Integrable semi-discretizations of the sine-Gordon equation in non-characteristic coordinates

TL;DR

This paper develops new integrable semi-discretizations of the sine-Gordon equation in non-characteristic coordinates, expanding the understanding of discretization methods beyond the well-studied characteristic case, with a focus on laboratory coordinates.

Contribution

It introduces novel integrable space discretizations for the sine-Gordon equation in non-characteristic coordinates, including a practical discretization in laboratory coordinates.

Findings

Proposes three integrable discretizations in non-characteristic coordinates.

Rewrites the sine-Gordon equation as a two-component system for discretization.

Provides an aesthetically acceptable discretization in laboratory coordinates.

Abstract

Integrable discretizations of the sine-Gordon equation in characteristic (or light-cone) coordinates have been extensively studied after the seminal works of Hirota and Orfanidis in the late 1970s. In contrast, integrable discretizations of the sine-Gordon equation in non-characteristic coordinates have been scarcely studied except the lattice sine-Gordon model proposed by Izergin and Korepin in the early 1980s. In this paper, using the zero-curvature representation, we propose integrable space discretizations of the sine-Gordon equation in three distinct cases of non-characteristic coordinates. For the most interesting case of the sine-Gordon equation in laboratory coordinates, the integrable space discretization is unwieldy; as a remedy, we rewrite the sine-Gordon equation as a two-component evolutionary system and present an aesthetically acceptable space discretization.