Third order difference equations with two rational integrals

Abstract

A systematic investigation to derive three-dimensional analogs of two-dimensional Quispel, Roberts and Thompson (QRT) mappings is presented. The question of integrability of the obtained three-dimensional mappings with two independent integrals is also analyzed. It is also shown that there exist three-dimensional QRT maps with three n-dependent integrals.

New Integrable and Linearizable Nonlinear Difference Equations

“…We have verified that equation (3.10) is measure preserving with measure 1 R 2 (n) [31]. Similarly higher order OΔE's can be obtained by taking different values for z 1 and z 2 in equation (3.7) which may admit more than one integral.…”

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.…”

“…Similarly, Matsukidaira and Takahashi [15] have shown how third-order integrable difference equations can be generated by a pair of second-order equations. In [26] we have reported several integrable three-dimensional analogs of QRT mappings. We mention here that one of the authors of this paper has considered in [6] a rational integral and identified six fourth-order autonomous mappings with two integrals, which can also be derived from the similarity reductions of discrete-discrete modified Korteweg de Vries and sine Gordon lattice equations.…”