Third-order integrable difference equations generated by a pair of second-order equations

Abstract

We show that the third-order difference equations proposed by Hirota, Kimura and Yahagi are generated by a pair of second-order difference equations. In some cases, the pair of the second-order equations are equivalent to the Quispel-Robert-Thomson(QRT) system, but in the other cases, they are irrelevant to the QRT system. We also discuss an ultradiscretization of the equations.

Some properties of thek-dimensional Lyness' map

“…The second and more specially the third order Lyness' difference equations y n+2 = a + y n+1 y n and y n+3 = a + y n+1 + y n+2 y n , with a ≥ 0, have been considered as emblematic examples of integrable discrete systems, see for instance [15,20,22,23]. The dynamics of the above equations, or their associated maps, has been the objective of recent intensive investigation.…”

“…In this result, the key point is the existence of a new (as far as we know) first integral for F 2 = F • F for any odd k ≥ 3. As we will see, our proof of the existence of this function is inspired in the paper [22], where this first integral is given for k = 3.…”

Geometric and integrability properties of Kahan’s method: the preservation of certain quadratic integrals

“…(ii) Note that the density (61) does not depend on the timestep h, and therefore is also preserved by the ODE (51). reader is referred to [7][8][9][10][11][12]. The maps in [7] are closest to the maps in the current paper.…”

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

“…ð3:39cÞ which was also identified by Quispel et al (2005) using a different approach (see also Matsukidaira & Takahashi 2006;Roberts & Quispel 2006).…”