### Abstract

We investigate the nature of particular solutions to the ultradiscrete Painlevé equations. We start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions. For the ultradiscrete Painlevé equations II and III we present special solutions reminiscent of the Casorati determinant ones which exist in the continuous and discrete cases. Finally we analyse the discrete Painlevé equation I and show how it contains both the continuous and the ultradiscrete ones as particular limits.

Original language | English |
---|---|

Pages (from-to) | 7953-7966 |

Number of pages | 14 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 30 |

Issue number | 22 |

DOIs | |

Publication status | Published - 1997 Nov 21 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Physics A: Mathematical and General*,

*30*(22), 7953-7966. https://doi.org/10.1088/0305-4470/30/22/029

**Constructing solutions to the ultradiscrete Painlevé equations.** / Takahashi, Daisuke; Tokihiro, T.; Grammaticos, B.; Ohta, Y.; Ramani, A.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 30, no. 22, pp. 7953-7966. https://doi.org/10.1088/0305-4470/30/22/029

}

TY - JOUR

T1 - Constructing solutions to the ultradiscrete Painlevé equations

AU - Takahashi, Daisuke

AU - Tokihiro, T.

AU - Grammaticos, B.

AU - Ohta, Y.

AU - Ramani, A.

PY - 1997/11/21

Y1 - 1997/11/21

N2 - We investigate the nature of particular solutions to the ultradiscrete Painlevé equations. We start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions. For the ultradiscrete Painlevé equations II and III we present special solutions reminiscent of the Casorati determinant ones which exist in the continuous and discrete cases. Finally we analyse the discrete Painlevé equation I and show how it contains both the continuous and the ultradiscrete ones as particular limits.

AB - We investigate the nature of particular solutions to the ultradiscrete Painlevé equations. We start by analysing the autonomous limit and show that the equations possess an explicit invariant which leads naturally to the ultradiscrete analogue of elliptic functions. For the ultradiscrete Painlevé equations II and III we present special solutions reminiscent of the Casorati determinant ones which exist in the continuous and discrete cases. Finally we analyse the discrete Painlevé equation I and show how it contains both the continuous and the ultradiscrete ones as particular limits.

UR - http://www.scopus.com/inward/record.url?scp=0031582915&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031582915&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/30/22/029

DO - 10.1088/0305-4470/30/22/029

M3 - Article

VL - 30

SP - 7953

EP - 7966

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 22

ER -