T Tamizhmani

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).

Special function solutions of the discrete painlevé equations

In this paper, we present a review of the special solutions of discrete (difference) and q-discrete Painleve equations in terms of discrete special functions. These solutions exist whenever the parameters of the Painleve equation satisfy a particular constraint. The discrete special functions belong to the hypergeometric family, although in some cases they seem to go beyond the known discrete and q-discrete hypergeometric functions. The equations studied in this paper are chosen on the basis of our recent classification of discrete Painleve equation with the help of affine Weyl groups.

Growth and integrability in discrete systems

We introduce a new discrete integrability criterion inspired from the recent findings of Ablowitz and collaborators. This criterion is based on the study of the growth of some characteristic of the solutions of a mapping, using Nevanlinna theory. Since the practical implementation of the latter does not always lead to a clear-cut answer, we complement the growth criterion by the singularity confinement property. This combination turns out to be particularly efficient. Its application allows us to recover the known forms of the discrete Painleve equations and to show that no new ones may exist within a given parametrization.

WRONSKIAN AND RATIONAL SOLUTIONS OF THE DIFFERENTIAL-DIFFERENCE KP EQUATION

We present the Wronskian form of the N-soliton solutions of the differential-difference Kadomtsev-Petviashvilli (DKP) equation. Also a wider class of rational solutions are derived using semi-discrete analogue of Schur polynomials and its generalization. Our approach is based on Sato theory formalism which gives all these solutions naturally.

Quadratic relations in continuous and discrete Painlevé equations

The quadratic relations between the solutions of a Painleve equation and that of a different one, or the same one with a different set of parameters, are investigated in the continuous and discrete cases. We show that the quadratic relations existing for the continuous PII , PIII , PV and PVI have analogues as well as consequences in the discrete case. Moreover, the discrete Painleve equations have quadratic relations of their own without any reference to the continuous case.

The road to the discrete analogue of the Painlevé property: Nevanlinna meets singularity confinement

The question of integrability of discrete systems is analysed in the light of the recent findings of Ablowitz et al., who have conjectured that a fast growth of the solutions of a differ- ence equation is an indication of nonintegrability. The study of the behaviour of the solutions of a mapping is based on the theory of Nevanlinna. In this paper, we show how this approach can be implemented in the csse of second-order mappings which include the discrete Painlevk equations. Since the Nevanlinna approach does offer only a necessary condition which is not restrictive enough, we complement it by the singularity confinement requirement, first in an autonomous setting and then for deautonomisation. We believe that this three-tiered approach is the closest one can get to a discrete analogue of the Painlevb property. @ 2003 Elsevier Science Ltd. All rights reserved.

On a transcendental equation related to Painlevé III, and its discrete forms

We examine the canonical forms of Painlev? equations and argue that the equation for PIII in which one parameter is taken to be equal to zero should be considered as a canonical form different from the standard PIII . Our argument is based on the fact that the value of this parameter cannot be modified through auto-B?cklund transformations. We investigate the possible discrete forms of this equation and produce two of them. One is of a difference type, where the independent variable enters linearly, while the second one is of q type where the independent variable enters in a multiplicative way. The properties of these discrete equations are also studied. ]]>

Do integrable cellular automata have the confinement property

We analyse a criterion, introduced by Joshi and Lafortune, for the integrability of cellular automata obtained from discrete systems through the ultradiscretization procedure. We show that while this criterion can be used in order to single out integrable ultradiscrete systems, there do exist cases where the system is nonintegrable and still the criterion is satisfied. Conversely we show that for ultradiscrete systems that are derived from linearizable mappings the criterion is not satisfied. We investigate this phenomenon further in the case of a mapping which includes a linearizable subcase and show how the violation of the criterion comes to be. Finally, we comment on the growth properties of ultradiscrete systems.

On a class of special solutions of the Painlevé equations

We present a class of special solutions for continuous, discrete and q-discrete Painleve equations. These solutions are expressed through quadratures or their discrete equivalents. The condition for their existence is a further constraint on the conditions for the existence of other special, linearisable, solutions of Painleve equations. Thus, these solutions exist for equations that have at least two parameters. In most cases, the special solutions are expressed in terms of functions defined through quadratures.

Special function solutions for asymmetric discrete Painlevé equations

We study the solutions of a family of discrete Painleve equations. The equations that we examine are given as a system of two first-order non-autonomous mappings. The solutions we are interested in are the ones obtained whenever the Painleve equation can be reduced to a discrete Riccati equation, which can be linearized through a Cole-Hopf transformation. The special solutions thus obtained involve generalizations or reductions of the hypergeometric (and q-hypergeometric) function.