Research ArticleOn a Theory for Analysing Second-Order Systems of OrdinaryDiscrete Equations

J J H Bashingwa 1 and A H Kara 2

1Comp B Group University of Cape Town South Africa2School of Mathematics University of e Witwatersrand Johannesburg South Africa

Correspondence should be addressed to A H Kara abdulkarawitsacza

Received 30 December 2018 Accepted 21 April 2019 Published 12 May 2019

Academic Editor Francisco Balibrea

Copyright copy 2019 J J H Bashingwa and A H Kara This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We present geometric based methods for solving systems of discrete or difference equations and introduce a technique for findingconservation laws for such systems

1 Introduction

Depending on amodel being studied somephysical lawsmaybe described by differential equations (DEs) Lie group theoryprovides us with powerful tools for obtaining analyticalsolutions of such equations [1] Over the last 30 years aconsiderable amount of work has been invested into applyingLiersquos theory to solve and classify difference equations (ΔEs)(see [2ndash6] and references therein)

The use of symmetry methods for ordinary differenceequations (OΔEs) has been introduced by Maeda [2] Heshowed that the resulting linearized symmetry condition(LSC) amounts to a set of functional equations which ishard to solve in general Hydon introduced a techniqueto solve LSC and obtain symmetries in closed form byrepetitive differentiations [5] For example a full classificationof second-order OΔEs according to their point symmetriesexists in the literature [6] However for systems of differenceequations (SΔEs) most of results are based on inductionmethods (see [7ndash9] and references therein) In this paperwe present a method for solving SΔEs using their underlyingsymmetry

It has been proved in [5] that every second-order lin-ear homogeneous OΔE has an eight-dimension Lie algebraisomorphic to sl(3) This is not valid for second-ordersystem of difference equations (SΔEs) We shall prove this inSection 3

2 Groundwork

Let us consider an119873-th order system of 119903 ΔEs119909119894119899+119873 = 120596119894 (119899 1199091119899 119909119903119899 1199091119899+1 119909119903119899+1 1199091119899+119873minus1 119909119903119899+119873minus1) 119894 = 1 119903 (1)

We assume that for each 120596119894 there exists at least one 119909119895119899 (119894 119895 =1 119903) such that 120597120596119894120597119909119895119899 = 0Consider a point transformation

Γ120598 119883 997891997888rarr 119883 (119883 120598) (2)

where 119883 = (1199091119899 119909119903119899) are continuous variables Γ willbe called one-parameter Lie group of transformations if itsatisfies the following properties

(i) Γ0 is the identity map ie119883 = 119883 for 120598 = 0(ii) Γ120583Γ] = Γ120583+] for every 120583 and ] close to 0

(iii) Each 119909119894119899 can be expanded as a Taylor series in aneighbourhood of 120598 = 0

Therefore we have

119909119894119899+119895 = 119909119894119899+119895 + 120598S119895119876119894 (119899 1199091119899 119909119903119899 1199091119899+1 119909119903119899+1 1199091119899+119873minus1 119909119903119899+119873minus1) + 119874 (1205982) (3)

HindawiJournal of MathematicsVolume 2019 Article ID 8256867 14 pageshttpsdoiorg10115520198256867

2 Journal of Mathematics

where 119876119894 are continuous functions which we shall refer to ascharacteristics 119894 = 1 119903 119895 = 1 119873 and S is the ldquoshiftrdquooperator It is defined as follows

S 119899 997891997888rarr 119899 + 1S119896 (119909119894119899) = 119909119894119899+119896 (4)

We define the discrete differentiation operator as follows

Δ = S minus 119868119889 (5)

where 119868119889 is the identity operatorThe symmetry condition for the SΔEs (1) is

119909119894119899+119873 = 120596119894 (119899 1199091119899 119909119903119899 1199091119899+1 119909119903119899+1 1199091119899+119873minus1 119909119903119899+119873minus1) 119894 = 1 119903 (6)

whenever (1) holdsLie symmetries are obtained by linearizing the symmetry

condition (6) about the identityWehave the following systemof linearized symmetry condition (SLSC)

S119873 (119876119894) minus 119883120596119894 = 0 119894 = 1 119903 (7)

where the symmetry generator 119883 is given by

119883 = 119873minus1sum119895=0

( 119903sum119894=1

S119895 (119876119894) 120597120597119909119894119899+119895) (8)

Definition 1 A function 119908119899 is invariant function under theLie group of transformations Γ if

119883(119908119899) = 0 (9)

where 119908119899 can be found by solving the characteristicequation

d11990911198991198761 = sdot sdot sdot =d119909119903119899119876119903 =

d1199091119899+1S (1198761) = sdot sdot sdot =

d119909119903119899+1S (119876119903) = sdot sdot sdot

= d1199091119899+119873minus1S119873minus1 (1198761) = sdot sdot sdot =

d119909119903119899+119873minus1S119873minus1 (119876119903) =

1199081198990(10)

Theorem 2 e discrete differential operator Δ in (5) and thegenerator of symmetry 119883 in (8) commute

Proof We prove the theorem for119873 = 1 any generalisation isstraightforward

[119883 Δ] 119865 (119899119883119899) = 119883 Δ (119865 (119899119883119899))minus Δ 119883 (119865 (119899 119909119894119899))

119883119899 = (1199091 1199092 119909119903)= 119883 119865 (119899 + 1119883119899+1) minus 119865 (119899119883119899)minus Δ( 119903sum

119894=1

119876119894 120597120597119909119894119899119865 (119899 119883119899))= 119903sum119894=1

S (119876119894) 120597120597119909119894119899+1119865 (119899 + 1119883119899+1) minus 119876119894120597120597119909119894119899119865 (119899119883119899)

minus 119903sum119894=1

S(119876119894 120597120597119909119894119899119865 (119899119883119899)) minus 119876119894120597120597119909119894119899119865 (119899 119883119899)

= 0

(11)

Corollary 3 For each invariant119908119899S119908119899 is also an invariantProof We have 119883Δ(119908119899) = Δ119883119908119899 = 0

Equivalently 119883(S minus 119868119889)119908119899 = 119883S(119908119899) minus 119883119908119899 = 0or 119883(S119908119899) = 0We shall use this corollary for reductions in Section 3A first integral for the system (1) is a quantity120601(119899 119909119899 119910119899 119909119899+1 119910119899+1) such that

Δ120601 (119899 119909119899 119910119899 119909119899+1 119910119899+1) = 0 (12)

whenever (1) holdsIn Section 5 we shall use the condition (12) to develop a

constructive technique for obtaining first integrals

Remark 4 In this paper we shall consider Lie point symme-try ie the characteristics are given by 119876119894(119899 1199091119899 119909119903119899)

We refer the reader to [1] for more information onsymmetry methods for differential equations

3 Symmetries and Reductions

31 Finding Characteristics Consider a second-order systemof 2 ΔEs

119909119899+2 = 1205961 (119899 119909119899 119910119899 119909119899+1 119910119899+1) 119910119899+2 = 1205962 (119899 119909119899 119910119899 119909119899+1 119910119899+1) (13)

We assume that 1205971205961120597119909119899 = 0 or 1205971205961120597119910119899 = 0 and 1205971205962120597119909119899 = 0or 1205971205962120597119910119899 = 0 so the system is of second order

Journal of Mathematics 3

The SLCS (7) reduces to

S2 (1198761) minus 11987611205961119909119899 minus 11987621205961119910119899 minusS (1198761) 1205961119909119899+1minusS (1198762) 1205961119910119899+1 = 0 (14)

S2 (1198762) minus 11987611205962119909119899 minus 11987621205962119910119899 minusS (1198761) 1205962119909119899+1minusS (1198762) 1205962119910119899+1 = 0 (15)

where 119892119909 = 120597119892120597119909 1198761 = 1198761(119899 119909119899 119910119899) and 1198762 =1198762(119899 119909119899 119910119899)The functional equations (14) and (15) contain functions1198761 and 1198762 with different pairs of arguments making them

difficult to solve For concreteness if for instance the discretevariable 119899 stands for ldquostaterdquo in physics 1198761(119899 119909119899 119910119899) andS(1198761) equiv 1198761(119899 + 1 119909119899+1 119910119899+1) belong to two different states

To overcome this we proceed as follows

Step 1 (elimination of S2(1198761) and S2(1198762)) We differentiate(total differentiation) (14) and (15) with respect to 119909119899 and119910119899 respectively keeping 1205961 and 1205962 fixed Here we take 119909119899+1as function of 119909119899 119910119899 119910119899+1 1205961 1205962 and 119910119899+1 as function of119909119899 119910119899 119909119899+1 1205961 1205962

The total derivative operators are given by

dd119909119899 =

120597120597119909119899 +120597119909119899+1120597119909119899

120597120597119909119899+1 +120597119910119899+1120597119909119899

120597120597119910119899+1 + sdot sdot sdotdd119910119899 =

120597120597119910119899 +120597119909119899+1120597119910119899

120597120597119909119899+1 +120597119910119899+1120597119910119899

120597120597119910119899+1 + sdot sdot sdot(16)

In this case this is simplified to

dd119909119899 =

120597120597119909119899 minus (12059611199091198991205961119909119899+1 +

12059621199091198991205962119909119899+1 )120597120597119909119899+1

minus ( 12059611199091198991205961119910119899+1 +12059621199091198991205962119910119899+1 )

120597120597119910119899+1(17)

dd119910119899 =

120597120597119910119899 minus (12059611199101198991205961119909119899+1 +

12059621199101198991205962119909119899+1 )120597120597119909119899+1

minus ( 12059611199101198991205961119910119899+1 +12059621199101198991205962119910119899+1 )

120597120597119910119899+1(18)

So we apply the operator (17) to (14) and (18) to (15) keeping1205961 and 1205962 fixed This leads to the determining system

[11987611205961119909119899 + 11987621205961119910119899]119909119899 +S (1198761) 1205961119909119899+1 119909119899 +S (1198762)sdot 1205961119910119899+1 119909119899 minus ( 12059611199091198991205961119909119899+1 +

12059621199091198991205962119909119899+1 ) [11987611205961119909119899+ 11987621205961119910119899]119909119899+1 minus ( 12059611199091198991205961119909119899+1 +

12059621199091198991205962119909119899+1 )sdot [S (1198761) 1205961119909119899+1 119909119899+1 + S (1198762) 1205961119910119899+1 119909119899+1+ [S (1198761)]119909119899+1 1205961119909119899+1 + [S (1198762)]119909119899+1 1205961119910119899+1]

minus ( 12059611199091198991205961119910119899+1 +12059621199091198991205962119910119899+1 ) [11987611205961119909119899 + 11987621205961119910119899]119910119899+1

minus ( 12059611199091198991205961119910119899+1 +12059621199091198991205962119910119899+1 ) [S (1198761) 1205961119909119899+1119910119899+1

+ S (1198762) 1205961119910119899+1 119910119899+1 + [S (1198761)]119910119899+1 1205961119909119899+1+ [S (1198762)]119910119899+1 1205961119910119899+1] = 0

(19)

[11987611205962119909119899 + 11987621205962119910119899]119910119899 +S (1198761) 1205962119909119899+1 119910119899 + S (1198762)sdot 1205962119910119899+1 119910119899 minus ( 12059611199101198991205961119909119899+1 +

12059621199101198991205962119909119899+1 ) [11987611205962119909119899+ 11987621205962119910119899]119909119899+1 minus ( 12059611199101198991205961119909119899+1 +

12059621199101198991205962119909119899+1 )sdot [S (1198761) 1205962119909119899+1 119909119899+1 +S (1198762) 1205962119910119899+1 119909119899+1+ [S (1198761)]119909119899+1 1205962119909119899+1 + [S (1198762)]119909119899+1 1205962119910119899+1]minus ( 12059611199101198991205961119910119899+1 +

12059621199101198991205962119910119899+1 ) [11987611205962119909119899 + 11987621205962119910119899]119910119899+1minus ( 12059611199101198991205961119910119899+1 +

12059621199101198991205962119910119899+1 ) [S (1198761) 1205962119909119899+1119910119899+1+ S (1198762) 1205962119910119899+1 119910119899+1 + [S (1198761)]119910119899+1 1205962119909119899+1+ [S (1198762)]119910119899+1 1205962119910119899+1] = 0

(20)

Step 2 (elimination of S(1198761) and S(1198762)) We now differen-tiate (19) and (20) with respect to 119909119899 and 119910119899 respectivelykeeping 119909119899+1 and 119910119899+1 fixed This means that we apply theoperator 120597120597119909119899 on (19) and 120597120597119910119899 on (20) For a second-order SΔEs we need at most to differentiate four timesAfter separating with respect to 119909119899+1 and 119910119899+1 the resultingequations we obtain a system of DEs in 1198761 and 1198762 which issolvable by hand or by using a computer algebra package

Step 3 (explicit form of constant of integration) Whenintegrating in Step 2 to obtain the characteristics 1198761 and 1198762we have constant of integration which appears to be functionsof 119899 To obtain their explicit form we need to substitute theresults obtained in Step 2 in (19) and (20) If we donot succeedin obtaining all the constant of integration we need furthersubstitution in the SLSC (14) and (15)

32 Reductions Consider a second-order SΔEs119909119899+2 = 1205961 (119899 119909119899 119910119899 119909119899+1 119910119899+1) 119910119899+2 = 1205962 (119899 119909119899 119910119899 119909119899+1 119910119899+1) (21)

4 Journal of Mathematics

and its symmetry generator

119883 = 1198761 120597120597119909119899 + 1198762120597120597119910119899 +S1198761 120597120597119909119899+1 +S1198762 120597120597119910119899+1 (22)

The method of characteristics for partial differential equa-tions (PDEs)

d1199091198991198761 =d1199101198991198762 =

d119909119899+11198761 = d119910119899+11198762 = 1198821198990 (23)

leads to three independent constants of integration1198701 1198702 1198703 Each invariant under 119883 is function of thoseconstant119882119899 = 119891(1198701 1198702 1198703)

For second-order systems two invariants suffice to doreduction of the systems

Let

119906119899 = 1198911 (119899 119909119899 119910119899 119909119899+1 119910119899+1) V119899 = 1198912 (119899 119909119899 119910119899 119909119899+1 119910119899+1) (24)

be the invariants functions under119883We choose them in awaythat the Jacobian is nonzero

100381610038161003816100381610038161003816100381610038161003816120597 (1198911 1198912)120597 (119909119899+1 119910119899+1)

100381610038161003816100381610038161003816100381610038161003816 = 0 (25)

That is (24) can be inverted as follows

119909119899+1 = 1198921 (119899 119909119899 119910119899 119906119899 V119899) 119910119899+1 = 1198922 (119899 119909119899 119910119899 119906119899 V119899) (26)

By Corollary 3 SV119899 and S119906119899 are also invariant functionsTherefore the solution of (21) satisfies

119906119899+1 = Ω1 (119906119899 V119899) V119899+1 = Ω2 (119906119899 V119899) (27)

(27) is a first-order SΔEs which can be solved by furtherreductions or by using computer algebra software (mapleMathematica ) for linear systems Note that there existsome first-order systems which cannot be solved analytically

The general solution is

119906119899 = 119906 (119899 1198621 1198622) V119899 = V (119899 1198621 1198622) (28)

for some constant 1198621 1198622So the second-order system (21) is equivalent to the first-

order system obtained by substituting (28) in (26)

119909119899+1 = 1198921 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622)) 119910119899+1 = 1198922 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622)) (29)

(29) also admits the symmetries generated by119883The best wayto integrate any first-order analytic ΔE is to use its canonicalcoordinates [10]

119879119899 = 119879 (119899 119909119899 119910119899) (30)

which satisfy

119883119879119899 = 1 (31)

The obvious choice of canonical coordinates is (see [10])

119905119899 = int d1199091198991198761 (119899 119909119899 119910119899 1198921 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622)) 1198922 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622))) 119904119899 = int d1199101198991198762 (119899 119909119899 119910119899 1198921 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622)) 1198922 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899C1 1198622)))

(32)

33 Applications

331 Example 1 Consider the most general homogeneoussecond-order linear system of difference equations

119909119899+2 = 1198861 (119899) 119909119899 + 1198862 (119899) 119910119899 + 1198863 (119899) 119909119899+1+ 1198864 (119899) 119910119899+1

119910119899+2 = 1198871 (119899) 119909119899 + 1198872 (119899) 119910119899 + 1198873 (119899) 119909119899+1 + 1198874 (119899) 119910119899+1(33)

where 119886119894(119899) 119887119894(119899) 119894 = 1 4 are arbitrary functions

One can readily verify that the determining system (19)and (20) amounts to

1198761119909119899119909119899 = 1198762119909119899119909119899 = 01198761119910119899119910119899 = 1198762119910119899119910119899 = 0 (34)

Therefore

1198761 (119899 119909119899 119910119899) = 1198621119909119899 + 1198622119910119899 + 1198651 (119899) 1198762 (119899 119909119899 119910119899) = 1198623119909119899 + 1198624119910119899 + 1198652 (119899) (35)

where 119862119894 119894 = 1 4 are constants

Journal of Mathematics 5

The characteristics in (35) must satisfy the SLSC (14) and(15) Hence we have

1198651 (119899 + 2) minus [1198861 (119899) 1198651 (119899) + 1198862 (119899) 1198652 (119899)+ 1198863 (119899) 1198651 (119899 + 1) + 1198864 (119899) 1198652 (119899 + 1)] = 0

1198652 (119899 + 2) minus [1198871 (119899) 1198651 (119899) + 1198872 (119899) 1198652 (119899)+ 1198873 (119899) 1198651 (119899 + 1) + 1198874 (119899) 1198652 (119899 + 1)] = 0

(36)

and

1198621 = 11986241198622 = 1198623 = 0 (37)

So (35) is simplified to

1198761 (119899 119909119899 119910119899) = 1198621119909119899 + 1198651 (119899) 1198762 (119899 119909119899 119910119899) = 1198621119910119899 + 1198652 (119899) (38)

The first generator of symmetry for a second-order homoge-neous linear system (36) is the scaling symmetry given by

119883 = 119909119899120597119909119899 + 119910119899120597119910119899 (39)

The system (36) which governs the remaining generators ofthe Lie point symmetry for the system (33) is of second orderin 1198651 and 1198652 Its general solution is

1198651 (119899) = 1198921 (1198991198701 1198702 1198703 1198704) 1198652 (119899) = 1198922 (1198991198701 1198702 1198703 1198704) (40)

where1198701 119894 = 1 4 are constantsSo the most large Lie algebra of symmetry generators

which can be obtained from a homogeneous second-ordersystem of 2 difference equations has dimension five

For clarification let us consider1198861(119899) = 1198863(119899) = 1198864(119899) = 0 1198862(119899) = 1 and 1198872(119899) = 1198873(119899) =1198874(119899) = 0 1198871(119899) = 1 The system (33) becomes

119909119899+2 = 119910119899119910119899+2 = 119909119899 (41)

The systemwhich governs the remaining generators of the Liepoint symmetry in this case is given by

1198651 (119899 + 2) minus 1198652 (119899) = 01198652 (119899 + 2) minus 1198651 (119899) = 0 (42)

The general solutions for this system will be

1198651 (119899) = 1198621 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198623 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4sdot 1198624 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1198652 (119899) = 1198623 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198621 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4sdot 1198622 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

(43)

Therefore we have 5 generators of the Lie point symmetryspanned by

X0 = 119909119899120597119909119899 + 119910119899120597119910119899X1 = [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 120597119909119899

+ [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 120597119910119899X2 = [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 120597119909119899

+ [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 120597119910119899X3 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 120597119909119899

+ [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 120597119910119899X4 = [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 120597119909119899

+ [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 120597119910119899

(44)

332 Example 2 Consider the system

119909119899+2 = 119909119899119910119899+1 + 1119909119899 + 119910119899+1119910119899+2 = 119910119899119909119899+1 + 1119910119899 + 119909119899+1

(45)

(45) is a special case of systems investigated in [11] where theauthor looked at the stability of the systems

6 Journal of Mathematics

We choose the ansatz 1198761(119899 119909119899) 1198762(119899 119910119899)The determining system (19) and (20) amounts to

minus 1198762119910119899119909119899+121199101198992 + 1198781198761119909119899+1119909119899+121199101198992 + 21198762119909119899+12119910119899minus 211987811987611199101198992119909119899+1 + 1198762119910119899119909119899+12 + 11987621199101198991199101198992minus 1198781198761119909119899+1119909119899+12 minus 1198781198761119909119899+11199101198992 minus 21198762119910119899+ 21198781198761119909119899+1 minus 1198762119910119899 + 1198781198761119909119899+1 = 0

(46)

minus 11987611199091198991199091198992119910119899+12 + 1198781198762119910119899+11199091198992119910119899+12 + 21198761119910119899+12119909119899minus 211987811987621199091198992119910119899+1 + 11987611199091198991199091198992 + 1198761119909119899119910119899+12minus 1198781198762119910119899+11199091198992 minus 1198781198762119910119899+1119910119899+12 minus 21198761119909119899+ 21198781198762119910119899+1 minus 1198761119909119899 + 1198781198762119910119899+1 = 0

(47)

Differentiating twice (46) with respect to 119909119899 and twice (47)with respect to 119910119899 keeping 119909119899+1 and 119910119899+1 fixed we obtain afterseparating with respect to119909119899+1 and 119910119899+1 the following systemof Des

119876101584010158401 + 1199091198991198761015840101584010158401 + 2 11987611199091198992 minus 211987610158401119909119899 +

1198761015840101584011199091198992 minus1198761015840101584010158401119909119899 = 0

119876101584010158402 + 1199101198991198761015840101584010158402 + 2 11987621199101198992 minus 211987610158402119910119899 +

1198761015840101584021199101198992 minus1198761015840101584010158402119910119899 = 0

(48)

whose most general solutions are

1198761 (119899 119909119899) = 1198651 (119899) 119909119899 + 1198652 (119899) (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1+ 1198653 (119899) (1199092119899 minus 1)

1198762 (119899 119910119899) = 1198654 (119899) 119910119899 + 1198655 (119899) (1199102119899 minus 1) ln 119910119899 + 1119910119899 minus 1+ 1198656 (119899) (1199102119899 minus 1)

(49)

To obtain the nature of functions 1198651 1198656 we substitute (49)in (46) and (47) After separating with respect to 119909119899 119909119899+1 119910119899and 119910119899+1 we get the following SΔEsminus41198652 (119899) minus 21198654 (119899 + 1) + 41198655 (119899 + 1) minus 21198651 (119899) = 041198652 (119899) minus 21198654 (119899 + 1) minus 41198655 (119899 + 1) + 21198651 (119899) = 0minus41198655 (119899) minus 21198651 (119899 + 1) + 41198652 (119899 + 1) minus 21198654 (119899) = 041198655 (119899) minus 21198651 (119899 + 1) minus 41198652 (119899 + 1) + 21198654 (119899) = 0

(50)

whose solutions are

1198651 (119899) = 1198654 (119899) = 01198652 (119899) = 1198621 + (minus1)119899 11986221198655 (119899) = 1198621 minus (minus1)119899 1198622

(51)

The remaining unknown functions 1198653(119899) and 1198656(119899) aredetermined by substituting (51) and (49) into the SLSC (14)and (15) This leads to the SΔEs

1198653 (119899) minus 1198653 (119899 + 2) + 1198656 (119899 + 1) = 01198656 (119899) + 1198653 (119899 + 1) minus 1198656 (119899 + 2) = 0 (52)

The general solutions to (52) are given by

1198653 (119899)= 1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986262119899+1

sdot minus1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986242119899+1

1198656 (119899)= 1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986242119899+1

sdot minus1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986262119899+1

(53)

where 1198621 1198626 are arbitrary constants It follows that thecharacteristics are given by

1198761 = (1198621 + (minus1)119899 1198622) (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1+

1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986262119899+1 ]]

]

Journal of Mathematics 7

sdot minus1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1] 11986252119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986242119899+1 ]]

](1199092119899 minus 1)

1198762 = (1198621 minus (minus1)119899 1198622) (1199102119899 minus 1) ln 119910119899 + 1119910119899 minus 1+

1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986242119899+1 ]]

]sdot minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1] 11986232119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986262119899+1 ]]

](1199102119899 minus 1)

(54)

Therefore we have six generators of Lie point symmetry

X1 = (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1120597119909119899 + (1199102119899 minus 1) ln119910119899 + 1119910119899 minus 1120597119910119899

X2 = (minus1)119899 (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1120597119909119899 minus (minus1)119899 (1199102119899 minus 1)sdot ln 119910119899 + 1119910119899 minus 1120597119910119899

X3 = 1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199092119899 minus 1) 120597119909119899+ minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199102119899 minus 1) 120597119910119899X4 = minus1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 + (1 + radic5)119899]

2119899+1 ]]

sdot (1199092119899 minus 1) 120597119909119899

+ 1 + (minus1)119899radic5 [[[(minus1 + radic5)119899 minus (1 + radic5)119899]

2119899+1 ]]

sdot (1199102119899 minus 1) 120597119910119899X5 = minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]

2119899 ]]]sdot (1199092119899 minus 1) 120597119909119899

+ 1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199102119899 minus 1) 120597119910119899X6 = 1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 minus (1 + radic5)119899]

2119899+1 ]]

sdot (1199092119899 minus 1) 120597119909119899+ minus1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 + (1 + radic5)119899]

2119899+1 ]]

sdot (1199102119899 minus 1) 120597119910119899

(55)

Each generator in (55) can be used to reduce the order of (45)Let us consider X1 By the characteristic method for

Partial Differential Equations the invariants are given byfollowing equation

d119909119899(1199092119899 minus 1) ln ((119909119899 + 1) (119909119899 minus 1))= d119910119899(1199102119899 minus 1) ln ((119910119899 + 1) (119910119899 minus 1))= d119909119899+1(1199092119899+1 minus 1) ln ((119909119899+1 + 1) (119909119899+1 minus 1))= d119910119899+1(1199102119899+1 minus 1) ln ((119910119899+1 + 1) (119910119899+1 minus 1)) =

1198811198990

(56)

We get

1198621 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119910119899+1 + 1) (119910119899+1 minus 1))

1198622 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119910119899 + 1) (119910119899 minus 1))

1198623 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119909119899+1 + 1) (119909119899+1 minus 1))

119881119899 = 119891 (1198621 1198622 1198623)

(57)

where 1198621 1198622 1198623 are constants

8 Journal of Mathematics

If we choose 119891(1198621 1198622 1198623) = 1198621 we have119906119899 = ln ((119909119899 + 1) (119909119899 minus 1))

ln ((119910119899+1 + 1) (119910119899+1 minus 1)) (58)

and if we choose 119891(1198621 1198622 1198623) = 11986231198622 we haveV119899 = ln ((119910119899 + 1) (119910119899 minus 1))

ln ((119909119899+1 + 1) (119909119899+1 minus 1)) (59)

From (58) and (59)we deduce

119906119899+1 = 11 + V119899

V119899+1 = 11 + 119906119899(60)

Let us now consider the generator X3 The resulting invari-ants are

V119899 = [(119909119899 minus 1) (119909119899 + 1)]120572119899(119910119899+1 minus 1) (119910119899+1 + 1) 119906119899 = [(119910119899 minus 1) (119910119899 + 1)]120573119899(119909119899+1 minus 1) (119909119899+1 + 1)

(61)

where

120572119899 = (1 + radic5)119899 minus (minus1 + radic5)1198992 [(1 + radic5)119899minus1 + (minus1 + radic5)119899minus1]

120573119899 = (1 + radic5)119899 + (minus1 + radic5)1198992 [(1 + radic5)119899minus1 minus (minus1 + radic5)119899minus1]

(62)

Note also the relationship between them

120572119899+1 minus 1 = 1120573119899 120573119899+1 minus 1 = 1120572119899

(63)

From (61) we deduce the following relation

V119899+1 = 11199061120573119899119899

119906119899+1 = 1V1120572119899119899

(64)

One can readily check that the general solution to (64) isgiven by

119906119899 = 1 + (minus1)1198992 [119906prod(119899minus2)2119896=0(112057321198961205722119896+1)

0 ]+ 1 minus (minus1)1198992 [Vprod(119899minus1)2119896=0

(11205722119896)prod(119899minus3)2

119896=0(11205732119896+1)

0 ]minus1

V119899 = 1 + (minus1)1198992 [Vprod(119899minus2)2119896=0(112057221198961205732119896+1)

0 ]+ 1 minus (minus1)1198992 [119906prod(119899minus1)2119896=0

(11205732119896)prod(119899minus3)2

119896=0(11205722119896+1)

0 ]minus1(65)

where 120572119899 and 120573119899 are defined in (62)

From (61) we obtain

V119899119910119899+1 minus 1119910119899+1 + 1 = [

119909119899 minus 1119909119899 + 1]120572119899

119906119899119909119899+1 minus 1119909119899+1 + 1 = [119910119899 minus 1119910119899 + 1]

120573119899(66)

which is a first-order system after substitution of 119906119899 V119899 by theresults given in (65) Its solutions can be obtained by usingthe following canonical coordinates

119904119899 = ln 119909119899 minus 1119909119899 + 1119905119899 = ln

119910119899 minus 1119910119899 + 1(67)

This leads to the the following linear system with variablecoefficients

119905119899+1 = 120572119899119904119899 + 120574119899119904119899+1 = 120573119899119905119899 + 120593119899 (68)

where minus120574119899 = ln 119906119899 and minus120593119899 = ln V119899The latter is a linear first-order system with variable

coefficients Its general solution is

1199042119899 = 119899minus1prod119896=0

12057221198961205732119896+11199040 + 119899minus1sum119903=0

(1205742119903 119899minus1prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899minus1prod119895=119903+1

12057221198951205732119895+1)

1199052119899 = 119899minus1prod119896=0

12057321198961205722119896+11199050 + 119899minus1sum119903=0

(1205932119903 119899minus1prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899minus1prod119895=119903+1

12057321198951205722119895+1)

1199052119899+1 = 119899prod119896=0

1205722119896119899minus1prod119896=0

1205732119896+11199040 + 119899sum119903=0

(1205742119903 119899prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899prod119895=119903+1

1205722119895 119899minus1prod119895=119903+1

1205732119895+1)

1199042119899+1 = 119899prod119896=0

1205732119896119899minus1prod119896=0

1205722119896+11199050 + 119899sum119903=0

(1205932119903 119899prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899prod119895=119903+1

1205732119895 119899minus1prod119895=119903+1

1205722119895+1)

(69)

The general solution of (45) is obtained by substituting (69)into (67)

Journal of Mathematics 9

4 Conservation Laws

In Section 2 we have defined a first integral associated with asecond-oreder SΔEs It is given by (12)

120601 (119899 119909119899 119910119899 119909119899+1 119910119899+1) = 120601 (119899 + 1 119909119899+1 119910119899+1 1205961 1205962) (70)

Let

1198751 = 120597120601120597119909119899 1198752 = 120597120601120597119909119899+1 1198761 = 120597120601120597119910119899 1198762 = 120597120601120597119910119899+1

(71)

By differentiating (70)with respect to119909119899 119910119899 119909119899+1 and 119910119899+1 weobtain

1198751 = S (1198752) 1205961119909119899 +S (1198762) 12059621199091198991198761 = S (1198752) 1205961119910119899 +S (1198762) 1205962119910119899 (72)

and

1198752 = S (1198751) +S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+11198762 = S (1198761) +S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 (73)

The substitution of (72) in (73) leads to the following second-order system of functional equations

S2 (1198752)S (1205961119909119899) + S

2 (1198762)S (1205962119909119899)+S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+1 minus 1198752 = 0

S2 (1198752)S (1205961119910119899) +S

2 (1198762)S (1205962119910119899)+S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 minus 1198762 = 0

(74)

As for SLSC we differentiate repeatedly to obtain a systemof DEs for 1198752 and 1198762 Given the solutions 1198752 1198762 of (74) weeasily construct 1198751 1198761 For consistency of our solutions wemust check the integrability conditions

1205971198751120597119909119899+1 =1205971198752120597119909119899 (75)

and

1205971198761120597119910119899+1 =1205971198762120597119910119899 (76)

The first integral is then given by

120601 = int (1198751d119909119899 + 1198752d119909119899+1 + 1198761d119910119899 + 1198762d119910119899+1) + 119865 (119899) (77)

The constant of integration 119865(119899) which is a function depend-ing on 119899 is determined by substituting (77) in (70)

41 Applications Let us consider the second-order SΔEs119909119899+2 = 119886 (119899) 119910119899119910119899+2 = 119887 (119899) 119909119899 (78)

By choosing the ansatz 1198752(119899 119909119899 119910119899) and1198762(119899 119909119899 119910119899) one canreadily check that the determining system (74) is simplifiedto

1198762 (119899 + 2 1205961 1205962) 119887 (119899 + 1) minus 1198752 (119899 119909119899 119910119899) = 01198752 (119899 + 2 1205961 1205962) 119886 (119899 + 1) minus 1198762 (119899 119909119899 119910119899) = 0 (79)

where 1205961 and 1205962 denote the right-hand side of (78)Differentiating (79) with respect to 119909119899 and 119910119899 leads to

1198752 = 1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899) 1198762 = 1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899) (80)

Thus we have from (72)

1198751= 119887 (119899) [119909119899+11205954 (119899 + 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)]1198761= 119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)]

(81)

Substituting (80) in (79) and separatingwith respect to119909119899 and119910119899 we obtain the system

119886 (119899) 119887 (119899 + 1)1205954 (119899 + 2) minus 1205952 (119899) = 0119887 (119899) 119886 (119899 + 1)1205952 (119899 + 2) minus 1205954 (119899) = 0119887 (119899) 119887 (119899 + 1)1205955 (119899 + 2) minus 1205951 (119899) = 0119886 (119899) 119886 (119899 + 1)1205951 (119899 + 2) minus 1205955 (119899) = 0

119887 (119899 + 1)1205956 (119899 + 2) minus 1205953 (119899) = 0119886 (119899 + 1)1205953 (119899 + 2) minus 1205956 (119899) = 0

(82)

The solutions to (82) will provide us with the explicit form of120595119894 119894 = 1 6The first integral is then given by

10 Journal of Mathematics

120601= int 119887 (119899) [119909119899+11205954 (119899+ 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)] d119909119899+ (1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899)) d119909119899+1119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)] d119910119899+ (1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899)) d119910119899+1+ 119870119894

(83)

for some constants 119870119894For clarification let us consider 119886(119899) = 119887(119899) = 1 that is

119909119899+2 = 119910119899119910119899+2 = 119909119899 (84)

The solutions to (82) will be

1205951 (119899) = 1198621 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198623 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205955 (119899) = 1198623 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198621 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205952 (119899) = 1198625 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198627 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205954 (119899) = 1198627 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198625 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205953 (119899) = 1198629 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 11986211 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205956 (119899) = 11986211 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198629 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

(85)

where 119862119894 119894 = 1 12 are constants We have twelvesolutions for 1198752 and 1198762 That is

(1) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899

(86)

Journal of Mathematics 11

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1

(87)

(2) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899

(88)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1

(89)

(3) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899

(90)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1

(91)

(4) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899

(92)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1

(93)

(5) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899

(94)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1

(95)

(6) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899

(96)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1

(97)

(7) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899

(98)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1

(99)

(8) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899

(100)

12 Journal of Mathematics

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1

(101)

(9) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 ) 1198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )

(102)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )

(103)

(10) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )

(104)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )

(105)

(11) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )

(106)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )

(107)

(12) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) (108)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) (109)

Therefore we obtain twelve conservation laws for the system(84) They are given by

1206011 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198701

1206012 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119909119899+ 1198702

1206013 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198703

1206014 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119909119899+ 1198704

Journal of Mathematics 13

1206015 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198705

1206016 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119909119899+ 1198706

1206017 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198707

1206018 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119909119899+ 1198708

1206019 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899 + 1198709

12060110 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899 + 11987010

12060111 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899 + 11987011

12060112 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899 + 11987012

(110)

5 Conclusion and Discussions

Wehave presented amethod for obtaining nontrivial symme-tries and how to use them for solving a second-order SΔEsEach symmetry can be used to reduce the order Howeverdifferent symmetries lead to different reductions (see (60) and(64)) but the same solution We also proposed a technique toconstruct first integral associated to second-order systems ofdifference equations

14 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1993

[2] S Maeda ldquoThe similarity method for difference equationsrdquoIMA Journal of Applied Mathematics vol 38 no 2 pp 129ndash1341987

[3] D Levi L Vinet and P Winternitz ldquoLie group formalism fordifference equationsrdquo Journal of Physics A Mathematical andGeneral vol 30 no 2 pp 633ndash649 1997

[4] G R W Quispel and R Sahadevan ldquoLie symmetries and theintegration of difference equationsrdquo Physics Letters A vol 184no 1 pp 64ndash70 1993

[5] P E Hydon Difference Equations by Differential EquationMethods Cambridge University Press Cambridge 2014

[6] V Dorodnitsyn R Kozlov and P Winternitz ldquoLie groupclassification of second-order ordinary difference equationsrdquoJournal of Mathematical Physics vol 41 no 1 pp 480ndash5042000

[7] N Touafek and E M Elsayed ldquoOn the solutions of systemsof rational difference equationsrdquo Mathematical and ComputerModelling vol 55 no 7-8 pp 1987ndash1997 2012

[8] E M Elsayed and T F Ibrahim ldquoPeriodicity and solutionsfor some systems of nonlinear rational difference equationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 6pp 1361ndash1390 2015

[9] A S Kurbanlı C Cinar and I Yalcinkaya ldquoOn the behavior ofpositive solutions of the system of rational difference equations119909119899+1 = 119909119899minus1(119910119899119909119899minus1 + 1) 119910119899+1 = 119910119899minus1(119909119899119910119899minus1 + 1)rdquoMathematical and Computer Modelling vol 53 no 5-6 pp1261ndash1267 2011

[10] N Joshi and P J Vassiliou ldquoThe existence of Lie symmetriesfor first-order analytic discrete dynamical systemsrdquo Journal ofMathematical Analysis and Applications vol 195 no 3 pp 872ndash887 1995

[11] I Yalcinkaya ldquoOn the global asymptotic stability of a secondorder system of difference equationrdquo Discrete Dynamics inNature and Society vol 2008 Article ID 860152 12 pages 2008

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Journal of Mathematics 3

The SLCS (7) reduces to

S2 (1198761) minus 11987611205961119909119899 minus 11987621205961119910119899 minusS (1198761) 1205961119909119899+1minusS (1198762) 1205961119910119899+1 = 0 (14)

S2 (1198762) minus 11987611205962119909119899 minus 11987621205962119910119899 minusS (1198761) 1205962119909119899+1minusS (1198762) 1205962119910119899+1 = 0 (15)

where 119892119909 = 120597119892120597119909 1198761 = 1198761(119899 119909119899 119910119899) and 1198762 =1198762(119899 119909119899 119910119899)The functional equations (14) and (15) contain functions1198761 and 1198762 with different pairs of arguments making them

difficult to solve For concreteness if for instance the discretevariable 119899 stands for ldquostaterdquo in physics 1198761(119899 119909119899 119910119899) andS(1198761) equiv 1198761(119899 + 1 119909119899+1 119910119899+1) belong to two different states

To overcome this we proceed as follows

Step 1 (elimination of S2(1198761) and S2(1198762)) We differentiate(total differentiation) (14) and (15) with respect to 119909119899 and119910119899 respectively keeping 1205961 and 1205962 fixed Here we take 119909119899+1as function of 119909119899 119910119899 119910119899+1 1205961 1205962 and 119910119899+1 as function of119909119899 119910119899 119909119899+1 1205961 1205962

The total derivative operators are given by

dd119909119899 =

120597120597119909119899 +120597119909119899+1120597119909119899

120597120597119909119899+1 +120597119910119899+1120597119909119899

120597120597119910119899+1 + sdot sdot sdotdd119910119899 =

120597120597119910119899 +120597119909119899+1120597119910119899

120597120597119909119899+1 +120597119910119899+1120597119910119899

120597120597119910119899+1 + sdot sdot sdot(16)

In this case this is simplified to

dd119909119899 =

120597120597119909119899 minus (12059611199091198991205961119909119899+1 +

12059621199091198991205962119909119899+1 )120597120597119909119899+1

minus ( 12059611199091198991205961119910119899+1 +12059621199091198991205962119910119899+1 )

120597120597119910119899+1(17)

dd119910119899 =

120597120597119910119899 minus (12059611199101198991205961119909119899+1 +

12059621199101198991205962119909119899+1 )120597120597119909119899+1

minus ( 12059611199101198991205961119910119899+1 +12059621199101198991205962119910119899+1 )

120597120597119910119899+1(18)

So we apply the operator (17) to (14) and (18) to (15) keeping1205961 and 1205962 fixed This leads to the determining system

[11987611205961119909119899 + 11987621205961119910119899]119909119899 +S (1198761) 1205961119909119899+1 119909119899 +S (1198762)sdot 1205961119910119899+1 119909119899 minus ( 12059611199091198991205961119909119899+1 +

12059621199091198991205962119909119899+1 ) [11987611205961119909119899+ 11987621205961119910119899]119909119899+1 minus ( 12059611199091198991205961119909119899+1 +

12059621199091198991205962119909119899+1 )sdot [S (1198761) 1205961119909119899+1 119909119899+1 + S (1198762) 1205961119910119899+1 119909119899+1+ [S (1198761)]119909119899+1 1205961119909119899+1 + [S (1198762)]119909119899+1 1205961119910119899+1]

minus ( 12059611199091198991205961119910119899+1 +12059621199091198991205962119910119899+1 ) [11987611205961119909119899 + 11987621205961119910119899]119910119899+1

minus ( 12059611199091198991205961119910119899+1 +12059621199091198991205962119910119899+1 ) [S (1198761) 1205961119909119899+1119910119899+1

+ S (1198762) 1205961119910119899+1 119910119899+1 + [S (1198761)]119910119899+1 1205961119909119899+1+ [S (1198762)]119910119899+1 1205961119910119899+1] = 0

(19)

[11987611205962119909119899 + 11987621205962119910119899]119910119899 +S (1198761) 1205962119909119899+1 119910119899 + S (1198762)sdot 1205962119910119899+1 119910119899 minus ( 12059611199101198991205961119909119899+1 +

12059621199101198991205962119909119899+1 ) [11987611205962119909119899+ 11987621205962119910119899]119909119899+1 minus ( 12059611199101198991205961119909119899+1 +

12059621199101198991205962119909119899+1 )sdot [S (1198761) 1205962119909119899+1 119909119899+1 +S (1198762) 1205962119910119899+1 119909119899+1+ [S (1198761)]119909119899+1 1205962119909119899+1 + [S (1198762)]119909119899+1 1205962119910119899+1]minus ( 12059611199101198991205961119910119899+1 +

12059621199101198991205962119910119899+1 ) [11987611205962119909119899 + 11987621205962119910119899]119910119899+1minus ( 12059611199101198991205961119910119899+1 +

12059621199101198991205962119910119899+1 ) [S (1198761) 1205962119909119899+1119910119899+1+ S (1198762) 1205962119910119899+1 119910119899+1 + [S (1198761)]119910119899+1 1205962119909119899+1+ [S (1198762)]119910119899+1 1205962119910119899+1] = 0

(20)

Step 2 (elimination of S(1198761) and S(1198762)) We now differen-tiate (19) and (20) with respect to 119909119899 and 119910119899 respectivelykeeping 119909119899+1 and 119910119899+1 fixed This means that we apply theoperator 120597120597119909119899 on (19) and 120597120597119910119899 on (20) For a second-order SΔEs we need at most to differentiate four timesAfter separating with respect to 119909119899+1 and 119910119899+1 the resultingequations we obtain a system of DEs in 1198761 and 1198762 which issolvable by hand or by using a computer algebra package

Step 3 (explicit form of constant of integration) Whenintegrating in Step 2 to obtain the characteristics 1198761 and 1198762we have constant of integration which appears to be functionsof 119899 To obtain their explicit form we need to substitute theresults obtained in Step 2 in (19) and (20) If we donot succeedin obtaining all the constant of integration we need furthersubstitution in the SLSC (14) and (15)

32 Reductions Consider a second-order SΔEs119909119899+2 = 1205961 (119899 119909119899 119910119899 119909119899+1 119910119899+1) 119910119899+2 = 1205962 (119899 119909119899 119910119899 119909119899+1 119910119899+1) (21)

4 Journal of Mathematics

and its symmetry generator

119883 = 1198761 120597120597119909119899 + 1198762120597120597119910119899 +S1198761 120597120597119909119899+1 +S1198762 120597120597119910119899+1 (22)

The method of characteristics for partial differential equa-tions (PDEs)

d1199091198991198761 =d1199101198991198762 =

d119909119899+11198761 = d119910119899+11198762 = 1198821198990 (23)

leads to three independent constants of integration1198701 1198702 1198703 Each invariant under 119883 is function of thoseconstant119882119899 = 119891(1198701 1198702 1198703)

For second-order systems two invariants suffice to doreduction of the systems

Let

119906119899 = 1198911 (119899 119909119899 119910119899 119909119899+1 119910119899+1) V119899 = 1198912 (119899 119909119899 119910119899 119909119899+1 119910119899+1) (24)

be the invariants functions under119883We choose them in awaythat the Jacobian is nonzero

100381610038161003816100381610038161003816100381610038161003816120597 (1198911 1198912)120597 (119909119899+1 119910119899+1)

100381610038161003816100381610038161003816100381610038161003816 = 0 (25)

That is (24) can be inverted as follows

119909119899+1 = 1198921 (119899 119909119899 119910119899 119906119899 V119899) 119910119899+1 = 1198922 (119899 119909119899 119910119899 119906119899 V119899) (26)

By Corollary 3 SV119899 and S119906119899 are also invariant functionsTherefore the solution of (21) satisfies

119906119899+1 = Ω1 (119906119899 V119899) V119899+1 = Ω2 (119906119899 V119899) (27)

(27) is a first-order SΔEs which can be solved by furtherreductions or by using computer algebra software (mapleMathematica ) for linear systems Note that there existsome first-order systems which cannot be solved analytically

The general solution is

119906119899 = 119906 (119899 1198621 1198622) V119899 = V (119899 1198621 1198622) (28)

for some constant 1198621 1198622So the second-order system (21) is equivalent to the first-

order system obtained by substituting (28) in (26)

119909119899+1 = 1198921 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622)) 119910119899+1 = 1198922 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622)) (29)

(29) also admits the symmetries generated by119883The best wayto integrate any first-order analytic ΔE is to use its canonicalcoordinates [10]

119879119899 = 119879 (119899 119909119899 119910119899) (30)

which satisfy

119883119879119899 = 1 (31)

The obvious choice of canonical coordinates is (see [10])

119905119899 = int d1199091198991198761 (119899 119909119899 119910119899 1198921 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622)) 1198922 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622))) 119904119899 = int d1199101198991198762 (119899 119909119899 119910119899 1198921 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622)) 1198922 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899C1 1198622)))

(32)

33 Applications

331 Example 1 Consider the most general homogeneoussecond-order linear system of difference equations

119909119899+2 = 1198861 (119899) 119909119899 + 1198862 (119899) 119910119899 + 1198863 (119899) 119909119899+1+ 1198864 (119899) 119910119899+1

119910119899+2 = 1198871 (119899) 119909119899 + 1198872 (119899) 119910119899 + 1198873 (119899) 119909119899+1 + 1198874 (119899) 119910119899+1(33)

where 119886119894(119899) 119887119894(119899) 119894 = 1 4 are arbitrary functions

One can readily verify that the determining system (19)and (20) amounts to

1198761119909119899119909119899 = 1198762119909119899119909119899 = 01198761119910119899119910119899 = 1198762119910119899119910119899 = 0 (34)

Therefore

1198761 (119899 119909119899 119910119899) = 1198621119909119899 + 1198622119910119899 + 1198651 (119899) 1198762 (119899 119909119899 119910119899) = 1198623119909119899 + 1198624119910119899 + 1198652 (119899) (35)

where 119862119894 119894 = 1 4 are constants

Journal of Mathematics 5

The characteristics in (35) must satisfy the SLSC (14) and(15) Hence we have

1198651 (119899 + 2) minus [1198861 (119899) 1198651 (119899) + 1198862 (119899) 1198652 (119899)+ 1198863 (119899) 1198651 (119899 + 1) + 1198864 (119899) 1198652 (119899 + 1)] = 0

1198652 (119899 + 2) minus [1198871 (119899) 1198651 (119899) + 1198872 (119899) 1198652 (119899)+ 1198873 (119899) 1198651 (119899 + 1) + 1198874 (119899) 1198652 (119899 + 1)] = 0

(36)

and

1198621 = 11986241198622 = 1198623 = 0 (37)

So (35) is simplified to

1198761 (119899 119909119899 119910119899) = 1198621119909119899 + 1198651 (119899) 1198762 (119899 119909119899 119910119899) = 1198621119910119899 + 1198652 (119899) (38)

The first generator of symmetry for a second-order homoge-neous linear system (36) is the scaling symmetry given by

119883 = 119909119899120597119909119899 + 119910119899120597119910119899 (39)

The system (36) which governs the remaining generators ofthe Lie point symmetry for the system (33) is of second orderin 1198651 and 1198652 Its general solution is

1198651 (119899) = 1198921 (1198991198701 1198702 1198703 1198704) 1198652 (119899) = 1198922 (1198991198701 1198702 1198703 1198704) (40)

where1198701 119894 = 1 4 are constantsSo the most large Lie algebra of symmetry generators

which can be obtained from a homogeneous second-ordersystem of 2 difference equations has dimension five

For clarification let us consider1198861(119899) = 1198863(119899) = 1198864(119899) = 0 1198862(119899) = 1 and 1198872(119899) = 1198873(119899) =1198874(119899) = 0 1198871(119899) = 1 The system (33) becomes

119909119899+2 = 119910119899119910119899+2 = 119909119899 (41)

The systemwhich governs the remaining generators of the Liepoint symmetry in this case is given by

1198651 (119899 + 2) minus 1198652 (119899) = 01198652 (119899 + 2) minus 1198651 (119899) = 0 (42)

The general solutions for this system will be

1198651 (119899) = 1198621 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198623 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4sdot 1198624 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1198652 (119899) = 1198623 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198621 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4sdot 1198622 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

(43)

Therefore we have 5 generators of the Lie point symmetryspanned by

X0 = 119909119899120597119909119899 + 119910119899120597119910119899X1 = [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 120597119909119899

+ [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 120597119910119899X2 = [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 120597119909119899

+ [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 120597119910119899X3 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 120597119909119899

+ [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 120597119910119899X4 = [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 120597119909119899

+ [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 120597119910119899

(44)

332 Example 2 Consider the system

119909119899+2 = 119909119899119910119899+1 + 1119909119899 + 119910119899+1119910119899+2 = 119910119899119909119899+1 + 1119910119899 + 119909119899+1

(45)

(45) is a special case of systems investigated in [11] where theauthor looked at the stability of the systems

6 Journal of Mathematics

We choose the ansatz 1198761(119899 119909119899) 1198762(119899 119910119899)The determining system (19) and (20) amounts to

minus 1198762119910119899119909119899+121199101198992 + 1198781198761119909119899+1119909119899+121199101198992 + 21198762119909119899+12119910119899minus 211987811987611199101198992119909119899+1 + 1198762119910119899119909119899+12 + 11987621199101198991199101198992minus 1198781198761119909119899+1119909119899+12 minus 1198781198761119909119899+11199101198992 minus 21198762119910119899+ 21198781198761119909119899+1 minus 1198762119910119899 + 1198781198761119909119899+1 = 0

(46)

minus 11987611199091198991199091198992119910119899+12 + 1198781198762119910119899+11199091198992119910119899+12 + 21198761119910119899+12119909119899minus 211987811987621199091198992119910119899+1 + 11987611199091198991199091198992 + 1198761119909119899119910119899+12minus 1198781198762119910119899+11199091198992 minus 1198781198762119910119899+1119910119899+12 minus 21198761119909119899+ 21198781198762119910119899+1 minus 1198761119909119899 + 1198781198762119910119899+1 = 0

(47)

Differentiating twice (46) with respect to 119909119899 and twice (47)with respect to 119910119899 keeping 119909119899+1 and 119910119899+1 fixed we obtain afterseparating with respect to119909119899+1 and 119910119899+1 the following systemof Des

119876101584010158401 + 1199091198991198761015840101584010158401 + 2 11987611199091198992 minus 211987610158401119909119899 +

1198761015840101584011199091198992 minus1198761015840101584010158401119909119899 = 0

119876101584010158402 + 1199101198991198761015840101584010158402 + 2 11987621199101198992 minus 211987610158402119910119899 +

1198761015840101584021199101198992 minus1198761015840101584010158402119910119899 = 0

(48)

whose most general solutions are

1198761 (119899 119909119899) = 1198651 (119899) 119909119899 + 1198652 (119899) (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1+ 1198653 (119899) (1199092119899 minus 1)

1198762 (119899 119910119899) = 1198654 (119899) 119910119899 + 1198655 (119899) (1199102119899 minus 1) ln 119910119899 + 1119910119899 minus 1+ 1198656 (119899) (1199102119899 minus 1)

(49)

To obtain the nature of functions 1198651 1198656 we substitute (49)in (46) and (47) After separating with respect to 119909119899 119909119899+1 119910119899and 119910119899+1 we get the following SΔEsminus41198652 (119899) minus 21198654 (119899 + 1) + 41198655 (119899 + 1) minus 21198651 (119899) = 041198652 (119899) minus 21198654 (119899 + 1) minus 41198655 (119899 + 1) + 21198651 (119899) = 0minus41198655 (119899) minus 21198651 (119899 + 1) + 41198652 (119899 + 1) minus 21198654 (119899) = 041198655 (119899) minus 21198651 (119899 + 1) minus 41198652 (119899 + 1) + 21198654 (119899) = 0

(50)

whose solutions are

1198651 (119899) = 1198654 (119899) = 01198652 (119899) = 1198621 + (minus1)119899 11986221198655 (119899) = 1198621 minus (minus1)119899 1198622

(51)

The remaining unknown functions 1198653(119899) and 1198656(119899) aredetermined by substituting (51) and (49) into the SLSC (14)and (15) This leads to the SΔEs

1198653 (119899) minus 1198653 (119899 + 2) + 1198656 (119899 + 1) = 01198656 (119899) + 1198653 (119899 + 1) minus 1198656 (119899 + 2) = 0 (52)

The general solutions to (52) are given by

1198653 (119899)= 1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986262119899+1

sdot minus1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986242119899+1

1198656 (119899)= 1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986242119899+1

sdot minus1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986262119899+1

(53)

where 1198621 1198626 are arbitrary constants It follows that thecharacteristics are given by

1198761 = (1198621 + (minus1)119899 1198622) (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1+

1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986262119899+1 ]]

]

Journal of Mathematics 7

sdot minus1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1] 11986252119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986242119899+1 ]]

](1199092119899 minus 1)

1198762 = (1198621 minus (minus1)119899 1198622) (1199102119899 minus 1) ln 119910119899 + 1119910119899 minus 1+

1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986242119899+1 ]]

]sdot minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1] 11986232119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986262119899+1 ]]

](1199102119899 minus 1)

(54)

Therefore we have six generators of Lie point symmetry

X1 = (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1120597119909119899 + (1199102119899 minus 1) ln119910119899 + 1119910119899 minus 1120597119910119899

X2 = (minus1)119899 (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1120597119909119899 minus (minus1)119899 (1199102119899 minus 1)sdot ln 119910119899 + 1119910119899 minus 1120597119910119899

X3 = 1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199092119899 minus 1) 120597119909119899+ minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199102119899 minus 1) 120597119910119899X4 = minus1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 + (1 + radic5)119899]

2119899+1 ]]

sdot (1199092119899 minus 1) 120597119909119899

+ 1 + (minus1)119899radic5 [[[(minus1 + radic5)119899 minus (1 + radic5)119899]

2119899+1 ]]

sdot (1199102119899 minus 1) 120597119910119899X5 = minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]

2119899 ]]]sdot (1199092119899 minus 1) 120597119909119899

+ 1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199102119899 minus 1) 120597119910119899X6 = 1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 minus (1 + radic5)119899]

2119899+1 ]]

sdot (1199092119899 minus 1) 120597119909119899+ minus1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 + (1 + radic5)119899]

2119899+1 ]]

sdot (1199102119899 minus 1) 120597119910119899

(55)

Each generator in (55) can be used to reduce the order of (45)Let us consider X1 By the characteristic method for

Partial Differential Equations the invariants are given byfollowing equation

d119909119899(1199092119899 minus 1) ln ((119909119899 + 1) (119909119899 minus 1))= d119910119899(1199102119899 minus 1) ln ((119910119899 + 1) (119910119899 minus 1))= d119909119899+1(1199092119899+1 minus 1) ln ((119909119899+1 + 1) (119909119899+1 minus 1))= d119910119899+1(1199102119899+1 minus 1) ln ((119910119899+1 + 1) (119910119899+1 minus 1)) =

1198811198990

(56)

We get

1198621 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119910119899+1 + 1) (119910119899+1 minus 1))

1198622 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119910119899 + 1) (119910119899 minus 1))

1198623 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119909119899+1 + 1) (119909119899+1 minus 1))

119881119899 = 119891 (1198621 1198622 1198623)

(57)

where 1198621 1198622 1198623 are constants

8 Journal of Mathematics

If we choose 119891(1198621 1198622 1198623) = 1198621 we have119906119899 = ln ((119909119899 + 1) (119909119899 minus 1))

ln ((119910119899+1 + 1) (119910119899+1 minus 1)) (58)

and if we choose 119891(1198621 1198622 1198623) = 11986231198622 we haveV119899 = ln ((119910119899 + 1) (119910119899 minus 1))

ln ((119909119899+1 + 1) (119909119899+1 minus 1)) (59)

From (58) and (59)we deduce

119906119899+1 = 11 + V119899

V119899+1 = 11 + 119906119899(60)

Let us now consider the generator X3 The resulting invari-ants are

V119899 = [(119909119899 minus 1) (119909119899 + 1)]120572119899(119910119899+1 minus 1) (119910119899+1 + 1) 119906119899 = [(119910119899 minus 1) (119910119899 + 1)]120573119899(119909119899+1 minus 1) (119909119899+1 + 1)

(61)

where

120572119899 = (1 + radic5)119899 minus (minus1 + radic5)1198992 [(1 + radic5)119899minus1 + (minus1 + radic5)119899minus1]

120573119899 = (1 + radic5)119899 + (minus1 + radic5)1198992 [(1 + radic5)119899minus1 minus (minus1 + radic5)119899minus1]

(62)

Note also the relationship between them

120572119899+1 minus 1 = 1120573119899 120573119899+1 minus 1 = 1120572119899

(63)

From (61) we deduce the following relation

V119899+1 = 11199061120573119899119899

119906119899+1 = 1V1120572119899119899

(64)

One can readily check that the general solution to (64) isgiven by

119906119899 = 1 + (minus1)1198992 [119906prod(119899minus2)2119896=0(112057321198961205722119896+1)

0 ]+ 1 minus (minus1)1198992 [Vprod(119899minus1)2119896=0

(11205722119896)prod(119899minus3)2

119896=0(11205732119896+1)

0 ]minus1

V119899 = 1 + (minus1)1198992 [Vprod(119899minus2)2119896=0(112057221198961205732119896+1)

0 ]+ 1 minus (minus1)1198992 [119906prod(119899minus1)2119896=0

(11205732119896)prod(119899minus3)2

119896=0(11205722119896+1)

0 ]minus1(65)

where 120572119899 and 120573119899 are defined in (62)

From (61) we obtain

V119899119910119899+1 minus 1119910119899+1 + 1 = [

119909119899 minus 1119909119899 + 1]120572119899

119906119899119909119899+1 minus 1119909119899+1 + 1 = [119910119899 minus 1119910119899 + 1]

120573119899(66)

which is a first-order system after substitution of 119906119899 V119899 by theresults given in (65) Its solutions can be obtained by usingthe following canonical coordinates

119904119899 = ln 119909119899 minus 1119909119899 + 1119905119899 = ln

119910119899 minus 1119910119899 + 1(67)

This leads to the the following linear system with variablecoefficients

119905119899+1 = 120572119899119904119899 + 120574119899119904119899+1 = 120573119899119905119899 + 120593119899 (68)

where minus120574119899 = ln 119906119899 and minus120593119899 = ln V119899The latter is a linear first-order system with variable

coefficients Its general solution is

1199042119899 = 119899minus1prod119896=0

12057221198961205732119896+11199040 + 119899minus1sum119903=0

(1205742119903 119899minus1prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899minus1prod119895=119903+1

12057221198951205732119895+1)

1199052119899 = 119899minus1prod119896=0

12057321198961205722119896+11199050 + 119899minus1sum119903=0

(1205932119903 119899minus1prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899minus1prod119895=119903+1

12057321198951205722119895+1)

1199052119899+1 = 119899prod119896=0

1205722119896119899minus1prod119896=0

1205732119896+11199040 + 119899sum119903=0

(1205742119903 119899prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899prod119895=119903+1

1205722119895 119899minus1prod119895=119903+1

1205732119895+1)

1199042119899+1 = 119899prod119896=0

1205732119896119899minus1prod119896=0

1205722119896+11199050 + 119899sum119903=0

(1205932119903 119899prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899prod119895=119903+1

1205732119895 119899minus1prod119895=119903+1

1205722119895+1)

(69)

The general solution of (45) is obtained by substituting (69)into (67)

Journal of Mathematics 9

4 Conservation Laws

In Section 2 we have defined a first integral associated with asecond-oreder SΔEs It is given by (12)

120601 (119899 119909119899 119910119899 119909119899+1 119910119899+1) = 120601 (119899 + 1 119909119899+1 119910119899+1 1205961 1205962) (70)

Let

1198751 = 120597120601120597119909119899 1198752 = 120597120601120597119909119899+1 1198761 = 120597120601120597119910119899 1198762 = 120597120601120597119910119899+1

(71)

By differentiating (70)with respect to119909119899 119910119899 119909119899+1 and 119910119899+1 weobtain

1198751 = S (1198752) 1205961119909119899 +S (1198762) 12059621199091198991198761 = S (1198752) 1205961119910119899 +S (1198762) 1205962119910119899 (72)

and

1198752 = S (1198751) +S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+11198762 = S (1198761) +S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 (73)

The substitution of (72) in (73) leads to the following second-order system of functional equations

S2 (1198752)S (1205961119909119899) + S

2 (1198762)S (1205962119909119899)+S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+1 minus 1198752 = 0

S2 (1198752)S (1205961119910119899) +S

2 (1198762)S (1205962119910119899)+S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 minus 1198762 = 0

(74)

As for SLSC we differentiate repeatedly to obtain a systemof DEs for 1198752 and 1198762 Given the solutions 1198752 1198762 of (74) weeasily construct 1198751 1198761 For consistency of our solutions wemust check the integrability conditions

1205971198751120597119909119899+1 =1205971198752120597119909119899 (75)

and

1205971198761120597119910119899+1 =1205971198762120597119910119899 (76)

The first integral is then given by

120601 = int (1198751d119909119899 + 1198752d119909119899+1 + 1198761d119910119899 + 1198762d119910119899+1) + 119865 (119899) (77)

The constant of integration 119865(119899) which is a function depend-ing on 119899 is determined by substituting (77) in (70)

41 Applications Let us consider the second-order SΔEs119909119899+2 = 119886 (119899) 119910119899119910119899+2 = 119887 (119899) 119909119899 (78)

By choosing the ansatz 1198752(119899 119909119899 119910119899) and1198762(119899 119909119899 119910119899) one canreadily check that the determining system (74) is simplifiedto

1198762 (119899 + 2 1205961 1205962) 119887 (119899 + 1) minus 1198752 (119899 119909119899 119910119899) = 01198752 (119899 + 2 1205961 1205962) 119886 (119899 + 1) minus 1198762 (119899 119909119899 119910119899) = 0 (79)

where 1205961 and 1205962 denote the right-hand side of (78)Differentiating (79) with respect to 119909119899 and 119910119899 leads to

1198752 = 1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899) 1198762 = 1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899) (80)

Thus we have from (72)

1198751= 119887 (119899) [119909119899+11205954 (119899 + 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)]1198761= 119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)]

(81)

Substituting (80) in (79) and separatingwith respect to119909119899 and119910119899 we obtain the system

119886 (119899) 119887 (119899 + 1)1205954 (119899 + 2) minus 1205952 (119899) = 0119887 (119899) 119886 (119899 + 1)1205952 (119899 + 2) minus 1205954 (119899) = 0119887 (119899) 119887 (119899 + 1)1205955 (119899 + 2) minus 1205951 (119899) = 0119886 (119899) 119886 (119899 + 1)1205951 (119899 + 2) minus 1205955 (119899) = 0

119887 (119899 + 1)1205956 (119899 + 2) minus 1205953 (119899) = 0119886 (119899 + 1)1205953 (119899 + 2) minus 1205956 (119899) = 0

(82)

The solutions to (82) will provide us with the explicit form of120595119894 119894 = 1 6The first integral is then given by

10 Journal of Mathematics

120601= int 119887 (119899) [119909119899+11205954 (119899+ 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)] d119909119899+ (1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899)) d119909119899+1119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)] d119910119899+ (1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899)) d119910119899+1+ 119870119894

(83)

for some constants 119870119894For clarification let us consider 119886(119899) = 119887(119899) = 1 that is

119909119899+2 = 119910119899119910119899+2 = 119909119899 (84)

The solutions to (82) will be

1205951 (119899) = 1198621 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198623 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205955 (119899) = 1198623 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198621 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205952 (119899) = 1198625 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198627 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205954 (119899) = 1198627 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198625 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205953 (119899) = 1198629 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 11986211 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205956 (119899) = 11986211 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198629 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

(85)

where 119862119894 119894 = 1 12 are constants We have twelvesolutions for 1198752 and 1198762 That is

(1) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899

(86)

Journal of Mathematics 11

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1

(87)

(2) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899

(88)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1

(89)

(3) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899

(90)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1

(91)

(4) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899

(92)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1

(93)

(5) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899

(94)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1

(95)

(6) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899

(96)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1

(97)

(7) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899

(98)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1

(99)

(8) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899

(100)

12 Journal of Mathematics

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1

(101)

(9) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 ) 1198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )

(102)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )

(103)

(10) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )

(104)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )

(105)

(11) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )

(106)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )

(107)

(12) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) (108)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) (109)

Therefore we obtain twelve conservation laws for the system(84) They are given by

1206011 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198701

1206012 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119909119899+ 1198702

1206013 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198703

1206014 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119909119899+ 1198704

Journal of Mathematics 13

1206015 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198705

1206016 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119909119899+ 1198706

1206017 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198707

1206018 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119909119899+ 1198708

1206019 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899 + 1198709

12060110 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899 + 11987010

12060111 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899 + 11987011

12060112 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899 + 11987012

(110)

5 Conclusion and Discussions

Wehave presented amethod for obtaining nontrivial symme-tries and how to use them for solving a second-order SΔEsEach symmetry can be used to reduce the order Howeverdifferent symmetries lead to different reductions (see (60) and(64)) but the same solution We also proposed a technique toconstruct first integral associated to second-order systems ofdifference equations

14 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1993

[2] S Maeda ldquoThe similarity method for difference equationsrdquoIMA Journal of Applied Mathematics vol 38 no 2 pp 129ndash1341987

[3] D Levi L Vinet and P Winternitz ldquoLie group formalism fordifference equationsrdquo Journal of Physics A Mathematical andGeneral vol 30 no 2 pp 633ndash649 1997

[4] G R W Quispel and R Sahadevan ldquoLie symmetries and theintegration of difference equationsrdquo Physics Letters A vol 184no 1 pp 64ndash70 1993

[5] P E Hydon Difference Equations by Differential EquationMethods Cambridge University Press Cambridge 2014

[6] V Dorodnitsyn R Kozlov and P Winternitz ldquoLie groupclassification of second-order ordinary difference equationsrdquoJournal of Mathematical Physics vol 41 no 1 pp 480ndash5042000

[7] N Touafek and E M Elsayed ldquoOn the solutions of systemsof rational difference equationsrdquo Mathematical and ComputerModelling vol 55 no 7-8 pp 1987ndash1997 2012

[8] E M Elsayed and T F Ibrahim ldquoPeriodicity and solutionsfor some systems of nonlinear rational difference equationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 6pp 1361ndash1390 2015

[9] A S Kurbanlı C Cinar and I Yalcinkaya ldquoOn the behavior ofpositive solutions of the system of rational difference equations119909119899+1 = 119909119899minus1(119910119899119909119899minus1 + 1) 119910119899+1 = 119910119899minus1(119909119899119910119899minus1 + 1)rdquoMathematical and Computer Modelling vol 53 no 5-6 pp1261ndash1267 2011

[10] N Joshi and P J Vassiliou ldquoThe existence of Lie symmetriesfor first-order analytic discrete dynamical systemsrdquo Journal ofMathematical Analysis and Applications vol 195 no 3 pp 872ndash887 1995

[11] I Yalcinkaya ldquoOn the global asymptotic stability of a secondorder system of difference equationrdquo Discrete Dynamics inNature and Society vol 2008 Article ID 860152 12 pages 2008

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4 Journal of Mathematics

and its symmetry generator

119883 = 1198761 120597120597119909119899 + 1198762120597120597119910119899 +S1198761 120597120597119909119899+1 +S1198762 120597120597119910119899+1 (22)

The method of characteristics for partial differential equa-tions (PDEs)

d1199091198991198761 =d1199101198991198762 =

d119909119899+11198761 = d119910119899+11198762 = 1198821198990 (23)

leads to three independent constants of integration1198701 1198702 1198703 Each invariant under 119883 is function of thoseconstant119882119899 = 119891(1198701 1198702 1198703)

For second-order systems two invariants suffice to doreduction of the systems

Let

119906119899 = 1198911 (119899 119909119899 119910119899 119909119899+1 119910119899+1) V119899 = 1198912 (119899 119909119899 119910119899 119909119899+1 119910119899+1) (24)

be the invariants functions under119883We choose them in awaythat the Jacobian is nonzero

100381610038161003816100381610038161003816100381610038161003816120597 (1198911 1198912)120597 (119909119899+1 119910119899+1)

100381610038161003816100381610038161003816100381610038161003816 = 0 (25)

That is (24) can be inverted as follows

119909119899+1 = 1198921 (119899 119909119899 119910119899 119906119899 V119899) 119910119899+1 = 1198922 (119899 119909119899 119910119899 119906119899 V119899) (26)

By Corollary 3 SV119899 and S119906119899 are also invariant functionsTherefore the solution of (21) satisfies

119906119899+1 = Ω1 (119906119899 V119899) V119899+1 = Ω2 (119906119899 V119899) (27)

(27) is a first-order SΔEs which can be solved by furtherreductions or by using computer algebra software (mapleMathematica ) for linear systems Note that there existsome first-order systems which cannot be solved analytically

The general solution is

119906119899 = 119906 (119899 1198621 1198622) V119899 = V (119899 1198621 1198622) (28)

for some constant 1198621 1198622So the second-order system (21) is equivalent to the first-

order system obtained by substituting (28) in (26)

119909119899+1 = 1198921 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622)) 119910119899+1 = 1198922 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622)) (29)

(29) also admits the symmetries generated by119883The best wayto integrate any first-order analytic ΔE is to use its canonicalcoordinates [10]

119879119899 = 119879 (119899 119909119899 119910119899) (30)

which satisfy

119883119879119899 = 1 (31)

The obvious choice of canonical coordinates is (see [10])

119905119899 = int d1199091198991198761 (119899 119909119899 119910119899 1198921 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622)) 1198922 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622))) 119904119899 = int d1199101198991198762 (119899 119909119899 119910119899 1198921 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899 1198621 1198622)) 1198922 (119899 119909119899 119910119899 119906 (119899 1198621 1198622) V (119899C1 1198622)))

(32)

33 Applications

331 Example 1 Consider the most general homogeneoussecond-order linear system of difference equations

119909119899+2 = 1198861 (119899) 119909119899 + 1198862 (119899) 119910119899 + 1198863 (119899) 119909119899+1+ 1198864 (119899) 119910119899+1

119910119899+2 = 1198871 (119899) 119909119899 + 1198872 (119899) 119910119899 + 1198873 (119899) 119909119899+1 + 1198874 (119899) 119910119899+1(33)

where 119886119894(119899) 119887119894(119899) 119894 = 1 4 are arbitrary functions

One can readily verify that the determining system (19)and (20) amounts to

1198761119909119899119909119899 = 1198762119909119899119909119899 = 01198761119910119899119910119899 = 1198762119910119899119910119899 = 0 (34)

Therefore

1198761 (119899 119909119899 119910119899) = 1198621119909119899 + 1198622119910119899 + 1198651 (119899) 1198762 (119899 119909119899 119910119899) = 1198623119909119899 + 1198624119910119899 + 1198652 (119899) (35)

where 119862119894 119894 = 1 4 are constants

Journal of Mathematics 5

The characteristics in (35) must satisfy the SLSC (14) and(15) Hence we have

1198651 (119899 + 2) minus [1198861 (119899) 1198651 (119899) + 1198862 (119899) 1198652 (119899)+ 1198863 (119899) 1198651 (119899 + 1) + 1198864 (119899) 1198652 (119899 + 1)] = 0

1198652 (119899 + 2) minus [1198871 (119899) 1198651 (119899) + 1198872 (119899) 1198652 (119899)+ 1198873 (119899) 1198651 (119899 + 1) + 1198874 (119899) 1198652 (119899 + 1)] = 0

(36)

and

1198621 = 11986241198622 = 1198623 = 0 (37)

So (35) is simplified to

1198761 (119899 119909119899 119910119899) = 1198621119909119899 + 1198651 (119899) 1198762 (119899 119909119899 119910119899) = 1198621119910119899 + 1198652 (119899) (38)

The first generator of symmetry for a second-order homoge-neous linear system (36) is the scaling symmetry given by

119883 = 119909119899120597119909119899 + 119910119899120597119910119899 (39)

The system (36) which governs the remaining generators ofthe Lie point symmetry for the system (33) is of second orderin 1198651 and 1198652 Its general solution is

1198651 (119899) = 1198921 (1198991198701 1198702 1198703 1198704) 1198652 (119899) = 1198922 (1198991198701 1198702 1198703 1198704) (40)

where1198701 119894 = 1 4 are constantsSo the most large Lie algebra of symmetry generators

which can be obtained from a homogeneous second-ordersystem of 2 difference equations has dimension five

For clarification let us consider1198861(119899) = 1198863(119899) = 1198864(119899) = 0 1198862(119899) = 1 and 1198872(119899) = 1198873(119899) =1198874(119899) = 0 1198871(119899) = 1 The system (33) becomes

119909119899+2 = 119910119899119910119899+2 = 119909119899 (41)

The systemwhich governs the remaining generators of the Liepoint symmetry in this case is given by

1198651 (119899 + 2) minus 1198652 (119899) = 01198652 (119899 + 2) minus 1198651 (119899) = 0 (42)

The general solutions for this system will be

1198651 (119899) = 1198621 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198623 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4sdot 1198624 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1198652 (119899) = 1198623 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198621 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4sdot 1198622 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

(43)

Therefore we have 5 generators of the Lie point symmetryspanned by

X0 = 119909119899120597119909119899 + 119910119899120597119910119899X1 = [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 120597119909119899

+ [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 120597119910119899X2 = [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 120597119909119899

+ [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 120597119910119899X3 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 120597119909119899

+ [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 120597119910119899X4 = [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 120597119909119899

+ [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 120597119910119899

(44)

332 Example 2 Consider the system

119909119899+2 = 119909119899119910119899+1 + 1119909119899 + 119910119899+1119910119899+2 = 119910119899119909119899+1 + 1119910119899 + 119909119899+1

(45)

(45) is a special case of systems investigated in [11] where theauthor looked at the stability of the systems

6 Journal of Mathematics

We choose the ansatz 1198761(119899 119909119899) 1198762(119899 119910119899)The determining system (19) and (20) amounts to

minus 1198762119910119899119909119899+121199101198992 + 1198781198761119909119899+1119909119899+121199101198992 + 21198762119909119899+12119910119899minus 211987811987611199101198992119909119899+1 + 1198762119910119899119909119899+12 + 11987621199101198991199101198992minus 1198781198761119909119899+1119909119899+12 minus 1198781198761119909119899+11199101198992 minus 21198762119910119899+ 21198781198761119909119899+1 minus 1198762119910119899 + 1198781198761119909119899+1 = 0

(46)

minus 11987611199091198991199091198992119910119899+12 + 1198781198762119910119899+11199091198992119910119899+12 + 21198761119910119899+12119909119899minus 211987811987621199091198992119910119899+1 + 11987611199091198991199091198992 + 1198761119909119899119910119899+12minus 1198781198762119910119899+11199091198992 minus 1198781198762119910119899+1119910119899+12 minus 21198761119909119899+ 21198781198762119910119899+1 minus 1198761119909119899 + 1198781198762119910119899+1 = 0

(47)

Differentiating twice (46) with respect to 119909119899 and twice (47)with respect to 119910119899 keeping 119909119899+1 and 119910119899+1 fixed we obtain afterseparating with respect to119909119899+1 and 119910119899+1 the following systemof Des

119876101584010158401 + 1199091198991198761015840101584010158401 + 2 11987611199091198992 minus 211987610158401119909119899 +

1198761015840101584011199091198992 minus1198761015840101584010158401119909119899 = 0

119876101584010158402 + 1199101198991198761015840101584010158402 + 2 11987621199101198992 minus 211987610158402119910119899 +

1198761015840101584021199101198992 minus1198761015840101584010158402119910119899 = 0

(48)

whose most general solutions are

1198761 (119899 119909119899) = 1198651 (119899) 119909119899 + 1198652 (119899) (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1+ 1198653 (119899) (1199092119899 minus 1)

1198762 (119899 119910119899) = 1198654 (119899) 119910119899 + 1198655 (119899) (1199102119899 minus 1) ln 119910119899 + 1119910119899 minus 1+ 1198656 (119899) (1199102119899 minus 1)

(49)

To obtain the nature of functions 1198651 1198656 we substitute (49)in (46) and (47) After separating with respect to 119909119899 119909119899+1 119910119899and 119910119899+1 we get the following SΔEsminus41198652 (119899) minus 21198654 (119899 + 1) + 41198655 (119899 + 1) minus 21198651 (119899) = 041198652 (119899) minus 21198654 (119899 + 1) minus 41198655 (119899 + 1) + 21198651 (119899) = 0minus41198655 (119899) minus 21198651 (119899 + 1) + 41198652 (119899 + 1) minus 21198654 (119899) = 041198655 (119899) minus 21198651 (119899 + 1) minus 41198652 (119899 + 1) + 21198654 (119899) = 0

(50)

whose solutions are

1198651 (119899) = 1198654 (119899) = 01198652 (119899) = 1198621 + (minus1)119899 11986221198655 (119899) = 1198621 minus (minus1)119899 1198622

(51)

The remaining unknown functions 1198653(119899) and 1198656(119899) aredetermined by substituting (51) and (49) into the SLSC (14)and (15) This leads to the SΔEs

1198653 (119899) minus 1198653 (119899 + 2) + 1198656 (119899 + 1) = 01198656 (119899) + 1198653 (119899 + 1) minus 1198656 (119899 + 2) = 0 (52)

The general solutions to (52) are given by

1198653 (119899)= 1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986262119899+1

sdot minus1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986242119899+1

1198656 (119899)= 1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986242119899+1

sdot minus1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986262119899+1

(53)

where 1198621 1198626 are arbitrary constants It follows that thecharacteristics are given by

1198761 = (1198621 + (minus1)119899 1198622) (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1+

1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986262119899+1 ]]

]

Journal of Mathematics 7

sdot minus1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1] 11986252119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986242119899+1 ]]

](1199092119899 minus 1)

1198762 = (1198621 minus (minus1)119899 1198622) (1199102119899 minus 1) ln 119910119899 + 1119910119899 minus 1+

1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986242119899+1 ]]

]sdot minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1] 11986232119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986262119899+1 ]]

](1199102119899 minus 1)

(54)

Therefore we have six generators of Lie point symmetry

X1 = (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1120597119909119899 + (1199102119899 minus 1) ln119910119899 + 1119910119899 minus 1120597119910119899

X2 = (minus1)119899 (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1120597119909119899 minus (minus1)119899 (1199102119899 minus 1)sdot ln 119910119899 + 1119910119899 minus 1120597119910119899

X3 = 1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199092119899 minus 1) 120597119909119899+ minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199102119899 minus 1) 120597119910119899X4 = minus1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 + (1 + radic5)119899]

2119899+1 ]]

sdot (1199092119899 minus 1) 120597119909119899

+ 1 + (minus1)119899radic5 [[[(minus1 + radic5)119899 minus (1 + radic5)119899]

2119899+1 ]]

sdot (1199102119899 minus 1) 120597119910119899X5 = minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]

2119899 ]]]sdot (1199092119899 minus 1) 120597119909119899

+ 1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199102119899 minus 1) 120597119910119899X6 = 1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 minus (1 + radic5)119899]

2119899+1 ]]

sdot (1199092119899 minus 1) 120597119909119899+ minus1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 + (1 + radic5)119899]

2119899+1 ]]

sdot (1199102119899 minus 1) 120597119910119899

(55)

Each generator in (55) can be used to reduce the order of (45)Let us consider X1 By the characteristic method for

Partial Differential Equations the invariants are given byfollowing equation

d119909119899(1199092119899 minus 1) ln ((119909119899 + 1) (119909119899 minus 1))= d119910119899(1199102119899 minus 1) ln ((119910119899 + 1) (119910119899 minus 1))= d119909119899+1(1199092119899+1 minus 1) ln ((119909119899+1 + 1) (119909119899+1 minus 1))= d119910119899+1(1199102119899+1 minus 1) ln ((119910119899+1 + 1) (119910119899+1 minus 1)) =

1198811198990

(56)

We get

1198621 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119910119899+1 + 1) (119910119899+1 minus 1))

1198622 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119910119899 + 1) (119910119899 minus 1))

1198623 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119909119899+1 + 1) (119909119899+1 minus 1))

119881119899 = 119891 (1198621 1198622 1198623)

(57)

where 1198621 1198622 1198623 are constants

8 Journal of Mathematics

If we choose 119891(1198621 1198622 1198623) = 1198621 we have119906119899 = ln ((119909119899 + 1) (119909119899 minus 1))

ln ((119910119899+1 + 1) (119910119899+1 minus 1)) (58)

and if we choose 119891(1198621 1198622 1198623) = 11986231198622 we haveV119899 = ln ((119910119899 + 1) (119910119899 minus 1))

ln ((119909119899+1 + 1) (119909119899+1 minus 1)) (59)

From (58) and (59)we deduce

119906119899+1 = 11 + V119899

V119899+1 = 11 + 119906119899(60)

Let us now consider the generator X3 The resulting invari-ants are

V119899 = [(119909119899 minus 1) (119909119899 + 1)]120572119899(119910119899+1 minus 1) (119910119899+1 + 1) 119906119899 = [(119910119899 minus 1) (119910119899 + 1)]120573119899(119909119899+1 minus 1) (119909119899+1 + 1)

(61)

where

120572119899 = (1 + radic5)119899 minus (minus1 + radic5)1198992 [(1 + radic5)119899minus1 + (minus1 + radic5)119899minus1]

120573119899 = (1 + radic5)119899 + (minus1 + radic5)1198992 [(1 + radic5)119899minus1 minus (minus1 + radic5)119899minus1]

(62)

Note also the relationship between them

120572119899+1 minus 1 = 1120573119899 120573119899+1 minus 1 = 1120572119899

(63)

From (61) we deduce the following relation

V119899+1 = 11199061120573119899119899

119906119899+1 = 1V1120572119899119899

(64)

One can readily check that the general solution to (64) isgiven by

119906119899 = 1 + (minus1)1198992 [119906prod(119899minus2)2119896=0(112057321198961205722119896+1)

0 ]+ 1 minus (minus1)1198992 [Vprod(119899minus1)2119896=0

(11205722119896)prod(119899minus3)2

119896=0(11205732119896+1)

0 ]minus1

V119899 = 1 + (minus1)1198992 [Vprod(119899minus2)2119896=0(112057221198961205732119896+1)

0 ]+ 1 minus (minus1)1198992 [119906prod(119899minus1)2119896=0

(11205732119896)prod(119899minus3)2

119896=0(11205722119896+1)

0 ]minus1(65)

where 120572119899 and 120573119899 are defined in (62)

From (61) we obtain

V119899119910119899+1 minus 1119910119899+1 + 1 = [

119909119899 minus 1119909119899 + 1]120572119899

119906119899119909119899+1 minus 1119909119899+1 + 1 = [119910119899 minus 1119910119899 + 1]

120573119899(66)

which is a first-order system after substitution of 119906119899 V119899 by theresults given in (65) Its solutions can be obtained by usingthe following canonical coordinates

119904119899 = ln 119909119899 minus 1119909119899 + 1119905119899 = ln

119910119899 minus 1119910119899 + 1(67)

This leads to the the following linear system with variablecoefficients

119905119899+1 = 120572119899119904119899 + 120574119899119904119899+1 = 120573119899119905119899 + 120593119899 (68)

where minus120574119899 = ln 119906119899 and minus120593119899 = ln V119899The latter is a linear first-order system with variable

coefficients Its general solution is

1199042119899 = 119899minus1prod119896=0

12057221198961205732119896+11199040 + 119899minus1sum119903=0

(1205742119903 119899minus1prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899minus1prod119895=119903+1

12057221198951205732119895+1)

1199052119899 = 119899minus1prod119896=0

12057321198961205722119896+11199050 + 119899minus1sum119903=0

(1205932119903 119899minus1prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899minus1prod119895=119903+1

12057321198951205722119895+1)

1199052119899+1 = 119899prod119896=0

1205722119896119899minus1prod119896=0

1205732119896+11199040 + 119899sum119903=0

(1205742119903 119899prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899prod119895=119903+1

1205722119895 119899minus1prod119895=119903+1

1205732119895+1)

1199042119899+1 = 119899prod119896=0

1205732119896119899minus1prod119896=0

1205722119896+11199050 + 119899sum119903=0

(1205932119903 119899prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899prod119895=119903+1

1205732119895 119899minus1prod119895=119903+1

1205722119895+1)

(69)

The general solution of (45) is obtained by substituting (69)into (67)

Journal of Mathematics 9

4 Conservation Laws

In Section 2 we have defined a first integral associated with asecond-oreder SΔEs It is given by (12)

120601 (119899 119909119899 119910119899 119909119899+1 119910119899+1) = 120601 (119899 + 1 119909119899+1 119910119899+1 1205961 1205962) (70)

Let

1198751 = 120597120601120597119909119899 1198752 = 120597120601120597119909119899+1 1198761 = 120597120601120597119910119899 1198762 = 120597120601120597119910119899+1

(71)

By differentiating (70)with respect to119909119899 119910119899 119909119899+1 and 119910119899+1 weobtain

1198751 = S (1198752) 1205961119909119899 +S (1198762) 12059621199091198991198761 = S (1198752) 1205961119910119899 +S (1198762) 1205962119910119899 (72)

and

1198752 = S (1198751) +S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+11198762 = S (1198761) +S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 (73)

The substitution of (72) in (73) leads to the following second-order system of functional equations

S2 (1198752)S (1205961119909119899) + S

2 (1198762)S (1205962119909119899)+S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+1 minus 1198752 = 0

S2 (1198752)S (1205961119910119899) +S

2 (1198762)S (1205962119910119899)+S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 minus 1198762 = 0

(74)

As for SLSC we differentiate repeatedly to obtain a systemof DEs for 1198752 and 1198762 Given the solutions 1198752 1198762 of (74) weeasily construct 1198751 1198761 For consistency of our solutions wemust check the integrability conditions

1205971198751120597119909119899+1 =1205971198752120597119909119899 (75)

and

1205971198761120597119910119899+1 =1205971198762120597119910119899 (76)

The first integral is then given by

120601 = int (1198751d119909119899 + 1198752d119909119899+1 + 1198761d119910119899 + 1198762d119910119899+1) + 119865 (119899) (77)

The constant of integration 119865(119899) which is a function depend-ing on 119899 is determined by substituting (77) in (70)

41 Applications Let us consider the second-order SΔEs119909119899+2 = 119886 (119899) 119910119899119910119899+2 = 119887 (119899) 119909119899 (78)

By choosing the ansatz 1198752(119899 119909119899 119910119899) and1198762(119899 119909119899 119910119899) one canreadily check that the determining system (74) is simplifiedto

1198762 (119899 + 2 1205961 1205962) 119887 (119899 + 1) minus 1198752 (119899 119909119899 119910119899) = 01198752 (119899 + 2 1205961 1205962) 119886 (119899 + 1) minus 1198762 (119899 119909119899 119910119899) = 0 (79)

where 1205961 and 1205962 denote the right-hand side of (78)Differentiating (79) with respect to 119909119899 and 119910119899 leads to

1198752 = 1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899) 1198762 = 1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899) (80)

Thus we have from (72)

1198751= 119887 (119899) [119909119899+11205954 (119899 + 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)]1198761= 119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)]

(81)

Substituting (80) in (79) and separatingwith respect to119909119899 and119910119899 we obtain the system

119886 (119899) 119887 (119899 + 1)1205954 (119899 + 2) minus 1205952 (119899) = 0119887 (119899) 119886 (119899 + 1)1205952 (119899 + 2) minus 1205954 (119899) = 0119887 (119899) 119887 (119899 + 1)1205955 (119899 + 2) minus 1205951 (119899) = 0119886 (119899) 119886 (119899 + 1)1205951 (119899 + 2) minus 1205955 (119899) = 0

119887 (119899 + 1)1205956 (119899 + 2) minus 1205953 (119899) = 0119886 (119899 + 1)1205953 (119899 + 2) minus 1205956 (119899) = 0

(82)

The solutions to (82) will provide us with the explicit form of120595119894 119894 = 1 6The first integral is then given by

10 Journal of Mathematics

120601= int 119887 (119899) [119909119899+11205954 (119899+ 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)] d119909119899+ (1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899)) d119909119899+1119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)] d119910119899+ (1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899)) d119910119899+1+ 119870119894

(83)

for some constants 119870119894For clarification let us consider 119886(119899) = 119887(119899) = 1 that is

119909119899+2 = 119910119899119910119899+2 = 119909119899 (84)

The solutions to (82) will be

1205951 (119899) = 1198621 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198623 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205955 (119899) = 1198623 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198621 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205952 (119899) = 1198625 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198627 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205954 (119899) = 1198627 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198625 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205953 (119899) = 1198629 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 11986211 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205956 (119899) = 11986211 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198629 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

(85)

where 119862119894 119894 = 1 12 are constants We have twelvesolutions for 1198752 and 1198762 That is

(1) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899

(86)

Journal of Mathematics 11

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1

(87)

(2) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899

(88)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1

(89)

(3) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899

(90)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1

(91)

(4) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899

(92)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1

(93)

(5) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899

(94)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1

(95)

(6) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899

(96)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1

(97)

(7) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899

(98)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1

(99)

(8) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899

(100)

12 Journal of Mathematics

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1

(101)

(9) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 ) 1198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )

(102)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )

(103)

(10) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )

(104)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )

(105)

(11) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )

(106)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )

(107)

(12) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) (108)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) (109)

Therefore we obtain twelve conservation laws for the system(84) They are given by

1206011 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198701

1206012 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119909119899+ 1198702

1206013 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198703

1206014 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119909119899+ 1198704

Journal of Mathematics 13

1206015 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198705

1206016 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119909119899+ 1198706

1206017 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198707

1206018 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119909119899+ 1198708

1206019 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899 + 1198709

12060110 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899 + 11987010

12060111 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899 + 11987011

12060112 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899 + 11987012

(110)

5 Conclusion and Discussions

Wehave presented amethod for obtaining nontrivial symme-tries and how to use them for solving a second-order SΔEsEach symmetry can be used to reduce the order Howeverdifferent symmetries lead to different reductions (see (60) and(64)) but the same solution We also proposed a technique toconstruct first integral associated to second-order systems ofdifference equations

14 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1993

[2] S Maeda ldquoThe similarity method for difference equationsrdquoIMA Journal of Applied Mathematics vol 38 no 2 pp 129ndash1341987

[3] D Levi L Vinet and P Winternitz ldquoLie group formalism fordifference equationsrdquo Journal of Physics A Mathematical andGeneral vol 30 no 2 pp 633ndash649 1997

[4] G R W Quispel and R Sahadevan ldquoLie symmetries and theintegration of difference equationsrdquo Physics Letters A vol 184no 1 pp 64ndash70 1993

[5] P E Hydon Difference Equations by Differential EquationMethods Cambridge University Press Cambridge 2014

[6] V Dorodnitsyn R Kozlov and P Winternitz ldquoLie groupclassification of second-order ordinary difference equationsrdquoJournal of Mathematical Physics vol 41 no 1 pp 480ndash5042000

[7] N Touafek and E M Elsayed ldquoOn the solutions of systemsof rational difference equationsrdquo Mathematical and ComputerModelling vol 55 no 7-8 pp 1987ndash1997 2012

[8] E M Elsayed and T F Ibrahim ldquoPeriodicity and solutionsfor some systems of nonlinear rational difference equationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 6pp 1361ndash1390 2015

[9] A S Kurbanlı C Cinar and I Yalcinkaya ldquoOn the behavior ofpositive solutions of the system of rational difference equations119909119899+1 = 119909119899minus1(119910119899119909119899minus1 + 1) 119910119899+1 = 119910119899minus1(119909119899119910119899minus1 + 1)rdquoMathematical and Computer Modelling vol 53 no 5-6 pp1261ndash1267 2011

[10] N Joshi and P J Vassiliou ldquoThe existence of Lie symmetriesfor first-order analytic discrete dynamical systemsrdquo Journal ofMathematical Analysis and Applications vol 195 no 3 pp 872ndash887 1995

[11] I Yalcinkaya ldquoOn the global asymptotic stability of a secondorder system of difference equationrdquo Discrete Dynamics inNature and Society vol 2008 Article ID 860152 12 pages 2008

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Journal of Mathematics 5

The characteristics in (35) must satisfy the SLSC (14) and(15) Hence we have

1198651 (119899 + 2) minus [1198861 (119899) 1198651 (119899) + 1198862 (119899) 1198652 (119899)+ 1198863 (119899) 1198651 (119899 + 1) + 1198864 (119899) 1198652 (119899 + 1)] = 0

1198652 (119899 + 2) minus [1198871 (119899) 1198651 (119899) + 1198872 (119899) 1198652 (119899)+ 1198873 (119899) 1198651 (119899 + 1) + 1198874 (119899) 1198652 (119899 + 1)] = 0

(36)

and

1198621 = 11986241198622 = 1198623 = 0 (37)

So (35) is simplified to

1198761 (119899 119909119899 119910119899) = 1198621119909119899 + 1198651 (119899) 1198762 (119899 119909119899 119910119899) = 1198621119910119899 + 1198652 (119899) (38)

The first generator of symmetry for a second-order homoge-neous linear system (36) is the scaling symmetry given by

119883 = 119909119899120597119909119899 + 119910119899120597119910119899 (39)

The system (36) which governs the remaining generators ofthe Lie point symmetry for the system (33) is of second orderin 1198651 and 1198652 Its general solution is

1198651 (119899) = 1198921 (1198991198701 1198702 1198703 1198704) 1198652 (119899) = 1198922 (1198991198701 1198702 1198703 1198704) (40)

where1198701 119894 = 1 4 are constantsSo the most large Lie algebra of symmetry generators

which can be obtained from a homogeneous second-ordersystem of 2 difference equations has dimension five

For clarification let us consider1198861(119899) = 1198863(119899) = 1198864(119899) = 0 1198862(119899) = 1 and 1198872(119899) = 1198873(119899) =1198874(119899) = 0 1198871(119899) = 1 The system (33) becomes

119909119899+2 = 119910119899119910119899+2 = 119909119899 (41)

The systemwhich governs the remaining generators of the Liepoint symmetry in this case is given by

1198651 (119899 + 2) minus 1198652 (119899) = 01198652 (119899 + 2) minus 1198651 (119899) = 0 (42)

The general solutions for this system will be

1198651 (119899) = 1198621 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198623 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4sdot 1198624 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1198652 (119899) = 1198623 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198621 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4sdot 1198622 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

(43)

Therefore we have 5 generators of the Lie point symmetryspanned by

X0 = 119909119899120597119909119899 + 119910119899120597119910119899X1 = [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 120597119909119899

+ [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 120597119910119899X2 = [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 120597119909119899

+ [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 120597119910119899X3 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 120597119909119899

+ [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 120597119910119899X4 = [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 120597119909119899

+ [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 120597119910119899

(44)

332 Example 2 Consider the system

119909119899+2 = 119909119899119910119899+1 + 1119909119899 + 119910119899+1119910119899+2 = 119910119899119909119899+1 + 1119910119899 + 119909119899+1

(45)

(45) is a special case of systems investigated in [11] where theauthor looked at the stability of the systems

6 Journal of Mathematics

We choose the ansatz 1198761(119899 119909119899) 1198762(119899 119910119899)The determining system (19) and (20) amounts to

minus 1198762119910119899119909119899+121199101198992 + 1198781198761119909119899+1119909119899+121199101198992 + 21198762119909119899+12119910119899minus 211987811987611199101198992119909119899+1 + 1198762119910119899119909119899+12 + 11987621199101198991199101198992minus 1198781198761119909119899+1119909119899+12 minus 1198781198761119909119899+11199101198992 minus 21198762119910119899+ 21198781198761119909119899+1 minus 1198762119910119899 + 1198781198761119909119899+1 = 0

(46)

minus 11987611199091198991199091198992119910119899+12 + 1198781198762119910119899+11199091198992119910119899+12 + 21198761119910119899+12119909119899minus 211987811987621199091198992119910119899+1 + 11987611199091198991199091198992 + 1198761119909119899119910119899+12minus 1198781198762119910119899+11199091198992 minus 1198781198762119910119899+1119910119899+12 minus 21198761119909119899+ 21198781198762119910119899+1 minus 1198761119909119899 + 1198781198762119910119899+1 = 0

(47)

Differentiating twice (46) with respect to 119909119899 and twice (47)with respect to 119910119899 keeping 119909119899+1 and 119910119899+1 fixed we obtain afterseparating with respect to119909119899+1 and 119910119899+1 the following systemof Des

119876101584010158401 + 1199091198991198761015840101584010158401 + 2 11987611199091198992 minus 211987610158401119909119899 +

1198761015840101584011199091198992 minus1198761015840101584010158401119909119899 = 0

119876101584010158402 + 1199101198991198761015840101584010158402 + 2 11987621199101198992 minus 211987610158402119910119899 +

1198761015840101584021199101198992 minus1198761015840101584010158402119910119899 = 0

(48)

whose most general solutions are

1198761 (119899 119909119899) = 1198651 (119899) 119909119899 + 1198652 (119899) (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1+ 1198653 (119899) (1199092119899 minus 1)

1198762 (119899 119910119899) = 1198654 (119899) 119910119899 + 1198655 (119899) (1199102119899 minus 1) ln 119910119899 + 1119910119899 minus 1+ 1198656 (119899) (1199102119899 minus 1)

(49)

To obtain the nature of functions 1198651 1198656 we substitute (49)in (46) and (47) After separating with respect to 119909119899 119909119899+1 119910119899and 119910119899+1 we get the following SΔEsminus41198652 (119899) minus 21198654 (119899 + 1) + 41198655 (119899 + 1) minus 21198651 (119899) = 041198652 (119899) minus 21198654 (119899 + 1) minus 41198655 (119899 + 1) + 21198651 (119899) = 0minus41198655 (119899) minus 21198651 (119899 + 1) + 41198652 (119899 + 1) minus 21198654 (119899) = 041198655 (119899) minus 21198651 (119899 + 1) minus 41198652 (119899 + 1) + 21198654 (119899) = 0

(50)

whose solutions are

1198651 (119899) = 1198654 (119899) = 01198652 (119899) = 1198621 + (minus1)119899 11986221198655 (119899) = 1198621 minus (minus1)119899 1198622

(51)

The remaining unknown functions 1198653(119899) and 1198656(119899) aredetermined by substituting (51) and (49) into the SLSC (14)and (15) This leads to the SΔEs

1198653 (119899) minus 1198653 (119899 + 2) + 1198656 (119899 + 1) = 01198656 (119899) + 1198653 (119899 + 1) minus 1198656 (119899 + 2) = 0 (52)

The general solutions to (52) are given by

1198653 (119899)= 1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986262119899+1

sdot minus1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986242119899+1

1198656 (119899)= 1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986242119899+1

sdot minus1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986262119899+1

(53)

where 1198621 1198626 are arbitrary constants It follows that thecharacteristics are given by

1198761 = (1198621 + (minus1)119899 1198622) (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1+

1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986262119899+1 ]]

]

Journal of Mathematics 7

sdot minus1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1] 11986252119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986242119899+1 ]]

](1199092119899 minus 1)

1198762 = (1198621 minus (minus1)119899 1198622) (1199102119899 minus 1) ln 119910119899 + 1119910119899 minus 1+

1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986242119899+1 ]]

]sdot minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1] 11986232119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986262119899+1 ]]

](1199102119899 minus 1)

(54)

Therefore we have six generators of Lie point symmetry

X1 = (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1120597119909119899 + (1199102119899 minus 1) ln119910119899 + 1119910119899 minus 1120597119910119899

X2 = (minus1)119899 (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1120597119909119899 minus (minus1)119899 (1199102119899 minus 1)sdot ln 119910119899 + 1119910119899 minus 1120597119910119899

X3 = 1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199092119899 minus 1) 120597119909119899+ minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199102119899 minus 1) 120597119910119899X4 = minus1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 + (1 + radic5)119899]

2119899+1 ]]

sdot (1199092119899 minus 1) 120597119909119899

+ 1 + (minus1)119899radic5 [[[(minus1 + radic5)119899 minus (1 + radic5)119899]

2119899+1 ]]

sdot (1199102119899 minus 1) 120597119910119899X5 = minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]

2119899 ]]]sdot (1199092119899 minus 1) 120597119909119899

+ 1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199102119899 minus 1) 120597119910119899X6 = 1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 minus (1 + radic5)119899]

2119899+1 ]]

sdot (1199092119899 minus 1) 120597119909119899+ minus1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 + (1 + radic5)119899]

2119899+1 ]]

sdot (1199102119899 minus 1) 120597119910119899

(55)

Each generator in (55) can be used to reduce the order of (45)Let us consider X1 By the characteristic method for

Partial Differential Equations the invariants are given byfollowing equation

d119909119899(1199092119899 minus 1) ln ((119909119899 + 1) (119909119899 minus 1))= d119910119899(1199102119899 minus 1) ln ((119910119899 + 1) (119910119899 minus 1))= d119909119899+1(1199092119899+1 minus 1) ln ((119909119899+1 + 1) (119909119899+1 minus 1))= d119910119899+1(1199102119899+1 minus 1) ln ((119910119899+1 + 1) (119910119899+1 minus 1)) =

1198811198990

(56)

We get

1198621 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119910119899+1 + 1) (119910119899+1 minus 1))

1198622 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119910119899 + 1) (119910119899 minus 1))

1198623 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119909119899+1 + 1) (119909119899+1 minus 1))

119881119899 = 119891 (1198621 1198622 1198623)

(57)

where 1198621 1198622 1198623 are constants

8 Journal of Mathematics

If we choose 119891(1198621 1198622 1198623) = 1198621 we have119906119899 = ln ((119909119899 + 1) (119909119899 minus 1))

ln ((119910119899+1 + 1) (119910119899+1 minus 1)) (58)

and if we choose 119891(1198621 1198622 1198623) = 11986231198622 we haveV119899 = ln ((119910119899 + 1) (119910119899 minus 1))

ln ((119909119899+1 + 1) (119909119899+1 minus 1)) (59)

From (58) and (59)we deduce

119906119899+1 = 11 + V119899

V119899+1 = 11 + 119906119899(60)

Let us now consider the generator X3 The resulting invari-ants are

V119899 = [(119909119899 minus 1) (119909119899 + 1)]120572119899(119910119899+1 minus 1) (119910119899+1 + 1) 119906119899 = [(119910119899 minus 1) (119910119899 + 1)]120573119899(119909119899+1 minus 1) (119909119899+1 + 1)

(61)

where

120572119899 = (1 + radic5)119899 minus (minus1 + radic5)1198992 [(1 + radic5)119899minus1 + (minus1 + radic5)119899minus1]

120573119899 = (1 + radic5)119899 + (minus1 + radic5)1198992 [(1 + radic5)119899minus1 minus (minus1 + radic5)119899minus1]

(62)

Note also the relationship between them

120572119899+1 minus 1 = 1120573119899 120573119899+1 minus 1 = 1120572119899

(63)

From (61) we deduce the following relation

V119899+1 = 11199061120573119899119899

119906119899+1 = 1V1120572119899119899

(64)

One can readily check that the general solution to (64) isgiven by

119906119899 = 1 + (minus1)1198992 [119906prod(119899minus2)2119896=0(112057321198961205722119896+1)

0 ]+ 1 minus (minus1)1198992 [Vprod(119899minus1)2119896=0

(11205722119896)prod(119899minus3)2

119896=0(11205732119896+1)

0 ]minus1

V119899 = 1 + (minus1)1198992 [Vprod(119899minus2)2119896=0(112057221198961205732119896+1)

0 ]+ 1 minus (minus1)1198992 [119906prod(119899minus1)2119896=0

(11205732119896)prod(119899minus3)2

119896=0(11205722119896+1)

0 ]minus1(65)

where 120572119899 and 120573119899 are defined in (62)

From (61) we obtain

V119899119910119899+1 minus 1119910119899+1 + 1 = [

119909119899 minus 1119909119899 + 1]120572119899

119906119899119909119899+1 minus 1119909119899+1 + 1 = [119910119899 minus 1119910119899 + 1]

120573119899(66)

which is a first-order system after substitution of 119906119899 V119899 by theresults given in (65) Its solutions can be obtained by usingthe following canonical coordinates

119904119899 = ln 119909119899 minus 1119909119899 + 1119905119899 = ln

119910119899 minus 1119910119899 + 1(67)

This leads to the the following linear system with variablecoefficients

119905119899+1 = 120572119899119904119899 + 120574119899119904119899+1 = 120573119899119905119899 + 120593119899 (68)

where minus120574119899 = ln 119906119899 and minus120593119899 = ln V119899The latter is a linear first-order system with variable

coefficients Its general solution is

1199042119899 = 119899minus1prod119896=0

12057221198961205732119896+11199040 + 119899minus1sum119903=0

(1205742119903 119899minus1prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899minus1prod119895=119903+1

12057221198951205732119895+1)

1199052119899 = 119899minus1prod119896=0

12057321198961205722119896+11199050 + 119899minus1sum119903=0

(1205932119903 119899minus1prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899minus1prod119895=119903+1

12057321198951205722119895+1)

1199052119899+1 = 119899prod119896=0

1205722119896119899minus1prod119896=0

1205732119896+11199040 + 119899sum119903=0

(1205742119903 119899prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899prod119895=119903+1

1205722119895 119899minus1prod119895=119903+1

1205732119895+1)

1199042119899+1 = 119899prod119896=0

1205732119896119899minus1prod119896=0

1205722119896+11199050 + 119899sum119903=0

(1205932119903 119899prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899prod119895=119903+1

1205732119895 119899minus1prod119895=119903+1

1205722119895+1)

(69)

The general solution of (45) is obtained by substituting (69)into (67)

Journal of Mathematics 9

4 Conservation Laws

In Section 2 we have defined a first integral associated with asecond-oreder SΔEs It is given by (12)

120601 (119899 119909119899 119910119899 119909119899+1 119910119899+1) = 120601 (119899 + 1 119909119899+1 119910119899+1 1205961 1205962) (70)

Let

1198751 = 120597120601120597119909119899 1198752 = 120597120601120597119909119899+1 1198761 = 120597120601120597119910119899 1198762 = 120597120601120597119910119899+1

(71)

By differentiating (70)with respect to119909119899 119910119899 119909119899+1 and 119910119899+1 weobtain

1198751 = S (1198752) 1205961119909119899 +S (1198762) 12059621199091198991198761 = S (1198752) 1205961119910119899 +S (1198762) 1205962119910119899 (72)

and

1198752 = S (1198751) +S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+11198762 = S (1198761) +S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 (73)

The substitution of (72) in (73) leads to the following second-order system of functional equations

S2 (1198752)S (1205961119909119899) + S

2 (1198762)S (1205962119909119899)+S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+1 minus 1198752 = 0

S2 (1198752)S (1205961119910119899) +S

2 (1198762)S (1205962119910119899)+S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 minus 1198762 = 0

(74)

As for SLSC we differentiate repeatedly to obtain a systemof DEs for 1198752 and 1198762 Given the solutions 1198752 1198762 of (74) weeasily construct 1198751 1198761 For consistency of our solutions wemust check the integrability conditions

1205971198751120597119909119899+1 =1205971198752120597119909119899 (75)

and

1205971198761120597119910119899+1 =1205971198762120597119910119899 (76)

The first integral is then given by

120601 = int (1198751d119909119899 + 1198752d119909119899+1 + 1198761d119910119899 + 1198762d119910119899+1) + 119865 (119899) (77)

The constant of integration 119865(119899) which is a function depend-ing on 119899 is determined by substituting (77) in (70)

41 Applications Let us consider the second-order SΔEs119909119899+2 = 119886 (119899) 119910119899119910119899+2 = 119887 (119899) 119909119899 (78)

By choosing the ansatz 1198752(119899 119909119899 119910119899) and1198762(119899 119909119899 119910119899) one canreadily check that the determining system (74) is simplifiedto

1198762 (119899 + 2 1205961 1205962) 119887 (119899 + 1) minus 1198752 (119899 119909119899 119910119899) = 01198752 (119899 + 2 1205961 1205962) 119886 (119899 + 1) minus 1198762 (119899 119909119899 119910119899) = 0 (79)

where 1205961 and 1205962 denote the right-hand side of (78)Differentiating (79) with respect to 119909119899 and 119910119899 leads to

1198752 = 1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899) 1198762 = 1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899) (80)

Thus we have from (72)

1198751= 119887 (119899) [119909119899+11205954 (119899 + 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)]1198761= 119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)]

(81)

Substituting (80) in (79) and separatingwith respect to119909119899 and119910119899 we obtain the system

119886 (119899) 119887 (119899 + 1)1205954 (119899 + 2) minus 1205952 (119899) = 0119887 (119899) 119886 (119899 + 1)1205952 (119899 + 2) minus 1205954 (119899) = 0119887 (119899) 119887 (119899 + 1)1205955 (119899 + 2) minus 1205951 (119899) = 0119886 (119899) 119886 (119899 + 1)1205951 (119899 + 2) minus 1205955 (119899) = 0

119887 (119899 + 1)1205956 (119899 + 2) minus 1205953 (119899) = 0119886 (119899 + 1)1205953 (119899 + 2) minus 1205956 (119899) = 0

(82)

The solutions to (82) will provide us with the explicit form of120595119894 119894 = 1 6The first integral is then given by

10 Journal of Mathematics

120601= int 119887 (119899) [119909119899+11205954 (119899+ 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)] d119909119899+ (1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899)) d119909119899+1119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)] d119910119899+ (1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899)) d119910119899+1+ 119870119894

(83)

for some constants 119870119894For clarification let us consider 119886(119899) = 119887(119899) = 1 that is

119909119899+2 = 119910119899119910119899+2 = 119909119899 (84)

The solutions to (82) will be

1205951 (119899) = 1198621 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198623 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205955 (119899) = 1198623 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198621 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205952 (119899) = 1198625 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198627 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205954 (119899) = 1198627 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198625 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205953 (119899) = 1198629 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 11986211 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205956 (119899) = 11986211 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198629 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

(85)

where 119862119894 119894 = 1 12 are constants We have twelvesolutions for 1198752 and 1198762 That is

(1) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899

(86)

Journal of Mathematics 11

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1

(87)

(2) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899

(88)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1

(89)

(3) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899

(90)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1

(91)

(4) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899

(92)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1

(93)

(5) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899

(94)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1

(95)

(6) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899

(96)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1

(97)

(7) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899

(98)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1

(99)

(8) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899

(100)

12 Journal of Mathematics

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1

(101)

(9) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 ) 1198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )

(102)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )

(103)

(10) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )

(104)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )

(105)

(11) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )

(106)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )

(107)

(12) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) (108)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) (109)

Therefore we obtain twelve conservation laws for the system(84) They are given by

1206011 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198701

1206012 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119909119899+ 1198702

1206013 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198703

1206014 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119909119899+ 1198704

Journal of Mathematics 13

1206015 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198705

1206016 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119909119899+ 1198706

1206017 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198707

1206018 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119909119899+ 1198708

1206019 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899 + 1198709

12060110 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899 + 11987010

12060111 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899 + 11987011

12060112 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899 + 11987012

(110)

5 Conclusion and Discussions

Wehave presented amethod for obtaining nontrivial symme-tries and how to use them for solving a second-order SΔEsEach symmetry can be used to reduce the order Howeverdifferent symmetries lead to different reductions (see (60) and(64)) but the same solution We also proposed a technique toconstruct first integral associated to second-order systems ofdifference equations

14 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1993

[2] S Maeda ldquoThe similarity method for difference equationsrdquoIMA Journal of Applied Mathematics vol 38 no 2 pp 129ndash1341987

[3] D Levi L Vinet and P Winternitz ldquoLie group formalism fordifference equationsrdquo Journal of Physics A Mathematical andGeneral vol 30 no 2 pp 633ndash649 1997

[4] G R W Quispel and R Sahadevan ldquoLie symmetries and theintegration of difference equationsrdquo Physics Letters A vol 184no 1 pp 64ndash70 1993

[5] P E Hydon Difference Equations by Differential EquationMethods Cambridge University Press Cambridge 2014

[6] V Dorodnitsyn R Kozlov and P Winternitz ldquoLie groupclassification of second-order ordinary difference equationsrdquoJournal of Mathematical Physics vol 41 no 1 pp 480ndash5042000

[7] N Touafek and E M Elsayed ldquoOn the solutions of systemsof rational difference equationsrdquo Mathematical and ComputerModelling vol 55 no 7-8 pp 1987ndash1997 2012

[8] E M Elsayed and T F Ibrahim ldquoPeriodicity and solutionsfor some systems of nonlinear rational difference equationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 6pp 1361ndash1390 2015

[9] A S Kurbanlı C Cinar and I Yalcinkaya ldquoOn the behavior ofpositive solutions of the system of rational difference equations119909119899+1 = 119909119899minus1(119910119899119909119899minus1 + 1) 119910119899+1 = 119910119899minus1(119909119899119910119899minus1 + 1)rdquoMathematical and Computer Modelling vol 53 no 5-6 pp1261ndash1267 2011

[10] N Joshi and P J Vassiliou ldquoThe existence of Lie symmetriesfor first-order analytic discrete dynamical systemsrdquo Journal ofMathematical Analysis and Applications vol 195 no 3 pp 872ndash887 1995

[11] I Yalcinkaya ldquoOn the global asymptotic stability of a secondorder system of difference equationrdquo Discrete Dynamics inNature and Society vol 2008 Article ID 860152 12 pages 2008

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6 Journal of Mathematics

We choose the ansatz 1198761(119899 119909119899) 1198762(119899 119910119899)The determining system (19) and (20) amounts to

minus 1198762119910119899119909119899+121199101198992 + 1198781198761119909119899+1119909119899+121199101198992 + 21198762119909119899+12119910119899minus 211987811987611199101198992119909119899+1 + 1198762119910119899119909119899+12 + 11987621199101198991199101198992minus 1198781198761119909119899+1119909119899+12 minus 1198781198761119909119899+11199101198992 minus 21198762119910119899+ 21198781198761119909119899+1 minus 1198762119910119899 + 1198781198761119909119899+1 = 0

(46)

minus 11987611199091198991199091198992119910119899+12 + 1198781198762119910119899+11199091198992119910119899+12 + 21198761119910119899+12119909119899minus 211987811987621199091198992119910119899+1 + 11987611199091198991199091198992 + 1198761119909119899119910119899+12minus 1198781198762119910119899+11199091198992 minus 1198781198762119910119899+1119910119899+12 minus 21198761119909119899+ 21198781198762119910119899+1 minus 1198761119909119899 + 1198781198762119910119899+1 = 0

(47)

Differentiating twice (46) with respect to 119909119899 and twice (47)with respect to 119910119899 keeping 119909119899+1 and 119910119899+1 fixed we obtain afterseparating with respect to119909119899+1 and 119910119899+1 the following systemof Des

119876101584010158401 + 1199091198991198761015840101584010158401 + 2 11987611199091198992 minus 211987610158401119909119899 +

1198761015840101584011199091198992 minus1198761015840101584010158401119909119899 = 0

119876101584010158402 + 1199101198991198761015840101584010158402 + 2 11987621199101198992 minus 211987610158402119910119899 +

1198761015840101584021199101198992 minus1198761015840101584010158402119910119899 = 0

(48)

whose most general solutions are

1198761 (119899 119909119899) = 1198651 (119899) 119909119899 + 1198652 (119899) (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1+ 1198653 (119899) (1199092119899 minus 1)

1198762 (119899 119910119899) = 1198654 (119899) 119910119899 + 1198655 (119899) (1199102119899 minus 1) ln 119910119899 + 1119910119899 minus 1+ 1198656 (119899) (1199102119899 minus 1)

(49)

To obtain the nature of functions 1198651 1198656 we substitute (49)in (46) and (47) After separating with respect to 119909119899 119909119899+1 119910119899and 119910119899+1 we get the following SΔEsminus41198652 (119899) minus 21198654 (119899 + 1) + 41198655 (119899 + 1) minus 21198651 (119899) = 041198652 (119899) minus 21198654 (119899 + 1) minus 41198655 (119899 + 1) + 21198651 (119899) = 0minus41198655 (119899) minus 21198651 (119899 + 1) + 41198652 (119899 + 1) minus 21198654 (119899) = 041198655 (119899) minus 21198651 (119899 + 1) minus 41198652 (119899 + 1) + 21198654 (119899) = 0

(50)

whose solutions are

1198651 (119899) = 1198654 (119899) = 01198652 (119899) = 1198621 + (minus1)119899 11986221198655 (119899) = 1198621 minus (minus1)119899 1198622

(51)

The remaining unknown functions 1198653(119899) and 1198656(119899) aredetermined by substituting (51) and (49) into the SLSC (14)and (15) This leads to the SΔEs

1198653 (119899) minus 1198653 (119899 + 2) + 1198656 (119899 + 1) = 01198656 (119899) + 1198653 (119899 + 1) minus 1198656 (119899 + 2) = 0 (52)

The general solutions to (52) are given by

1198653 (119899)= 1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986262119899+1

sdot minus1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986242119899+1

1198656 (119899)= 1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986242119899+1

sdot minus1 + (minus1)119899radic5

[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986262119899+1

(53)

where 1198621 1198626 are arbitrary constants It follows that thecharacteristics are given by

1198761 = (1198621 + (minus1)119899 1198622) (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1+

1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986232119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986262119899+1 ]]

]

Journal of Mathematics 7

sdot minus1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1] 11986252119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986242119899+1 ]]

](1199092119899 minus 1)

1198762 = (1198621 minus (minus1)119899 1198622) (1199102119899 minus 1) ln 119910119899 + 1119910119899 minus 1+

1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986242119899+1 ]]

]sdot minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1] 11986232119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986262119899+1 ]]

](1199102119899 minus 1)

(54)

Therefore we have six generators of Lie point symmetry

X1 = (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1120597119909119899 + (1199102119899 minus 1) ln119910119899 + 1119910119899 minus 1120597119910119899

X2 = (minus1)119899 (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1120597119909119899 minus (minus1)119899 (1199102119899 minus 1)sdot ln 119910119899 + 1119910119899 minus 1120597119910119899

X3 = 1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199092119899 minus 1) 120597119909119899+ minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199102119899 minus 1) 120597119910119899X4 = minus1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 + (1 + radic5)119899]

2119899+1 ]]

sdot (1199092119899 minus 1) 120597119909119899

+ 1 + (minus1)119899radic5 [[[(minus1 + radic5)119899 minus (1 + radic5)119899]

2119899+1 ]]

sdot (1199102119899 minus 1) 120597119910119899X5 = minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]

2119899 ]]]sdot (1199092119899 minus 1) 120597119909119899

+ 1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199102119899 minus 1) 120597119910119899X6 = 1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 minus (1 + radic5)119899]

2119899+1 ]]

sdot (1199092119899 minus 1) 120597119909119899+ minus1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 + (1 + radic5)119899]

2119899+1 ]]

sdot (1199102119899 minus 1) 120597119910119899

(55)

Each generator in (55) can be used to reduce the order of (45)Let us consider X1 By the characteristic method for

Partial Differential Equations the invariants are given byfollowing equation

d119909119899(1199092119899 minus 1) ln ((119909119899 + 1) (119909119899 minus 1))= d119910119899(1199102119899 minus 1) ln ((119910119899 + 1) (119910119899 minus 1))= d119909119899+1(1199092119899+1 minus 1) ln ((119909119899+1 + 1) (119909119899+1 minus 1))= d119910119899+1(1199102119899+1 minus 1) ln ((119910119899+1 + 1) (119910119899+1 minus 1)) =

1198811198990

(56)

We get

1198621 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119910119899+1 + 1) (119910119899+1 minus 1))

1198622 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119910119899 + 1) (119910119899 minus 1))

1198623 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119909119899+1 + 1) (119909119899+1 minus 1))

119881119899 = 119891 (1198621 1198622 1198623)

(57)

where 1198621 1198622 1198623 are constants

8 Journal of Mathematics

If we choose 119891(1198621 1198622 1198623) = 1198621 we have119906119899 = ln ((119909119899 + 1) (119909119899 minus 1))

ln ((119910119899+1 + 1) (119910119899+1 minus 1)) (58)

and if we choose 119891(1198621 1198622 1198623) = 11986231198622 we haveV119899 = ln ((119910119899 + 1) (119910119899 minus 1))

ln ((119909119899+1 + 1) (119909119899+1 minus 1)) (59)

From (58) and (59)we deduce

119906119899+1 = 11 + V119899

V119899+1 = 11 + 119906119899(60)

Let us now consider the generator X3 The resulting invari-ants are

V119899 = [(119909119899 minus 1) (119909119899 + 1)]120572119899(119910119899+1 minus 1) (119910119899+1 + 1) 119906119899 = [(119910119899 minus 1) (119910119899 + 1)]120573119899(119909119899+1 minus 1) (119909119899+1 + 1)

(61)

where

120572119899 = (1 + radic5)119899 minus (minus1 + radic5)1198992 [(1 + radic5)119899minus1 + (minus1 + radic5)119899minus1]

120573119899 = (1 + radic5)119899 + (minus1 + radic5)1198992 [(1 + radic5)119899minus1 minus (minus1 + radic5)119899minus1]

(62)

Note also the relationship between them

120572119899+1 minus 1 = 1120573119899 120573119899+1 minus 1 = 1120572119899

(63)

From (61) we deduce the following relation

V119899+1 = 11199061120573119899119899

119906119899+1 = 1V1120572119899119899

(64)

One can readily check that the general solution to (64) isgiven by

119906119899 = 1 + (minus1)1198992 [119906prod(119899minus2)2119896=0(112057321198961205722119896+1)

0 ]+ 1 minus (minus1)1198992 [Vprod(119899minus1)2119896=0

(11205722119896)prod(119899minus3)2

119896=0(11205732119896+1)

0 ]minus1

V119899 = 1 + (minus1)1198992 [Vprod(119899minus2)2119896=0(112057221198961205732119896+1)

0 ]+ 1 minus (minus1)1198992 [119906prod(119899minus1)2119896=0

(11205732119896)prod(119899minus3)2

119896=0(11205722119896+1)

0 ]minus1(65)

where 120572119899 and 120573119899 are defined in (62)

From (61) we obtain

V119899119910119899+1 minus 1119910119899+1 + 1 = [

119909119899 minus 1119909119899 + 1]120572119899

119906119899119909119899+1 minus 1119909119899+1 + 1 = [119910119899 minus 1119910119899 + 1]

120573119899(66)

which is a first-order system after substitution of 119906119899 V119899 by theresults given in (65) Its solutions can be obtained by usingthe following canonical coordinates

119904119899 = ln 119909119899 minus 1119909119899 + 1119905119899 = ln

119910119899 minus 1119910119899 + 1(67)

This leads to the the following linear system with variablecoefficients

119905119899+1 = 120572119899119904119899 + 120574119899119904119899+1 = 120573119899119905119899 + 120593119899 (68)

where minus120574119899 = ln 119906119899 and minus120593119899 = ln V119899The latter is a linear first-order system with variable

coefficients Its general solution is

1199042119899 = 119899minus1prod119896=0

12057221198961205732119896+11199040 + 119899minus1sum119903=0

(1205742119903 119899minus1prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899minus1prod119895=119903+1

12057221198951205732119895+1)

1199052119899 = 119899minus1prod119896=0

12057321198961205722119896+11199050 + 119899minus1sum119903=0

(1205932119903 119899minus1prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899minus1prod119895=119903+1

12057321198951205722119895+1)

1199052119899+1 = 119899prod119896=0

1205722119896119899minus1prod119896=0

1205732119896+11199040 + 119899sum119903=0

(1205742119903 119899prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899prod119895=119903+1

1205722119895 119899minus1prod119895=119903+1

1205732119895+1)

1199042119899+1 = 119899prod119896=0

1205732119896119899minus1prod119896=0

1205722119896+11199050 + 119899sum119903=0

(1205932119903 119899prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899prod119895=119903+1

1205732119895 119899minus1prod119895=119903+1

1205722119895+1)

(69)

The general solution of (45) is obtained by substituting (69)into (67)

Journal of Mathematics 9

4 Conservation Laws

In Section 2 we have defined a first integral associated with asecond-oreder SΔEs It is given by (12)

120601 (119899 119909119899 119910119899 119909119899+1 119910119899+1) = 120601 (119899 + 1 119909119899+1 119910119899+1 1205961 1205962) (70)

Let

1198751 = 120597120601120597119909119899 1198752 = 120597120601120597119909119899+1 1198761 = 120597120601120597119910119899 1198762 = 120597120601120597119910119899+1

(71)

By differentiating (70)with respect to119909119899 119910119899 119909119899+1 and 119910119899+1 weobtain

1198751 = S (1198752) 1205961119909119899 +S (1198762) 12059621199091198991198761 = S (1198752) 1205961119910119899 +S (1198762) 1205962119910119899 (72)

and

1198752 = S (1198751) +S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+11198762 = S (1198761) +S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 (73)

The substitution of (72) in (73) leads to the following second-order system of functional equations

S2 (1198752)S (1205961119909119899) + S

2 (1198762)S (1205962119909119899)+S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+1 minus 1198752 = 0

S2 (1198752)S (1205961119910119899) +S

2 (1198762)S (1205962119910119899)+S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 minus 1198762 = 0

(74)

As for SLSC we differentiate repeatedly to obtain a systemof DEs for 1198752 and 1198762 Given the solutions 1198752 1198762 of (74) weeasily construct 1198751 1198761 For consistency of our solutions wemust check the integrability conditions

1205971198751120597119909119899+1 =1205971198752120597119909119899 (75)

and

1205971198761120597119910119899+1 =1205971198762120597119910119899 (76)

The first integral is then given by

120601 = int (1198751d119909119899 + 1198752d119909119899+1 + 1198761d119910119899 + 1198762d119910119899+1) + 119865 (119899) (77)

The constant of integration 119865(119899) which is a function depend-ing on 119899 is determined by substituting (77) in (70)

41 Applications Let us consider the second-order SΔEs119909119899+2 = 119886 (119899) 119910119899119910119899+2 = 119887 (119899) 119909119899 (78)

By choosing the ansatz 1198752(119899 119909119899 119910119899) and1198762(119899 119909119899 119910119899) one canreadily check that the determining system (74) is simplifiedto

1198762 (119899 + 2 1205961 1205962) 119887 (119899 + 1) minus 1198752 (119899 119909119899 119910119899) = 01198752 (119899 + 2 1205961 1205962) 119886 (119899 + 1) minus 1198762 (119899 119909119899 119910119899) = 0 (79)

where 1205961 and 1205962 denote the right-hand side of (78)Differentiating (79) with respect to 119909119899 and 119910119899 leads to

1198752 = 1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899) 1198762 = 1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899) (80)

Thus we have from (72)

1198751= 119887 (119899) [119909119899+11205954 (119899 + 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)]1198761= 119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)]

(81)

Substituting (80) in (79) and separatingwith respect to119909119899 and119910119899 we obtain the system

119886 (119899) 119887 (119899 + 1)1205954 (119899 + 2) minus 1205952 (119899) = 0119887 (119899) 119886 (119899 + 1)1205952 (119899 + 2) minus 1205954 (119899) = 0119887 (119899) 119887 (119899 + 1)1205955 (119899 + 2) minus 1205951 (119899) = 0119886 (119899) 119886 (119899 + 1)1205951 (119899 + 2) minus 1205955 (119899) = 0

119887 (119899 + 1)1205956 (119899 + 2) minus 1205953 (119899) = 0119886 (119899 + 1)1205953 (119899 + 2) minus 1205956 (119899) = 0

(82)

The solutions to (82) will provide us with the explicit form of120595119894 119894 = 1 6The first integral is then given by

10 Journal of Mathematics

120601= int 119887 (119899) [119909119899+11205954 (119899+ 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)] d119909119899+ (1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899)) d119909119899+1119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)] d119910119899+ (1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899)) d119910119899+1+ 119870119894

(83)

for some constants 119870119894For clarification let us consider 119886(119899) = 119887(119899) = 1 that is

119909119899+2 = 119910119899119910119899+2 = 119909119899 (84)

The solutions to (82) will be

1205951 (119899) = 1198621 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198623 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205955 (119899) = 1198623 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198621 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205952 (119899) = 1198625 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198627 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205954 (119899) = 1198627 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198625 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205953 (119899) = 1198629 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 11986211 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205956 (119899) = 11986211 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198629 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

(85)

where 119862119894 119894 = 1 12 are constants We have twelvesolutions for 1198752 and 1198762 That is

(1) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899

(86)

Journal of Mathematics 11

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1

(87)

(2) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899

(88)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1

(89)

(3) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899

(90)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1

(91)

(4) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899

(92)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1

(93)

(5) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899

(94)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1

(95)

(6) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899

(96)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1

(97)

(7) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899

(98)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1

(99)

(8) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899

(100)

12 Journal of Mathematics

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1

(101)

(9) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 ) 1198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )

(102)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )

(103)

(10) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )

(104)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )

(105)

(11) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )

(106)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )

(107)

(12) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) (108)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) (109)

Therefore we obtain twelve conservation laws for the system(84) They are given by

1206011 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198701

1206012 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119909119899+ 1198702

1206013 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198703

1206014 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119909119899+ 1198704

Journal of Mathematics 13

1206015 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198705

1206016 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119909119899+ 1198706

1206017 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198707

1206018 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119909119899+ 1198708

1206019 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899 + 1198709

12060110 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899 + 11987010

12060111 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899 + 11987011

12060112 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899 + 11987012

(110)

5 Conclusion and Discussions

Wehave presented amethod for obtaining nontrivial symme-tries and how to use them for solving a second-order SΔEsEach symmetry can be used to reduce the order Howeverdifferent symmetries lead to different reductions (see (60) and(64)) but the same solution We also proposed a technique toconstruct first integral associated to second-order systems ofdifference equations

14 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1993

[2] S Maeda ldquoThe similarity method for difference equationsrdquoIMA Journal of Applied Mathematics vol 38 no 2 pp 129ndash1341987

[3] D Levi L Vinet and P Winternitz ldquoLie group formalism fordifference equationsrdquo Journal of Physics A Mathematical andGeneral vol 30 no 2 pp 633ndash649 1997

[4] G R W Quispel and R Sahadevan ldquoLie symmetries and theintegration of difference equationsrdquo Physics Letters A vol 184no 1 pp 64ndash70 1993

[5] P E Hydon Difference Equations by Differential EquationMethods Cambridge University Press Cambridge 2014

[6] V Dorodnitsyn R Kozlov and P Winternitz ldquoLie groupclassification of second-order ordinary difference equationsrdquoJournal of Mathematical Physics vol 41 no 1 pp 480ndash5042000

[7] N Touafek and E M Elsayed ldquoOn the solutions of systemsof rational difference equationsrdquo Mathematical and ComputerModelling vol 55 no 7-8 pp 1987ndash1997 2012

[8] E M Elsayed and T F Ibrahim ldquoPeriodicity and solutionsfor some systems of nonlinear rational difference equationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 6pp 1361ndash1390 2015

[9] A S Kurbanlı C Cinar and I Yalcinkaya ldquoOn the behavior ofpositive solutions of the system of rational difference equations119909119899+1 = 119909119899minus1(119910119899119909119899minus1 + 1) 119910119899+1 = 119910119899minus1(119909119899119910119899minus1 + 1)rdquoMathematical and Computer Modelling vol 53 no 5-6 pp1261ndash1267 2011

[10] N Joshi and P J Vassiliou ldquoThe existence of Lie symmetriesfor first-order analytic discrete dynamical systemsrdquo Journal ofMathematical Analysis and Applications vol 195 no 3 pp 872ndash887 1995

[11] I Yalcinkaya ldquoOn the global asymptotic stability of a secondorder system of difference equationrdquo Discrete Dynamics inNature and Society vol 2008 Article ID 860152 12 pages 2008

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Journal of Mathematics 7

sdot minus1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1] 11986252119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986242119899+1 ]]

](1199092119899 minus 1)

1198762 = (1198621 minus (minus1)119899 1198622) (1199102119899 minus 1) ln 119910119899 + 1119910119899 minus 1+

1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]11986252119899

minus [(minus1 + radic5)119899 minus (1 + radic5)119899] 11986242119899+1 ]]

]sdot minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1] 11986232119899

minus [(minus1 + radic5)119899 + (1 + radic5)119899] 11986262119899+1 ]]

](1199102119899 minus 1)

(54)

Therefore we have six generators of Lie point symmetry

X1 = (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1120597119909119899 + (1199102119899 minus 1) ln119910119899 + 1119910119899 minus 1120597119910119899

X2 = (minus1)119899 (1199092119899 minus 1) ln 119909119899 + 1119909119899 minus 1120597119909119899 minus (minus1)119899 (1199102119899 minus 1)sdot ln 119910119899 + 1119910119899 minus 1120597119910119899

X3 = 1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199092119899 minus 1) 120597119909119899+ minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199102119899 minus 1) 120597119910119899X4 = minus1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 + (1 + radic5)119899]

2119899+1 ]]

sdot (1199092119899 minus 1) 120597119909119899

+ 1 + (minus1)119899radic5 [[[(minus1 + radic5)119899 minus (1 + radic5)119899]

2119899+1 ]]

sdot (1199102119899 minus 1) 120597119910119899X5 = minus1 + (minus1)119899radic5 [[

[[(minus1 + radic5)119899minus1 minus (1 + radic5)119899minus1]

2119899 ]]]sdot (1199092119899 minus 1) 120597119909119899

+ 1 + (minus1)119899radic5 [[[[(minus1 + radic5)119899minus1 + (1 + radic5)119899minus1]

2119899 ]]]

sdot (1199102119899 minus 1) 120597119910119899X6 = 1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 minus (1 + radic5)119899]

2119899+1 ]]

sdot (1199092119899 minus 1) 120597119909119899+ minus1 + (minus1)119899radic5 [

[[(minus1 + radic5)119899 + (1 + radic5)119899]

2119899+1 ]]

sdot (1199102119899 minus 1) 120597119910119899

(55)

Each generator in (55) can be used to reduce the order of (45)Let us consider X1 By the characteristic method for

Partial Differential Equations the invariants are given byfollowing equation

d119909119899(1199092119899 minus 1) ln ((119909119899 + 1) (119909119899 minus 1))= d119910119899(1199102119899 minus 1) ln ((119910119899 + 1) (119910119899 minus 1))= d119909119899+1(1199092119899+1 minus 1) ln ((119909119899+1 + 1) (119909119899+1 minus 1))= d119910119899+1(1199102119899+1 minus 1) ln ((119910119899+1 + 1) (119910119899+1 minus 1)) =

1198811198990

(56)

We get

1198621 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119910119899+1 + 1) (119910119899+1 minus 1))

1198622 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119910119899 + 1) (119910119899 minus 1))

1198623 = ln ((119909119899 + 1) (119909119899 minus 1))ln ((119909119899+1 + 1) (119909119899+1 minus 1))

119881119899 = 119891 (1198621 1198622 1198623)

(57)

where 1198621 1198622 1198623 are constants

8 Journal of Mathematics

If we choose 119891(1198621 1198622 1198623) = 1198621 we have119906119899 = ln ((119909119899 + 1) (119909119899 minus 1))

ln ((119910119899+1 + 1) (119910119899+1 minus 1)) (58)

and if we choose 119891(1198621 1198622 1198623) = 11986231198622 we haveV119899 = ln ((119910119899 + 1) (119910119899 minus 1))

ln ((119909119899+1 + 1) (119909119899+1 minus 1)) (59)

From (58) and (59)we deduce

119906119899+1 = 11 + V119899

V119899+1 = 11 + 119906119899(60)

Let us now consider the generator X3 The resulting invari-ants are

V119899 = [(119909119899 minus 1) (119909119899 + 1)]120572119899(119910119899+1 minus 1) (119910119899+1 + 1) 119906119899 = [(119910119899 minus 1) (119910119899 + 1)]120573119899(119909119899+1 minus 1) (119909119899+1 + 1)

(61)

where

120572119899 = (1 + radic5)119899 minus (minus1 + radic5)1198992 [(1 + radic5)119899minus1 + (minus1 + radic5)119899minus1]

120573119899 = (1 + radic5)119899 + (minus1 + radic5)1198992 [(1 + radic5)119899minus1 minus (minus1 + radic5)119899minus1]

(62)

Note also the relationship between them

120572119899+1 minus 1 = 1120573119899 120573119899+1 minus 1 = 1120572119899

(63)

From (61) we deduce the following relation

V119899+1 = 11199061120573119899119899

119906119899+1 = 1V1120572119899119899

(64)

One can readily check that the general solution to (64) isgiven by

119906119899 = 1 + (minus1)1198992 [119906prod(119899minus2)2119896=0(112057321198961205722119896+1)

0 ]+ 1 minus (minus1)1198992 [Vprod(119899minus1)2119896=0

(11205722119896)prod(119899minus3)2

119896=0(11205732119896+1)

0 ]minus1

V119899 = 1 + (minus1)1198992 [Vprod(119899minus2)2119896=0(112057221198961205732119896+1)

0 ]+ 1 minus (minus1)1198992 [119906prod(119899minus1)2119896=0

(11205732119896)prod(119899minus3)2

119896=0(11205722119896+1)

0 ]minus1(65)

where 120572119899 and 120573119899 are defined in (62)

From (61) we obtain

V119899119910119899+1 minus 1119910119899+1 + 1 = [

119909119899 minus 1119909119899 + 1]120572119899

119906119899119909119899+1 minus 1119909119899+1 + 1 = [119910119899 minus 1119910119899 + 1]

120573119899(66)

which is a first-order system after substitution of 119906119899 V119899 by theresults given in (65) Its solutions can be obtained by usingthe following canonical coordinates

119904119899 = ln 119909119899 minus 1119909119899 + 1119905119899 = ln

119910119899 minus 1119910119899 + 1(67)

This leads to the the following linear system with variablecoefficients

119905119899+1 = 120572119899119904119899 + 120574119899119904119899+1 = 120573119899119905119899 + 120593119899 (68)

where minus120574119899 = ln 119906119899 and minus120593119899 = ln V119899The latter is a linear first-order system with variable

coefficients Its general solution is

1199042119899 = 119899minus1prod119896=0

12057221198961205732119896+11199040 + 119899minus1sum119903=0

(1205742119903 119899minus1prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899minus1prod119895=119903+1

12057221198951205732119895+1)

1199052119899 = 119899minus1prod119896=0

12057321198961205722119896+11199050 + 119899minus1sum119903=0

(1205932119903 119899minus1prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899minus1prod119895=119903+1

12057321198951205722119895+1)

1199052119899+1 = 119899prod119896=0

1205722119896119899minus1prod119896=0

1205732119896+11199040 + 119899sum119903=0

(1205742119903 119899prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899prod119895=119903+1

1205722119895 119899minus1prod119895=119903+1

1205732119895+1)

1199042119899+1 = 119899prod119896=0

1205732119896119899minus1prod119896=0

1205722119896+11199050 + 119899sum119903=0

(1205932119903 119899prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899prod119895=119903+1

1205732119895 119899minus1prod119895=119903+1

1205722119895+1)

(69)

The general solution of (45) is obtained by substituting (69)into (67)

Journal of Mathematics 9

4 Conservation Laws

In Section 2 we have defined a first integral associated with asecond-oreder SΔEs It is given by (12)

120601 (119899 119909119899 119910119899 119909119899+1 119910119899+1) = 120601 (119899 + 1 119909119899+1 119910119899+1 1205961 1205962) (70)

Let

1198751 = 120597120601120597119909119899 1198752 = 120597120601120597119909119899+1 1198761 = 120597120601120597119910119899 1198762 = 120597120601120597119910119899+1

(71)

By differentiating (70)with respect to119909119899 119910119899 119909119899+1 and 119910119899+1 weobtain

1198751 = S (1198752) 1205961119909119899 +S (1198762) 12059621199091198991198761 = S (1198752) 1205961119910119899 +S (1198762) 1205962119910119899 (72)

and

1198752 = S (1198751) +S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+11198762 = S (1198761) +S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 (73)

The substitution of (72) in (73) leads to the following second-order system of functional equations

S2 (1198752)S (1205961119909119899) + S

2 (1198762)S (1205962119909119899)+S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+1 minus 1198752 = 0

S2 (1198752)S (1205961119910119899) +S

2 (1198762)S (1205962119910119899)+S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 minus 1198762 = 0

(74)

As for SLSC we differentiate repeatedly to obtain a systemof DEs for 1198752 and 1198762 Given the solutions 1198752 1198762 of (74) weeasily construct 1198751 1198761 For consistency of our solutions wemust check the integrability conditions

1205971198751120597119909119899+1 =1205971198752120597119909119899 (75)

and

1205971198761120597119910119899+1 =1205971198762120597119910119899 (76)

The first integral is then given by

120601 = int (1198751d119909119899 + 1198752d119909119899+1 + 1198761d119910119899 + 1198762d119910119899+1) + 119865 (119899) (77)

The constant of integration 119865(119899) which is a function depend-ing on 119899 is determined by substituting (77) in (70)

41 Applications Let us consider the second-order SΔEs119909119899+2 = 119886 (119899) 119910119899119910119899+2 = 119887 (119899) 119909119899 (78)

By choosing the ansatz 1198752(119899 119909119899 119910119899) and1198762(119899 119909119899 119910119899) one canreadily check that the determining system (74) is simplifiedto

1198762 (119899 + 2 1205961 1205962) 119887 (119899 + 1) minus 1198752 (119899 119909119899 119910119899) = 01198752 (119899 + 2 1205961 1205962) 119886 (119899 + 1) minus 1198762 (119899 119909119899 119910119899) = 0 (79)

where 1205961 and 1205962 denote the right-hand side of (78)Differentiating (79) with respect to 119909119899 and 119910119899 leads to

1198752 = 1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899) 1198762 = 1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899) (80)

Thus we have from (72)

1198751= 119887 (119899) [119909119899+11205954 (119899 + 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)]1198761= 119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)]

(81)

Substituting (80) in (79) and separatingwith respect to119909119899 and119910119899 we obtain the system

119886 (119899) 119887 (119899 + 1)1205954 (119899 + 2) minus 1205952 (119899) = 0119887 (119899) 119886 (119899 + 1)1205952 (119899 + 2) minus 1205954 (119899) = 0119887 (119899) 119887 (119899 + 1)1205955 (119899 + 2) minus 1205951 (119899) = 0119886 (119899) 119886 (119899 + 1)1205951 (119899 + 2) minus 1205955 (119899) = 0

119887 (119899 + 1)1205956 (119899 + 2) minus 1205953 (119899) = 0119886 (119899 + 1)1205953 (119899 + 2) minus 1205956 (119899) = 0

(82)

The solutions to (82) will provide us with the explicit form of120595119894 119894 = 1 6The first integral is then given by

10 Journal of Mathematics

120601= int 119887 (119899) [119909119899+11205954 (119899+ 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)] d119909119899+ (1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899)) d119909119899+1119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)] d119910119899+ (1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899)) d119910119899+1+ 119870119894

(83)

for some constants 119870119894For clarification let us consider 119886(119899) = 119887(119899) = 1 that is

119909119899+2 = 119910119899119910119899+2 = 119909119899 (84)

The solutions to (82) will be

1205951 (119899) = 1198621 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198623 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205955 (119899) = 1198623 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198621 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205952 (119899) = 1198625 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198627 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205954 (119899) = 1198627 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198625 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205953 (119899) = 1198629 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 11986211 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205956 (119899) = 11986211 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198629 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

(85)

where 119862119894 119894 = 1 12 are constants We have twelvesolutions for 1198752 and 1198762 That is

(1) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899

(86)

Journal of Mathematics 11

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1

(87)

(2) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899

(88)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1

(89)

(3) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899

(90)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1

(91)

(4) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899

(92)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1

(93)

(5) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899

(94)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1

(95)

(6) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899

(96)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1

(97)

(7) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899

(98)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1

(99)

(8) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899

(100)

12 Journal of Mathematics

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1

(101)

(9) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 ) 1198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )

(102)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )

(103)

(10) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )

(104)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )

(105)

(11) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )

(106)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )

(107)

(12) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) (108)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) (109)

Therefore we obtain twelve conservation laws for the system(84) They are given by

1206011 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198701

1206012 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119909119899+ 1198702

1206013 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198703

1206014 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119909119899+ 1198704

Journal of Mathematics 13

1206015 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198705

1206016 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119909119899+ 1198706

1206017 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198707

1206018 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119909119899+ 1198708

1206019 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899 + 1198709

12060110 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899 + 11987010

12060111 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899 + 11987011

12060112 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899 + 11987012

(110)

5 Conclusion and Discussions

Wehave presented amethod for obtaining nontrivial symme-tries and how to use them for solving a second-order SΔEsEach symmetry can be used to reduce the order Howeverdifferent symmetries lead to different reductions (see (60) and(64)) but the same solution We also proposed a technique toconstruct first integral associated to second-order systems ofdifference equations

14 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1993

[2] S Maeda ldquoThe similarity method for difference equationsrdquoIMA Journal of Applied Mathematics vol 38 no 2 pp 129ndash1341987

[3] D Levi L Vinet and P Winternitz ldquoLie group formalism fordifference equationsrdquo Journal of Physics A Mathematical andGeneral vol 30 no 2 pp 633ndash649 1997

[4] G R W Quispel and R Sahadevan ldquoLie symmetries and theintegration of difference equationsrdquo Physics Letters A vol 184no 1 pp 64ndash70 1993

[5] P E Hydon Difference Equations by Differential EquationMethods Cambridge University Press Cambridge 2014

[6] V Dorodnitsyn R Kozlov and P Winternitz ldquoLie groupclassification of second-order ordinary difference equationsrdquoJournal of Mathematical Physics vol 41 no 1 pp 480ndash5042000

[7] N Touafek and E M Elsayed ldquoOn the solutions of systemsof rational difference equationsrdquo Mathematical and ComputerModelling vol 55 no 7-8 pp 1987ndash1997 2012

[8] E M Elsayed and T F Ibrahim ldquoPeriodicity and solutionsfor some systems of nonlinear rational difference equationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 6pp 1361ndash1390 2015

[9] A S Kurbanlı C Cinar and I Yalcinkaya ldquoOn the behavior ofpositive solutions of the system of rational difference equations119909119899+1 = 119909119899minus1(119910119899119909119899minus1 + 1) 119910119899+1 = 119910119899minus1(119909119899119910119899minus1 + 1)rdquoMathematical and Computer Modelling vol 53 no 5-6 pp1261ndash1267 2011

[10] N Joshi and P J Vassiliou ldquoThe existence of Lie symmetriesfor first-order analytic discrete dynamical systemsrdquo Journal ofMathematical Analysis and Applications vol 195 no 3 pp 872ndash887 1995

[11] I Yalcinkaya ldquoOn the global asymptotic stability of a secondorder system of difference equationrdquo Discrete Dynamics inNature and Society vol 2008 Article ID 860152 12 pages 2008

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8 Journal of Mathematics

If we choose 119891(1198621 1198622 1198623) = 1198621 we have119906119899 = ln ((119909119899 + 1) (119909119899 minus 1))

ln ((119910119899+1 + 1) (119910119899+1 minus 1)) (58)

and if we choose 119891(1198621 1198622 1198623) = 11986231198622 we haveV119899 = ln ((119910119899 + 1) (119910119899 minus 1))

ln ((119909119899+1 + 1) (119909119899+1 minus 1)) (59)

From (58) and (59)we deduce

119906119899+1 = 11 + V119899

V119899+1 = 11 + 119906119899(60)

Let us now consider the generator X3 The resulting invari-ants are

V119899 = [(119909119899 minus 1) (119909119899 + 1)]120572119899(119910119899+1 minus 1) (119910119899+1 + 1) 119906119899 = [(119910119899 minus 1) (119910119899 + 1)]120573119899(119909119899+1 minus 1) (119909119899+1 + 1)

(61)

where

120572119899 = (1 + radic5)119899 minus (minus1 + radic5)1198992 [(1 + radic5)119899minus1 + (minus1 + radic5)119899minus1]

120573119899 = (1 + radic5)119899 + (minus1 + radic5)1198992 [(1 + radic5)119899minus1 minus (minus1 + radic5)119899minus1]

(62)

Note also the relationship between them

120572119899+1 minus 1 = 1120573119899 120573119899+1 minus 1 = 1120572119899

(63)

From (61) we deduce the following relation

V119899+1 = 11199061120573119899119899

119906119899+1 = 1V1120572119899119899

(64)

One can readily check that the general solution to (64) isgiven by

119906119899 = 1 + (minus1)1198992 [119906prod(119899minus2)2119896=0(112057321198961205722119896+1)

0 ]+ 1 minus (minus1)1198992 [Vprod(119899minus1)2119896=0

(11205722119896)prod(119899minus3)2

119896=0(11205732119896+1)

0 ]minus1

V119899 = 1 + (minus1)1198992 [Vprod(119899minus2)2119896=0(112057221198961205732119896+1)

0 ]+ 1 minus (minus1)1198992 [119906prod(119899minus1)2119896=0

(11205732119896)prod(119899minus3)2

119896=0(11205722119896+1)

0 ]minus1(65)

where 120572119899 and 120573119899 are defined in (62)

From (61) we obtain

V119899119910119899+1 minus 1119910119899+1 + 1 = [

119909119899 minus 1119909119899 + 1]120572119899

119906119899119909119899+1 minus 1119909119899+1 + 1 = [119910119899 minus 1119910119899 + 1]

120573119899(66)

which is a first-order system after substitution of 119906119899 V119899 by theresults given in (65) Its solutions can be obtained by usingthe following canonical coordinates

119904119899 = ln 119909119899 minus 1119909119899 + 1119905119899 = ln

119910119899 minus 1119910119899 + 1(67)

This leads to the the following linear system with variablecoefficients

119905119899+1 = 120572119899119904119899 + 120574119899119904119899+1 = 120573119899119905119899 + 120593119899 (68)

where minus120574119899 = ln 119906119899 and minus120593119899 = ln V119899The latter is a linear first-order system with variable

coefficients Its general solution is

1199042119899 = 119899minus1prod119896=0

12057221198961205732119896+11199040 + 119899minus1sum119903=0

(1205742119903 119899minus1prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899minus1prod119895=119903+1

12057221198951205732119895+1)

1199052119899 = 119899minus1prod119896=0

12057321198961205722119896+11199050 + 119899minus1sum119903=0

(1205932119903 119899minus1prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899minus1prod119895=119903+1

12057321198951205722119895+1)

1199052119899+1 = 119899prod119896=0

1205722119896119899minus1prod119896=0

1205732119896+11199040 + 119899sum119903=0

(1205742119903 119899prod119895=119903+1

1205722119895119899minus1prod119895=119903

1205732119895+1)

+ 119899minus1sum119903=0

(1205932119903+1 119899prod119895=119903+1

1205722119895 119899minus1prod119895=119903+1

1205732119895+1)

1199042119899+1 = 119899prod119896=0

1205732119896119899minus1prod119896=0

1205722119896+11199050 + 119899sum119903=0

(1205932119903 119899prod119895=119903+1

1205732119895119899minus1prod119895=119903

1205722119895+1)

+ 119899minus1sum119903=0

(1205742119903+1 119899prod119895=119903+1

1205732119895 119899minus1prod119895=119903+1

1205722119895+1)

(69)

The general solution of (45) is obtained by substituting (69)into (67)

Journal of Mathematics 9

4 Conservation Laws

In Section 2 we have defined a first integral associated with asecond-oreder SΔEs It is given by (12)

120601 (119899 119909119899 119910119899 119909119899+1 119910119899+1) = 120601 (119899 + 1 119909119899+1 119910119899+1 1205961 1205962) (70)

Let

1198751 = 120597120601120597119909119899 1198752 = 120597120601120597119909119899+1 1198761 = 120597120601120597119910119899 1198762 = 120597120601120597119910119899+1

(71)

By differentiating (70)with respect to119909119899 119910119899 119909119899+1 and 119910119899+1 weobtain

1198751 = S (1198752) 1205961119909119899 +S (1198762) 12059621199091198991198761 = S (1198752) 1205961119910119899 +S (1198762) 1205962119910119899 (72)

and

1198752 = S (1198751) +S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+11198762 = S (1198761) +S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 (73)

The substitution of (72) in (73) leads to the following second-order system of functional equations

S2 (1198752)S (1205961119909119899) + S

2 (1198762)S (1205962119909119899)+S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+1 minus 1198752 = 0

S2 (1198752)S (1205961119910119899) +S

2 (1198762)S (1205962119910119899)+S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 minus 1198762 = 0

(74)

As for SLSC we differentiate repeatedly to obtain a systemof DEs for 1198752 and 1198762 Given the solutions 1198752 1198762 of (74) weeasily construct 1198751 1198761 For consistency of our solutions wemust check the integrability conditions

1205971198751120597119909119899+1 =1205971198752120597119909119899 (75)

and

1205971198761120597119910119899+1 =1205971198762120597119910119899 (76)

The first integral is then given by

120601 = int (1198751d119909119899 + 1198752d119909119899+1 + 1198761d119910119899 + 1198762d119910119899+1) + 119865 (119899) (77)

The constant of integration 119865(119899) which is a function depend-ing on 119899 is determined by substituting (77) in (70)

41 Applications Let us consider the second-order SΔEs119909119899+2 = 119886 (119899) 119910119899119910119899+2 = 119887 (119899) 119909119899 (78)

By choosing the ansatz 1198752(119899 119909119899 119910119899) and1198762(119899 119909119899 119910119899) one canreadily check that the determining system (74) is simplifiedto

1198762 (119899 + 2 1205961 1205962) 119887 (119899 + 1) minus 1198752 (119899 119909119899 119910119899) = 01198752 (119899 + 2 1205961 1205962) 119886 (119899 + 1) minus 1198762 (119899 119909119899 119910119899) = 0 (79)

where 1205961 and 1205962 denote the right-hand side of (78)Differentiating (79) with respect to 119909119899 and 119910119899 leads to

1198752 = 1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899) 1198762 = 1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899) (80)

Thus we have from (72)

1198751= 119887 (119899) [119909119899+11205954 (119899 + 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)]1198761= 119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)]

(81)

Substituting (80) in (79) and separatingwith respect to119909119899 and119910119899 we obtain the system

119886 (119899) 119887 (119899 + 1)1205954 (119899 + 2) minus 1205952 (119899) = 0119887 (119899) 119886 (119899 + 1)1205952 (119899 + 2) minus 1205954 (119899) = 0119887 (119899) 119887 (119899 + 1)1205955 (119899 + 2) minus 1205951 (119899) = 0119886 (119899) 119886 (119899 + 1)1205951 (119899 + 2) minus 1205955 (119899) = 0

119887 (119899 + 1)1205956 (119899 + 2) minus 1205953 (119899) = 0119886 (119899 + 1)1205953 (119899 + 2) minus 1205956 (119899) = 0

(82)

The solutions to (82) will provide us with the explicit form of120595119894 119894 = 1 6The first integral is then given by

10 Journal of Mathematics

120601= int 119887 (119899) [119909119899+11205954 (119899+ 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)] d119909119899+ (1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899)) d119909119899+1119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)] d119910119899+ (1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899)) d119910119899+1+ 119870119894

(83)

for some constants 119870119894For clarification let us consider 119886(119899) = 119887(119899) = 1 that is

119909119899+2 = 119910119899119910119899+2 = 119909119899 (84)

The solutions to (82) will be

1205951 (119899) = 1198621 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198623 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205955 (119899) = 1198623 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198621 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205952 (119899) = 1198625 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198627 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205954 (119899) = 1198627 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198625 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205953 (119899) = 1198629 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 11986211 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205956 (119899) = 11986211 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198629 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

(85)

where 119862119894 119894 = 1 12 are constants We have twelvesolutions for 1198752 and 1198762 That is

(1) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899

(86)

Journal of Mathematics 11

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1

(87)

(2) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899

(88)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1

(89)

(3) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899

(90)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1

(91)

(4) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899

(92)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1

(93)

(5) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899

(94)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1

(95)

(6) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899

(96)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1

(97)

(7) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899

(98)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1

(99)

(8) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899

(100)

12 Journal of Mathematics

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1

(101)

(9) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 ) 1198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )

(102)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )

(103)

(10) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )

(104)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )

(105)

(11) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )

(106)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )

(107)

(12) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) (108)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) (109)

Therefore we obtain twelve conservation laws for the system(84) They are given by

1206011 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198701

1206012 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119909119899+ 1198702

1206013 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198703

1206014 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119909119899+ 1198704

Journal of Mathematics 13

1206015 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198705

1206016 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119909119899+ 1198706

1206017 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198707

1206018 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119909119899+ 1198708

1206019 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899 + 1198709

12060110 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899 + 11987010

12060111 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899 + 11987011

12060112 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899 + 11987012

(110)

5 Conclusion and Discussions

Wehave presented amethod for obtaining nontrivial symme-tries and how to use them for solving a second-order SΔEsEach symmetry can be used to reduce the order Howeverdifferent symmetries lead to different reductions (see (60) and(64)) but the same solution We also proposed a technique toconstruct first integral associated to second-order systems ofdifference equations

14 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1993

[2] S Maeda ldquoThe similarity method for difference equationsrdquoIMA Journal of Applied Mathematics vol 38 no 2 pp 129ndash1341987

[3] D Levi L Vinet and P Winternitz ldquoLie group formalism fordifference equationsrdquo Journal of Physics A Mathematical andGeneral vol 30 no 2 pp 633ndash649 1997

[4] G R W Quispel and R Sahadevan ldquoLie symmetries and theintegration of difference equationsrdquo Physics Letters A vol 184no 1 pp 64ndash70 1993

[5] P E Hydon Difference Equations by Differential EquationMethods Cambridge University Press Cambridge 2014

[6] V Dorodnitsyn R Kozlov and P Winternitz ldquoLie groupclassification of second-order ordinary difference equationsrdquoJournal of Mathematical Physics vol 41 no 1 pp 480ndash5042000

[7] N Touafek and E M Elsayed ldquoOn the solutions of systemsof rational difference equationsrdquo Mathematical and ComputerModelling vol 55 no 7-8 pp 1987ndash1997 2012

[8] E M Elsayed and T F Ibrahim ldquoPeriodicity and solutionsfor some systems of nonlinear rational difference equationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 6pp 1361ndash1390 2015

[9] A S Kurbanlı C Cinar and I Yalcinkaya ldquoOn the behavior ofpositive solutions of the system of rational difference equations119909119899+1 = 119909119899minus1(119910119899119909119899minus1 + 1) 119910119899+1 = 119910119899minus1(119909119899119910119899minus1 + 1)rdquoMathematical and Computer Modelling vol 53 no 5-6 pp1261ndash1267 2011

[10] N Joshi and P J Vassiliou ldquoThe existence of Lie symmetriesfor first-order analytic discrete dynamical systemsrdquo Journal ofMathematical Analysis and Applications vol 195 no 3 pp 872ndash887 1995

[11] I Yalcinkaya ldquoOn the global asymptotic stability of a secondorder system of difference equationrdquo Discrete Dynamics inNature and Society vol 2008 Article ID 860152 12 pages 2008

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Journal of Mathematics 9

4 Conservation Laws

In Section 2 we have defined a first integral associated with asecond-oreder SΔEs It is given by (12)

120601 (119899 119909119899 119910119899 119909119899+1 119910119899+1) = 120601 (119899 + 1 119909119899+1 119910119899+1 1205961 1205962) (70)

Let

1198751 = 120597120601120597119909119899 1198752 = 120597120601120597119909119899+1 1198761 = 120597120601120597119910119899 1198762 = 120597120601120597119910119899+1

(71)

By differentiating (70)with respect to119909119899 119910119899 119909119899+1 and 119910119899+1 weobtain

1198751 = S (1198752) 1205961119909119899 +S (1198762) 12059621199091198991198761 = S (1198752) 1205961119910119899 +S (1198762) 1205962119910119899 (72)

and

1198752 = S (1198751) +S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+11198762 = S (1198761) +S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 (73)

The substitution of (72) in (73) leads to the following second-order system of functional equations

S2 (1198752)S (1205961119909119899) + S

2 (1198762)S (1205962119909119899)+S (1198752) 1205961119909119899+1 + S (1198762) 1205962119909119899+1 minus 1198752 = 0

S2 (1198752)S (1205961119910119899) +S

2 (1198762)S (1205962119910119899)+S (1198752) 1205961119910119899+1 +S (1198762) 1205962119910119899+1 minus 1198762 = 0

(74)

As for SLSC we differentiate repeatedly to obtain a systemof DEs for 1198752 and 1198762 Given the solutions 1198752 1198762 of (74) weeasily construct 1198751 1198761 For consistency of our solutions wemust check the integrability conditions

1205971198751120597119909119899+1 =1205971198752120597119909119899 (75)

and

1205971198761120597119910119899+1 =1205971198762120597119910119899 (76)

The first integral is then given by

120601 = int (1198751d119909119899 + 1198752d119909119899+1 + 1198761d119910119899 + 1198762d119910119899+1) + 119865 (119899) (77)

The constant of integration 119865(119899) which is a function depend-ing on 119899 is determined by substituting (77) in (70)

41 Applications Let us consider the second-order SΔEs119909119899+2 = 119886 (119899) 119910119899119910119899+2 = 119887 (119899) 119909119899 (78)

By choosing the ansatz 1198752(119899 119909119899 119910119899) and1198762(119899 119909119899 119910119899) one canreadily check that the determining system (74) is simplifiedto

1198762 (119899 + 2 1205961 1205962) 119887 (119899 + 1) minus 1198752 (119899 119909119899 119910119899) = 01198752 (119899 + 2 1205961 1205962) 119886 (119899 + 1) minus 1198762 (119899 119909119899 119910119899) = 0 (79)

where 1205961 and 1205962 denote the right-hand side of (78)Differentiating (79) with respect to 119909119899 and 119910119899 leads to

1198752 = 1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899) 1198762 = 1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899) (80)

Thus we have from (72)

1198751= 119887 (119899) [119909119899+11205954 (119899 + 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)]1198761= 119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)]

(81)

Substituting (80) in (79) and separatingwith respect to119909119899 and119910119899 we obtain the system

119886 (119899) 119887 (119899 + 1)1205954 (119899 + 2) minus 1205952 (119899) = 0119887 (119899) 119886 (119899 + 1)1205952 (119899 + 2) minus 1205954 (119899) = 0119887 (119899) 119887 (119899 + 1)1205955 (119899 + 2) minus 1205951 (119899) = 0119886 (119899) 119886 (119899 + 1)1205951 (119899 + 2) minus 1205955 (119899) = 0

119887 (119899 + 1)1205956 (119899 + 2) minus 1205953 (119899) = 0119886 (119899 + 1)1205953 (119899 + 2) minus 1205956 (119899) = 0

(82)

The solutions to (82) will provide us with the explicit form of120595119894 119894 = 1 6The first integral is then given by

10 Journal of Mathematics

120601= int 119887 (119899) [119909119899+11205954 (119899+ 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)] d119909119899+ (1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899)) d119909119899+1119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)] d119910119899+ (1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899)) d119910119899+1+ 119870119894

(83)

for some constants 119870119894For clarification let us consider 119886(119899) = 119887(119899) = 1 that is

119909119899+2 = 119910119899119910119899+2 = 119909119899 (84)

The solutions to (82) will be

1205951 (119899) = 1198621 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198623 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205955 (119899) = 1198623 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198621 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205952 (119899) = 1198625 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198627 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205954 (119899) = 1198627 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198625 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205953 (119899) = 1198629 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 11986211 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205956 (119899) = 11986211 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198629 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

(85)

where 119862119894 119894 = 1 12 are constants We have twelvesolutions for 1198752 and 1198762 That is

(1) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899

(86)

Journal of Mathematics 11

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1

(87)

(2) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899

(88)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1

(89)

(3) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899

(90)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1

(91)

(4) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899

(92)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1

(93)

(5) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899

(94)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1

(95)

(6) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899

(96)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1

(97)

(7) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899

(98)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1

(99)

(8) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899

(100)

12 Journal of Mathematics

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1

(101)

(9) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 ) 1198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )

(102)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )

(103)

(10) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )

(104)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )

(105)

(11) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )

(106)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )

(107)

(12) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) (108)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) (109)

Therefore we obtain twelve conservation laws for the system(84) They are given by

1206011 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198701

1206012 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119909119899+ 1198702

1206013 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198703

1206014 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119909119899+ 1198704

Journal of Mathematics 13

1206015 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198705

1206016 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119909119899+ 1198706

1206017 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198707

1206018 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119909119899+ 1198708

1206019 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899 + 1198709

12060110 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899 + 11987010

12060111 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899 + 11987011

12060112 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899 + 11987012

(110)

5 Conclusion and Discussions

Wehave presented amethod for obtaining nontrivial symme-tries and how to use them for solving a second-order SΔEsEach symmetry can be used to reduce the order Howeverdifferent symmetries lead to different reductions (see (60) and(64)) but the same solution We also proposed a technique toconstruct first integral associated to second-order systems ofdifference equations

14 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1993

[2] S Maeda ldquoThe similarity method for difference equationsrdquoIMA Journal of Applied Mathematics vol 38 no 2 pp 129ndash1341987

[3] D Levi L Vinet and P Winternitz ldquoLie group formalism fordifference equationsrdquo Journal of Physics A Mathematical andGeneral vol 30 no 2 pp 633ndash649 1997

[4] G R W Quispel and R Sahadevan ldquoLie symmetries and theintegration of difference equationsrdquo Physics Letters A vol 184no 1 pp 64ndash70 1993

[5] P E Hydon Difference Equations by Differential EquationMethods Cambridge University Press Cambridge 2014

[6] V Dorodnitsyn R Kozlov and P Winternitz ldquoLie groupclassification of second-order ordinary difference equationsrdquoJournal of Mathematical Physics vol 41 no 1 pp 480ndash5042000

[7] N Touafek and E M Elsayed ldquoOn the solutions of systemsof rational difference equationsrdquo Mathematical and ComputerModelling vol 55 no 7-8 pp 1987ndash1997 2012

[8] E M Elsayed and T F Ibrahim ldquoPeriodicity and solutionsfor some systems of nonlinear rational difference equationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 6pp 1361ndash1390 2015

[9] A S Kurbanlı C Cinar and I Yalcinkaya ldquoOn the behavior ofpositive solutions of the system of rational difference equations119909119899+1 = 119909119899minus1(119910119899119909119899minus1 + 1) 119910119899+1 = 119910119899minus1(119909119899119910119899minus1 + 1)rdquoMathematical and Computer Modelling vol 53 no 5-6 pp1261ndash1267 2011

[10] N Joshi and P J Vassiliou ldquoThe existence of Lie symmetriesfor first-order analytic discrete dynamical systemsrdquo Journal ofMathematical Analysis and Applications vol 195 no 3 pp 872ndash887 1995

[11] I Yalcinkaya ldquoOn the global asymptotic stability of a secondorder system of difference equationrdquo Discrete Dynamics inNature and Society vol 2008 Article ID 860152 12 pages 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Journal of Mathematics

120601= int 119887 (119899) [119909119899+11205954 (119899+ 1) + 119910119899+11205955 (119899 + 1) + 1205956 (119899 + 1)] d119909119899+ (1199091198991205951 (119899) + 1199101198991205952 (119899) + 1205953 (119899)) d119909119899+1119886 (119899) [119909119899+11205951 (119899 + 1) + 119910119899+11205952 (119899 + 1) + 1205953 (119899 + 1)] d119910119899+ (1199091198991205954 (119899) + 1199101198991205955 (119899) + 1205956 (119899)) d119910119899+1+ 119870119894

(83)

for some constants 119870119894For clarification let us consider 119886(119899) = 119887(119899) = 1 that is

119909119899+2 = 119910119899119910119899+2 = 119909119899 (84)

The solutions to (82) will be

1205951 (119899) = 1198621 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198623 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205955 (119899) = 1198623 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198624 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198621 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198622 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205952 (119899) = 1198625 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198627 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205954 (119899) = 1198627 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 1198628 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198625 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 1198626 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205953 (119899) = 1198629 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 11986211 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

1205956 (119899) = 11986211 [1 + (minus1)119899 + 119894119899 + (minus119894)119899]4+ 11986212 [1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4+ 1198629 [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4+ 11986210 [1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4

(85)

where 119862119894 119894 = 1 12 are constants We have twelvesolutions for 1198752 and 1198762 That is

(1) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899

(86)

Journal of Mathematics 11

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1

(87)

(2) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899

(88)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1

(89)

(3) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899

(90)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1

(91)

(4) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899

(92)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1

(93)

(5) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899

(94)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1

(95)

(6) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899

(96)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1

(97)

(7) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899

(98)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1

(99)

(8) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899

(100)

12 Journal of Mathematics

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1

(101)

(9) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 ) 1198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )

(102)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )

(103)

(10) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )

(104)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )

(105)

(11) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )

(106)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )

(107)

(12) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) (108)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) (109)

Therefore we obtain twelve conservation laws for the system(84) They are given by

1206011 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198701

1206012 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119909119899+ 1198702

1206013 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198703

1206014 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119909119899+ 1198704

Journal of Mathematics 13

1206015 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198705

1206016 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119909119899+ 1198706

1206017 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198707

1206018 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119909119899+ 1198708

1206019 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899 + 1198709

12060110 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899 + 11987010

12060111 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899 + 11987011

12060112 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899 + 11987012

(110)

5 Conclusion and Discussions

Wehave presented amethod for obtaining nontrivial symme-tries and how to use them for solving a second-order SΔEsEach symmetry can be used to reduce the order Howeverdifferent symmetries lead to different reductions (see (60) and(64)) but the same solution We also proposed a technique toconstruct first integral associated to second-order systems ofdifference equations

14 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1993

[2] S Maeda ldquoThe similarity method for difference equationsrdquoIMA Journal of Applied Mathematics vol 38 no 2 pp 129ndash1341987

[3] D Levi L Vinet and P Winternitz ldquoLie group formalism fordifference equationsrdquo Journal of Physics A Mathematical andGeneral vol 30 no 2 pp 633ndash649 1997

[4] G R W Quispel and R Sahadevan ldquoLie symmetries and theintegration of difference equationsrdquo Physics Letters A vol 184no 1 pp 64ndash70 1993

[5] P E Hydon Difference Equations by Differential EquationMethods Cambridge University Press Cambridge 2014

[6] V Dorodnitsyn R Kozlov and P Winternitz ldquoLie groupclassification of second-order ordinary difference equationsrdquoJournal of Mathematical Physics vol 41 no 1 pp 480ndash5042000

[7] N Touafek and E M Elsayed ldquoOn the solutions of systemsof rational difference equationsrdquo Mathematical and ComputerModelling vol 55 no 7-8 pp 1987ndash1997 2012

[8] E M Elsayed and T F Ibrahim ldquoPeriodicity and solutionsfor some systems of nonlinear rational difference equationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 6pp 1361ndash1390 2015

[9] A S Kurbanlı C Cinar and I Yalcinkaya ldquoOn the behavior ofpositive solutions of the system of rational difference equations119909119899+1 = 119909119899minus1(119910119899119909119899minus1 + 1) 119910119899+1 = 119910119899minus1(119909119899119910119899minus1 + 1)rdquoMathematical and Computer Modelling vol 53 no 5-6 pp1261ndash1267 2011

[10] N Joshi and P J Vassiliou ldquoThe existence of Lie symmetriesfor first-order analytic discrete dynamical systemsrdquo Journal ofMathematical Analysis and Applications vol 195 no 3 pp 872ndash887 1995

[11] I Yalcinkaya ldquoOn the global asymptotic stability of a secondorder system of difference equationrdquo Discrete Dynamics inNature and Society vol 2008 Article ID 860152 12 pages 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Journal of Mathematics 11

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1

(87)

(2) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899

(88)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1

(89)

(3) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199091198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899

(90)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1

(91)

(4) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199091198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899

(92)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1

(93)

(5) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899

(94)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1

(95)

(6) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899

(96)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1

(97)

(7) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1199101198991198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899

(98)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+11198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1

(99)

(8) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )1199101198991198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899

(100)

12 Journal of Mathematics

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1

(101)

(9) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 ) 1198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )

(102)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )

(103)

(10) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )

(104)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )

(105)

(11) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )

(106)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )

(107)

(12) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) (108)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) (109)

Therefore we obtain twelve conservation laws for the system(84) They are given by

1206011 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198701

1206012 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119909119899+ 1198702

1206013 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198703

1206014 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119909119899+ 1198704

Journal of Mathematics 13

1206015 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198705

1206016 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119909119899+ 1198706

1206017 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198707

1206018 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119909119899+ 1198708

1206019 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899 + 1198709

12060110 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899 + 11987010

12060111 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899 + 11987011

12060112 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899 + 11987012

(110)

5 Conclusion and Discussions

Wehave presented amethod for obtaining nontrivial symme-tries and how to use them for solving a second-order SΔEsEach symmetry can be used to reduce the order Howeverdifferent symmetries lead to different reductions (see (60) and(64)) but the same solution We also proposed a technique toconstruct first integral associated to second-order systems ofdifference equations

14 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1993

[2] S Maeda ldquoThe similarity method for difference equationsrdquoIMA Journal of Applied Mathematics vol 38 no 2 pp 129ndash1341987

[3] D Levi L Vinet and P Winternitz ldquoLie group formalism fordifference equationsrdquo Journal of Physics A Mathematical andGeneral vol 30 no 2 pp 633ndash649 1997

[4] G R W Quispel and R Sahadevan ldquoLie symmetries and theintegration of difference equationsrdquo Physics Letters A vol 184no 1 pp 64ndash70 1993

[5] P E Hydon Difference Equations by Differential EquationMethods Cambridge University Press Cambridge 2014

[6] V Dorodnitsyn R Kozlov and P Winternitz ldquoLie groupclassification of second-order ordinary difference equationsrdquoJournal of Mathematical Physics vol 41 no 1 pp 480ndash5042000

[7] N Touafek and E M Elsayed ldquoOn the solutions of systemsof rational difference equationsrdquo Mathematical and ComputerModelling vol 55 no 7-8 pp 1987ndash1997 2012

[8] E M Elsayed and T F Ibrahim ldquoPeriodicity and solutionsfor some systems of nonlinear rational difference equationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 6pp 1361ndash1390 2015

[9] A S Kurbanlı C Cinar and I Yalcinkaya ldquoOn the behavior ofpositive solutions of the system of rational difference equations119909119899+1 = 119909119899minus1(119910119899119909119899minus1 + 1) 119910119899+1 = 119910119899minus1(119909119899119910119899minus1 + 1)rdquoMathematical and Computer Modelling vol 53 no 5-6 pp1261ndash1267 2011

[10] N Joshi and P J Vassiliou ldquoThe existence of Lie symmetriesfor first-order analytic discrete dynamical systemsrdquo Journal ofMathematical Analysis and Applications vol 195 no 3 pp 872ndash887 1995

[11] I Yalcinkaya ldquoOn the global asymptotic stability of a secondorder system of difference equationrdquo Discrete Dynamics inNature and Society vol 2008 Article ID 860152 12 pages 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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AnalysisInternational Journal of

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Submit your manuscripts atwwwhindawicom

12 Journal of Mathematics

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+11198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1

(101)

(9) If

1198752 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 ) 1198762 = ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )

(102)

then

1198761 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )

(103)

(10) If

1198752 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )

(104)

then

1198761 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )

(105)

(11) If

1198752 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )1198762 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )

(106)

then

1198761 = ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 ) 1198751 = ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )

(107)

(12) If

1198752 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 ) 1198762 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 ) (108)

then

1198761 = ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 ) 1198751 = ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 ) (109)

Therefore we obtain twelve conservation laws for the system(84) They are given by

1206011 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198701

1206012 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119909119899+ 1198702

1206013 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119909119899+ 1198703

1206014 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119909119899+ 1198704

Journal of Mathematics 13

1206015 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198705

1206016 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119909119899+ 1198706

1206017 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198707

1206018 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119909119899+ 1198708

1206019 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899 + 1198709

12060110 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899 + 11987010

12060111 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899 + 11987011

12060112 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899 + 11987012

(110)

5 Conclusion and Discussions

Wehave presented amethod for obtaining nontrivial symme-tries and how to use them for solving a second-order SΔEsEach symmetry can be used to reduce the order Howeverdifferent symmetries lead to different reductions (see (60) and(64)) but the same solution We also proposed a technique toconstruct first integral associated to second-order systems ofdifference equations

14 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1993

[2] S Maeda ldquoThe similarity method for difference equationsrdquoIMA Journal of Applied Mathematics vol 38 no 2 pp 129ndash1341987

[3] D Levi L Vinet and P Winternitz ldquoLie group formalism fordifference equationsrdquo Journal of Physics A Mathematical andGeneral vol 30 no 2 pp 633ndash649 1997

[4] G R W Quispel and R Sahadevan ldquoLie symmetries and theintegration of difference equationsrdquo Physics Letters A vol 184no 1 pp 64ndash70 1993

[5] P E Hydon Difference Equations by Differential EquationMethods Cambridge University Press Cambridge 2014

[6] V Dorodnitsyn R Kozlov and P Winternitz ldquoLie groupclassification of second-order ordinary difference equationsrdquoJournal of Mathematical Physics vol 41 no 1 pp 480ndash5042000

[7] N Touafek and E M Elsayed ldquoOn the solutions of systemsof rational difference equationsrdquo Mathematical and ComputerModelling vol 55 no 7-8 pp 1987ndash1997 2012

[8] E M Elsayed and T F Ibrahim ldquoPeriodicity and solutionsfor some systems of nonlinear rational difference equationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 6pp 1361ndash1390 2015

[9] A S Kurbanlı C Cinar and I Yalcinkaya ldquoOn the behavior ofpositive solutions of the system of rational difference equations119909119899+1 = 119909119899minus1(119910119899119909119899minus1 + 1) 119910119899+1 = 119910119899minus1(119909119899119910119899minus1 + 1)rdquoMathematical and Computer Modelling vol 53 no 5-6 pp1261ndash1267 2011

[10] N Joshi and P J Vassiliou ldquoThe existence of Lie symmetriesfor first-order analytic discrete dynamical systemsrdquo Journal ofMathematical Analysis and Applications vol 195 no 3 pp 872ndash887 1995

[11] I Yalcinkaya ldquoOn the global asymptotic stability of a secondorder system of difference equationrdquo Discrete Dynamics inNature and Society vol 2008 Article ID 860152 12 pages 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Journal of Mathematics 13

1206015 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198705

1206016 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899+1119909119899+ 1198706

1206017 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+1119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899+1119909119899+ 1198707

1206018 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+1119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899+1119909119899+ 1198708

1206019 = ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119909119899 + 1198709

12060110 = ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119909119899 + 11987010

12060111 = [1 + (minus1)119899 minus 119894119899 minus (minus119894)119899]4 )119909119899+1+ ([1 + (minus1)119899 + 119894119899 + (minus119894)119899]4 )119910119899+1+ ([1 minus (minus1)119899 minus 119894119894119899 + 119894 (minus119894)119899]4 )119910119899+ ([1 minus (minus1)119899 + 119894119894119899 minus 119894 (minus119894)119899]4 )119909119899 + 11987011

12060112 = ([1 minus (minus1)119899 + 119894 (119894)119899 minus 119894 (minus119894)119899]4 )119909119899+1+ ([1 minus (minus1)119899 minus 119894 (119894)119899 + 119894 (minus119894)119899]4 )119910119899+1+ ([1 + (minus1)119899 minus (119894)119899 minus (minus119894)119899]4 )119910119899+ ([1 + (minus1)119899 + (119894)119899 + (minus119894)119899]4 )119909119899 + 11987012

(110)

5 Conclusion and Discussions

Wehave presented amethod for obtaining nontrivial symme-tries and how to use them for solving a second-order SΔEsEach symmetry can be used to reduce the order Howeverdifferent symmetries lead to different reductions (see (60) and(64)) but the same solution We also proposed a technique toconstruct first integral associated to second-order systems ofdifference equations

14 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1993

[2] S Maeda ldquoThe similarity method for difference equationsrdquoIMA Journal of Applied Mathematics vol 38 no 2 pp 129ndash1341987

[3] D Levi L Vinet and P Winternitz ldquoLie group formalism fordifference equationsrdquo Journal of Physics A Mathematical andGeneral vol 30 no 2 pp 633ndash649 1997

[4] G R W Quispel and R Sahadevan ldquoLie symmetries and theintegration of difference equationsrdquo Physics Letters A vol 184no 1 pp 64ndash70 1993

[5] P E Hydon Difference Equations by Differential EquationMethods Cambridge University Press Cambridge 2014

[6] V Dorodnitsyn R Kozlov and P Winternitz ldquoLie groupclassification of second-order ordinary difference equationsrdquoJournal of Mathematical Physics vol 41 no 1 pp 480ndash5042000

[7] N Touafek and E M Elsayed ldquoOn the solutions of systemsof rational difference equationsrdquo Mathematical and ComputerModelling vol 55 no 7-8 pp 1987ndash1997 2012

[8] E M Elsayed and T F Ibrahim ldquoPeriodicity and solutionsfor some systems of nonlinear rational difference equationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 6pp 1361ndash1390 2015

[9] A S Kurbanlı C Cinar and I Yalcinkaya ldquoOn the behavior ofpositive solutions of the system of rational difference equations119909119899+1 = 119909119899minus1(119910119899119909119899minus1 + 1) 119910119899+1 = 119910119899minus1(119909119899119910119899minus1 + 1)rdquoMathematical and Computer Modelling vol 53 no 5-6 pp1261ndash1267 2011

[10] N Joshi and P J Vassiliou ldquoThe existence of Lie symmetriesfor first-order analytic discrete dynamical systemsrdquo Journal ofMathematical Analysis and Applications vol 195 no 3 pp 872ndash887 1995

[11] I Yalcinkaya ldquoOn the global asymptotic stability of a secondorder system of difference equationrdquo Discrete Dynamics inNature and Society vol 2008 Article ID 860152 12 pages 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

14 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] P J Olver Applications of Lie Groups to Differential EquationsSpringer New York NY USA 1993

[2] S Maeda ldquoThe similarity method for difference equationsrdquoIMA Journal of Applied Mathematics vol 38 no 2 pp 129ndash1341987

[3] D Levi L Vinet and P Winternitz ldquoLie group formalism fordifference equationsrdquo Journal of Physics A Mathematical andGeneral vol 30 no 2 pp 633ndash649 1997

[4] G R W Quispel and R Sahadevan ldquoLie symmetries and theintegration of difference equationsrdquo Physics Letters A vol 184no 1 pp 64ndash70 1993

[5] P E Hydon Difference Equations by Differential EquationMethods Cambridge University Press Cambridge 2014

[6] V Dorodnitsyn R Kozlov and P Winternitz ldquoLie groupclassification of second-order ordinary difference equationsrdquoJournal of Mathematical Physics vol 41 no 1 pp 480ndash5042000

[7] N Touafek and E M Elsayed ldquoOn the solutions of systemsof rational difference equationsrdquo Mathematical and ComputerModelling vol 55 no 7-8 pp 1987ndash1997 2012

[8] E M Elsayed and T F Ibrahim ldquoPeriodicity and solutionsfor some systems of nonlinear rational difference equationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 6pp 1361ndash1390 2015

[9] A S Kurbanlı C Cinar and I Yalcinkaya ldquoOn the behavior ofpositive solutions of the system of rational difference equations119909119899+1 = 119909119899minus1(119910119899119909119899minus1 + 1) 119910119899+1 = 119910119899minus1(119909119899119910119899minus1 + 1)rdquoMathematical and Computer Modelling vol 53 no 5-6 pp1261ndash1267 2011

[10] N Joshi and P J Vassiliou ldquoThe existence of Lie symmetriesfor first-order analytic discrete dynamical systemsrdquo Journal ofMathematical Analysis and Applications vol 195 no 3 pp 872ndash887 1995

[11] I Yalcinkaya ldquoOn the global asymptotic stability of a secondorder system of difference equationrdquo Discrete Dynamics inNature and Society vol 2008 Article ID 860152 12 pages 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom