Alfred Ramani

We study second-order, second-degree systems related to the Painlevé equations which possess one and two parameters. In every case we show that by introducing a quantity related to the canonical Hamiltonian variables it is possible to derive such a second-degree equation. We investigate also the contiguity relations of the solutions of these higher-degree equations. In most cases these relations have the form of correspondences, which would make them non-integrable in general. However, as we show, in our case these contiguity relations are indeed integrable mappings, with a single ambiguity in their evolution (due to the sign of a square root).

We present a cascade of quantum Painlevé equations consisting in successive contiguity relations, whereupon starting form a continuous equations we obtain a discrete one, and continuous limits of the latter. We start from the quantum Painlevé V and in the process derive the quantum form of continuous P_III which was missing in previous studies.

We show how, starting from the geometrical description of discrete Painlevé equations in terms of affine Weyl groups, one can generate new second-order systems. We use this approach to introduce a new definition of the discrete Painlevé equations which eschews the reference to continuous systems.

We present a detailed study of the properties of two q-discrete Painlevé IV equations: singularity structure, bilinear forms, auto-Bäcklund/Schlesinger transformations, rational solutions as well as special solutions obtained through second- or first-order linear equations.

We study the degree growth of the iterates of the initial conditions for a class of third-order integrable mappings which result from the coupling of a discrete Painlevé equation to an homographic mapping. We show that the degree grows like $n^3$. In the special cases where the mapping satisfies the singularity confinement requirement we find a slower, quadratic growth. Finally we present a method for the construction of integrable $N$th-order mappings with degree growth $n^N$.

We investigate the possible integrable nonautonomous forms of a given class of mappings involving more than one dependent variable. These integrable discrete systems define "asymmetric" Painlevé equations. Our main tool of investigation is the application of the singularity confinement discrete integrability criterion. A new way of implementing it, first proposed for the singularity analysis of continuous systems, is also introduced.

We review our findings on discrete Painleve equations with emphasis on the two direct methods we have proposed for their derivation: deautonomisation through singularity confinement and geometrical approach based on affine Weyl groups. The question of integrability of discrete Painleve equations is also addressed in terms of the existence of a Lax pair or of a description in the frame of the Grand Scheme. A list of discrete Painleve equations, as complete as possible but still not exhaustive, is also presented.