Direct "Delay" Reductions of the Toda Equation

TL;DR

This paper introduces a new direct reduction method for the Toda equation, resulting in delay-differential equations, including a potential new Painlevé-type equation, with an associated Lax pair.

Contribution

It presents a canonical, complete class of reductions of the Toda equation to delay-differential equations and derives their Lax pairs, proposing a new Painlevé-like equation.

Findings

Derived a complete class of reductions to delay-differential equations.

Identified a potential new Painlevé-type equation.

Obtained the Lax pair for the reduced equations.

Abstract

A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painlev\'e equations. The Lax pair associated to this equation is obtained, also by reduction.