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What is the discrete analogue of the Painlevé property?

Published online by Cambridge University Press:  17 February 2009

A. Ramani  and
B. Grammaticos
A. Ramani
Affiliation:
CPT, Ecole Polytechnique, CNRS, UMR 7644, 91128 Palaiseau, France; e-mail: [email protected].
B. Grammaticos
Affiliation:
GMPIB, Université Paris VII, Tour 24-14, 5e étage, case 7021, 75251 Paris, France.
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Abstract

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We analyse the various integrability criteria which have been proposed for discrete systems, focusing on the singularity confinement method. We present the exact procedure used for the derivation of discrete Painlevé equations based on the deautonomisation of integrable autonomous mappings. This procedure is then examined in the light of more recent criteria based on the notion of the complexity of the mapping. We show that the low-growth requirements lead, in the case of the discrete Painlevé equations, to exactly the same results as singularity confinement. The analysis of linearisable mappings shows that they have special growth properties which can be used in order to identify them. A working strategy for the study of discrete integrability based on singularity confinement and low-growth considerations is also proposed.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002 , pp. 21 - 32
Copyright
Copyright © Australian Mathematical Society 2002

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