Direct delay reductions of the Toda hierarchy

TL;DR

This paper introduces a novel hierarchy of delay-differential equations derived from the Toda hierarchy using direct reduction methods, including their Lax pairs, marking a first in the literature.

Contribution

It presents the first known hierarchy of delay-differential equations obtained from the Toda hierarchy through direct reduction, along with their associated Lax pairs.

Findings

Established a hierarchy of delay-differential equations from the Toda hierarchy.

Derived all possible reductions under certain assumptions.

Constructed the Lax pair for the reduced hierarchy.

Abstract

We apply the direct method of obtaining reductions to the Toda hierarchy of equations. The resulting equations form a hierarchy of ordinary differential difference equations, also known as delay-differential equations. Such a hierarchy appears to be the first of its kind in the literature. All possible reductions, under certain assumptions, are obtained. The Lax pair associated to this reduced hierarchy is obtained.