Basil Grammaticos

Isomonodromic deformation problems for discrete analogues of Painlevé equations

Abstract Starting from isospectral problems of two-dimensional integrable mappings, isomonodromic deformation problems for the new descrete versions of the Painleve I–III equations are constructed. It is argued that they lead to differential and q-difference deformations (“quantizations”) of the corresponding spectral curves.

Discrete Integrable Systems

Three Lessons on the Painleve Property and the Painleve Equations (M.D. Kruskal, B. Grammaticos, T. Tamizhmani).- Sato Theory and Transformation Groups. A Unified Approach to Integrable Systems (R. Willox, J. Satsuma).- Special Solutions of Discrete Integrable Systems (Y. Ohta).- Discrete Differential Geometry. Integrability as Consistency (A.I. Bobenko).- Discrete Lagrangian Models (Yu.B. Suris).- Symmetries of Discrete Systems (P. Winternitz).- Discrete Painleve Equations: A Review (B. Grammaticos, A. Ramani).- Special Solutions for Discrete Painleve Equations (K.M. Tamizhmani, T. Tamizhmani, B. Grammaticos, A. Ramani).- Ultradiscrete Systems (Cellular Automata) (T. Tokihiro).- Time in Science: Reversibility vs. Irreversibility (Y. Pomeau).

CONSTRUCTING SOLUTIONS TO THE ULTRADISCRETE PAINLEVE EQUATIONS

We investigate the nature of particular solutions to the ultradiscrete Painleve equations. We start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions. For the ultradiscrete Painleve equations II and III we present special solutions reminiscent of the Casorati determinant ones which exist in the continuous and discrete cases. Finally we analyse the discrete Painleve equation I and show how it contains both the continuous and the ultradiscrete ones as particular limits.

The ultimate discretisation of the Painleve´ equations

Abstract We present a systematic way to construct ultra-discrete versions of the Painleve equations starting from know discrete forms. These ultra-discrete equations are generalised cellular automata in the sense that the dependent variable takes only integer values. The ultra-discrete Painleve equations have the properties characteristic of the continuous and discrete Painleves, namely coalescence cascades, particular solutions and auto-Backlund relations.

How to detect the integrability of discrete systems

Several integrability tests for discrete equations will be reviewed. All tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as Lax pairs. Specifically, singularity confinement, algebraic entropy, Nevanlinna theory, Diophantine integrability and discrete systems over finite fields will be described.

Ultradiscretization without positivity

We present a new ultradiscretization approach which can be applied to discrete systems, the solutions of which are not positive definite. This was made possible, thanks to an ansatz involving the hyperbolic-sine function. We apply this new procedure to simple mappings. For the linear and homographic mappings, we obtain ultradiscrete forms and explicitly construct their solutions. Two discrete Painleve II equations are also analysed and ultradiscretized. We show how to construct the ultradiscrete analogues of their rational and Airy-type solutions.

Singularity Analysis and its Relation to Complete, Partial and Non-Integrability

The aim of this course is to present and illustrate the connection between integrability and the singularity structure of the solutions of nonlinear dynamical systems. We start by reviewing the various aspects of integrability and then introduce the notions of partial and constrained integrability. Next we present the methodology of the singularity (“Painleve”) analysis and apply it to the study of various systems. A similar approach, due essentially to Yoshida, is presented by the course of D. Bessis which is complementary to ours. Finally we present a detailed review of the recent progress in the domain of nonintegrability. Based on Ziglin’s theorem, we prove rigorously the nonexistence of integrals for several systems of physical interest. As a non-linear extension of Ziglin’s approach we present the “poly-Painleve” criterion of (non-)integrability, illustrate it through some example, and propose a practical method for its implementation.

Deautonomization by singularity confinement: an algebro-geometric justification

The ‘deautonomization’ of an integrable mapping of the plane consists in treating the free parameters in the mapping as functions of the independent variable, the precise expressions of which are to be determined with the help of a suitable criterion for integrability. Standard practice is to use the singularity confinement criterion and to require that singularities be confined at the very first opportunity. An algebro-geometrical analysis will show that confinement at a later stage leads to a non-integrable deautonomized system, thus justifying the standard singularity confinement approach. In particular, it will be shown on some selected examples of discrete Painlevé equations, how their regularization through blow-up yields exactly the same conditions on the parameters in the mapping as the singularity confinement criterion. Moreover, for all these examples, it will be shown that the conditions on the parameters are in fact equivalent to a linear transformation on part of the Picard group, obtained from the blow-up.

The redemption of singularity confinement

We present a novel way to apply the singularity confinement property as a discrete integrability criterion. We shall use what we call a full deautonomisation approach, which consists in treating the free parameters in the mapping as functions of the independent variable, applied to a mapping complemented with terms that are absent in the original mapping but which do not change the singularity structure. We shall show, on a host of examples including the well-known mapping of Hietarinta-Viallet, that our approach offers a way to compute the algebraic entropy for these mappings exactly, thereby allowing one to distinguish between the integrable and non-integrable cases even when both have confined singularities.

Singularity confinement and full-deautonomisation: A discrete integrability criterion

Abstract We present a new approach to singularity confinement which makes it an efficient and reliable discrete integrability detector. Our method is based on the full-deautonomisation procedure, which consists in analysing non-autonomous extensions of a given discrete system obtained by adding terms that are initially absent, but whose presence does not alter the singularity pattern. A justification for this approach is given through an algebro-geometric analysis. We also introduce the notions of early and late confinement. While the former is a confinement that may exist already for the autonomous system, the latter corresponds to a singularity pattern longer than that of the autonomous case. Late confinement will be shown to play an important role in the singularity analysis of systems with non-trivial gauge freedom, for which the existence of an undetected gauge in conjunction with a sketchy analysis, might lead to erroneous conclusions as to their integrability. An algebro-geometric analysis of the role of late confinement in this context is also offered. This novel type of singularity confinement analysis will be shown to allow for the exact calculation of the algebraic entropy of a given mapping.