Generalized discrete integrable operator and integrable hierarchy

TL;DR

This paper introduces two new classes of discrete integrable operators with matrix and differential kernels, revealing their connection to higher-order pole solutions of integrable hierarchies and providing explicit resolvent formulas.

Contribution

It generalizes previous discrete integrable operators to include matrix and differential kernels, establishing their link to higher-order pole solutions and deriving explicit resolvent formulas.

Findings

Connection to higher-order pole solutions of integrable hierarchies

Explicit resolvent formulas for new classes of operators

Generalization of discrete analogue of integrable operators

Abstract

We introduce and systematically develop two classes of discrete integrable operators: those with $2\times 2$ matrix kernels and those possessing general differential kernels, thereby generalizing the discrete analogue previously studied. A central finding is their inherent connection to higher-order pole solutions of integrable hierarchies, contrasting sharply with standard operators linked to simple poles. This work not only provides explicit resolvent formulas for matrix kernels and differential operator analogues but also offers discrete integrable structures that encode higher-order behaviour.