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

Abstract

A direct method to construct polynomial integrals for third order ordinary difference equation (O∆E) w(n + 3) = F (w(n), w(n + 1), w(n + 2)) and fourth order O∆E w(n + 4) = F (w(n), w(n + 1), w(n + 2), w(n + 3)) is presented. The effectiveness of the method to construct more than one polynomial integral for N-th order O∆E is also briefly discussed.

Third order difference equations with two rational integrals

“…(2.1) A non-trivial function I (w n , w n+1 , w n+2 ) is said to be an integral for (2.1) if I (w n , w n+1 , w n+2 ) = I (w n+1 , w n+2 , w n+3 ) holds. In [23], the authors have proposed a method to construct a polynomial integral for (2.1) having the form…”

“…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.…”

On the conservation laws for nonlinear partial difference equations

“…Recently, we have proposed a direct method to construct rational and polynomial integrals for an Nth order autonomous nonlinear O E and illustrated the effectiveness of the method by deriving many new autonomous third and fourth order O E with sufficient number of integrals [23][24][25]27]. Considering nonlinear partial difference equations, the available techniques in analyzing their integrability is scarce [8-14, 19-22, 26, 29].…”

Integrable fourth-order difference equations

“…This path-breaking result has led to the construction of the well-known discrete Painlevé equations. Systematic efforts to analyze a third-order autonomous ordinary nonlinear difference equation particularly from the point of view of integrability have been made by several researchers in recent years [3,10,12,15,19,23,25,26]. For instance, Iatrou [12] has identified few third-order integrable difference equations which admit two independent polynomial integrals.…”

“…The second integral of the identified mappings has been derived from the reductions of the discrete-discrete modified Korteweg de Vries and sine-Gordon lattice equations. The purpose of this paper is to explain how to construct two or more integrals directly for fourth-order difference equations leading to integrable four-dimensional analogs of the two-dimensional QRT mappings [5,6,13,24,25].…”