Symmetries , Associated First Integrals , and Double Reduction of Difference Equations

and Applied Analysis 3 3. Application The aim of this section is to consider two examples and find their symmetries, first integrals, and general solution. We also briefly discuss what is meant by double reduction and association. 3.1. Example 1. Consider the second-order OΔE [11]: ω = u n+2 = n n + 1 u n + 1 u n+1 . (25) 3.1.1. Symmetry Generator. Suppose that we seek characteristics of the formQ = Q(n, u n ). To do this, we use the symmetry condition and solve for Q = Q(n, u n ). Here, the symmetry condition, given by (7), becomes Q (n + 2, ω) + Q (n + 1, u n+1 ) 1 u 2 n+1 − Q (n, u n ) ( n


Introduction
The theory, reasoning, and algebraic structures dealing with the construction of symmetries for differential equations (DEs) are now well established and documented.Moreover, the application of these in the analysis of DEs, in particular, for finding exact solutions, is widely used in a variety of areas from relativity to fluid mechanics (see [1][2][3][4]).Secondly, the relationship between symmetries and conservation laws has been a subject of interest since Noether's celebrated work [5] for variational DEs.The extension of this relationship to DEs which may not be variational has been done more recently [6,7].The first consequence of this interplay has led to the double reduction of DEs [8][9][10].
A vast amount of work has been done to extend the ideas and applications of symmetries to difference equations (ΔEs) in a number of ways-see [11][12][13][14][15] and references therein.In some cases, the ΔEs are constructed from the DEs in such a way that the algebra of Lie symmetries remains the same [16].As far as conservation laws of ΔEs go, the work is more recent-see [12,17].Here, we construct symmetries and conservation laws for some ordinary ΔEs, utilise the symmetries to obtain reductions of the equations, and show, in fact, that the notion of "association" between these concepts can be analogously extended to ordinary ΔEs.That is, an association between a symmetry and first integral exists if and only if the first integral is invariant under the symmetry.Thus, a "double reduction" of the ΔE is possible.

Preliminaries and Definitions
Consider the following th-order OΔE: where  is a smooth function such that (/  ) ̸ = 0 and integer  is an independent variable.The general solution of (1) can be written in the form and depends on  arbitrary independent constants   .
Definition 1.We define S to be the shift operator acting on  as follows: That is, if   = (,  1 , . . .,   ) then In the same way, where S is the shift operator defined in (3).

First Integral.
In [11], Hydon presents a methodology to construct the first integrals of OΔEs directly.For this method, the symmetries of the OΔE need not be known.Here, we will only consider second-order OΔE's.We construct first integrals using ( 8) and an additional condition; that is, Now let Next we differentiate ( 9) with respect to   ; we obtain Differentiating (9) with respect to  +1 we get Thus,  2 satisfies the second-order linear functional equation or first integral condition, After solving for  2 and constructing  1 , we need to check that the integrability condition is satisfied.Hence if (14) holds, the first integral takes the form To solve for (), we substitute (15) into ( 9) and solve for the resulting first-order OΔE.

Using Symmetries to Obtain the General Solution of an OΔE.
We begin this section by providing some useful definitions.We consider the theory and example provided by Hydon in [11].
Definition 4. The commutator of two symmetry generators   and   is denoted by [  ,   ] and defined by Definition 5. Given a symmetry generator for a second-order OΔE, there exists an invariant, satisfying To determine the invariant, we use the method of characteristics.Note that the invariant satisfies We make the assumption that (18) can be inverted to obtain for some function .Solving (21) requires finding a canonical coordinate which satisfies   = 1.The most obvious choice [11] of canonical coordinate is with a general solution of the form where  0 is any integer.

Application
The aim of this section is to consider two examples and find their symmetries, first integrals, and general solution.We also briefly discuss what is meant by double reduction and association.

Example 1.
Consider the second-order OΔE [11]: 3.1.1.Symmetry Generator.Suppose that we seek characteristics of the form  = (,   ).To do this, we use the symmetry condition and solve for  = (,   ).Here, the symmetry condition, given by ( 7), becomes Firstly, we differentiate (26) with respect to   (keeping  fixed) and we consider  +1 to be a function of ,   , and .By the implicit function theorem differentiating  +1 with respect to   yields Secondly, we apply the differential operator, given by to (26) to get To solve (29), we differentiate it with respect to   keeping  +1 fixed.As a result we obtain the ODE: whose solution is given by We suppose that () = 0 for ease of computation.Next we substitute (31) into (29) and we simplify the resulting equation to obtain Thus, where  is a constant.Substituting (33) into (31) leads to Therefore, the symmetry generator is given by 3.1.2.First Integral.Suppose that  2 =  2 (,   ); then (13) can be rewritten to give We apply the differential operator , given by (28), to (36) to get Next we differentiate (37) with respect to   keeping  +1 constant to obtain (/  )(  2 (,   )) = 0 whose solution is given by if we take  = 0. We substitute (38) into (37) to obtain the difference equation We choose (1) = 1 to get The next step consists of substituting (40) into (38) to get From (11) we get Since the integrability condition holds, we can calculate the first integral .From (41) and (42) we have To find () we substitute (43) into (9).We obtain whose solution is given by Finally we substitute (45) into (43) to obtain the first integral Note.The symmetry generator given by (35) acts on the first integral, , to produce the following equation: We say  and  are associated and this property has far reaching consequences on "further" reduction of the equation.

Symmetry Reduction.
Recall that, in Section 3.1.1,we calculated the symmetry generator, , to be given by (35).Suppose V  = V(,   ,  +1 ) is an invariant of . Then We can use the characteristics to solve for V  and construct the equation.The independent and dependent variables are given by respectively.Therefore by (51), the dependent variable, V  , is given by Applying the shift operator on V  and solving the resulting equation we get where  is a constant.Then by ( 52) and (53) and solving for  +1 we obtain Note.Equation (25) has been reduced by one order into (54).Solving (54) for  gives The first integral , given by (46), and the reduction are the same.This is another indication of a relationship between  and .In fact, this is the association; that is,  is invariant under .