The ultimate discretisation of the Painlevé equations

A. Ramani*, D. Takahashi, B. Grammaticos, Y. Ohta

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)


We present a systematic way to construct ultra-discrete versions of the Painlevé 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 Painlevé equations have the properties characteristic of the continuous and discrete Painlevé's, namely coalescence cascades, particular solutions and auto-Bäcklund relations.

Original languageEnglish
Pages (from-to)185-196
Number of pages12
JournalPhysica D: Nonlinear Phenomena
Issue number3-4
Publication statusPublished - 1998
Externally publishedYes


  • Cellular automota
  • Difference equations
  • Equations
  • Integrability
  • Painlevé

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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