Properties of the Non-Autonomous Lattice Sine-Gordon Equation: Consistency around a Broken Cube Property

TL;DR

This paper investigates the non-autonomous lattice sine-Gordon equation, demonstrating its consistency around a broken cube and constructing new Lax pairs, thus extending understanding of its integrability properties.

Contribution

It establishes the consistency around a broken cube property for the non-autonomous lattice sine-Gordon equation and constructs two new Lax pairs.

Findings

Non-autonomous lattice sine-Gordon equation has the broken cube property.

Two new Lax pairs are constructed for the non-autonomous case.

The property extends the integrability framework to non-autonomous equations.

Abstract

The lattice sine-Gordon equation is an integrable partial difference equation on ${\mathbb Z}^2$, which approaches the sine-Gordon equation in a continuum limit. In this paper, we show that the non-autonomous lattice sine-Gordon equation has the consistency around a broken cube property as well as its autonomous version. Moreover, we construct two new Lax pairs of the non-autonomous case by using the consistency property.