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A new family of discrete Painlevé equations and associated linearisable systems

“…In that sense (12) is dual to (13), (14) dual to (16), (17) dual to (20), (18) dual to (19) while (15) is a self-dual equation. Equation (12) appears to be new, at least to the authors' knowledge, while the case of (15) will be examined in the discussion. On the other hand (14), (17) and (18) were identified [2], in their autonomous forms, as being linearisable.…”

Integrability detectors

“…We obtain the constraints a n+1 − 2a n + a n−1 = 0. (58) Solving (58), we get a n = αn + β. Hence, we obtain the non-autonomous mapping…”

“…However, it was noticed that d-P's can involve more than one dependent variables. The systematic derivation of the asymmetric form of discrete equations was carried out by Kruskal et al [58]. Now, we consider asymmetric d-P II and take its continuous limit.…”

“…For the past few years, it has been productively used and has identified many integrable discrete systems [55]. In particular, using singularity confinement method, the discrete Painlevé equations [55,56] and asymmetric discrete Painlevé equations have been derived [55,[57][58][59][60]. Singularity confinement test is used to analyse the behaviour around movable singularity of the map.…”

Contiguity relations for discrete and ultradiscrete Painleve equations

“…System (4.5) is the dual equation of (4.3) or, equivalently, the contiguity relation of its solutions. Equation (4.5) was first obtained in [18] and is in fact a q-discrete form of P III . The geometry of this q Painlevé equation is, of course, described by the affine Weyl group (A 2 + A 1 ) (1) and thus in general this equation is nonself-dual.…”