Three Lessons on the Painlevé Property and the Painlevé Equations

  • M.D. Kruskal
  • B. Grammaticos
  • T. Tamizhmani
Part of the Lecture Notes in Physics book series (LNP, volume 644)


While this school focuses on discrete integrable systems we feel it necessary, if only for reasons of comparison, to go back to fundamentals and introduce the basic notion of the Painlevé property for continuous systems together with a critical analysis of what is called the Painlevé test. The extension of the latter to what is called the poly-Painlevé test is also introduced. Finally we devote a lesson to the proof that the Painlevé equations do have the Painlevé property.


Singular Point Double Pole Essential Singularity Laurent Expansion Dominant Behaviour 
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Authors and Affiliations

  • M.D. Kruskal
    • 1
  • B. Grammaticos
    • 2
  • T. Tamizhmani
    • 3
  1. 1.Department of Mathematics, Rutgers University, New Brunswick, NJ 08903USA
  2. 2.GMPIB, Université Paris VII, Tour 24-14, 5e étage, case 7021, 75251 ParisFrance
  3. 3.Department of Mathematics, Kanchi Mamunivar Centre for Postgraduate Studies, Pondicherry 605008India

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