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Discrete Painlevé Equations: A Review

  • B. Grammaticos
  • A. Ramani
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 644)

Abstract

We present a review of what is current knowledge about discrete Painlevé equations. We start with a historical introduction which explains how the discrete Painlevé equations made their appearance. Since the most widely used derivation method was the one based on integrability detectors, we start by presenting a brief review of the existing discrete integrability detectors, and proceed to show how they were indeed used in deriving the discrete Painlevé equations. Given the profusion of different forms of these mappings, we examine the various approaches to their classification, from the first, crude ones, to the complete classification based on affine Weyl groups. The properties of the discrete Painlevé equations, which make them so special, are also reviewed with ample details. Finally, we present a series of results which are peripherally related to the discrete Painlevé equations but help cast more light on these (very) special systems.

Keywords

Weyl Group Continuous Limit Discrete System Discrete Form Contiguity Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Authors and Affiliations

  • B. Grammaticos
    • 1
  • A. Ramani
    • 2
  1. 1.GMPIB, Université Paris VII, Tour 24-14, 5e étage, case 7021, 75251 Paris, France
  2. 2.CPT, Ecole Polytechnique, CNRS, UMR 7644, 91128 PalaiseauFrance

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