# Rational solutions of the discrete time Toda lattice and the alternate discrete Painleve II equation

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 1751-8113. (doi:10.1088/1751-8113/41/48/485203)

## Abstract

The Yablonskii-Vorob'ev polynomials yn(t), which are defined by a second order bilinear differential-difference equation, provide rational solutions of the Toda lattice. They are also polynomial tau-functions for the rational solutions of the second Painlev\'{e} equation (PII). Here we define two-variable 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 Yablonskii-Vorob'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 10.1088/1751-8113/41/48/485203 Q Science > QA Mathematics (inc Computing science) > QA351 Special functionsQ Science > QA Mathematics (inc Computing science) > QA372 Ordinary differential equations Faculties > Sciences > School of Mathematics Statistics and Actuarial Science Andrew N W Hone 21 Jun 2014 00:08 UTC 29 May 2019 12:41 UTC https://kar.kent.ac.uk/id/eprint/41488 (The current URI for this page, for reference purposes)