Consistency around a cube property of Hirota's discrete KdV equation and the lattice sine-Gordon equation

TL;DR

This paper proves that Hirota's discrete KdV and lattice sine-Gordon equations possess the consistency around a cube property and can be extended to three-dimensional lattice systems.

Contribution

It establishes the CAC property for these equations and demonstrates their extension to 3D lattice systems, resolving an open question.

Findings

Hirota's discrete KdV equation has the CAC property.

The lattice sine-Gordon equation has the CAC property.

Both equations can be extended to 3D lattice systems.

Abstract

It has been unknown whether Hirota's discrete Korteweg-de Vries equation and the lattice sine-Gordon equation have the consistency around a cube (CAC) property. In this paper, we show that they have the CAC property. Moreover, we also show that they can be extended to systems on the 3-dimensional integer lattice.