) and (10) We substitute (13) (9)

and (10) into (12) and collect all terms with the same powerin [120593(120585

119899)]119894

(119894 = 0 1 2 ) Setting each coefficient of thispolynomial to zero we derive a set of algebraic equationsfor 1198860 1198861 119889 and 119862

1 Solving the set of algebraic equations by

using Maple or Mathematica we have the following results

1198860= minus

1

2

120573119890119889 + 120573 plusmn radicminus (119890119889 minus 1)2

(minus1205732+ 4120572120574)

(119890119889 + 1) 120574

1198861= plusmn

119860radicminus (119890119889 minus 1)2

(minus1205732+ 4120572120574)

(119890119889 + 1) 120574

1198621=(119890119889 minus 1) (4120572120574 minus 1205732)

(119890119889 + 1) 120574

(14)

where 120572 119860 119889 120573 and 120574 are arbitrary constants In this casethe exact wave solution takes the following form

119880 (120585119899) = minus

1

2

120573119890119889 + 120573 plusmn radicminus (119890119889 minus 1)2

(minus1205732+ 4120572120574)

(119890119889 + 1) 120574

plusmn119860radicminus (119890119889 minus 1)

2

(minus1205732+ 4120572120574)

(119890119889 + 1) 120574 (119860 + 119890120585119899)

(15)

where 120585119899= ((119890119889 minus 1)(4120572120574 minus 1205732)(119890119889 + 1)120574)119905 + 119899119889 + 120585

0

32 Example 2 The Discrete Nonlinear Schrodinger Equationwith a Saturable Nonlinearity If we take the transformation

120595119899= 119884 (120585

119899) 119890minus119894(120590119905+120588)

120585119899= 119889119899 + 120573 (16)

where 119889 120590 120573 and 120588 are arbitrary constants to be determinedlater

The transformation (16) leads to write (2) into the follow-ing form

(120590 minus 2) 119884 (120585119899) + 119884 (120585

119899+ 119889) + 119884 (120585

119899minus 119889) +

1205781198843 (120585119899)

1 + 1205831198842 (120585119899)= 0

(17)

We suppose the solution of (17) takes the form

119884 (120585119899) = 1198860+ 1198861120593 (120585119899) (18)

where 1198860and 1198861are arbitrary constants to be determined later

and 120593(120585119899) satisfies (9) and (10)

We substitute (18) (9) and (10) into (17) and collect allterms with the same power in [120593(120585

119899)]119894

(119894 = 0 1 2 )Setting each coefficient of this polynomial to zero we derivea set of algebraic equations for 119886

0 1198861 119889 and 119862

1 Solving the

set of algebraic equations by usingMaple orMathematica wehave the following results

1198860= plusmn

(119890120572 minus 1)

radicminus120583 (119890120572 + 1) 119886

1= plusmn

2119860 (1 minus 119890120572)

radicminus120583 (119890120572 + 1)

120590 =2(119890120572 minus 1)

2

(119890120572 + 1)2 120578 =

8119890120572120583

(119890120572 + 1)2

(19)

where 120572 120583 119860 and 119889 are arbitrary constants In this case theexact wave solution of (2) takes the following form

120595119899= (plusmn

(119890120572 minus 1)

radicminus120583 (119890120572 + 1)plusmn

2119860 (1 minus 119890120572)

radicminus120583 (119890120572 + 1)119860 + 119890120585119899)

times 119890minus119894((2(119890

120572minus1)2(119890120572+1)2)119905+120588)

(20)

where 120585119899= 119889119899 + 120573

33 Example 3 The Quintic Discrete Nonlinear SchrodingerEquation In this subsection we study the quintic discretenonlinear Schrodinger equation (3) by using the modifiedtruncated expansion method

If we take the transformation

120595119899= 119884119899119890119894120596119905 (21)

where 120596 is arbitrary constant to be determined laterThe transformation (21) leads to write (3) into the follow-

ing form

119884119899+1+ 119884119899minus1

=(2120572 minus 120596)119884

119899minus 1205731198843119899

120572 + 1205741198842119899+ 1205751198844119899

(22)

If we suppose

119884119899= 119880 (120585

119899) 120585

119899= 119889119899 + 119896 (23)

4 Abstract and Applied Analysis

Equation (23) leads to write (22) into the following form

119880 (120585119899+ 119889) + 119880 (120585

119899minus 119889) =

(2120572 minus 120596)119880 (120585119899) minus 1205731198803 (120585

119899)

120572 + 1205741198802 (120585119899) + 1205751198804 (120585

119899)

(24)

We suppose that the solution of (24) takes the form

119880 (120585119899) = 1198860+ 1198861120593 (120585119899) (25)

where 1198860and 119886

1are arbitrary constants to be determined

later and 120593(120585119899) satisfies (9) and (10) We substitute (25) (9)

and (10) into (24) and collect all terms with the same powerin [120593(120585

119899)]119894 (119894 = 0 1 2 ) Setting each coefficient of this

polynomial to zero we derive a set of algebraic equations for1198860 1198861 119889 and 119862

1 Solving the set of algebraic equations by

using Maple or Mathematica we have the following results

1198860= plusmn

1

4

radic2radic120575119890119889120573 (119890119889 minus 1)

120575119890119889

1198861= plusmn

1

2

radic2radic120575119890119889120573 (119890119889 minus 1)119860

120575119890119889

120596 = minus120573

32

[8120574119890119889(119890119889 minus 1)2

+ 120573((119890119889)2

minus 1)2

]

120575(119890119889)2

120572 = minus120573

64

(119890119889 + 1)

2

(8120574119890119889 + 120573(119890119889 + 1)2

)

120575(119890119889)2

(26)

where 120573119860 119889 120574 and 120575 are arbitrary constants In this case thesolitary wave solution takes the following form

119880 (120585119899) = plusmn

1

4

radic2radic120575119890119889120573 (119890119889 minus 1)

120575119890119889plusmn1

2

radic2radic120575119890119889120573 (119890119889 minus 1)119860

120575119890119889 (119860 + 119890120585119899)

(27)

Consequently the exact solution of (3) is given by

120595119899= (plusmn

1

4

radic2radic120575119890119889120573 (119890119889 minus 1)

120575119890119889plusmn1

2

radic2radic120575119890119889120573 (119890119889 minus 1)119860

120575119890119889 (119860 + 119890120585119899))

times 119890119894(minus(12057332)([8120574119890

119889(119890119889minus1)

2

+120573((119890119889)

2

minus1)

2

]120575(119890119889)

2

))119905

(28)

where 120585119899= 119899119889 + 119896

34 Example 4 The Relativistic Toda Lattice System In thissubsection we use themodified truncated expansionmethodto study the relativistic Toda lattice system (4) The travelingwave variables 119906

119899= 119880(120585

119899) 119907119899= 119881(120585

119899) and 120585

119899= 1198621119905 + 119899119889+ 120585

0

permit us to reduce (4) to the following nonlinear differencedifferential equations

11986211198801015840(120585119899) minus (1 + 120572119880 (120585

119899))

times (119881 (120585119899) minus 119881 (120585

119899minus 119889)) = 0

11986211198811015840(120585119899) minus 119881 (120585

119899) (119880 (120585

119899+ 119889) minus 119880 (120585

119899)

+120572119881 (120585119899+ 119889) minus 120572119881 (120585

119899minus 119889)) = 0

(29)

Considering the homogeneous balance between the highestorder derivatives and nonlinear terms in (30) we get 119896 = 1So we look for the solutions of (30) in the form

119880(120585119899) = 1198860+ 1198861120593 (120585119899) 119881 (120585

119899) = 1198870+ 1198871120593 (120585119899) (30)

where 1198860 1198870 1198861 and 119887

1are arbitrary constants to be deter-

mined later and 120593(120585119899) satisfies (9) and (10) We substitute

(31) (9) and (10) into (30) and collect all terms with the samepower in [120593(120585

119899)]119894 (119894 = 0 1 2 ) Setting each coefficient of

this polynomial to zero we derive a set of algebraic equationsfor 1198860 1198861 119889 and 119862

1 Solving the set of algebraic equations by

using Maple or Mathematica we have the following results

1198860= minus

(119890119889 minus 1) + 1205721198621

120572 (119890119889 minus 1) 119886

1= minus1198621119860

1198870=

1198621

120572 (119890119889 minus 1) 119887

1=1198621119860

120572

(31)

where 119860 119889 1198621 and 120572 are arbitrary constants In this case the

solitary wave solutions take the following form

119880 (120585119899) = minus

119890119889 minus 1 + 1205721198621

120572 (119890119889 minus 1)+minus1198621119860

119860 + 119890120585119899

119881 (120585119899) =

1198621

120572 (119890119889 minus 1)+

1198621119860

120572 (119860 + 119890120585119899)

(32)

where 120585119899= 1198621119905 + 119899119889 + 120585

0

4 Description of the Rational Solitary WaveFunctions Method

In this section we would like to outline the algorithm forusing the rational solitary wave functions method to solvenonlinear DDEs Consider a given system of119872 polynomialNDDEs

Δ (119906119899+1199011

(119883) 119906119899+119901119896

(119883) 1199061015840

119899+1199011

(119883) 1199061015840

119899+119901119896

(119883)

119906(119903)

119899+1199011

(119883) 119906(119903)

119899+119901119896

(119883)) = 0

(33)

where the dependent variable 119906 has 119872 components 119906119894 the

continuous variable 119909 has 119873 components 119909119895 the discrete

variable 119899 has 119876 components 119899119894 the 119896 shift vectors 119875

119904isin 119885119876

Abstract and Applied Analysis 5

and 119906(119903) denotes the collection of mixed derivative terms oforder 119903

The main steps of the algorithm for the rational solitarywave functions method to solve nonlinear DDEs are outlinedas follows

Step 1 We suppose the wave transformation in the followingform

119906119899+119901119904

(119883) = 119880 (120585119899+119901119904

) 120585119899=

119876

sum119894=1

119889119894119899119894+

119898

sum119895=1

119888119895119909119895+ 1205850

119904 = 1 2 119896

(34)

where the coefficients 119889119894(119894 = 1 119876) 119888

119895(119895 = 1 119873) and

the phase 1205850are constants The transformations (34) lead to

write (33) into the following form

Δ (119880119899+1199011

(120585119899) 119880

119899+119901119896

(120585119899) 1198801015840

119899+1199011

(120585119899)

1198801015840

119899+119901119896

(120585119899) 119880

(119903)

119899+1199011

(120585119899) 119880

(119903)

119899+119901119896

(120585119899)) = 0

(35)

Step 2 We suppose the rational solitary wave series expan-sion solutions of (35) in the following form

119880 (120585119899) =

119873

sum119894=0

119886119894[119892 (120585119899)]119894

+

119873

sum119895=1

119887119895[119892 (120585119899)]119895minus1

119891 (120585119899) (36)

with

119891 (120585119899) =

1

119860 tanh (120585119899) + 119861 sec ℎ (120585

119899)

119892 (120585119899) =

sec ℎ (120585119899)

119860 tanh (120585119899) + 119861 sec ℎ (120585

119899)

(37)

which satisfy

1198911015840(120585119899) = minus119860119892

2(120585119899) +

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) +

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) plusmn [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(38)

where 119886119894 119887119895 119860 and 119861 are arbitrary constants to be deter-

mined later and

120590119904= 1199011199041

1198891+ 1199011199042

1198892+ sdot sdot sdot + 119901

119904119876119889119876 (39)

Also we can assume that

119891 (120585119899) =

1

119860 tan (120585119899) + 119861 sec (120585

119899)

119892 (120585119899) =

sec (120585119899)

119860 tan (120585119899) + 119861 sec (120585

119899)

(40)

which satisfy

1198911015840(120585119899) = minus119860119892

2(120585119899) minus

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) minus

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) ∓ [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(41)

Equations (38) and (40) can be written into unified form

1198911015840(120585119899) = minus119860119892

2(120585119899) + 120588

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) + 120588

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) plusmn 120588 [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(42)

where 120588 = plusmn1

6 Abstract and Applied Analysis

Step 3 Determine the degree 119873 of (35) by balancing thehighest order nonlinear term(s) and the highest order deriva-tives of 119880(120585

119899) in (35)

Step 4 Substituting (36)ndash(41) and given the value of 119873determined in Step 3 into (35) and collecting all terms withthe same degree of 119891(120585

119899) and 119892(120585

119899) together the left-hand

side of (35) is converted into polynomial in 119891(120585119899) and 119892(120585

119899)

Then setting each coefficient 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) of this polynomial to zero we derive a set ofalgebraic equations for 119886

119894 119887119895 119862119894 119860 119861

Step 5 Solving the overdetermined system of nonlinearalgebraic equations by using Maple or Mathematica soft-ware package we end up with explicit expressions for119886119894 119887119895 119862119894 119860 and 119861

Step 6 Substituting 119886119894 119887119895 119862119894 119860 and 119861 into (36) along with

(37) and (39) we can finally obtain the rational solitary wavesolutions for nonlinear difference differential equations (33)

5 Applications

In this section we apply the proposed rational solitary wavefunctions method to construct the rational solitary wavesolutions for some nonlinear DDEs via the discrete non-linear Schrodinger equation with a saturable nonlinearitythe quintic discrete nonlinear Schrodinger equation and therelativistic Toda lattice system which are very important inthe mathematical physics and modern physics

51 Example 1 The Discrete Nonlinear Schrodinger Equationwith a Saturable Nonlinearity We suppose that the solutionof (17) takes the form

119880 (120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (43)

where 1198860 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (43)(41) into (17) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585

119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) to zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198861= plusmn119860radic

minus2

120578

sinh (1198892)cosh2 (1198892)

1198862= plusmnradic

minus2 (1198602 + 1198612)

120578

sinh (1198892)cosh2 (1198892)

120583 =1

4(1 + cosh (119889)) 120578

120590 =2 (cosh (119889) minus 1)1 + cos (119889)

1198860= 0

(44)

where 119860 119861 and 120578 are arbitrary constants and 120578 lt 0 In thiscase the rational hyperbolic solitary wave solution of (17)takes the following form

119880(120585119899) = plusmn 119860radic

minus2

120578

sinh (1198892)cosh2 (1198892) [119860 tanh (120585

119899) + 119861 sech (120585

119899)]

plusmn radicminus2 (1198602 + 1198612)

120578

sinh (1198892) sech (120585119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(45)

Consequently the rational hyperbolic solitary wave solu-tion of (2) has the following form

120595119899=[[

[

= plusmn119860radicminus2

120578

sinh (1198892)cosh2 (1198892) [119860 tanh (120585

119899) + 119861 sech (120585

119899)]

plusmnradicminus2 (1198602 + 1198612)

120578

sinh (1198892) sech (120585119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

]]

]

times 119890minus119894((2(cosh(119889)minus1)(1+cos(119889)))119905+120588)

(46)

where 120585119899= 119899119889 + 120573

Case 2 (120588 = minus1)

1198861= plusmn119860radic

minus2

120578

sin (1198892)cos2 (1198892)

1198862= plusmnradic

minus2 (1198602 minus 1198612)

120578

sin (1198892)cos2 (1198892)

120583 =1

4(cos (119889) + 1) 120578

120590 =2 (minus1 + 2 cos (119889))

cos (119889) + 1 119886

0= 0

(47)

where 119860 119861 and 120578 are arbitrary constants and 120578 lt 0 In thiscase the rational trigonometric solitary wave solution of (17)takes the following form

119880(120585119899) = plusmn 119860radic

minus2

120578

sin (1198892)cos2 (1198892) [119860 tan (120585

119899) + 119861 sec (120585

119899)]

plusmn radicminus2 (1198602 minus 1198612)

120578

timessin (1198892) sec (120585

119899)

cos2 (1198892) [119860 tan (120585119899) + 119861 sec (120585

119899)]

(48)

Abstract and Applied Analysis 7

Consequently the rational trigonometric solitary wave solu-tion of (2) has the following form

120595119899= [plusmn119860radic

minus2

120578

sin (1198892)cos2 (1198892) [119860 tan (120585

119899) + 119861 sec (120585

119899)]

plusmn radicminus2 (1198602 minus 1198612)

120578

timessin (1198892) sec (120585

119899)

cos2 (1198892) [119860 tan (120585119899) + 119861 sec (120585

119899)]]

times 119890minus119894((2(minus1+2 cos(119889))(cos(119889)+1))119905+120588)

(49)

where 120585119899= 119899119889 + 120573

52 Example 2 The Quintic Discrete Nonlinear SchrodingerEquation In this subsection we study the quintic discretenonlinear Schrodinger equation (3) by using the rationalsolitary wave functions method

We suppose that the solution of (24) takes the form

119880 (120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (50)

where 1198860 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (50) and(41) into (24) collect all terms with the same power in119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 = 0 1 2 ) and setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) to zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198861= plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

1198862= plusmn

1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) )

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

120572 =minus1

16

[4120574 (cos ℎ (119889) + 1)+120573 (1+2 cosh (119889) + cosh2 (119889))]cosh (119889) minus 1

120596 =minus1

8

120573 [4120574 (cos ℎ (119889) minus 1) + 120573 (cosh2 (119889) minus 1)]120575

1198860= 0

(51)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational hyperbolic solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

)

(52)

Consequently the rational hyperbolic solitary wave solutionof (3) has the following form

120595119899= [plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

) ]

times 119890119894((minus18)(120573[4120574(cos ℎ(119889)minus1)+120573(cosh2(119889)minus1)]120575))119905

(53)

where 120585119899= 119899119889 + 119896

Case 2 (120588 = minus1)

1198861= plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575

1198862= plusmn

1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))120575

8 Abstract and Applied Analysis

1198860= 0

120572 =minus1

16

120573 [(cos (119889) + 1) (4120574 + 2120573) minus 120573 sin2 (119889)]120575

120596 =minus1

8

120573 [4120574 (cos (119889) minus 1) minus 120573 sin2 (119889)]120575

(54)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational trigonometric solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

(55)

Consequently the rational trigonometric solitary wave solu-tion of (3) has the following form

120595119899= ( plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

)

times119890119894((minus18)(120573[4120574(cos(119889)minus1)minus120573 sin2(119889)]120575))119905

(56)

where 120585119899= 119899119889 + 119896

53 Example 3 The Relativistic Toda Lattice System In thissubsection we study the relativistic Toda lattice system (4)by using the rational solitary wave functions method

If we take the transformation

119907119899= minus

1

120572119906119899minus1

1205722 (57)

the transformation (57) reduced the relativistic Toda latticesystem (4) into the following difference differential equation

119906119899119905

= (119906119899+1

120572) (119906119899minus1minus 119906119899) (58)

According to the main steps of rational solitary wave func-tions method we seek traveling wave solutions of (58) in thefollowing form

119906119899(119905) = 119880 (120585

119899) 120585

119899= 1198621119905 + 119899119889 + 120585

0 (59)

where 119889 1198621 and 120585

0are constants The transformation (59)

permits us converting (58) into the following form

11986211198801015840(120585119899) = (119880 (120585

119899) +

1

120572) [119880 (120585

119899minus 119889) minus 119880 (120585

119899)] (60)

Considering the homogeneous balance between the highestorder derivatives and nonlinear terms in (60) we get119873 = 1Thus the solution of (60) has the following form

119880(120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (61)

where 1198860 1198870 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (61) and(41) into (60) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585119899) (119894 = 0 1 119895 =

0 1 2 ) to be zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198860=

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

1198861=

plusmn1198601198862

radic1198602 + 1198612 119862

1=

plusmn21198862

radic1198602 + 1198612

(62)

where 119860 119861 1198862 and 120572 are arbitrary constants In this case

the rational hyperbolic solitary wave solution of (58) has thefollowing form

119880 (120585119899) =

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

plusmn1198601198862

radic1198602 + 1198612 [119860 tanh (120585119899) + 119861 sech (120585

119899)]

+1198862sech (120585

119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(63)

where

120585119899= plusmn

21198862

radic1198602 + 1198612119905 + 119899119889 + 120585

0 (64)

Case 2 (120588 = minus1)

1198860=

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

1198861=

plusmn1198601198862

radic1198602 minus 1198612 119862

1=

plusmn21198862

radic1198602 minus 1198612

(65)

In this case the rational trigonometric solitary wave solutionof (58) has the following form

119880(120585119899) =

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

plusmn1198601198862

radic1198602 minus 1198612 [119860 tan (120585119899) + 119861 sec (120585

119899)]

+1198862sec (120585119899)

[119860 tan (120585119899) + 119861 sec (120585

119899)]

(66)

where

120585119899= plusmn

21198862

radic1198602 minus 1198612119905 + 119899119889 + 120585

0 (67)

Abstract and Applied Analysis 9

6 Discussion

Whenwe compare between the results which obtained in thispaper and other exact solutions we get the following

The solutions obtained in the modified truncated expan-sion functionsmethod are equivalent to the solution obtainedby the exp-functions method but the modified truncatedexpansion is simple and allowed us to solvemore complicatednonlinear difference differential equations such as the dis-crete nonlinear Schrodinger equation with a saturable non-linearity the quintic discrete nonlinear Schrodinger equa-tion and the relativistic Toda lattice system For example thesolution (15) equivalent is to the solution (40) in [20]

In the special case when 119861 = 0 in the rational solitarywave function method we get this method which is equiv-alent to the tanh-function method which discussed in [29]The rational solitary wave functionmethod is extended to thenew rational formal solution which is discussed by [30] when119887119895= 0 in (36)

Remarks These methods which are discussed in this paperallowed us to obtain some new rational solitary wave solu-tions for some complicated nonlinear differential differenceequations

These methods prefer to another methods to convertthe complicated rational methods into a direct nonrationalmethod

7 Conclusions

In this paper we use the modified truncated expansionmethod to obtain the exact solutions for some nonlineardifferential difference equations in the mathematical physicsAlso we calculate the rational solitary wave solutions for thenonlinear differential difference equations As a result manynew and more rational solitary wave solutions are obtainedfrom the hyperbolic function solutions and trigonometricfunction

Acknowledgment

The authors wish to thank the referees for their suggestionsand very useful comments

References

[1] E Fermi J Pasta and S Ulam Collected Papers of Enrico FermiII University of Chicago Press Chicago Ill USA 1965

[2] W P Su J R Schrieffer and A J Heeger ldquoSolitons inpolyacetylenerdquo Physical Review Letters vol 42 no 25 pp 1698ndash1701 1979

[3] A S Davydov ldquoThe theory of contraction of proteins undertheir excitationrdquo Journal ofTheoretical Biology vol 38 no 3 pp559ndash569 1973

[4] P Marquie J M Bilbault and M Remoissenet ldquoObservationof nonlinear localized modes in an electrical latticerdquo PhysicalReview E vol 51 no 6 pp 6127ndash6133 1995

[5] M Toda Theory of Nonlinear Lattices Springer Berlin Ger-many 1989

[6] M Wadati ldquoTransformation theories for nonlinear discretesystemsrdquo Progress ofTheoretical Physics vol 59 pp 36ndash63 1976

[7] YOhta andRHirota ldquoA discrete KdV equation and its Casoratideterminant solutionrdquo Journal of the Physical Society of Japanvol 60 p 2095 1991

[8] M J Ablowitz and J F Ladik ldquoNonlinear differential-differenceequationsrdquo Journal of Mathematical Physics vol 16 no 3 pp598ndash603 1974

[9] X B Hu and W X Ma ldquoApplication of Hirotarsquos bilinearformalism to the Toeplitz latticemdashsome special soliton-likesolutionsrdquo Physics Letters Section A vol 293 no 3-4 pp 161ndash165 2002

[10] D Baldwin U Goktas and W Hereman ldquoSymbolic computa-tion of hyperbolic tangent solutions for nonlinear differential-difference equationsrdquo Computer Physics Communications vol162 no 3 pp 203ndash217 2004

[11] S K Liu Z T Fu Z GWang and S D Liu ldquoPeriodic solutionsfor a class of nonlinear differential-difference equationsrdquo Com-munications in Theoretical Physics vol 49 no 5 pp 1155ndash11582008

[12] C Qiong and L Bin ldquoApplications of jacobi elliptic functionexpansion method for nonlinear differential-difference equa-tionsrdquo Communications inTheoretical Physics vol 43 no 3 pp385ndash388 2005

[13] F Xie M Ji and H Zhao ldquoSome solutions of discrete sine-Gordon equationrdquo Chaos Solitons and Fractals vol 33 no 5pp 1791ndash1795 2007

[14] S D Zhu ldquoExp-function method for the Hybrid-Lattice sys-temrdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 no 3 pp 461ndash464 2007

[15] I Aslan ldquoA discrete generalization of the extended simplestequation methodrdquo Communications in Nonlinear Science andNumerical Simulation vol 15 no 8 pp 1967ndash1973 2010

[16] P Yang Y Chen and Z B Li ldquoADM-Pade technique for thenonlinear lattice equationsrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 362ndash375 2009

[17] SD Zhu YMChu and S LQiu ldquoThehomotopy perturbationmethod for discontinued problems arising in nanotechnologyrdquoComputers andMathematics with Applications vol 58 no 11-12pp 2398ndash2401 2009

[18] F Xie ldquoNew travellingwave solutions of the generalized coupledHirota-Satsuma KdV systemrdquo Chaos Solitons and Fractals vol20 no 5 pp 1005ndash1012 2004

[19] E M E Zayed A M Abourabia K A Gepreel and MM El Horbaty ldquoOn the rational solitary wave solutions forthe nonlinear Hirota-Satsuma coupled KdV systemrdquoApplicableAnalysis vol 85 pp 751ndash768 2006

[20] C S Liu ldquoExponential function rational expansion method fornonlinear differential-difference equationsrdquoChaos Solitons andFractals vol 40 no 2 pp 708ndash716 2009

[21] S Zhang L Dong J M Ba and Y N Sun ldquoThe (frac(1198661015840119866))-expansion method for nonlinear differential-difference equa-tionsrdquo Physics Letters Section A vol 373 no 10 pp 905ndash9102009

[22] I Aslan ldquoThe Discrete (1198661015840119866)-expansion method applied tothe differential-difference Burgers equation and the relativisticToda lattice systemrdquo Numerical Methods for Partial DifferentialEquations vol 28 pp 127ndash137 2012

[23] I Aslan ldquoExact and explicit solutions to the discrete nonlinearSchrodinger equation with asaturable nonlinearityrdquo PhysicsLetters A vol 375 pp 4214ndash4217 2011

10 Abstract and Applied Analysis

[24] K A Gepreel ldquoRational Jacobi Elliptic solutions for nonlineardifference differential equationsrdquo Nonlinear Science Letters Avol 2 pp 151ndash158 2011

[25] K A Gepreel and A R Shehata ldquoRational Jacobi ellipticsolutions for nonlinear differential-difference lattice equationsrdquoApplied Mathematics Letters vol 25 pp 1173ndash1178 2012

[26] K A Gepreel T A Nofal and A A AlThobaiti ldquoThe modi-fied rational Jacobi elliptic solutions for nonlinear differentialdifference equationsrdquo Journal of Applied Mathematics vol 2012Article ID 427479 30 pages 2012

[27] G CWu and T C Xia ldquoA newmethod for constructing solitonsolutions to differential-difference equationwith symbolic com-putationrdquo Chaos Solitons and Fractals vol 39 no 5 pp 2245ndash2248 2009

[28] J Zhang Z Liu S Li and M Wang ldquoSolitary waves andstable analysis for the quintic discrete nonlinear Schrodingerequationrdquo Physica Scripta vol 86 pp 015401ndash015409 2012

[29] F Xie and J Wang ldquoA new method for solving nonlineardifferential-difference equationrdquo Chaos Solitons and Fractalsvol 27 no 4 pp 1067ndash1071 2006

[30] Q Wang and Y Yu ldquoNew rational formal solutions for (1 + 1)-dimensional Toda equation and another Toda equationrdquoChaosSolitons and Fractals vol 29 no 4 pp 904ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Equation (23) leads to write (22) into the following form

119880 (120585119899+ 119889) + 119880 (120585

119899minus 119889) =

(2120572 minus 120596)119880 (120585119899) minus 1205731198803 (120585

119899)

120572 + 1205741198802 (120585119899) + 1205751198804 (120585

119899)

(24)

We suppose that the solution of (24) takes the form

119880 (120585119899) = 1198860+ 1198861120593 (120585119899) (25)

where 1198860and 119886

1are arbitrary constants to be determined

later and 120593(120585119899) satisfies (9) and (10) We substitute (25) (9)

and (10) into (24) and collect all terms with the same powerin [120593(120585

119899)]119894 (119894 = 0 1 2 ) Setting each coefficient of this

polynomial to zero we derive a set of algebraic equations for1198860 1198861 119889 and 119862

1 Solving the set of algebraic equations by

using Maple or Mathematica we have the following results

1198860= plusmn

1

4

radic2radic120575119890119889120573 (119890119889 minus 1)

120575119890119889

1198861= plusmn

1

2

radic2radic120575119890119889120573 (119890119889 minus 1)119860

120575119890119889

120596 = minus120573

32

[8120574119890119889(119890119889 minus 1)2

+ 120573((119890119889)2

minus 1)2

]

120575(119890119889)2

120572 = minus120573

64

(119890119889 + 1)

2

(8120574119890119889 + 120573(119890119889 + 1)2

)

120575(119890119889)2

(26)

where 120573119860 119889 120574 and 120575 are arbitrary constants In this case thesolitary wave solution takes the following form

119880 (120585119899) = plusmn

1

4

radic2radic120575119890119889120573 (119890119889 minus 1)

120575119890119889plusmn1

2

radic2radic120575119890119889120573 (119890119889 minus 1)119860

120575119890119889 (119860 + 119890120585119899)

(27)

Consequently the exact solution of (3) is given by

120595119899= (plusmn

1

4

radic2radic120575119890119889120573 (119890119889 minus 1)

120575119890119889plusmn1

2

radic2radic120575119890119889120573 (119890119889 minus 1)119860

120575119890119889 (119860 + 119890120585119899))

times 119890119894(minus(12057332)([8120574119890

119889(119890119889minus1)

2

+120573((119890119889)

2

minus1)

2

]120575(119890119889)

2

))119905

(28)

where 120585119899= 119899119889 + 119896

34 Example 4 The Relativistic Toda Lattice System In thissubsection we use themodified truncated expansionmethodto study the relativistic Toda lattice system (4) The travelingwave variables 119906

119899= 119880(120585

119899) 119907119899= 119881(120585

119899) and 120585

119899= 1198621119905 + 119899119889+ 120585

0

permit us to reduce (4) to the following nonlinear differencedifferential equations

11986211198801015840(120585119899) minus (1 + 120572119880 (120585

119899))

times (119881 (120585119899) minus 119881 (120585

119899minus 119889)) = 0

11986211198811015840(120585119899) minus 119881 (120585

119899) (119880 (120585

119899+ 119889) minus 119880 (120585

119899)

+120572119881 (120585119899+ 119889) minus 120572119881 (120585

119899minus 119889)) = 0

(29)

Considering the homogeneous balance between the highestorder derivatives and nonlinear terms in (30) we get 119896 = 1So we look for the solutions of (30) in the form

119880(120585119899) = 1198860+ 1198861120593 (120585119899) 119881 (120585

119899) = 1198870+ 1198871120593 (120585119899) (30)

where 1198860 1198870 1198861 and 119887

1are arbitrary constants to be deter-

mined later and 120593(120585119899) satisfies (9) and (10) We substitute

(31) (9) and (10) into (30) and collect all terms with the samepower in [120593(120585

119899)]119894 (119894 = 0 1 2 ) Setting each coefficient of

this polynomial to zero we derive a set of algebraic equationsfor 1198860 1198861 119889 and 119862

1 Solving the set of algebraic equations by

using Maple or Mathematica we have the following results

1198860= minus

(119890119889 minus 1) + 1205721198621

120572 (119890119889 minus 1) 119886

1= minus1198621119860

1198870=

1198621

120572 (119890119889 minus 1) 119887

1=1198621119860

120572

(31)

where 119860 119889 1198621 and 120572 are arbitrary constants In this case the

solitary wave solutions take the following form

119880 (120585119899) = minus

119890119889 minus 1 + 1205721198621

120572 (119890119889 minus 1)+minus1198621119860

119860 + 119890120585119899

119881 (120585119899) =

1198621

120572 (119890119889 minus 1)+

1198621119860

120572 (119860 + 119890120585119899)

(32)

where 120585119899= 1198621119905 + 119899119889 + 120585

0

4 Description of the Rational Solitary WaveFunctions Method

In this section we would like to outline the algorithm forusing the rational solitary wave functions method to solvenonlinear DDEs Consider a given system of119872 polynomialNDDEs

Δ (119906119899+1199011

(119883) 119906119899+119901119896

(119883) 1199061015840

119899+1199011

(119883) 1199061015840

119899+119901119896

(119883)

119906(119903)

119899+1199011

(119883) 119906(119903)

119899+119901119896

(119883)) = 0

(33)

where the dependent variable 119906 has 119872 components 119906119894 the

continuous variable 119909 has 119873 components 119909119895 the discrete

variable 119899 has 119876 components 119899119894 the 119896 shift vectors 119875

119904isin 119885119876

Abstract and Applied Analysis 5

and 119906(119903) denotes the collection of mixed derivative terms oforder 119903

The main steps of the algorithm for the rational solitarywave functions method to solve nonlinear DDEs are outlinedas follows

Step 1 We suppose the wave transformation in the followingform

119906119899+119901119904

(119883) = 119880 (120585119899+119901119904

) 120585119899=

119876

sum119894=1

119889119894119899119894+

119898

sum119895=1

119888119895119909119895+ 1205850

119904 = 1 2 119896

(34)

where the coefficients 119889119894(119894 = 1 119876) 119888

119895(119895 = 1 119873) and

the phase 1205850are constants The transformations (34) lead to

write (33) into the following form

Δ (119880119899+1199011

(120585119899) 119880

119899+119901119896

(120585119899) 1198801015840

119899+1199011

(120585119899)

1198801015840

119899+119901119896

(120585119899) 119880

(119903)

119899+1199011

(120585119899) 119880

(119903)

119899+119901119896

(120585119899)) = 0

(35)

Step 2 We suppose the rational solitary wave series expan-sion solutions of (35) in the following form

119880 (120585119899) =

119873

sum119894=0

119886119894[119892 (120585119899)]119894

+

119873

sum119895=1

119887119895[119892 (120585119899)]119895minus1

119891 (120585119899) (36)

with

119891 (120585119899) =

1

119860 tanh (120585119899) + 119861 sec ℎ (120585

119899)

119892 (120585119899) =

sec ℎ (120585119899)

119860 tanh (120585119899) + 119861 sec ℎ (120585

119899)

(37)

which satisfy

1198911015840(120585119899) = minus119860119892

2(120585119899) +

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) +

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) plusmn [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(38)

where 119886119894 119887119895 119860 and 119861 are arbitrary constants to be deter-

mined later and

120590119904= 1199011199041

1198891+ 1199011199042

1198892+ sdot sdot sdot + 119901

119904119876119889119876 (39)

Also we can assume that

119891 (120585119899) =

1

119860 tan (120585119899) + 119861 sec (120585

119899)

119892 (120585119899) =

sec (120585119899)

119860 tan (120585119899) + 119861 sec (120585

119899)

(40)

which satisfy

1198911015840(120585119899) = minus119860119892

2(120585119899) minus

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) minus

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) ∓ [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(41)

Equations (38) and (40) can be written into unified form

1198911015840(120585119899) = minus119860119892

2(120585119899) + 120588

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) + 120588

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) plusmn 120588 [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(42)

where 120588 = plusmn1

6 Abstract and Applied Analysis

Step 3 Determine the degree 119873 of (35) by balancing thehighest order nonlinear term(s) and the highest order deriva-tives of 119880(120585

119899) in (35)

Step 4 Substituting (36)ndash(41) and given the value of 119873determined in Step 3 into (35) and collecting all terms withthe same degree of 119891(120585

119899) and 119892(120585

119899) together the left-hand

side of (35) is converted into polynomial in 119891(120585119899) and 119892(120585

119899)

Then setting each coefficient 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) of this polynomial to zero we derive a set ofalgebraic equations for 119886

119894 119887119895 119862119894 119860 119861

Step 5 Solving the overdetermined system of nonlinearalgebraic equations by using Maple or Mathematica soft-ware package we end up with explicit expressions for119886119894 119887119895 119862119894 119860 and 119861

Step 6 Substituting 119886119894 119887119895 119862119894 119860 and 119861 into (36) along with

(37) and (39) we can finally obtain the rational solitary wavesolutions for nonlinear difference differential equations (33)

5 Applications

In this section we apply the proposed rational solitary wavefunctions method to construct the rational solitary wavesolutions for some nonlinear DDEs via the discrete non-linear Schrodinger equation with a saturable nonlinearitythe quintic discrete nonlinear Schrodinger equation and therelativistic Toda lattice system which are very important inthe mathematical physics and modern physics

51 Example 1 The Discrete Nonlinear Schrodinger Equationwith a Saturable Nonlinearity We suppose that the solutionof (17) takes the form

119880 (120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (43)

where 1198860 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (43)(41) into (17) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585

119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) to zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198861= plusmn119860radic

minus2

120578

sinh (1198892)cosh2 (1198892)

1198862= plusmnradic

minus2 (1198602 + 1198612)

120578

sinh (1198892)cosh2 (1198892)

120583 =1

4(1 + cosh (119889)) 120578

120590 =2 (cosh (119889) minus 1)1 + cos (119889)

1198860= 0

(44)

where 119860 119861 and 120578 are arbitrary constants and 120578 lt 0 In thiscase the rational hyperbolic solitary wave solution of (17)takes the following form

119880(120585119899) = plusmn 119860radic

minus2

120578

sinh (1198892)cosh2 (1198892) [119860 tanh (120585

119899) + 119861 sech (120585

119899)]

plusmn radicminus2 (1198602 + 1198612)

120578

sinh (1198892) sech (120585119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(45)

Consequently the rational hyperbolic solitary wave solu-tion of (2) has the following form

120595119899=[[

[

= plusmn119860radicminus2

120578

sinh (1198892)cosh2 (1198892) [119860 tanh (120585

119899) + 119861 sech (120585

119899)]

plusmnradicminus2 (1198602 + 1198612)

120578

sinh (1198892) sech (120585119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

]]

]

times 119890minus119894((2(cosh(119889)minus1)(1+cos(119889)))119905+120588)

(46)

where 120585119899= 119899119889 + 120573

Case 2 (120588 = minus1)

1198861= plusmn119860radic

minus2

120578

sin (1198892)cos2 (1198892)

1198862= plusmnradic

minus2 (1198602 minus 1198612)

120578

sin (1198892)cos2 (1198892)

120583 =1

4(cos (119889) + 1) 120578

120590 =2 (minus1 + 2 cos (119889))

cos (119889) + 1 119886

0= 0

(47)

where 119860 119861 and 120578 are arbitrary constants and 120578 lt 0 In thiscase the rational trigonometric solitary wave solution of (17)takes the following form

119880(120585119899) = plusmn 119860radic

minus2

120578

sin (1198892)cos2 (1198892) [119860 tan (120585

119899) + 119861 sec (120585

119899)]

plusmn radicminus2 (1198602 minus 1198612)

120578

timessin (1198892) sec (120585

119899)

cos2 (1198892) [119860 tan (120585119899) + 119861 sec (120585

119899)]

(48)

Abstract and Applied Analysis 7

Consequently the rational trigonometric solitary wave solu-tion of (2) has the following form

120595119899= [plusmn119860radic

minus2

120578

sin (1198892)cos2 (1198892) [119860 tan (120585

119899) + 119861 sec (120585

119899)]

plusmn radicminus2 (1198602 minus 1198612)

120578

timessin (1198892) sec (120585

119899)

cos2 (1198892) [119860 tan (120585119899) + 119861 sec (120585

119899)]]

times 119890minus119894((2(minus1+2 cos(119889))(cos(119889)+1))119905+120588)

(49)

where 120585119899= 119899119889 + 120573

52 Example 2 The Quintic Discrete Nonlinear SchrodingerEquation In this subsection we study the quintic discretenonlinear Schrodinger equation (3) by using the rationalsolitary wave functions method

We suppose that the solution of (24) takes the form

119880 (120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (50)

where 1198860 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (50) and(41) into (24) collect all terms with the same power in119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 = 0 1 2 ) and setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) to zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198861= plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

1198862= plusmn

1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) )

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

120572 =minus1

16

[4120574 (cos ℎ (119889) + 1)+120573 (1+2 cosh (119889) + cosh2 (119889))]cosh (119889) minus 1

120596 =minus1

8

120573 [4120574 (cos ℎ (119889) minus 1) + 120573 (cosh2 (119889) minus 1)]120575

1198860= 0

(51)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational hyperbolic solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

)

(52)

Consequently the rational hyperbolic solitary wave solutionof (3) has the following form

120595119899= [plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

) ]

times 119890119894((minus18)(120573[4120574(cos ℎ(119889)minus1)+120573(cosh2(119889)minus1)]120575))119905

(53)

where 120585119899= 119899119889 + 119896

Case 2 (120588 = minus1)

1198861= plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575

1198862= plusmn

1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))120575

8 Abstract and Applied Analysis

1198860= 0

120572 =minus1

16

120573 [(cos (119889) + 1) (4120574 + 2120573) minus 120573 sin2 (119889)]120575

120596 =minus1

8

120573 [4120574 (cos (119889) minus 1) minus 120573 sin2 (119889)]120575

(54)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational trigonometric solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

(55)

Consequently the rational trigonometric solitary wave solu-tion of (3) has the following form

120595119899= ( plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

)

times119890119894((minus18)(120573[4120574(cos(119889)minus1)minus120573 sin2(119889)]120575))119905

(56)

where 120585119899= 119899119889 + 119896

53 Example 3 The Relativistic Toda Lattice System In thissubsection we study the relativistic Toda lattice system (4)by using the rational solitary wave functions method

If we take the transformation

119907119899= minus

1

120572119906119899minus1

1205722 (57)

the transformation (57) reduced the relativistic Toda latticesystem (4) into the following difference differential equation

119906119899119905

= (119906119899+1

120572) (119906119899minus1minus 119906119899) (58)

According to the main steps of rational solitary wave func-tions method we seek traveling wave solutions of (58) in thefollowing form

119906119899(119905) = 119880 (120585

119899) 120585

119899= 1198621119905 + 119899119889 + 120585

0 (59)

where 119889 1198621 and 120585

0are constants The transformation (59)

permits us converting (58) into the following form

11986211198801015840(120585119899) = (119880 (120585

119899) +

1

120572) [119880 (120585

119899minus 119889) minus 119880 (120585

119899)] (60)

Considering the homogeneous balance between the highestorder derivatives and nonlinear terms in (60) we get119873 = 1Thus the solution of (60) has the following form

119880(120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (61)

where 1198860 1198870 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (61) and(41) into (60) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585119899) (119894 = 0 1 119895 =

0 1 2 ) to be zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198860=

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

1198861=

plusmn1198601198862

radic1198602 + 1198612 119862

1=

plusmn21198862

radic1198602 + 1198612

(62)

where 119860 119861 1198862 and 120572 are arbitrary constants In this case

the rational hyperbolic solitary wave solution of (58) has thefollowing form

119880 (120585119899) =

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

plusmn1198601198862

radic1198602 + 1198612 [119860 tanh (120585119899) + 119861 sech (120585

119899)]

+1198862sech (120585

119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(63)

where

120585119899= plusmn

21198862

radic1198602 + 1198612119905 + 119899119889 + 120585

0 (64)

Case 2 (120588 = minus1)

1198860=

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

1198861=

plusmn1198601198862

radic1198602 minus 1198612 119862

1=

plusmn21198862

radic1198602 minus 1198612

(65)

In this case the rational trigonometric solitary wave solutionof (58) has the following form

119880(120585119899) =

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

plusmn1198601198862

radic1198602 minus 1198612 [119860 tan (120585119899) + 119861 sec (120585

119899)]

+1198862sec (120585119899)

[119860 tan (120585119899) + 119861 sec (120585

119899)]

(66)

where

120585119899= plusmn

21198862

radic1198602 minus 1198612119905 + 119899119889 + 120585

0 (67)

Abstract and Applied Analysis 9

6 Discussion

Whenwe compare between the results which obtained in thispaper and other exact solutions we get the following

The solutions obtained in the modified truncated expan-sion functionsmethod are equivalent to the solution obtainedby the exp-functions method but the modified truncatedexpansion is simple and allowed us to solvemore complicatednonlinear difference differential equations such as the dis-crete nonlinear Schrodinger equation with a saturable non-linearity the quintic discrete nonlinear Schrodinger equa-tion and the relativistic Toda lattice system For example thesolution (15) equivalent is to the solution (40) in [20]

In the special case when 119861 = 0 in the rational solitarywave function method we get this method which is equiv-alent to the tanh-function method which discussed in [29]The rational solitary wave functionmethod is extended to thenew rational formal solution which is discussed by [30] when119887119895= 0 in (36)

Remarks These methods which are discussed in this paperallowed us to obtain some new rational solitary wave solu-tions for some complicated nonlinear differential differenceequations

These methods prefer to another methods to convertthe complicated rational methods into a direct nonrationalmethod

7 Conclusions

In this paper we use the modified truncated expansionmethod to obtain the exact solutions for some nonlineardifferential difference equations in the mathematical physicsAlso we calculate the rational solitary wave solutions for thenonlinear differential difference equations As a result manynew and more rational solitary wave solutions are obtainedfrom the hyperbolic function solutions and trigonometricfunction

Acknowledgment

The authors wish to thank the referees for their suggestionsand very useful comments

References

[1] E Fermi J Pasta and S Ulam Collected Papers of Enrico FermiII University of Chicago Press Chicago Ill USA 1965

[2] W P Su J R Schrieffer and A J Heeger ldquoSolitons inpolyacetylenerdquo Physical Review Letters vol 42 no 25 pp 1698ndash1701 1979

[3] A S Davydov ldquoThe theory of contraction of proteins undertheir excitationrdquo Journal ofTheoretical Biology vol 38 no 3 pp559ndash569 1973

[4] P Marquie J M Bilbault and M Remoissenet ldquoObservationof nonlinear localized modes in an electrical latticerdquo PhysicalReview E vol 51 no 6 pp 6127ndash6133 1995

[5] M Toda Theory of Nonlinear Lattices Springer Berlin Ger-many 1989

[6] M Wadati ldquoTransformation theories for nonlinear discretesystemsrdquo Progress ofTheoretical Physics vol 59 pp 36ndash63 1976

[7] YOhta andRHirota ldquoA discrete KdV equation and its Casoratideterminant solutionrdquo Journal of the Physical Society of Japanvol 60 p 2095 1991

[8] M J Ablowitz and J F Ladik ldquoNonlinear differential-differenceequationsrdquo Journal of Mathematical Physics vol 16 no 3 pp598ndash603 1974

[9] X B Hu and W X Ma ldquoApplication of Hirotarsquos bilinearformalism to the Toeplitz latticemdashsome special soliton-likesolutionsrdquo Physics Letters Section A vol 293 no 3-4 pp 161ndash165 2002

[10] D Baldwin U Goktas and W Hereman ldquoSymbolic computa-tion of hyperbolic tangent solutions for nonlinear differential-difference equationsrdquo Computer Physics Communications vol162 no 3 pp 203ndash217 2004

[11] S K Liu Z T Fu Z GWang and S D Liu ldquoPeriodic solutionsfor a class of nonlinear differential-difference equationsrdquo Com-munications in Theoretical Physics vol 49 no 5 pp 1155ndash11582008

[12] C Qiong and L Bin ldquoApplications of jacobi elliptic functionexpansion method for nonlinear differential-difference equa-tionsrdquo Communications inTheoretical Physics vol 43 no 3 pp385ndash388 2005

[13] F Xie M Ji and H Zhao ldquoSome solutions of discrete sine-Gordon equationrdquo Chaos Solitons and Fractals vol 33 no 5pp 1791ndash1795 2007

[14] S D Zhu ldquoExp-function method for the Hybrid-Lattice sys-temrdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 no 3 pp 461ndash464 2007

[15] I Aslan ldquoA discrete generalization of the extended simplestequation methodrdquo Communications in Nonlinear Science andNumerical Simulation vol 15 no 8 pp 1967ndash1973 2010

[16] P Yang Y Chen and Z B Li ldquoADM-Pade technique for thenonlinear lattice equationsrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 362ndash375 2009

[17] SD Zhu YMChu and S LQiu ldquoThehomotopy perturbationmethod for discontinued problems arising in nanotechnologyrdquoComputers andMathematics with Applications vol 58 no 11-12pp 2398ndash2401 2009

[18] F Xie ldquoNew travellingwave solutions of the generalized coupledHirota-Satsuma KdV systemrdquo Chaos Solitons and Fractals vol20 no 5 pp 1005ndash1012 2004

[19] E M E Zayed A M Abourabia K A Gepreel and MM El Horbaty ldquoOn the rational solitary wave solutions forthe nonlinear Hirota-Satsuma coupled KdV systemrdquoApplicableAnalysis vol 85 pp 751ndash768 2006

[20] C S Liu ldquoExponential function rational expansion method fornonlinear differential-difference equationsrdquoChaos Solitons andFractals vol 40 no 2 pp 708ndash716 2009

[21] S Zhang L Dong J M Ba and Y N Sun ldquoThe (frac(1198661015840119866))-expansion method for nonlinear differential-difference equa-tionsrdquo Physics Letters Section A vol 373 no 10 pp 905ndash9102009

[22] I Aslan ldquoThe Discrete (1198661015840119866)-expansion method applied tothe differential-difference Burgers equation and the relativisticToda lattice systemrdquo Numerical Methods for Partial DifferentialEquations vol 28 pp 127ndash137 2012

[23] I Aslan ldquoExact and explicit solutions to the discrete nonlinearSchrodinger equation with asaturable nonlinearityrdquo PhysicsLetters A vol 375 pp 4214ndash4217 2011

10 Abstract and Applied Analysis

[24] K A Gepreel ldquoRational Jacobi Elliptic solutions for nonlineardifference differential equationsrdquo Nonlinear Science Letters Avol 2 pp 151ndash158 2011

[25] K A Gepreel and A R Shehata ldquoRational Jacobi ellipticsolutions for nonlinear differential-difference lattice equationsrdquoApplied Mathematics Letters vol 25 pp 1173ndash1178 2012

[26] K A Gepreel T A Nofal and A A AlThobaiti ldquoThe modi-fied rational Jacobi elliptic solutions for nonlinear differentialdifference equationsrdquo Journal of Applied Mathematics vol 2012Article ID 427479 30 pages 2012

[27] G CWu and T C Xia ldquoA newmethod for constructing solitonsolutions to differential-difference equationwith symbolic com-putationrdquo Chaos Solitons and Fractals vol 39 no 5 pp 2245ndash2248 2009

[28] J Zhang Z Liu S Li and M Wang ldquoSolitary waves andstable analysis for the quintic discrete nonlinear Schrodingerequationrdquo Physica Scripta vol 86 pp 015401ndash015409 2012

[29] F Xie and J Wang ldquoA new method for solving nonlineardifferential-difference equationrdquo Chaos Solitons and Fractalsvol 27 no 4 pp 1067ndash1071 2006

[30] Q Wang and Y Yu ldquoNew rational formal solutions for (1 + 1)-dimensional Toda equation and another Toda equationrdquoChaosSolitons and Fractals vol 29 no 4 pp 904ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

and 119906(119903) denotes the collection of mixed derivative terms oforder 119903

The main steps of the algorithm for the rational solitarywave functions method to solve nonlinear DDEs are outlinedas follows

Step 1 We suppose the wave transformation in the followingform

119906119899+119901119904

(119883) = 119880 (120585119899+119901119904

) 120585119899=

119876

sum119894=1

119889119894119899119894+

119898

sum119895=1

119888119895119909119895+ 1205850

119904 = 1 2 119896

(34)

where the coefficients 119889119894(119894 = 1 119876) 119888

119895(119895 = 1 119873) and

the phase 1205850are constants The transformations (34) lead to

write (33) into the following form

Δ (119880119899+1199011

(120585119899) 119880

119899+119901119896

(120585119899) 1198801015840

119899+1199011

(120585119899)

1198801015840

119899+119901119896

(120585119899) 119880

(119903)

119899+1199011

(120585119899) 119880

(119903)

119899+119901119896

(120585119899)) = 0

(35)

Step 2 We suppose the rational solitary wave series expan-sion solutions of (35) in the following form

119880 (120585119899) =

119873

sum119894=0

119886119894[119892 (120585119899)]119894

+

119873

sum119895=1

119887119895[119892 (120585119899)]119895minus1

119891 (120585119899) (36)

with

119891 (120585119899) =

1

119860 tanh (120585119899) + 119861 sec ℎ (120585

119899)

119892 (120585119899) =

sec ℎ (120585119899)

119860 tanh (120585119899) + 119861 sec ℎ (120585

119899)

(37)

which satisfy

1198911015840(120585119899) = minus119860119892

2(120585119899) +

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) +

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) plusmn [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(38)

where 119886119894 119887119895 119860 and 119861 are arbitrary constants to be deter-

mined later and

120590119904= 1199011199041

1198891+ 1199011199042

1198892+ sdot sdot sdot + 119901

119904119876119889119876 (39)

Also we can assume that

119891 (120585119899) =

1

119860 tan (120585119899) + 119861 sec (120585

119899)

119892 (120585119899) =

sec (120585119899)

119860 tan (120585119899) + 119861 sec (120585

119899)

(40)

which satisfy

1198911015840(120585119899) = minus119860119892

2(120585119899) minus

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) minus

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) ∓ [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(41)

Equations (38) and (40) can be written into unified form

1198911015840(120585119899) = minus119860119892

2(120585119899) + 120588

119861119892 (120585119899)

119860[1 minus 119861119892 (120585

119899)]

1198921015840(120585119899) = minus119860119891 (120585

119899) 119892 (120585119899)

1198912(120585119899) = 1198922(120585119899) + 120588

1

1198602[1 minus 119861119892 (120585

119899)]2

119891 (120585119899plusmn 120590119904)

= (1198602119891 (120590119904) 119891 (120585119899) plusmn 120588 [1 minus 119861119892 (120585

119899)] [1 minus 119861119892 (120590

119904)])

times (1198602119891 (120590119904) [1 minus 119861119892 (120585

119899)] plusmn 119860

2119891 (120585119899) [1 minus 119861119892 (120590

119904)]

+1198611198602119892 (120585119899) 119892 (120590119904))minus1

119892 (120585119899plusmn 120590119904)

= (119892 (120585119899) 119892 (120590119904)) (119891 (120590

119904) [1 minus 119861119892 (120585

119899)] plusmn 119891 (120585

119899)

times [1 minus 119861119892 (120590119904)] + 119861119892 (120585

119899) 119892 (120590119904))minus1

(42)

where 120588 = plusmn1

6 Abstract and Applied Analysis

Step 3 Determine the degree 119873 of (35) by balancing thehighest order nonlinear term(s) and the highest order deriva-tives of 119880(120585

119899) in (35)

Step 4 Substituting (36)ndash(41) and given the value of 119873determined in Step 3 into (35) and collecting all terms withthe same degree of 119891(120585

119899) and 119892(120585

119899) together the left-hand

side of (35) is converted into polynomial in 119891(120585119899) and 119892(120585

119899)

Then setting each coefficient 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) of this polynomial to zero we derive a set ofalgebraic equations for 119886

119894 119887119895 119862119894 119860 119861

Step 5 Solving the overdetermined system of nonlinearalgebraic equations by using Maple or Mathematica soft-ware package we end up with explicit expressions for119886119894 119887119895 119862119894 119860 and 119861

Step 6 Substituting 119886119894 119887119895 119862119894 119860 and 119861 into (36) along with

(37) and (39) we can finally obtain the rational solitary wavesolutions for nonlinear difference differential equations (33)

5 Applications

In this section we apply the proposed rational solitary wavefunctions method to construct the rational solitary wavesolutions for some nonlinear DDEs via the discrete non-linear Schrodinger equation with a saturable nonlinearitythe quintic discrete nonlinear Schrodinger equation and therelativistic Toda lattice system which are very important inthe mathematical physics and modern physics

51 Example 1 The Discrete Nonlinear Schrodinger Equationwith a Saturable Nonlinearity We suppose that the solutionof (17) takes the form

119880 (120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (43)

where 1198860 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (43)(41) into (17) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585

119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) to zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198861= plusmn119860radic

minus2

120578

sinh (1198892)cosh2 (1198892)

1198862= plusmnradic

minus2 (1198602 + 1198612)

120578

sinh (1198892)cosh2 (1198892)

120583 =1

4(1 + cosh (119889)) 120578

120590 =2 (cosh (119889) minus 1)1 + cos (119889)

1198860= 0

(44)

where 119860 119861 and 120578 are arbitrary constants and 120578 lt 0 In thiscase the rational hyperbolic solitary wave solution of (17)takes the following form

119880(120585119899) = plusmn 119860radic

minus2

120578

sinh (1198892)cosh2 (1198892) [119860 tanh (120585

119899) + 119861 sech (120585

119899)]

plusmn radicminus2 (1198602 + 1198612)

120578

sinh (1198892) sech (120585119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(45)

Consequently the rational hyperbolic solitary wave solu-tion of (2) has the following form

120595119899=[[

[

= plusmn119860radicminus2

120578

sinh (1198892)cosh2 (1198892) [119860 tanh (120585

119899) + 119861 sech (120585

119899)]

plusmnradicminus2 (1198602 + 1198612)

120578

sinh (1198892) sech (120585119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

]]

]

times 119890minus119894((2(cosh(119889)minus1)(1+cos(119889)))119905+120588)

(46)

where 120585119899= 119899119889 + 120573

Case 2 (120588 = minus1)

1198861= plusmn119860radic

minus2

120578

sin (1198892)cos2 (1198892)

1198862= plusmnradic

minus2 (1198602 minus 1198612)

120578

sin (1198892)cos2 (1198892)

120583 =1

4(cos (119889) + 1) 120578

120590 =2 (minus1 + 2 cos (119889))

cos (119889) + 1 119886

0= 0

(47)

where 119860 119861 and 120578 are arbitrary constants and 120578 lt 0 In thiscase the rational trigonometric solitary wave solution of (17)takes the following form

119880(120585119899) = plusmn 119860radic

minus2

120578

sin (1198892)cos2 (1198892) [119860 tan (120585

119899) + 119861 sec (120585

119899)]

plusmn radicminus2 (1198602 minus 1198612)

120578

timessin (1198892) sec (120585

119899)

cos2 (1198892) [119860 tan (120585119899) + 119861 sec (120585

119899)]

(48)

Abstract and Applied Analysis 7

Consequently the rational trigonometric solitary wave solu-tion of (2) has the following form

120595119899= [plusmn119860radic

minus2

120578

sin (1198892)cos2 (1198892) [119860 tan (120585

119899) + 119861 sec (120585

119899)]

plusmn radicminus2 (1198602 minus 1198612)

120578

timessin (1198892) sec (120585

119899)

cos2 (1198892) [119860 tan (120585119899) + 119861 sec (120585

119899)]]

times 119890minus119894((2(minus1+2 cos(119889))(cos(119889)+1))119905+120588)

(49)

where 120585119899= 119899119889 + 120573

52 Example 2 The Quintic Discrete Nonlinear SchrodingerEquation In this subsection we study the quintic discretenonlinear Schrodinger equation (3) by using the rationalsolitary wave functions method

We suppose that the solution of (24) takes the form

119880 (120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (50)

where 1198860 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (50) and(41) into (24) collect all terms with the same power in119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 = 0 1 2 ) and setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) to zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198861= plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

1198862= plusmn

1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) )

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

120572 =minus1

16

[4120574 (cos ℎ (119889) + 1)+120573 (1+2 cosh (119889) + cosh2 (119889))]cosh (119889) minus 1

120596 =minus1

8

120573 [4120574 (cos ℎ (119889) minus 1) + 120573 (cosh2 (119889) minus 1)]120575

1198860= 0

(51)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational hyperbolic solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

)

(52)

Consequently the rational hyperbolic solitary wave solutionof (3) has the following form

120595119899= [plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

) ]

times 119890119894((minus18)(120573[4120574(cos ℎ(119889)minus1)+120573(cosh2(119889)minus1)]120575))119905

(53)

where 120585119899= 119899119889 + 119896

Case 2 (120588 = minus1)

1198861= plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575

1198862= plusmn

1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))120575

8 Abstract and Applied Analysis

1198860= 0

120572 =minus1

16

120573 [(cos (119889) + 1) (4120574 + 2120573) minus 120573 sin2 (119889)]120575

120596 =minus1

8

120573 [4120574 (cos (119889) minus 1) minus 120573 sin2 (119889)]120575

(54)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational trigonometric solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

(55)

Consequently the rational trigonometric solitary wave solu-tion of (3) has the following form

120595119899= ( plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

)

times119890119894((minus18)(120573[4120574(cos(119889)minus1)minus120573 sin2(119889)]120575))119905

(56)

where 120585119899= 119899119889 + 119896

53 Example 3 The Relativistic Toda Lattice System In thissubsection we study the relativistic Toda lattice system (4)by using the rational solitary wave functions method

If we take the transformation

119907119899= minus

1

120572119906119899minus1

1205722 (57)

the transformation (57) reduced the relativistic Toda latticesystem (4) into the following difference differential equation

119906119899119905

= (119906119899+1

120572) (119906119899minus1minus 119906119899) (58)

According to the main steps of rational solitary wave func-tions method we seek traveling wave solutions of (58) in thefollowing form

119906119899(119905) = 119880 (120585

119899) 120585

119899= 1198621119905 + 119899119889 + 120585

0 (59)

where 119889 1198621 and 120585

0are constants The transformation (59)

permits us converting (58) into the following form

11986211198801015840(120585119899) = (119880 (120585

119899) +

1

120572) [119880 (120585

119899minus 119889) minus 119880 (120585

119899)] (60)

Considering the homogeneous balance between the highestorder derivatives and nonlinear terms in (60) we get119873 = 1Thus the solution of (60) has the following form

119880(120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (61)

where 1198860 1198870 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (61) and(41) into (60) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585119899) (119894 = 0 1 119895 =

0 1 2 ) to be zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198860=

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

1198861=

plusmn1198601198862

radic1198602 + 1198612 119862

1=

plusmn21198862

radic1198602 + 1198612

(62)

where 119860 119861 1198862 and 120572 are arbitrary constants In this case

the rational hyperbolic solitary wave solution of (58) has thefollowing form

119880 (120585119899) =

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

plusmn1198601198862

radic1198602 + 1198612 [119860 tanh (120585119899) + 119861 sech (120585

119899)]

+1198862sech (120585

119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(63)

where

120585119899= plusmn

21198862

radic1198602 + 1198612119905 + 119899119889 + 120585

0 (64)

Case 2 (120588 = minus1)

1198860=

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

1198861=

plusmn1198601198862

radic1198602 minus 1198612 119862

1=

plusmn21198862

radic1198602 minus 1198612

(65)

In this case the rational trigonometric solitary wave solutionof (58) has the following form

119880(120585119899) =

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

plusmn1198601198862

radic1198602 minus 1198612 [119860 tan (120585119899) + 119861 sec (120585

119899)]

+1198862sec (120585119899)

[119860 tan (120585119899) + 119861 sec (120585

119899)]

(66)

where

120585119899= plusmn

21198862

radic1198602 minus 1198612119905 + 119899119889 + 120585

0 (67)

Abstract and Applied Analysis 9

6 Discussion

Whenwe compare between the results which obtained in thispaper and other exact solutions we get the following

The solutions obtained in the modified truncated expan-sion functionsmethod are equivalent to the solution obtainedby the exp-functions method but the modified truncatedexpansion is simple and allowed us to solvemore complicatednonlinear difference differential equations such as the dis-crete nonlinear Schrodinger equation with a saturable non-linearity the quintic discrete nonlinear Schrodinger equa-tion and the relativistic Toda lattice system For example thesolution (15) equivalent is to the solution (40) in [20]

In the special case when 119861 = 0 in the rational solitarywave function method we get this method which is equiv-alent to the tanh-function method which discussed in [29]The rational solitary wave functionmethod is extended to thenew rational formal solution which is discussed by [30] when119887119895= 0 in (36)

Remarks These methods which are discussed in this paperallowed us to obtain some new rational solitary wave solu-tions for some complicated nonlinear differential differenceequations

These methods prefer to another methods to convertthe complicated rational methods into a direct nonrationalmethod

7 Conclusions

In this paper we use the modified truncated expansionmethod to obtain the exact solutions for some nonlineardifferential difference equations in the mathematical physicsAlso we calculate the rational solitary wave solutions for thenonlinear differential difference equations As a result manynew and more rational solitary wave solutions are obtainedfrom the hyperbolic function solutions and trigonometricfunction

Acknowledgment

The authors wish to thank the referees for their suggestionsand very useful comments

References

[1] E Fermi J Pasta and S Ulam Collected Papers of Enrico FermiII University of Chicago Press Chicago Ill USA 1965

[2] W P Su J R Schrieffer and A J Heeger ldquoSolitons inpolyacetylenerdquo Physical Review Letters vol 42 no 25 pp 1698ndash1701 1979

[3] A S Davydov ldquoThe theory of contraction of proteins undertheir excitationrdquo Journal ofTheoretical Biology vol 38 no 3 pp559ndash569 1973

[4] P Marquie J M Bilbault and M Remoissenet ldquoObservationof nonlinear localized modes in an electrical latticerdquo PhysicalReview E vol 51 no 6 pp 6127ndash6133 1995

[5] M Toda Theory of Nonlinear Lattices Springer Berlin Ger-many 1989

[6] M Wadati ldquoTransformation theories for nonlinear discretesystemsrdquo Progress ofTheoretical Physics vol 59 pp 36ndash63 1976

[7] YOhta andRHirota ldquoA discrete KdV equation and its Casoratideterminant solutionrdquo Journal of the Physical Society of Japanvol 60 p 2095 1991

[8] M J Ablowitz and J F Ladik ldquoNonlinear differential-differenceequationsrdquo Journal of Mathematical Physics vol 16 no 3 pp598ndash603 1974

[9] X B Hu and W X Ma ldquoApplication of Hirotarsquos bilinearformalism to the Toeplitz latticemdashsome special soliton-likesolutionsrdquo Physics Letters Section A vol 293 no 3-4 pp 161ndash165 2002

[10] D Baldwin U Goktas and W Hereman ldquoSymbolic computa-tion of hyperbolic tangent solutions for nonlinear differential-difference equationsrdquo Computer Physics Communications vol162 no 3 pp 203ndash217 2004

[11] S K Liu Z T Fu Z GWang and S D Liu ldquoPeriodic solutionsfor a class of nonlinear differential-difference equationsrdquo Com-munications in Theoretical Physics vol 49 no 5 pp 1155ndash11582008

[12] C Qiong and L Bin ldquoApplications of jacobi elliptic functionexpansion method for nonlinear differential-difference equa-tionsrdquo Communications inTheoretical Physics vol 43 no 3 pp385ndash388 2005

[13] F Xie M Ji and H Zhao ldquoSome solutions of discrete sine-Gordon equationrdquo Chaos Solitons and Fractals vol 33 no 5pp 1791ndash1795 2007

[14] S D Zhu ldquoExp-function method for the Hybrid-Lattice sys-temrdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 no 3 pp 461ndash464 2007

[15] I Aslan ldquoA discrete generalization of the extended simplestequation methodrdquo Communications in Nonlinear Science andNumerical Simulation vol 15 no 8 pp 1967ndash1973 2010

[16] P Yang Y Chen and Z B Li ldquoADM-Pade technique for thenonlinear lattice equationsrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 362ndash375 2009

[17] SD Zhu YMChu and S LQiu ldquoThehomotopy perturbationmethod for discontinued problems arising in nanotechnologyrdquoComputers andMathematics with Applications vol 58 no 11-12pp 2398ndash2401 2009

[18] F Xie ldquoNew travellingwave solutions of the generalized coupledHirota-Satsuma KdV systemrdquo Chaos Solitons and Fractals vol20 no 5 pp 1005ndash1012 2004

[19] E M E Zayed A M Abourabia K A Gepreel and MM El Horbaty ldquoOn the rational solitary wave solutions forthe nonlinear Hirota-Satsuma coupled KdV systemrdquoApplicableAnalysis vol 85 pp 751ndash768 2006

[20] C S Liu ldquoExponential function rational expansion method fornonlinear differential-difference equationsrdquoChaos Solitons andFractals vol 40 no 2 pp 708ndash716 2009

[21] S Zhang L Dong J M Ba and Y N Sun ldquoThe (frac(1198661015840119866))-expansion method for nonlinear differential-difference equa-tionsrdquo Physics Letters Section A vol 373 no 10 pp 905ndash9102009

[22] I Aslan ldquoThe Discrete (1198661015840119866)-expansion method applied tothe differential-difference Burgers equation and the relativisticToda lattice systemrdquo Numerical Methods for Partial DifferentialEquations vol 28 pp 127ndash137 2012

[23] I Aslan ldquoExact and explicit solutions to the discrete nonlinearSchrodinger equation with asaturable nonlinearityrdquo PhysicsLetters A vol 375 pp 4214ndash4217 2011

10 Abstract and Applied Analysis

[24] K A Gepreel ldquoRational Jacobi Elliptic solutions for nonlineardifference differential equationsrdquo Nonlinear Science Letters Avol 2 pp 151ndash158 2011

[25] K A Gepreel and A R Shehata ldquoRational Jacobi ellipticsolutions for nonlinear differential-difference lattice equationsrdquoApplied Mathematics Letters vol 25 pp 1173ndash1178 2012

[26] K A Gepreel T A Nofal and A A AlThobaiti ldquoThe modi-fied rational Jacobi elliptic solutions for nonlinear differentialdifference equationsrdquo Journal of Applied Mathematics vol 2012Article ID 427479 30 pages 2012

[27] G CWu and T C Xia ldquoA newmethod for constructing solitonsolutions to differential-difference equationwith symbolic com-putationrdquo Chaos Solitons and Fractals vol 39 no 5 pp 2245ndash2248 2009

[28] J Zhang Z Liu S Li and M Wang ldquoSolitary waves andstable analysis for the quintic discrete nonlinear Schrodingerequationrdquo Physica Scripta vol 86 pp 015401ndash015409 2012

[29] F Xie and J Wang ldquoA new method for solving nonlineardifferential-difference equationrdquo Chaos Solitons and Fractalsvol 27 no 4 pp 1067ndash1071 2006

[30] Q Wang and Y Yu ldquoNew rational formal solutions for (1 + 1)-dimensional Toda equation and another Toda equationrdquoChaosSolitons and Fractals vol 29 no 4 pp 904ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Step 3 Determine the degree 119873 of (35) by balancing thehighest order nonlinear term(s) and the highest order deriva-tives of 119880(120585

119899) in (35)

Step 4 Substituting (36)ndash(41) and given the value of 119873determined in Step 3 into (35) and collecting all terms withthe same degree of 119891(120585

119899) and 119892(120585

119899) together the left-hand

side of (35) is converted into polynomial in 119891(120585119899) and 119892(120585

119899)

Then setting each coefficient 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) of this polynomial to zero we derive a set ofalgebraic equations for 119886

119894 119887119895 119862119894 119860 119861

Step 5 Solving the overdetermined system of nonlinearalgebraic equations by using Maple or Mathematica soft-ware package we end up with explicit expressions for119886119894 119887119895 119862119894 119860 and 119861

Step 6 Substituting 119886119894 119887119895 119862119894 119860 and 119861 into (36) along with

(37) and (39) we can finally obtain the rational solitary wavesolutions for nonlinear difference differential equations (33)

5 Applications

In this section we apply the proposed rational solitary wavefunctions method to construct the rational solitary wavesolutions for some nonlinear DDEs via the discrete non-linear Schrodinger equation with a saturable nonlinearitythe quintic discrete nonlinear Schrodinger equation and therelativistic Toda lattice system which are very important inthe mathematical physics and modern physics

51 Example 1 The Discrete Nonlinear Schrodinger Equationwith a Saturable Nonlinearity We suppose that the solutionof (17) takes the form

119880 (120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (43)

where 1198860 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (43)(41) into (17) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585

119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) to zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198861= plusmn119860radic

minus2

120578

sinh (1198892)cosh2 (1198892)

1198862= plusmnradic

minus2 (1198602 + 1198612)

120578

sinh (1198892)cosh2 (1198892)

120583 =1

4(1 + cosh (119889)) 120578

120590 =2 (cosh (119889) minus 1)1 + cos (119889)

1198860= 0

(44)

where 119860 119861 and 120578 are arbitrary constants and 120578 lt 0 In thiscase the rational hyperbolic solitary wave solution of (17)takes the following form

119880(120585119899) = plusmn 119860radic

minus2

120578

sinh (1198892)cosh2 (1198892) [119860 tanh (120585

119899) + 119861 sech (120585

119899)]

plusmn radicminus2 (1198602 + 1198612)

120578

sinh (1198892) sech (120585119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(45)

Consequently the rational hyperbolic solitary wave solu-tion of (2) has the following form

120595119899=[[

[

= plusmn119860radicminus2

120578

sinh (1198892)cosh2 (1198892) [119860 tanh (120585

119899) + 119861 sech (120585

119899)]

plusmnradicminus2 (1198602 + 1198612)

120578

sinh (1198892) sech (120585119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

]]

]

times 119890minus119894((2(cosh(119889)minus1)(1+cos(119889)))119905+120588)

(46)

where 120585119899= 119899119889 + 120573

Case 2 (120588 = minus1)

1198861= plusmn119860radic

minus2

120578

sin (1198892)cos2 (1198892)

1198862= plusmnradic

minus2 (1198602 minus 1198612)

120578

sin (1198892)cos2 (1198892)

120583 =1

4(cos (119889) + 1) 120578

120590 =2 (minus1 + 2 cos (119889))

cos (119889) + 1 119886

0= 0

(47)

where 119860 119861 and 120578 are arbitrary constants and 120578 lt 0 In thiscase the rational trigonometric solitary wave solution of (17)takes the following form

119880(120585119899) = plusmn 119860radic

minus2

120578

sin (1198892)cos2 (1198892) [119860 tan (120585

119899) + 119861 sec (120585

119899)]

plusmn radicminus2 (1198602 minus 1198612)

120578

timessin (1198892) sec (120585

119899)

cos2 (1198892) [119860 tan (120585119899) + 119861 sec (120585

119899)]

(48)

Abstract and Applied Analysis 7

Consequently the rational trigonometric solitary wave solu-tion of (2) has the following form

120595119899= [plusmn119860radic

minus2

120578

sin (1198892)cos2 (1198892) [119860 tan (120585

119899) + 119861 sec (120585

119899)]

plusmn radicminus2 (1198602 minus 1198612)

120578

timessin (1198892) sec (120585

119899)

cos2 (1198892) [119860 tan (120585119899) + 119861 sec (120585

119899)]]

times 119890minus119894((2(minus1+2 cos(119889))(cos(119889)+1))119905+120588)

(49)

where 120585119899= 119899119889 + 120573

52 Example 2 The Quintic Discrete Nonlinear SchrodingerEquation In this subsection we study the quintic discretenonlinear Schrodinger equation (3) by using the rationalsolitary wave functions method

We suppose that the solution of (24) takes the form

119880 (120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (50)

where 1198860 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (50) and(41) into (24) collect all terms with the same power in119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 = 0 1 2 ) and setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) to zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198861= plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

1198862= plusmn

1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) )

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

120572 =minus1

16

[4120574 (cos ℎ (119889) + 1)+120573 (1+2 cosh (119889) + cosh2 (119889))]cosh (119889) minus 1

120596 =minus1

8

120573 [4120574 (cos ℎ (119889) minus 1) + 120573 (cosh2 (119889) minus 1)]120575

1198860= 0

(51)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational hyperbolic solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

)

(52)

Consequently the rational hyperbolic solitary wave solutionof (3) has the following form

120595119899= [plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

) ]

times 119890119894((minus18)(120573[4120574(cos ℎ(119889)minus1)+120573(cosh2(119889)minus1)]120575))119905

(53)

where 120585119899= 119899119889 + 119896

Case 2 (120588 = minus1)

1198861= plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575

1198862= plusmn

1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))120575

8 Abstract and Applied Analysis

1198860= 0

120572 =minus1

16

120573 [(cos (119889) + 1) (4120574 + 2120573) minus 120573 sin2 (119889)]120575

120596 =minus1

8

120573 [4120574 (cos (119889) minus 1) minus 120573 sin2 (119889)]120575

(54)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational trigonometric solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

(55)

Consequently the rational trigonometric solitary wave solu-tion of (3) has the following form

120595119899= ( plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

)

times119890119894((minus18)(120573[4120574(cos(119889)minus1)minus120573 sin2(119889)]120575))119905

(56)

where 120585119899= 119899119889 + 119896

53 Example 3 The Relativistic Toda Lattice System In thissubsection we study the relativistic Toda lattice system (4)by using the rational solitary wave functions method

If we take the transformation

119907119899= minus

1

120572119906119899minus1

1205722 (57)

the transformation (57) reduced the relativistic Toda latticesystem (4) into the following difference differential equation

119906119899119905

= (119906119899+1

120572) (119906119899minus1minus 119906119899) (58)

According to the main steps of rational solitary wave func-tions method we seek traveling wave solutions of (58) in thefollowing form

119906119899(119905) = 119880 (120585

119899) 120585

119899= 1198621119905 + 119899119889 + 120585

0 (59)

where 119889 1198621 and 120585

0are constants The transformation (59)

permits us converting (58) into the following form

11986211198801015840(120585119899) = (119880 (120585

119899) +

1

120572) [119880 (120585

119899minus 119889) minus 119880 (120585

119899)] (60)

Considering the homogeneous balance between the highestorder derivatives and nonlinear terms in (60) we get119873 = 1Thus the solution of (60) has the following form

119880(120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (61)

where 1198860 1198870 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (61) and(41) into (60) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585119899) (119894 = 0 1 119895 =

0 1 2 ) to be zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198860=

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

1198861=

plusmn1198601198862

radic1198602 + 1198612 119862

1=

plusmn21198862

radic1198602 + 1198612

(62)

where 119860 119861 1198862 and 120572 are arbitrary constants In this case

the rational hyperbolic solitary wave solution of (58) has thefollowing form

119880 (120585119899) =

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

plusmn1198601198862

radic1198602 + 1198612 [119860 tanh (120585119899) + 119861 sech (120585

119899)]

+1198862sech (120585

119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(63)

where

120585119899= plusmn

21198862

radic1198602 + 1198612119905 + 119899119889 + 120585

0 (64)

Case 2 (120588 = minus1)

1198860=

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

1198861=

plusmn1198601198862

radic1198602 minus 1198612 119862

1=

plusmn21198862

radic1198602 minus 1198612

(65)

In this case the rational trigonometric solitary wave solutionof (58) has the following form

119880(120585119899) =

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

plusmn1198601198862

radic1198602 minus 1198612 [119860 tan (120585119899) + 119861 sec (120585

119899)]

+1198862sec (120585119899)

[119860 tan (120585119899) + 119861 sec (120585

119899)]

(66)

where

120585119899= plusmn

21198862

radic1198602 minus 1198612119905 + 119899119889 + 120585

0 (67)

Abstract and Applied Analysis 9

6 Discussion

Whenwe compare between the results which obtained in thispaper and other exact solutions we get the following

The solutions obtained in the modified truncated expan-sion functionsmethod are equivalent to the solution obtainedby the exp-functions method but the modified truncatedexpansion is simple and allowed us to solvemore complicatednonlinear difference differential equations such as the dis-crete nonlinear Schrodinger equation with a saturable non-linearity the quintic discrete nonlinear Schrodinger equa-tion and the relativistic Toda lattice system For example thesolution (15) equivalent is to the solution (40) in [20]

In the special case when 119861 = 0 in the rational solitarywave function method we get this method which is equiv-alent to the tanh-function method which discussed in [29]The rational solitary wave functionmethod is extended to thenew rational formal solution which is discussed by [30] when119887119895= 0 in (36)

Remarks These methods which are discussed in this paperallowed us to obtain some new rational solitary wave solu-tions for some complicated nonlinear differential differenceequations

These methods prefer to another methods to convertthe complicated rational methods into a direct nonrationalmethod

7 Conclusions

In this paper we use the modified truncated expansionmethod to obtain the exact solutions for some nonlineardifferential difference equations in the mathematical physicsAlso we calculate the rational solitary wave solutions for thenonlinear differential difference equations As a result manynew and more rational solitary wave solutions are obtainedfrom the hyperbolic function solutions and trigonometricfunction

Acknowledgment

The authors wish to thank the referees for their suggestionsand very useful comments

References

[1] E Fermi J Pasta and S Ulam Collected Papers of Enrico FermiII University of Chicago Press Chicago Ill USA 1965

[2] W P Su J R Schrieffer and A J Heeger ldquoSolitons inpolyacetylenerdquo Physical Review Letters vol 42 no 25 pp 1698ndash1701 1979

[3] A S Davydov ldquoThe theory of contraction of proteins undertheir excitationrdquo Journal ofTheoretical Biology vol 38 no 3 pp559ndash569 1973

[4] P Marquie J M Bilbault and M Remoissenet ldquoObservationof nonlinear localized modes in an electrical latticerdquo PhysicalReview E vol 51 no 6 pp 6127ndash6133 1995

[5] M Toda Theory of Nonlinear Lattices Springer Berlin Ger-many 1989

[6] M Wadati ldquoTransformation theories for nonlinear discretesystemsrdquo Progress ofTheoretical Physics vol 59 pp 36ndash63 1976

[7] YOhta andRHirota ldquoA discrete KdV equation and its Casoratideterminant solutionrdquo Journal of the Physical Society of Japanvol 60 p 2095 1991

[8] M J Ablowitz and J F Ladik ldquoNonlinear differential-differenceequationsrdquo Journal of Mathematical Physics vol 16 no 3 pp598ndash603 1974

[9] X B Hu and W X Ma ldquoApplication of Hirotarsquos bilinearformalism to the Toeplitz latticemdashsome special soliton-likesolutionsrdquo Physics Letters Section A vol 293 no 3-4 pp 161ndash165 2002

[10] D Baldwin U Goktas and W Hereman ldquoSymbolic computa-tion of hyperbolic tangent solutions for nonlinear differential-difference equationsrdquo Computer Physics Communications vol162 no 3 pp 203ndash217 2004

[11] S K Liu Z T Fu Z GWang and S D Liu ldquoPeriodic solutionsfor a class of nonlinear differential-difference equationsrdquo Com-munications in Theoretical Physics vol 49 no 5 pp 1155ndash11582008

[12] C Qiong and L Bin ldquoApplications of jacobi elliptic functionexpansion method for nonlinear differential-difference equa-tionsrdquo Communications inTheoretical Physics vol 43 no 3 pp385ndash388 2005

[13] F Xie M Ji and H Zhao ldquoSome solutions of discrete sine-Gordon equationrdquo Chaos Solitons and Fractals vol 33 no 5pp 1791ndash1795 2007

[14] S D Zhu ldquoExp-function method for the Hybrid-Lattice sys-temrdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 no 3 pp 461ndash464 2007

[15] I Aslan ldquoA discrete generalization of the extended simplestequation methodrdquo Communications in Nonlinear Science andNumerical Simulation vol 15 no 8 pp 1967ndash1973 2010

[16] P Yang Y Chen and Z B Li ldquoADM-Pade technique for thenonlinear lattice equationsrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 362ndash375 2009

[17] SD Zhu YMChu and S LQiu ldquoThehomotopy perturbationmethod for discontinued problems arising in nanotechnologyrdquoComputers andMathematics with Applications vol 58 no 11-12pp 2398ndash2401 2009

[18] F Xie ldquoNew travellingwave solutions of the generalized coupledHirota-Satsuma KdV systemrdquo Chaos Solitons and Fractals vol20 no 5 pp 1005ndash1012 2004

[19] E M E Zayed A M Abourabia K A Gepreel and MM El Horbaty ldquoOn the rational solitary wave solutions forthe nonlinear Hirota-Satsuma coupled KdV systemrdquoApplicableAnalysis vol 85 pp 751ndash768 2006

[20] C S Liu ldquoExponential function rational expansion method fornonlinear differential-difference equationsrdquoChaos Solitons andFractals vol 40 no 2 pp 708ndash716 2009

[21] S Zhang L Dong J M Ba and Y N Sun ldquoThe (frac(1198661015840119866))-expansion method for nonlinear differential-difference equa-tionsrdquo Physics Letters Section A vol 373 no 10 pp 905ndash9102009

[22] I Aslan ldquoThe Discrete (1198661015840119866)-expansion method applied tothe differential-difference Burgers equation and the relativisticToda lattice systemrdquo Numerical Methods for Partial DifferentialEquations vol 28 pp 127ndash137 2012

[23] I Aslan ldquoExact and explicit solutions to the discrete nonlinearSchrodinger equation with asaturable nonlinearityrdquo PhysicsLetters A vol 375 pp 4214ndash4217 2011

10 Abstract and Applied Analysis

[24] K A Gepreel ldquoRational Jacobi Elliptic solutions for nonlineardifference differential equationsrdquo Nonlinear Science Letters Avol 2 pp 151ndash158 2011

[25] K A Gepreel and A R Shehata ldquoRational Jacobi ellipticsolutions for nonlinear differential-difference lattice equationsrdquoApplied Mathematics Letters vol 25 pp 1173ndash1178 2012

[26] K A Gepreel T A Nofal and A A AlThobaiti ldquoThe modi-fied rational Jacobi elliptic solutions for nonlinear differentialdifference equationsrdquo Journal of Applied Mathematics vol 2012Article ID 427479 30 pages 2012

[27] G CWu and T C Xia ldquoA newmethod for constructing solitonsolutions to differential-difference equationwith symbolic com-putationrdquo Chaos Solitons and Fractals vol 39 no 5 pp 2245ndash2248 2009

[28] J Zhang Z Liu S Li and M Wang ldquoSolitary waves andstable analysis for the quintic discrete nonlinear Schrodingerequationrdquo Physica Scripta vol 86 pp 015401ndash015409 2012

[29] F Xie and J Wang ldquoA new method for solving nonlineardifferential-difference equationrdquo Chaos Solitons and Fractalsvol 27 no 4 pp 1067ndash1071 2006

[30] Q Wang and Y Yu ldquoNew rational formal solutions for (1 + 1)-dimensional Toda equation and another Toda equationrdquoChaosSolitons and Fractals vol 29 no 4 pp 904ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Consequently the rational trigonometric solitary wave solu-tion of (2) has the following form

120595119899= [plusmn119860radic

minus2

120578

sin (1198892)cos2 (1198892) [119860 tan (120585

119899) + 119861 sec (120585

119899)]

plusmn radicminus2 (1198602 minus 1198612)

120578

timessin (1198892) sec (120585

119899)

cos2 (1198892) [119860 tan (120585119899) + 119861 sec (120585

119899)]]

times 119890minus119894((2(minus1+2 cos(119889))(cos(119889)+1))119905+120588)

(49)

where 120585119899= 119899119889 + 120573

52 Example 2 The Quintic Discrete Nonlinear SchrodingerEquation In this subsection we study the quintic discretenonlinear Schrodinger equation (3) by using the rationalsolitary wave functions method

We suppose that the solution of (24) takes the form

119880 (120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (50)

where 1198860 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (50) and(41) into (24) collect all terms with the same power in119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 = 0 1 2 ) and setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585

119899) (119894 = 0 1 119895 =

0 1 2 ) to zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198861= plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

1198862= plusmn

1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) )

times (120575 (cos ℎ (119889) + sin ℎ (119889)))minus1)

120572 =minus1

16

[4120574 (cos ℎ (119889) + 1)+120573 (1+2 cosh (119889) + cosh2 (119889))]cosh (119889) minus 1

120596 =minus1

8

120573 [4120574 (cos ℎ (119889) minus 1) + 120573 (cosh2 (119889) minus 1)]120575

1198860= 0

(51)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational hyperbolic solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

)

(52)

Consequently the rational hyperbolic solitary wave solutionof (3) has the following form

120595119899= [plusmn

1

4((radic2radic120575120573 (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1)119860)

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times (119860 tanh (120585119899) + 119861 sech (120585

119899)))minus1

)

plusmn1

4((radic2radic120575120573 (1198602 + 1198612) (cos ℎ (119889) + sin ℎ (119889))

times (cos ℎ (119889) + sin ℎ (119889) minus 1) sech (120585119899) )

times (120575 (cos ℎ (119889) + sin ℎ (119889))

times [119860 tanh (120585119899) + 119861 sech (120585

119899)])minus1

) ]

times 119890119894((minus18)(120573[4120574(cos ℎ(119889)minus1)+120573(cosh2(119889)minus1)]120575))119905

(53)

where 120585119899= 119899119889 + 119896

Case 2 (120588 = minus1)

1198861= plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575

1198862= plusmn

1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))120575

8 Abstract and Applied Analysis

1198860= 0

120572 =minus1

16

120573 [(cos (119889) + 1) (4120574 + 2120573) minus 120573 sin2 (119889)]120575

120596 =minus1

8

120573 [4120574 (cos (119889) minus 1) minus 120573 sin2 (119889)]120575

(54)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational trigonometric solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

(55)

Consequently the rational trigonometric solitary wave solu-tion of (3) has the following form

120595119899= ( plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

)

times119890119894((minus18)(120573[4120574(cos(119889)minus1)minus120573 sin2(119889)]120575))119905

(56)

where 120585119899= 119899119889 + 119896

53 Example 3 The Relativistic Toda Lattice System In thissubsection we study the relativistic Toda lattice system (4)by using the rational solitary wave functions method

If we take the transformation

119907119899= minus

1

120572119906119899minus1

1205722 (57)

the transformation (57) reduced the relativistic Toda latticesystem (4) into the following difference differential equation

119906119899119905

= (119906119899+1

120572) (119906119899minus1minus 119906119899) (58)

According to the main steps of rational solitary wave func-tions method we seek traveling wave solutions of (58) in thefollowing form

119906119899(119905) = 119880 (120585

119899) 120585

119899= 1198621119905 + 119899119889 + 120585

0 (59)

where 119889 1198621 and 120585

0are constants The transformation (59)

permits us converting (58) into the following form

11986211198801015840(120585119899) = (119880 (120585

119899) +

1

120572) [119880 (120585

119899minus 119889) minus 119880 (120585

119899)] (60)

Considering the homogeneous balance between the highestorder derivatives and nonlinear terms in (60) we get119873 = 1Thus the solution of (60) has the following form

119880(120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (61)

where 1198860 1198870 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (61) and(41) into (60) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585119899) (119894 = 0 1 119895 =

0 1 2 ) to be zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198860=

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

1198861=

plusmn1198601198862

radic1198602 + 1198612 119862

1=

plusmn21198862

radic1198602 + 1198612

(62)

where 119860 119861 1198862 and 120572 are arbitrary constants In this case

the rational hyperbolic solitary wave solution of (58) has thefollowing form

119880 (120585119899) =

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

plusmn1198601198862

radic1198602 + 1198612 [119860 tanh (120585119899) + 119861 sech (120585

119899)]

+1198862sech (120585

119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(63)

where

120585119899= plusmn

21198862

radic1198602 + 1198612119905 + 119899119889 + 120585

0 (64)

Case 2 (120588 = minus1)

1198860=

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

1198861=

plusmn1198601198862

radic1198602 minus 1198612 119862

1=

plusmn21198862

radic1198602 minus 1198612

(65)

In this case the rational trigonometric solitary wave solutionof (58) has the following form

119880(120585119899) =

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

plusmn1198601198862

radic1198602 minus 1198612 [119860 tan (120585119899) + 119861 sec (120585

119899)]

+1198862sec (120585119899)

[119860 tan (120585119899) + 119861 sec (120585

119899)]

(66)

where

120585119899= plusmn

21198862

radic1198602 minus 1198612119905 + 119899119889 + 120585

0 (67)

Abstract and Applied Analysis 9

6 Discussion

Whenwe compare between the results which obtained in thispaper and other exact solutions we get the following

The solutions obtained in the modified truncated expan-sion functionsmethod are equivalent to the solution obtainedby the exp-functions method but the modified truncatedexpansion is simple and allowed us to solvemore complicatednonlinear difference differential equations such as the dis-crete nonlinear Schrodinger equation with a saturable non-linearity the quintic discrete nonlinear Schrodinger equa-tion and the relativistic Toda lattice system For example thesolution (15) equivalent is to the solution (40) in [20]

In the special case when 119861 = 0 in the rational solitarywave function method we get this method which is equiv-alent to the tanh-function method which discussed in [29]The rational solitary wave functionmethod is extended to thenew rational formal solution which is discussed by [30] when119887119895= 0 in (36)

Remarks These methods which are discussed in this paperallowed us to obtain some new rational solitary wave solu-tions for some complicated nonlinear differential differenceequations

These methods prefer to another methods to convertthe complicated rational methods into a direct nonrationalmethod

7 Conclusions

In this paper we use the modified truncated expansionmethod to obtain the exact solutions for some nonlineardifferential difference equations in the mathematical physicsAlso we calculate the rational solitary wave solutions for thenonlinear differential difference equations As a result manynew and more rational solitary wave solutions are obtainedfrom the hyperbolic function solutions and trigonometricfunction

Acknowledgment

The authors wish to thank the referees for their suggestionsand very useful comments

References

[1] E Fermi J Pasta and S Ulam Collected Papers of Enrico FermiII University of Chicago Press Chicago Ill USA 1965

[2] W P Su J R Schrieffer and A J Heeger ldquoSolitons inpolyacetylenerdquo Physical Review Letters vol 42 no 25 pp 1698ndash1701 1979

[3] A S Davydov ldquoThe theory of contraction of proteins undertheir excitationrdquo Journal ofTheoretical Biology vol 38 no 3 pp559ndash569 1973

[4] P Marquie J M Bilbault and M Remoissenet ldquoObservationof nonlinear localized modes in an electrical latticerdquo PhysicalReview E vol 51 no 6 pp 6127ndash6133 1995

[5] M Toda Theory of Nonlinear Lattices Springer Berlin Ger-many 1989

[6] M Wadati ldquoTransformation theories for nonlinear discretesystemsrdquo Progress ofTheoretical Physics vol 59 pp 36ndash63 1976

[7] YOhta andRHirota ldquoA discrete KdV equation and its Casoratideterminant solutionrdquo Journal of the Physical Society of Japanvol 60 p 2095 1991

[8] M J Ablowitz and J F Ladik ldquoNonlinear differential-differenceequationsrdquo Journal of Mathematical Physics vol 16 no 3 pp598ndash603 1974

[9] X B Hu and W X Ma ldquoApplication of Hirotarsquos bilinearformalism to the Toeplitz latticemdashsome special soliton-likesolutionsrdquo Physics Letters Section A vol 293 no 3-4 pp 161ndash165 2002

[10] D Baldwin U Goktas and W Hereman ldquoSymbolic computa-tion of hyperbolic tangent solutions for nonlinear differential-difference equationsrdquo Computer Physics Communications vol162 no 3 pp 203ndash217 2004

[11] S K Liu Z T Fu Z GWang and S D Liu ldquoPeriodic solutionsfor a class of nonlinear differential-difference equationsrdquo Com-munications in Theoretical Physics vol 49 no 5 pp 1155ndash11582008

[12] C Qiong and L Bin ldquoApplications of jacobi elliptic functionexpansion method for nonlinear differential-difference equa-tionsrdquo Communications inTheoretical Physics vol 43 no 3 pp385ndash388 2005

[13] F Xie M Ji and H Zhao ldquoSome solutions of discrete sine-Gordon equationrdquo Chaos Solitons and Fractals vol 33 no 5pp 1791ndash1795 2007

[14] S D Zhu ldquoExp-function method for the Hybrid-Lattice sys-temrdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 no 3 pp 461ndash464 2007

[15] I Aslan ldquoA discrete generalization of the extended simplestequation methodrdquo Communications in Nonlinear Science andNumerical Simulation vol 15 no 8 pp 1967ndash1973 2010

[16] P Yang Y Chen and Z B Li ldquoADM-Pade technique for thenonlinear lattice equationsrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 362ndash375 2009

[17] SD Zhu YMChu and S LQiu ldquoThehomotopy perturbationmethod for discontinued problems arising in nanotechnologyrdquoComputers andMathematics with Applications vol 58 no 11-12pp 2398ndash2401 2009

[18] F Xie ldquoNew travellingwave solutions of the generalized coupledHirota-Satsuma KdV systemrdquo Chaos Solitons and Fractals vol20 no 5 pp 1005ndash1012 2004

[19] E M E Zayed A M Abourabia K A Gepreel and MM El Horbaty ldquoOn the rational solitary wave solutions forthe nonlinear Hirota-Satsuma coupled KdV systemrdquoApplicableAnalysis vol 85 pp 751ndash768 2006

[20] C S Liu ldquoExponential function rational expansion method fornonlinear differential-difference equationsrdquoChaos Solitons andFractals vol 40 no 2 pp 708ndash716 2009

[21] S Zhang L Dong J M Ba and Y N Sun ldquoThe (frac(1198661015840119866))-expansion method for nonlinear differential-difference equa-tionsrdquo Physics Letters Section A vol 373 no 10 pp 905ndash9102009

[22] I Aslan ldquoThe Discrete (1198661015840119866)-expansion method applied tothe differential-difference Burgers equation and the relativisticToda lattice systemrdquo Numerical Methods for Partial DifferentialEquations vol 28 pp 127ndash137 2012

[23] I Aslan ldquoExact and explicit solutions to the discrete nonlinearSchrodinger equation with asaturable nonlinearityrdquo PhysicsLetters A vol 375 pp 4214ndash4217 2011

10 Abstract and Applied Analysis

[24] K A Gepreel ldquoRational Jacobi Elliptic solutions for nonlineardifference differential equationsrdquo Nonlinear Science Letters Avol 2 pp 151ndash158 2011

[25] K A Gepreel and A R Shehata ldquoRational Jacobi ellipticsolutions for nonlinear differential-difference lattice equationsrdquoApplied Mathematics Letters vol 25 pp 1173ndash1178 2012

[26] K A Gepreel T A Nofal and A A AlThobaiti ldquoThe modi-fied rational Jacobi elliptic solutions for nonlinear differentialdifference equationsrdquo Journal of Applied Mathematics vol 2012Article ID 427479 30 pages 2012

[27] G CWu and T C Xia ldquoA newmethod for constructing solitonsolutions to differential-difference equationwith symbolic com-putationrdquo Chaos Solitons and Fractals vol 39 no 5 pp 2245ndash2248 2009

[28] J Zhang Z Liu S Li and M Wang ldquoSolitary waves andstable analysis for the quintic discrete nonlinear Schrodingerequationrdquo Physica Scripta vol 86 pp 015401ndash015409 2012

[29] F Xie and J Wang ldquoA new method for solving nonlineardifferential-difference equationrdquo Chaos Solitons and Fractalsvol 27 no 4 pp 1067ndash1071 2006

[30] Q Wang and Y Yu ldquoNew rational formal solutions for (1 + 1)-dimensional Toda equation and another Toda equationrdquoChaosSolitons and Fractals vol 29 no 4 pp 904ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

1198860= 0

120572 =minus1

16

120573 [(cos (119889) + 1) (4120574 + 2120573) minus 120573 sin2 (119889)]120575

120596 =minus1

8

120573 [4120574 (cos (119889) minus 1) minus 120573 sin2 (119889)]120575

(54)

where 119860 119861 120573 120574 and 120575 are arbitrary constants In this casethe rational trigonometric solitary wave solution of (24) takesthe following form

119880(120585119899) = plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

(55)

Consequently the rational trigonometric solitary wave solu-tion of (3) has the following form

120595119899= ( plusmn

1

2

radicminus120575120573 (cos (119889) minus 1)119860120575 (119860 tan (120585

119899) + 119861 sec (120585

119899))

plusmn1

2

radic120575120573 (1198602 minus 1198612) (1 minus cos (119889))sec (120585119899)

120575 [119860 tan (120585119899) + 119861 sec (120585

119899)]

)

times119890119894((minus18)(120573[4120574(cos(119889)minus1)minus120573 sin2(119889)]120575))119905

(56)

where 120585119899= 119899119889 + 119896

53 Example 3 The Relativistic Toda Lattice System In thissubsection we study the relativistic Toda lattice system (4)by using the rational solitary wave functions method

If we take the transformation

119907119899= minus

1

120572119906119899minus1

1205722 (57)

the transformation (57) reduced the relativistic Toda latticesystem (4) into the following difference differential equation

119906119899119905

= (119906119899+1

120572) (119906119899minus1minus 119906119899) (58)

According to the main steps of rational solitary wave func-tions method we seek traveling wave solutions of (58) in thefollowing form

119906119899(119905) = 119880 (120585

119899) 120585

119899= 1198621119905 + 119899119889 + 120585

0 (59)

where 119889 1198621 and 120585

0are constants The transformation (59)

permits us converting (58) into the following form

11986211198801015840(120585119899) = (119880 (120585

119899) +

1

120572) [119880 (120585

119899minus 119889) minus 119880 (120585

119899)] (60)

Considering the homogeneous balance between the highestorder derivatives and nonlinear terms in (60) we get119873 = 1Thus the solution of (60) has the following form

119880(120585119899) = 1198860+ 1198861119891 (120585119899) + 1198862119892 (120585119899) (61)

where 1198860 1198870 1198861 and 119887

1are arbitrary constants to be deter-

mined later With the aid of Maple we substitute (61) and(41) into (60) and collect all terms with the same powerin 119891119894(120585

119899) 119892119895(120585119899) (119894 = 0 1 119895 = 0 1 2 ) Setting each

coefficient of these terms 119891119894(120585119899) 119892119895(120585119899) (119894 = 0 1 119895 =

0 1 2 ) to be zero yields a set of algebraic equations whichhave the following solutions

Case 1 (120588 = 1)

1198860=

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

1198861=

plusmn1198601198862

radic1198602 + 1198612 119862

1=

plusmn21198862

radic1198602 + 1198612

(62)

where 119860 119861 1198862 and 120572 are arbitrary constants In this case

the rational hyperbolic solitary wave solution of (58) has thefollowing form

119880 (120585119899) =

1

120572 (119890119889 minus 1)[plusmn1205721198862(119890119889 + 1)

radic1198602 + 1198612+ (1 minus 119890

119889)]

plusmn1198601198862

radic1198602 + 1198612 [119860 tanh (120585119899) + 119861 sech (120585

119899)]

+1198862sech (120585

119899)

[119860 tanh (120585119899) + 119861 sech (120585

119899)]

(63)

where

120585119899= plusmn

21198862

radic1198602 + 1198612119905 + 119899119889 + 120585

0 (64)

Case 2 (120588 = minus1)

1198860=

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

1198861=

plusmn1198601198862

radic1198602 minus 1198612 119862

1=

plusmn21198862

radic1198602 minus 1198612

(65)

In this case the rational trigonometric solitary wave solutionof (58) has the following form

119880(120585119899) =

1

120572 sin (119889)[plusmn1205721198862(cos (119889) + 1)radic1198602 minus 1198612

+ sin (119889)]

plusmn1198601198862

radic1198602 minus 1198612 [119860 tan (120585119899) + 119861 sec (120585

119899)]

+1198862sec (120585119899)

[119860 tan (120585119899) + 119861 sec (120585

119899)]

(66)

where

120585119899= plusmn

21198862

radic1198602 minus 1198612119905 + 119899119889 + 120585

0 (67)

Abstract and Applied Analysis 9

6 Discussion

Whenwe compare between the results which obtained in thispaper and other exact solutions we get the following

The solutions obtained in the modified truncated expan-sion functionsmethod are equivalent to the solution obtainedby the exp-functions method but the modified truncatedexpansion is simple and allowed us to solvemore complicatednonlinear difference differential equations such as the dis-crete nonlinear Schrodinger equation with a saturable non-linearity the quintic discrete nonlinear Schrodinger equa-tion and the relativistic Toda lattice system For example thesolution (15) equivalent is to the solution (40) in [20]

In the special case when 119861 = 0 in the rational solitarywave function method we get this method which is equiv-alent to the tanh-function method which discussed in [29]The rational solitary wave functionmethod is extended to thenew rational formal solution which is discussed by [30] when119887119895= 0 in (36)

Remarks These methods which are discussed in this paperallowed us to obtain some new rational solitary wave solu-tions for some complicated nonlinear differential differenceequations

These methods prefer to another methods to convertthe complicated rational methods into a direct nonrationalmethod

7 Conclusions

In this paper we use the modified truncated expansionmethod to obtain the exact solutions for some nonlineardifferential difference equations in the mathematical physicsAlso we calculate the rational solitary wave solutions for thenonlinear differential difference equations As a result manynew and more rational solitary wave solutions are obtainedfrom the hyperbolic function solutions and trigonometricfunction

Acknowledgment

The authors wish to thank the referees for their suggestionsand very useful comments

References

[1] E Fermi J Pasta and S Ulam Collected Papers of Enrico FermiII University of Chicago Press Chicago Ill USA 1965

[2] W P Su J R Schrieffer and A J Heeger ldquoSolitons inpolyacetylenerdquo Physical Review Letters vol 42 no 25 pp 1698ndash1701 1979

[3] A S Davydov ldquoThe theory of contraction of proteins undertheir excitationrdquo Journal ofTheoretical Biology vol 38 no 3 pp559ndash569 1973

[4] P Marquie J M Bilbault and M Remoissenet ldquoObservationof nonlinear localized modes in an electrical latticerdquo PhysicalReview E vol 51 no 6 pp 6127ndash6133 1995

[5] M Toda Theory of Nonlinear Lattices Springer Berlin Ger-many 1989

[6] M Wadati ldquoTransformation theories for nonlinear discretesystemsrdquo Progress ofTheoretical Physics vol 59 pp 36ndash63 1976

[7] YOhta andRHirota ldquoA discrete KdV equation and its Casoratideterminant solutionrdquo Journal of the Physical Society of Japanvol 60 p 2095 1991

[8] M J Ablowitz and J F Ladik ldquoNonlinear differential-differenceequationsrdquo Journal of Mathematical Physics vol 16 no 3 pp598ndash603 1974

[9] X B Hu and W X Ma ldquoApplication of Hirotarsquos bilinearformalism to the Toeplitz latticemdashsome special soliton-likesolutionsrdquo Physics Letters Section A vol 293 no 3-4 pp 161ndash165 2002

[10] D Baldwin U Goktas and W Hereman ldquoSymbolic computa-tion of hyperbolic tangent solutions for nonlinear differential-difference equationsrdquo Computer Physics Communications vol162 no 3 pp 203ndash217 2004

[11] S K Liu Z T Fu Z GWang and S D Liu ldquoPeriodic solutionsfor a class of nonlinear differential-difference equationsrdquo Com-munications in Theoretical Physics vol 49 no 5 pp 1155ndash11582008

[12] C Qiong and L Bin ldquoApplications of jacobi elliptic functionexpansion method for nonlinear differential-difference equa-tionsrdquo Communications inTheoretical Physics vol 43 no 3 pp385ndash388 2005

[13] F Xie M Ji and H Zhao ldquoSome solutions of discrete sine-Gordon equationrdquo Chaos Solitons and Fractals vol 33 no 5pp 1791ndash1795 2007

[14] S D Zhu ldquoExp-function method for the Hybrid-Lattice sys-temrdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 no 3 pp 461ndash464 2007

[15] I Aslan ldquoA discrete generalization of the extended simplestequation methodrdquo Communications in Nonlinear Science andNumerical Simulation vol 15 no 8 pp 1967ndash1973 2010

[16] P Yang Y Chen and Z B Li ldquoADM-Pade technique for thenonlinear lattice equationsrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 362ndash375 2009

[17] SD Zhu YMChu and S LQiu ldquoThehomotopy perturbationmethod for discontinued problems arising in nanotechnologyrdquoComputers andMathematics with Applications vol 58 no 11-12pp 2398ndash2401 2009

[18] F Xie ldquoNew travellingwave solutions of the generalized coupledHirota-Satsuma KdV systemrdquo Chaos Solitons and Fractals vol20 no 5 pp 1005ndash1012 2004

[19] E M E Zayed A M Abourabia K A Gepreel and MM El Horbaty ldquoOn the rational solitary wave solutions forthe nonlinear Hirota-Satsuma coupled KdV systemrdquoApplicableAnalysis vol 85 pp 751ndash768 2006

[20] C S Liu ldquoExponential function rational expansion method fornonlinear differential-difference equationsrdquoChaos Solitons andFractals vol 40 no 2 pp 708ndash716 2009

[21] S Zhang L Dong J M Ba and Y N Sun ldquoThe (frac(1198661015840119866))-expansion method for nonlinear differential-difference equa-tionsrdquo Physics Letters Section A vol 373 no 10 pp 905ndash9102009

[22] I Aslan ldquoThe Discrete (1198661015840119866)-expansion method applied tothe differential-difference Burgers equation and the relativisticToda lattice systemrdquo Numerical Methods for Partial DifferentialEquations vol 28 pp 127ndash137 2012

[23] I Aslan ldquoExact and explicit solutions to the discrete nonlinearSchrodinger equation with asaturable nonlinearityrdquo PhysicsLetters A vol 375 pp 4214ndash4217 2011

10 Abstract and Applied Analysis

[24] K A Gepreel ldquoRational Jacobi Elliptic solutions for nonlineardifference differential equationsrdquo Nonlinear Science Letters Avol 2 pp 151ndash158 2011

[25] K A Gepreel and A R Shehata ldquoRational Jacobi ellipticsolutions for nonlinear differential-difference lattice equationsrdquoApplied Mathematics Letters vol 25 pp 1173ndash1178 2012

[26] K A Gepreel T A Nofal and A A AlThobaiti ldquoThe modi-fied rational Jacobi elliptic solutions for nonlinear differentialdifference equationsrdquo Journal of Applied Mathematics vol 2012Article ID 427479 30 pages 2012

[27] G CWu and T C Xia ldquoA newmethod for constructing solitonsolutions to differential-difference equationwith symbolic com-putationrdquo Chaos Solitons and Fractals vol 39 no 5 pp 2245ndash2248 2009

[28] J Zhang Z Liu S Li and M Wang ldquoSolitary waves andstable analysis for the quintic discrete nonlinear Schrodingerequationrdquo Physica Scripta vol 86 pp 015401ndash015409 2012

[29] F Xie and J Wang ldquoA new method for solving nonlineardifferential-difference equationrdquo Chaos Solitons and Fractalsvol 27 no 4 pp 1067ndash1071 2006

[30] Q Wang and Y Yu ldquoNew rational formal solutions for (1 + 1)-dimensional Toda equation and another Toda equationrdquoChaosSolitons and Fractals vol 29 no 4 pp 904ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

6 Discussion

Whenwe compare between the results which obtained in thispaper and other exact solutions we get the following

The solutions obtained in the modified truncated expan-sion functionsmethod are equivalent to the solution obtainedby the exp-functions method but the modified truncatedexpansion is simple and allowed us to solvemore complicatednonlinear difference differential equations such as the dis-crete nonlinear Schrodinger equation with a saturable non-linearity the quintic discrete nonlinear Schrodinger equa-tion and the relativistic Toda lattice system For example thesolution (15) equivalent is to the solution (40) in [20]

In the special case when 119861 = 0 in the rational solitarywave function method we get this method which is equiv-alent to the tanh-function method which discussed in [29]The rational solitary wave functionmethod is extended to thenew rational formal solution which is discussed by [30] when119887119895= 0 in (36)

Remarks These methods which are discussed in this paperallowed us to obtain some new rational solitary wave solu-tions for some complicated nonlinear differential differenceequations

These methods prefer to another methods to convertthe complicated rational methods into a direct nonrationalmethod

7 Conclusions

In this paper we use the modified truncated expansionmethod to obtain the exact solutions for some nonlineardifferential difference equations in the mathematical physicsAlso we calculate the rational solitary wave solutions for thenonlinear differential difference equations As a result manynew and more rational solitary wave solutions are obtainedfrom the hyperbolic function solutions and trigonometricfunction

Acknowledgment

The authors wish to thank the referees for their suggestionsand very useful comments

References

[1] E Fermi J Pasta and S Ulam Collected Papers of Enrico FermiII University of Chicago Press Chicago Ill USA 1965

[2] W P Su J R Schrieffer and A J Heeger ldquoSolitons inpolyacetylenerdquo Physical Review Letters vol 42 no 25 pp 1698ndash1701 1979

[3] A S Davydov ldquoThe theory of contraction of proteins undertheir excitationrdquo Journal ofTheoretical Biology vol 38 no 3 pp559ndash569 1973

[4] P Marquie J M Bilbault and M Remoissenet ldquoObservationof nonlinear localized modes in an electrical latticerdquo PhysicalReview E vol 51 no 6 pp 6127ndash6133 1995

[5] M Toda Theory of Nonlinear Lattices Springer Berlin Ger-many 1989

[6] M Wadati ldquoTransformation theories for nonlinear discretesystemsrdquo Progress ofTheoretical Physics vol 59 pp 36ndash63 1976

[7] YOhta andRHirota ldquoA discrete KdV equation and its Casoratideterminant solutionrdquo Journal of the Physical Society of Japanvol 60 p 2095 1991

[8] M J Ablowitz and J F Ladik ldquoNonlinear differential-differenceequationsrdquo Journal of Mathematical Physics vol 16 no 3 pp598ndash603 1974

[9] X B Hu and W X Ma ldquoApplication of Hirotarsquos bilinearformalism to the Toeplitz latticemdashsome special soliton-likesolutionsrdquo Physics Letters Section A vol 293 no 3-4 pp 161ndash165 2002

[10] D Baldwin U Goktas and W Hereman ldquoSymbolic computa-tion of hyperbolic tangent solutions for nonlinear differential-difference equationsrdquo Computer Physics Communications vol162 no 3 pp 203ndash217 2004

[11] S K Liu Z T Fu Z GWang and S D Liu ldquoPeriodic solutionsfor a class of nonlinear differential-difference equationsrdquo Com-munications in Theoretical Physics vol 49 no 5 pp 1155ndash11582008

[12] C Qiong and L Bin ldquoApplications of jacobi elliptic functionexpansion method for nonlinear differential-difference equa-tionsrdquo Communications inTheoretical Physics vol 43 no 3 pp385ndash388 2005

[13] F Xie M Ji and H Zhao ldquoSome solutions of discrete sine-Gordon equationrdquo Chaos Solitons and Fractals vol 33 no 5pp 1791ndash1795 2007

[14] S D Zhu ldquoExp-function method for the Hybrid-Lattice sys-temrdquo International Journal of Nonlinear Sciences and NumericalSimulation vol 8 no 3 pp 461ndash464 2007

[15] I Aslan ldquoA discrete generalization of the extended simplestequation methodrdquo Communications in Nonlinear Science andNumerical Simulation vol 15 no 8 pp 1967ndash1973 2010

[16] P Yang Y Chen and Z B Li ldquoADM-Pade technique for thenonlinear lattice equationsrdquo Applied Mathematics and Compu-tation vol 210 no 2 pp 362ndash375 2009

[17] SD Zhu YMChu and S LQiu ldquoThehomotopy perturbationmethod for discontinued problems arising in nanotechnologyrdquoComputers andMathematics with Applications vol 58 no 11-12pp 2398ndash2401 2009

[18] F Xie ldquoNew travellingwave solutions of the generalized coupledHirota-Satsuma KdV systemrdquo Chaos Solitons and Fractals vol20 no 5 pp 1005ndash1012 2004

[19] E M E Zayed A M Abourabia K A Gepreel and MM El Horbaty ldquoOn the rational solitary wave solutions forthe nonlinear Hirota-Satsuma coupled KdV systemrdquoApplicableAnalysis vol 85 pp 751ndash768 2006

[20] C S Liu ldquoExponential function rational expansion method fornonlinear differential-difference equationsrdquoChaos Solitons andFractals vol 40 no 2 pp 708ndash716 2009

[21] S Zhang L Dong J M Ba and Y N Sun ldquoThe (frac(1198661015840119866))-expansion method for nonlinear differential-difference equa-tionsrdquo Physics Letters Section A vol 373 no 10 pp 905ndash9102009

[22] I Aslan ldquoThe Discrete (1198661015840119866)-expansion method applied tothe differential-difference Burgers equation and the relativisticToda lattice systemrdquo Numerical Methods for Partial DifferentialEquations vol 28 pp 127ndash137 2012

[23] I Aslan ldquoExact and explicit solutions to the discrete nonlinearSchrodinger equation with asaturable nonlinearityrdquo PhysicsLetters A vol 375 pp 4214ndash4217 2011

10 Abstract and Applied Analysis

[24] K A Gepreel ldquoRational Jacobi Elliptic solutions for nonlineardifference differential equationsrdquo Nonlinear Science Letters Avol 2 pp 151ndash158 2011

[25] K A Gepreel and A R Shehata ldquoRational Jacobi ellipticsolutions for nonlinear differential-difference lattice equationsrdquoApplied Mathematics Letters vol 25 pp 1173ndash1178 2012

[26] K A Gepreel T A Nofal and A A AlThobaiti ldquoThe modi-fied rational Jacobi elliptic solutions for nonlinear differentialdifference equationsrdquo Journal of Applied Mathematics vol 2012Article ID 427479 30 pages 2012

[27] G CWu and T C Xia ldquoA newmethod for constructing solitonsolutions to differential-difference equationwith symbolic com-putationrdquo Chaos Solitons and Fractals vol 39 no 5 pp 2245ndash2248 2009

[28] J Zhang Z Liu S Li and M Wang ldquoSolitary waves andstable analysis for the quintic discrete nonlinear Schrodingerequationrdquo Physica Scripta vol 86 pp 015401ndash015409 2012

[29] F Xie and J Wang ldquoA new method for solving nonlineardifferential-difference equationrdquo Chaos Solitons and Fractalsvol 27 no 4 pp 1067ndash1071 2006

[30] Q Wang and Y Yu ldquoNew rational formal solutions for (1 + 1)-dimensional Toda equation and another Toda equationrdquoChaosSolitons and Fractals vol 29 no 4 pp 904ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

[24] K A Gepreel ldquoRational Jacobi Elliptic solutions for nonlineardifference differential equationsrdquo Nonlinear Science Letters Avol 2 pp 151ndash158 2011

[25] K A Gepreel and A R Shehata ldquoRational Jacobi ellipticsolutions for nonlinear differential-difference lattice equationsrdquoApplied Mathematics Letters vol 25 pp 1173ndash1178 2012

[26] K A Gepreel T A Nofal and A A AlThobaiti ldquoThe modi-fied rational Jacobi elliptic solutions for nonlinear differentialdifference equationsrdquo Journal of Applied Mathematics vol 2012Article ID 427479 30 pages 2012

[27] G CWu and T C Xia ldquoA newmethod for constructing solitonsolutions to differential-difference equationwith symbolic com-putationrdquo Chaos Solitons and Fractals vol 39 no 5 pp 2245ndash2248 2009

[28] J Zhang Z Liu S Li and M Wang ldquoSolitary waves andstable analysis for the quintic discrete nonlinear Schrodingerequationrdquo Physica Scripta vol 86 pp 015401ndash015409 2012

[29] F Xie and J Wang ldquoA new method for solving nonlineardifferential-difference equationrdquo Chaos Solitons and Fractalsvol 27 no 4 pp 1067ndash1071 2006

[30] Q Wang and Y Yu ldquoNew rational formal solutions for (1 + 1)-dimensional Toda equation and another Toda equationrdquoChaosSolitons and Fractals vol 29 no 4 pp 904ndash915 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of