Alfred Ramani

Discretising the Painlevé equations à la Hirota-Mickens

We present a systematic method for discretising the Painleve equations inspired by the method of Hirota (while extending it) and by that of Mickens (by specifying it to the case at hand). We derive various discrete analogues of Painleve I and II. We obtain forms that have been previously derived as well as new ones, in particular, equations with a geometry described by the affine Weyl group E8(1). As a by-product we obtain also linearisable equations, some of which are new.

Miura transformations for discrete Painlevé equations coming from the affine E8 Weyl group

We derive integrable equations starting from autonomous mappings with a general form inspired by the additive systems associated to the affine Weyl group E8(1). By deautonomisation we obtain two hitherto unknown systems, one of which turns out to be a linearisable one, and we show that both these systems arise from the deautonomisation of a non-QRT (Quispel-Roberts-Thompson) mapping. In order to unambiguously prove the integrability of these nonautonomous systems, we introduce a series of Miura transformations which allows us to prove that one of these systems is indeed a discrete Painleve equation, related to the affine Weyl group E7(1), and to cast it in canonical form. A similar sequence of Miura transformations allows us to effectively linearise the second system we obtain. An interesting off-shoot of our calculations is that the series of Miura transformations, when applied at the autonomous limit, allows one to transform a non-QRT invariant into a QRT one.

Strongly asymmetric discrete Painlevé equations: The multiplicative case

We examine a class of multiplicative discrete Painleve equations which may possess a strongly asymmetric form. When the latter occurs, the equation is written as a system of two equations the right hand sides of which have different functional forms. The present investigation focuses upon two canonical families of the Quispel-Roberts-Thompson classification which contain equations associated with the affine Weyl groups D5(1) and E6(1) (or groups appearing lower in the degeneration cascade of these two). Many new discrete Painleve equations with strongly asymmetric forms are obtained.

On the limits of discrete Painlevé equations associated with the affine Weyl group E8

We study the discrete Painleve equations that can be obtained as limits from the equations associated with the affine Weyl group E8(1). We obtain equations associated with the groups E7(1) and E6(1) as well as linearisable systems. In the E7(1) and E6(1) cases, we obtain several new discrete Painleve equations along with equations which can be related to the ones already known. The same is true for linearisable systems. In the case of new linearisable mappings, we present their explicit linearisation.

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

We present a systematic and quite elementary method for constructing discrete Painleve equations in the degeneration cascade for E8(1). Starting from the invariant for the autonomous limit of the E8(1) 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.

Multiplicative equations related to the affine Weyl group E8

We derive integrable equations starting from autonomous mappings with a general form inspired by the multiplicative systems associated with the affine Weyl group E8(1). Five such systems are obtained, three of which turn out to be linearisable and the remaining two are integrable in terms of elliptic functions. In the case of the linearisable mappings, we derive non-autonomous forms which contain a free function of the independent variable and we present the linearisation in each case. The two remaining systems are deautonomised to new discrete Painleve equations. We show that these equations are in fact special forms of much richer systems associated with the affine Weyl groups E7(1) and E8(1), respectively.

Full-deautonomisation of a lattice equation

In this letter we report on the unexpected possibility of applying the full-deautonomisation approach we recently proposed for predicting the algebraic entropy of second-order birational mappings, to discrete lattice equations. Moreover, we show, on two examples, that the full-deautonomisation technique can in fact also be successfully applied to reductions of these lattice equations to mappings with orders higher than 2. In particular, we apply this technique to a recently discovered lattice equation that has confined singularities while being nonintegrable, and we show that our approach accurately predicts this nonintegrable character. Finally, we demonstrate how our method can even be used to predict the algebraic entropy for some nonconfining higher order mappings.

Linearisable mappings, revisited

We examine the growth properties of second-order mappings which are integrable by linearisation and which generically exhibit a linear growth of the homogeneous degree of initial conditions. We show that for Gambier-type mappings for which the growth proceeds generically with a step of 1 there exist cases where the degree increase by unity every two steps. We examine also mappings belonging to the family known as “of third kind” in relation to the approach of Diller and Favre concerning the regularisable or not character of mappings and show that the anticonfined singularities of these mappings exhibit a linear growth with step 1. (The term anticonfined is used for singularities where the singular values extend all the way to infinity on both sides with just a few regular values in the middle). Moreover we construct specific examples of Gambier-type mappings which have anticonfined singularities and where the degree of the singularity increases linearly but where the average slope can be adjusted so as to be arbitrarily small.