The ultimate discretisation of the Painlevé equations

A. Ramani, Daisuke Takahashi, B. Grammaticos, Y. Ohta

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

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
Volume114
Issue number3-4
Publication statusPublished - 1998
Externally publishedYes

Fingerprint

Cascades (fluid mechanics)
Cellular automata
Coalescence
Discretization
Discrete Equations
dependent variables
Particular Solution
cellular automata
Cellular Automata
coalescing
Cascade
integers
cascades
Integer
Dependent

Keywords

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

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistical and Nonlinear Physics

Cite this

Ramani, A., Takahashi, D., Grammaticos, B., & Ohta, Y. (1998). The ultimate discretisation of the Painlevé equations. Physica D: Nonlinear Phenomena, 114(3-4), 185-196.

The ultimate discretisation of the Painlevé equations. / Ramani, A.; Takahashi, Daisuke; Grammaticos, B.; Ohta, Y.

In: Physica D: Nonlinear Phenomena, Vol. 114, No. 3-4, 1998, p. 185-196.

Research output: Contribution to journalArticle

Ramani, A, Takahashi, D, Grammaticos, B & Ohta, Y 1998, 'The ultimate discretisation of the Painlevé equations', Physica D: Nonlinear Phenomena, vol. 114, no. 3-4, pp. 185-196.
Ramani, A. ; Takahashi, Daisuke ; Grammaticos, B. ; Ohta, Y. / The ultimate discretisation of the Painlevé equations. In: Physica D: Nonlinear Phenomena. 1998 ; Vol. 114, No. 3-4. pp. 185-196.
@article{9fb11dff112d4569976270d1ff251876,
title = "The ultimate discretisation of the Painlev{\'e} equations",
abstract = "We present a systematic way to construct ultra-discrete versions of the Painlev{\'e} 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 Painlev{\'e} equations have the properties characteristic of the continuous and discrete Painlev{\'e}'s, namely coalescence cascades, particular solutions and auto-B{\"a}cklund relations.",
keywords = "Cellular automota, Difference equations, Equations, Integrability, Painlev{\'e}",
author = "A. Ramani and Daisuke Takahashi and B. Grammaticos and Y. Ohta",
year = "1998",
language = "English",
volume = "114",
pages = "185--196",
journal = "Physica D: Nonlinear Phenomena",
issn = "0167-2789",
publisher = "Elsevier",
number = "3-4",

}

TY - JOUR

T1 - The ultimate discretisation of the Painlevé equations

AU - Ramani, A.

AU - Takahashi, Daisuke

AU - Grammaticos, B.

AU - Ohta, Y.

PY - 1998

Y1 - 1998

N2 - 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.

AB - 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.

KW - Cellular automota

KW - Difference equations

KW - Equations

KW - Integrability

KW - Painlevé

UR - http://www.scopus.com/inward/record.url?scp=0040227335&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040227335&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0040227335

VL - 114

SP - 185

EP - 196

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 3-4

ER -