Direct method to construct integrals for N th-order autonomous ordinary difference equations

Abstract

A direct method to construct integrals for an Nth-order autonomous ordinary difference equation (ODE ): w nCN ZF(w n , ., w nCNK1 ) is presented. As an illustration we first consider third-order autonomous ODE w nC3 ZF(w n , w nC1 , w nC2 ) 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 ODEs is also demonstrated. The question of integrability of each of the identified difference equations with more than one integral is also discussed.

Third order difference equations with two rational integrals

“…we have identified several third order O E with two rational integrals [22] (see also [8,10,21]). In this paper, we consider a more general rational integral given in (2.2).…”

“…It is known that equation (2.10) arises as a reduction of sine-Gordon lattice equation. It is appropriate to mention that the O E having the form (2.10) have been analyzed by several authors from different points of view including its complete integrability [8,10,13,17,21,22].…”

“…Considerable progress have been made for second order ordinary difference equations (O E) or mappings toward finding its solution and analyzing its integrability [4,12,18,19]. Systematic efforts to analyze third order O E particularly from the point of view of integrability have been made by several researchers in recent years [1,8,10,13,17,[20][21][22][23][24]. But a more general form of third order O E of the Quispel, Roberts and Thompson (QRT) form similar to the one in the second order case is elusive.…”

Super integrable four-dimensional autonomous mappings

“…On the basis of the factorization property, these mapping have two or more integrals which can be evaluated explicitly (and possibly in special cases (N − 1) integrals to find new super integrable cases). The results of the above will be reported elsewhere [25]. On the other hand the factorization property is a natural expression of duality [9] leading to integrable mappings of type G, cf equation (1.3) which for even N include all periodic reductions of the integrable sine-Gordon and modified Korteweg-de Vries equations on the two-dimensional lattice [4].…”

Polynomial integrals for third- and fourth-order ordinary difference equations

“…Given an autonomous N-th order nonlinear O∆E there exists no systematic analytic technique to derive its integrals enabling one to investigate its integrability. Recently a direct method was proposed Copyright c 2008 by R Sahadevan and C Uma Maheswari for N -th order autonomous difference equation to construct rational integrals and several new integrable difference equations of higher order were identified [23]. In this article we present a method to construct polynomial integrals through third and fourth order O∆E [ see 14,15 for a different method].…”