On the three-dimensional consistency of Hirota's discrete Korteweg-de Vries Equation

TL;DR

This paper investigates the three-dimensional consistency of Hirota's discrete Korteweg-de Vries equation, revealing new transformations and embeddings into higher-dimensional lattices, which extend the understanding of integrability in discrete systems.

Contribution

It introduces novel transformations and embeddings of Hirota's dKdV into three-dimensional lattices, enhancing the analysis of multidimensional consistency in integrable difference equations.

Findings

New transformations to other equations including second-degree second-order difference equations

Unusual embedding into a three-dimensional lattice

Extension of the consistency property to higher dimensions

Abstract

Hirota's discrete Korteweg-de Vries equation (dKdV) is an integrable partial difference equation on 2-dimensional integer lattice, which approaches the Korteweg-de Vries equation in a continuum limit. We find new transformations to other equations, including a second-degree second-order partial difference equation, which provide an unusual embedding into a three-dimensional lattice. The consistency of the resulting system extends a property that has been widely used to study partial difference equations on multidimensional lattices.