A systematic method for constructing discrete Painlevé equations in the degeneration cascade of the E8 group

Abstract

Abstract. We present a systematic and quite elementary method for constructing discrete Painlevé equations in the degeneration cascade for E (1) 8 . Starting from the invariant for the autonomous limit of the E (1) 8 equation one wishes to study, the method relies on choosing simple homographies that will cast this invariant into certain judiciously chosen canonical forms. These new invariants lead to mappings the deautonomisations of which allow us to build up the entire degeneration cascade of the original mapping. We explain the method on three examples, two symmetric mappings and an asymmetric one, and we discuss the link between our results and the known geometric structure of these mappings.

Detecting discrete integrability: the singularity approach

“…This turned out to be indeed the case, allowing us to complement the list of the E (1) 8 -related discrete Painlevé equations [27]. Also, when deautonomising QRT mappings, the cyclic patterns of the original autonomous mapping yield important information on the geometric structure of the discrete Painlevé equations one obtains (as explained in [45]). The algebro-geometric underpinnings of the deautonomisation of QRT mappings have been developed in [5].…”

“…(The fact that the cyclic pattern remains cyclic after deautonomisation is not a general feature: in many cases a cyclic pattern becomes a genuinely confined one when deautonomised, see e.g. [45]).…”

Discrete Painlevé equations from singularity patterns: The asymmetric trihomographic case

“…It turns out that this is true (taking into account that 5.2.9 is, in fact, identical to 5.2.10). What is also interesting is that for the patterns which lead to equations corresponding to an artificial asymmetrisation of cases already identified in [20], namely (6,2,6,2), (4,4,4,4), (7,1,7,1) and (5,3,5,3), an elimination of either of the two variables leads to an equation corresponding to a double-step evolution as shown in [23], [24] and [25]. This paper was devoted to the study of additive equation exclusively.…”

“…Eliminating either of the two variables leads to an equation already derived in [21], equation 5.2.7. As shown in [25] this equation is a double-step evolution obtained from case IV.…”

Constructing discrete Painlevé equations: from E8(1) to A1(1) and back

“…Another interesting construction of discrete Painlevé equations, which we introduced in [11] and [12] is one based on multistep evolutions. Depending on the equation at hand it may be possible, for example, to skip one out of two indices and obtain a mapping relating variables at a distance of two.…”