The ultimate discretisation of the Painlevé equations

A. Ramani; D. Takahashi; B. Grammaticos; Y. Ohta

doi:10.1016/S0167-2789(97)00192-9

The ultimate discretisation of the Painlevé equations

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

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

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 language	English
Pages (from-to)	185-196
Number of pages	12
Journal	Physica D: Nonlinear Phenomena
Volume	114
Issue number	3-4
DOIs	https://doi.org/10.1016/S0167-2789(97)00192-9
Publication status	Published - 1998
Externally published	Yes

Keywords

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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10.1016/S0167-2789(97)00192-9

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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 D. Takahashi and B. Grammaticos and Y. Ohta",

note = "Funding Information: acknowledges the financial support of the Japan Ministry of Education through the Foreign Study Program.",

year = "1998",

doi = "10.1016/S0167-2789(97)00192-9",

language = "English",

volume = "114",

pages = "185--196",

journal = "Physica D: Nonlinear Phenomena",

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T1 - The ultimate discretisation of the Painlevé equations

AU - Ramani, A.

AU - Takahashi, D.

AU - Grammaticos, B.

AU - Ohta, Y.

N1 - Funding Information: acknowledges the financial support of the Japan Ministry of Education through the Foreign Study Program.

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

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DO - 10.1016/S0167-2789(97)00192-9

M3 - Article

AN - SCOPUS:0040227335

SN - 0167-2789

VL - 114

SP - 185

EP - 196

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 3-4

ER -

The ultimate discretisation of the Painlevé equations

Abstract

Keywords

ASJC Scopus subject areas

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