## Abstract

A direct method to construct integrals for an *N*th-order autonomous ordinary difference equation (*O*Δ*E*): *w*_{n+N}=*F*(*w*_{n}, …, *w*_{n+N−1}) is presented. As an illustration we first consider third-order autonomous *O*Δ*E w*_{n+3}=*F*(*w*_{n}, *w*_{n+1}, *w*_{n+2}) and identify the forms of *F* for which two independent integrals exist. The effectiveness of the method to construct two or more integrals for fourth- and fifth-order *O*Δ*E*s is also demonstrated. The question of integrability of each of the identified difference equations with more than one integral is also discussed.

## 1. Introduction

In recent years the study of integrable discrete nonlinear systems governed by both ordinary difference equations (or mappings) and partial difference equations (or lattice equations) has undergone an impressive development from several points of view (Ablowitz & Ladik 1976*a*,*b*; Arnold 1978; Quispel *et al*. 1988, 1989, 1991; Bruschi *et al*. 1991; Grammaticos *et al*. 1991, 2004; Veselov 1991). There are several reasons to study discrete systems more particularly integrable nonlinear difference equations. For example, the discrete systems governed by difference equations are more fundamental than the continuous ones described by differential equations. Also the study of discrete systems may yield new insight into the analytic theory of difference equations (Ruijsenaars 1997), the development of which has greatly lagged behind its continuous counterpart—the theory of differential equations. The investigation of integrable discrete nonlinear systems may open a new avenue on what ultimately becomes a general theory of discrete and nonlinear difference equations. In recent years, much effort has been spent by several research groups to develop analytical techniques to determine whether the given autonomous nonlinear *O*Δ*E* is integrable or not (Quispel *et al*. 1988, 1989, 1991; Bruschi *et al*. 1991; Grammaticos *et al*. 1991, 2004; Veselov 1991). In fact a rich class of interesting examples of integrable nonlinear *O*Δ*E* of order 2 which are amenable to exact and rigorous approaches for their solution has been identified in recent years (Quispel *et al*. 1988; Capel *et al*. 1991; Grammaticos *et al*. 1991, 2004; Hydon 2001). It is appropriate to mention here that there exists no unique definition of integrability of a given nonlinear *O*Δ*E* like there is for nonlinear differential equations. However, there exists considerable number of analytical methods formulated by different groups (Quispel *et al*. 1988, 1989; Grammaticos *et al*. 1991; Veselov 1991; Hietarinta & Viallet 1998; Bellon & Viallet 1999; Tremblay *et al*. 2001; Lafortune & Goriely 2004; Halburd 2005) to determine the integrability nature of *O*Δ*E* but we will not get into the technicalities of them here. In this paper we restrict our attention to the definitions of integrability related with the existence of a sufficient number of integrals of an *O*Δ*E*. We use the same terminology for integral as for ordinary differential equations (ODE): an integral (also referred to as conserved quantities) is a function that is not identically constant but is constant on all solutions of the *O*Δ*E*.

An autonomous *N*th-order nonlinear *O*Δ*E* is said to be integrable if it admits (*N*−1) functionally independent integrals. Note that if a difference equation is measure preserving in dimension *N* and has (at least) (*N*−2) independent integrals then it has a (degenerate) Poisson structure, so defines a symplectic map on each two-dimensional level set of these integrals (Byrnes *et al*. 1999). A set of *N* integrals of an *N*th-order *O*Δ*E* is said to be functionally independent if the Jacobian of the integrals does not vanish. A mapping is said to be a measure preserving with density *m*(*w*_{1}, …, *w*_{n}) if the Jacobian determinant can be written as .

Given an autonomous *N*th-order nonlinear ODE, there exists no systematic analytic technique to derive its integrals enabling one to investigate its integrability. Of course for Hamiltonian systems with finite degrees of freedom, Darboux and others have developed a direct method to construct its integrals which are polynomial in momenta (Whittaker 1937; Chandrasekhar 1942). The primary aim of this article is to present a direct and algorithmic method and show how it provides an effective tool to derive two or more integrals for an autonomous *N*th-order *O*Δ*E* (see also Hirota *et al*. 2001).

The plan of the article is as follows. In §2 we present brief details of a direct and algorithmic method to derive integrals of rational form for an autonomous *N*th-order nonlinear *O*Δ*E*. In §3 we consider third-order autonomous *O*Δ*E w*_{n+3}=*F*(*w*_{n}, *w*_{n+1}, *w*_{n+2}) and identify the forms of F for which two independent integrals exist. In §4 we illustrate the usefulness of the method to construct two or more integrals for fourth- and fifth-order nonlinear *O*Δ*E*. In §5 we give a brief summary of our investigations.

## 2. Construction of integrals: autonomous *N*th-order nonlinear *O*Δ*E*

Consider an autonomous *N*th-order *O*Δ*E* given by(2.1)where *F* is an arbitrary function. A non-trivial function *I*(*n*) given byis said to be an integral for (2.1) if that is

Given an autonomous nonlinear *O*Δ*E*, it is not clear what form of integral *I*(*n*) one has to choose. Recent investigations show that the integrals (or conserved quantities) of autonomous nonlinear difference equations or mappings can be expressed as a ratio of two polynomials in dependent variables (Quispel *et al*. 1988; Capel & Sahadevan 2001) which has motivated us to consider an integral *I*(*n*) of equation (2.1) having the form(2.2)where(2.3)We below explain the details of constructing integrals having rational form for (2.1). The construction of polynomial integrals for (2.1) will be presented elsewhere (Uma Maheswari & Sahadevan in preparation).

For clarity of presentation we consider(2.4)We wish to note that one can construct rational integrals for *a*_{1j}(*n*)≠0, *a*_{2j}(*n*)≠0 and *a*_{3j}(*n*)≠0, *j*=1, 2, 3, but the calculations will be lengthy and unmanageable for arbitrary *N*. Let us assume that (2.1) admits an integral *I*(*n*), (2.2) with the restrictions given in (2.4). It is straightforward to check that the integrability condition *I*(*n*+1)−*I*(*n*)=0 leads to a quadratic equation in *w*_{n+N}(2.5)Equation (2.5) can be factorized into(2.6)provided the following conditions hold:(2.7)

(2.8)From (2.6) and using (2.8) we have(2.9a)

(2.9b)In a similar manner we obtain(2.10a)

(2.10b)We consider below (2.9*a*), (2.9*b*), (2.10*a*) and (2.10*b*) for further discussion separately.

Now (2.9*a*) and (2.9*b*) can be written as(2.11)where(2.12a)(2.12b)Here *γ* and *δ* are integer exponents. Then from (2.9*a*) we have(2.13a)

(2.13b)(2.13c)(2.13d)

(2.13e)(2.13f)Also from (2.9*b*) we obtain(2.14a)(2.14b)(2.14c)(2.14d)(2.14e)(2.14f)Here(2.15)From (2.12*a*) we observe that(2.16)and so(2.17a)

(2.17b)

(2.17c)

(2.17d)satisfying (2.12*a*). In a similar manner we obtain(2.18a)

(2.18b)

(2.18c)

(2.18d)satisfying (2.12*b*). Here , , *N*≥4. Note that for *N*=3, the arbitrary functions , and reduce into constants.

Next, from equations (2.13*a*)–(2.13*f*) and (2.14*a*)–(2.14*f*) we have(2.19a)

(2.19b)

(2.19c)

(2.19d)Taking for simplicity it is clear that if we have solutions of equations (2.17)–(2.19) satisfying the condition (2.7) as well, then the difference equation (2.11) has more than one integral corresponding to the arbitrary constants appearing in *h*_{i}(*n*) in the expressions for *I*(*n*) given in (2.2). For example, ifand sowe find that for *h*_{3}(*n*)=*h*_{1}(*n*) (2.19*a*) is satisfied identically for the following forms of *f*_{i}(*n*):(2.20)

(2.21)

(2.22)

(2.23)Substituting the above forms of *f*_{i}(*n*) along with *h*_{i}(*n*) in *A*_{12}(*n*) and *A*_{21}(*n*) in (2.7), we obtain(2.24)Equation (2.24) suggests an ansatz and . As a consequence (2.24) becomes(2.25)which can be solved and the explicit forms of *f*_{2}(*n*), *f*_{4}(*n*), *f*_{7}(*n*), *f*_{8}(*n*) and *Ã*_{22}(*n*) are given in appendix A. Thus we conclude that the *N*th-order *O*Δ*E*(2.26)admits an integral(2.27)where *f*_{1}(*n*), *f*_{3}(*n*), *f*_{5}(*n*), *f*_{6}(*n*) are as given in (2.20)–(2.23) and is an arbitrary function satisfying (2.25). We would like to mention that the second integral *I*_{2}(*n*) for (2.26) can be constructed through the change of variables by in *I*_{1}(*n*), that is .

If *I*_{2}(*n*) is required to have the same form as *I*_{1}(*n*) with variables replaced by their reciprocals, then the number of arbitrary constants in (2.26) will be reduced. Proceeding further with the restrictions on , one can find more distinct integrals for the same difference equation (2.26) which has been illustrated explicitly for *N*=3, 4 and 5.

The difference equation (2.10*a*) and (2.10*b*) can also be written as(2.28)and the expressions for are given in (2.12*a*) and (2.12*b*). From (2.10*a*) we have(2.29a)(2.29b)(2.29c)

(2.29d)(2.29e)

(2.29f)Also from (2.10*b*) we obtain(2.30a)(2.30b)

(2.30c)

(2.30d)

(2.30e)

(2.30f)From equations (2.29*a*)–(2.29*f*) and (2.30*a*)–(2.30*f*) we have(2.31a)

(2.31b)

(2.31c)

(2.31d)

Following the procedure outlined in case 2.1, we can construct more than one integral to (2.28) which has been illustrated for *N*=3, 4 and 5.

## 3. Third-order difference equation with two integrals

Let us consider an autonomous third-order difference equation having the form(3.1)and assume that it admits an integral *I*(*n*) given in (2.2) with *N*=3, that is(3.2)Here after we denote *w*_{0}=*w*_{n}, *w*_{1}=*w*_{n+1}, …, *w*_{n+N−1}=*w*_{N−1} unless otherwise specified. It is easy to check that the equation resulting from the integrability condition *I*(*w*_{0}, *w*_{1}, *w*_{2})=*I*(*w*_{1}, *w*_{2}, *w*_{3}) can be factorized into(3.3)provided the following conditions hold:(3.4)(3.5)As shown in §2 we obtain two distinct third-order difference equations given by(3.6)

(3.7)(3.8)(3.9)For clarity of presentation we next consider the above difference equations given by (3.6), (3.7), (3.8) and (3.9), respectively, for further discussion separately.

Equation (3.6) and (3.7) can be written as(3.10)where(3.11a)(3.11b)and *γ*, *δ* are integer exponents while *a*_{1}, *a*_{2}, …, *a*_{8} are arbitrary constants. Then from (3.6) we find(3.12a)(3.12b)(3.12c)(3.12d)(3.12e)(3.12f)Similarly from (3.7) we find(3.13a)(3.13b)(3.13c)(3.13d)(3.13e)

(3.13f)Solving (3.11*a*) and (3.11*b*) we obtain(3.14a)(3.14b)(3.14c)(3.14d)(3.15a)(3.15b)(3.15c)(3.15d)where , , , , , , and are arbitrary constants.

Then from equations (3.12*a*)–(3.12*f*) and (3.13*a*)–(3.13*f*) we get(3.16)(3.17)(3.18)(3.19)Since the above four equations (3.16–3.19) involve more than four arbitrary constants and , one can derive more than one solution corresponding to the constants and which ultimately leads to the construction of more than one integral *I*(*n*) for a single difference equation.

Solving (3.4) we find(3.20a)(3.20b)(3.20c)(3.20d)(3.20e)where *α*_{i}, *i*=1, …, 7 are arbitrary constants. From equations (3.20*a*), (3.20*b*), (3.12*c*) and (3.12*d*) we have

which suggests that the integer exponents (*γ*, *δ*)∈[−2,2]. Similar conclusions can be arrived at by comparing equations (3.20*c*) and (3.20*d*) with (3.13*e*) and (3.13*f*). However, a detailed calculation shows that non-trivial third-order difference equation of the form (3.10) admitting two independent integrals is possible only if(3.21)This fact was verified by considering each of the ordered pair in [−2,2]. Details of each case in (3.21) are given below briefly.

(*γ*, *δ*)=(0, 0)

In order to find *I*_{1}(*n*), we consider and the remaining constants are zero, that is(3.22)and so(3.23)Then from (3.16) we find that *a*_{5}=*a*_{1}, *a*_{6}=*a*_{3}. Proceeding further with the above parametric restrictions we find that (3.4) is satisfied if(3.24)

Thus we obtain a non-trivial third-order difference equation(3.25)possessing an integral(3.26)

The second integral *I*_{2}(*n*) for the above difference equation (3.25) can be constructed by choosing in (3.12)–(3.19) and the explicit form of *I*_{2}(*n*)(3.27)Note that if

(*γ*, *δ*)=(−1, −1)

Proceeding as before we obtain another third-order difference equation(3.28a)admitting two independent integrals *I*_{1}(*n*) and *I*_{2}(*n*)(3.28b)and(3.28c)It is appropriate to mention here that the above difference equation (3.28*a*) has also been derived by Quispel *et al*. (2005) but with *a*_{3}=*a*_{1}

(*γ*, *δ*)=(1, 1)

This case also leads to a non-trivial third-order equation(3.29a)admitting two integrals(3.29b)

and(3.29c)

Next we consider (3.8) and (3.9) which can also be written as(3.30)and the expressions for are given in (3.11*a*) and (3.11*b*). Then from (3.9) we find(3.31a)(3.31b)(3.31c)(3.31d)(3.31e)(3.31f)Similarly from equation (3.9) we find(3.32a)(3.32b)(3.32c)(3.32d)(3.32e)(3.32f)Then from equations (3.31*a*)–(3.31*f*) and (3.32*a*)–(3.32*f*) we get(3.33)(3.34)(3.35)(3.36)Proceeding further along the lines described for case 3.1, we find that the integer exponents (*γ*, *δ*)∈[−2,2]. The list of integrable difference equations with two integrals identified by Hirota *et al*. (2001) belongs to the set {(0,0), (1,0), (0,1)}. However, we have identified a non-trivial third-order equation with two integrals for (*γ*, *δ*)=(−1, 1). Moreover, we have considered here both types of third-order difference equations and while Hirota *et al*. (2001) have considered third-order difference equations of the former form only. Some of the identified difference equations and their integrals are given below.

(*γ*, *δ*)=(0, 0)

In this case there exist two third-order difference equations admitting two independent integrals. They are(3.37a)(3.37b)(3.37c)

(3.38a)(3.38b)(3.38c)

(*γ*, *δ*)=(1, −1)

(3.39a)(3.39b)(3.39c)which was also identified by Quispel *et al*. (2005) using a different approach (see also Matsukidaira & Takahashi 2006; Roberts & Quispel 2006).

## 4. Higher-order difference equations with two or more integrals

We would like to mention here that the method described in §2 to construct integrals for *N*th-order *O*Δ*E* and illustrated for third-order *O*Δ*E* in §3 can also be applied to higher-order difference equations using Maple or Mathematica. The calculations to derive integrals for higher-order difference equations are tedious and the explicit expressions of *A*_{ij}(*n*) in *I*(*n*) are lengthy. To illustrate the effectiveness of the method, we provide below only the results for *N*=4 and 5 (the results are not exhaustive).

### (a) Fourth-order autonomous *O*Δ*E*

A detailed calculation shows that the fourth-order difference equation(4.1a)with

admits the following three independent integrals(4.1b)

(4.1c)

(4.1d)where . We would like to mention that the fourth-order equation (4.1*a*) is a generalization of (3.25).

In this case we obtain another fourth-order difference equation(4.2a)withadmitting two independent integrals, *I*_{1}(*n*) and *I*_{2}(*n*) having the form(4.2b)Explicit expression of *A*_{ij}(*w*_{n+1},*w*_{n+2}) for the integral *I*_{1}(*n*) is as follows:

The expression of *A*_{ij}(*w*_{n+1},*w*_{n+2}) for the integral *I*_{2}(*n*) is given by

whereIn both the integrals *I*_{1}(*n*) and *I*_{2}(*n*) the following relation holds:

### (b) Fifth-order autonomous *OΔ**E*

Here also we obtain a non-trivial fifth-order difference equation(4.3a)with

admitting four independent integrals *I*_{1}(*n*), *I*_{2}(*n*), *I*_{3}(*n*) and *I*_{4}(*n*)(4.3b)

(4.3c)whereThe other two integrals *I*_{3}(*n*) and *I*_{4}(*n*) are given by(4.3d)The expression of *A*_{ij}(*w*_{n+1}, *w*_{n+2}, *w*_{n+3}) for *I*_{3}(*n*) and *I*_{4}(*n*) is given in appendix B. Note that the fifth-order difference equation (4.3*a*) is a generalization of the fourth-order difference equation (4.1*a*).

In this case we obtain another non-trivial fifth-order difference equation(4.4)withadmitting only two independent integrals *I*_{1}(*n*) and *I*_{2}(*n*) given by the form in (4.3*d*). The expressions of *A*_{ij}(*w*_{n+1,} *w*_{n+2}, *w*_{n+3}) for *I*_{1}(*n*) and *I*_{2}(*n*) are given in appendix C.

## 5. Summary and discussion

In this article we have proposed a direct and algorithmic method to derive more than one integral of motion of rational form for an autonomous *N*th-order *O*Δ*E*. The investigation presented in this article is only an indicator but not an exhaustive one. In order to illustrate the effectiveness of the devised method, we first consider third-order autonomous *O*Δ*E*, in §3, and identify the equations (3.25), (3.28*a*), (3.29*a*), (3.37*a*), (3.38*a*) and (3.39*a*) possessing two independent integrals and hence they are integrable. In §4 we have considered an autonomous fourth-order *O*Δ*E* and identified two difference equations in which one of them, equation (4.1*a*), admits three independent integrals and so integrable and other, equation (4.2*a*), with two integrals. Equation (4.2*a*) is a measure preserving one with measure 1/(*w*_{0}*w*_{1}*w*_{2}*w*_{3}) and so defines a symplectic map on each two-dimensional level set of these integrals (Byrnes *et al*. 1999; more super integrable as well as mappings with two integrals have been reported in Capel *et al*. 2007). Also, we have identified two fifth-order *O*Δ*E* difference equations out of which one, equation (4.3*a*), possesses four independent integrals and hence is integrable and the other, equation (4.4), admits two independent integrals and is also measure preserving with measure 1/(*w*_{0}*w*_{1}*w*_{2}*w*_{3}*w*_{4}) and so is non-integrable in the sense of the working definition. Since the calculations of finding integrals of order greater than or equal to 6 are cumbersome, involving very lengthy expression, we have not presented the details of it here. However, the calculations reveal that one can identify more than one integrable difference equations of higher order greater than or equal to 6 using mathematical software, such as Maple or Mathematica. We would like to mention that Hirota *et al*. (2001) have also devised a procedure to construct conserved densities for discrete equations in which they made some assumptions, such as symmetry properties of conserved quantities.

We would like to mention that the autonomous higher-order *O*Δ*E*s arise in different subjects and various contexts. For example, the stationary version of coupled discrete nonlinear Schrödinger equations leads to higher-order *O*Δ*E*s (Bruschi *et al*. 1991). Also, the periodic reductions of integrable lattice Korteweg–de Vries equation (KdV), modified KdV and Sine Gordon equations lead to higher-order *O*Δ*E* (Capel & Sahadevan 2001). Thus, the proposed method will be very useful to construct integrals enabling the analyses of the nature of these equations. Having isolated autonomous integrable *O*Δ*E*, it would be very useful to find the corresponding non-autonomous version, at least for third-order difference equations and this is under investigation.

## Acknowledgements

The authors wish to thank the anonymous referees for their helpful and critical comments. The work of R.S. forms part of the research project funded by CSIR, New Delhi. The work of C.U. is supported by UGC-JRF, New Delhi.

## Footnotes

- Received April 2, 2007.
- Accepted October 30, 2007.

- © 2007 The Royal Society