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

TL;DR

This paper introduces two-variable polynomials related to the discrete time Toda lattice and the alternate discrete Painleve II equation, revealing their properties, solutions, and symplectic structure, and connecting them to classical Painleve equations.

Contribution

It defines new polynomials for the discrete Toda lattice and discrete Painleve II, establishing their properties, solutions, and symplectic transformations, and links to classical Painleve equations.

Findings

Polynomials $Y_{n}(t,h)$ generalize Yablonskii-Vorob'ev polynomials for discrete systems.

These polynomials provide rational solutions to the discrete Painleve II equation.

A Lax pair for the discrete Painleve II is constructed, recovering the continuous case as $h o 0$.

Abstract

The Yablonskii-Vorob'ev polynomials $y_{n}(t)$, 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 ($P_{II}$). Here we define two-variable polynomials $Y_{n}(t,h)$ 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 $P_{II}$, namely the so called {\it alternate discrete} $P_{II}$, and this connection leads to an expression in terms of the Umemura polynomials for the third Painlev\'{e} equation ($P_{III}$). 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 $P_{II}$, which recovers Jimbo and Miwa's Lax pair for $P_{II}$ in the continuum limit $h\to 0$.