On the classification of discrete Hirota-type equations in 3D

TL;DR

This paper extends a classification method for 3D integrable equations to the fully discrete case, using hydrodynamic reductions, offering an alternative to multi-dimensional consistency.

Contribution

It introduces a novel classification approach for discrete Hirota-type equations based on hydrodynamic reductions, expanding previous differential/difference equation classifications.

Findings

Classified discrete 3D integrable Hirota-type equations within specific subclasses.

Extended the hydrodynamic reduction method to fully discrete equations.

Provided an alternative to the multi-dimensional consistency approach.

Abstract

In the series of recent publications we have proposed a novel approach to the classification of integrable differential/difference equations in 3D based on the requirement that hydrodynamic reductions of the corresponding dispersionless limits are `inherited' by the dispersive equations. In this paper we extend this to the fully discrete case. Our only constraint is that the initial ansatz possesses a non-degenerate dispersionless limit (this is the case for all known Hirota-type equations). Based on the method of deformations of hydrodynamic reductions, we classify discrete 3D integrable Hirota-type equations within various particularly interesting subclasses. Our method can be viewed as an alternative to the conventional multi-dimensional consistency approach.