Common, Alan K., Hone, Andrew N.W. (2008) Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation. Journal of Physics A: Mathematical and Theoretical, 41 (48). p. 485203. ISSN 17518113. (doi:10.1088/17518113/41/48/485203) (KAR id:41488)
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Official URL http://dx.doi.org/10.1088/17518113/41/48/485203 
Abstract
The YablonskiiVorob'ev polynomials yn(t), which are defined by a second order bilinear differentialdifference equation, provide rational solutions of the Toda lattice. They are also polynomial taufunctions for the rational solutions of the second Painlev\'{e} equation (PII). Here we define twovariable polynomials Yn(t,h) on a lattice with spacing h, by considering rational solutions of the discrete time Toda lattice as introduced by Suris. These polynomials are shown to have many properties that are analogous to those of the YablonskiiVorob'ev polynomials, to which they reduce when h=0. They also provide rational solutions for a particular discretisation of PII, namely the so called {\it alternate discrete} PII, and this connection leads to an expression in terms of the Umemura polynomials for the third Painlev\'{e} equation (PIII). It is shown that B\"{a}cklund transformation for the alternate discrete Painlev\'{e} equation is a symplectic map, and the shift in time is also symplectic. Finally we present a Lax pair for the alternate discrete PII, which recovers Jimbo and Miwa's Lax pair for PII in the continuum limit h?0.
Item Type:  Article 

DOI/Identification number:  10.1088/17518113/41/48/485203 
Subjects: 
Q Science > QA Mathematics (inc Computing science) > QA351 Special functions Q Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equations 
Divisions:  Faculties > Sciences > School of Mathematics Statistics and Actuarial Science 
Depositing User:  Andrew Hone 
Date Deposited:  21 Jun 2014 00:08 UTC 
Last Modified:  06 Feb 2020 04:09 UTC 
Resource URI:  https://kar.kent.ac.uk/id/eprint/41488 (The current URI for this page, for reference purposes) 
Hone, Andrew N.W.:  https://orcid.org/0000000197807369 
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