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

TL;DR

The paper addresses integrable semi-discretizations of the sine-Gordon equation in non-characteristic coordinates, including the laboratory form $u_{tt}-u_{xx}+sin u=0$. It develops a zero-curvature framework with multiple Lax pairs to produce three distinct space-discretizations: one for $u_{xt}-u_{xx}=sin u$, one for $u_{tt}+u_{xt}=sin u$, and a two-component discretization for lab coordinates (plus a scalar discretization whose continuous limit is the lab SG). These constructions preserve integrability via Lax pairs and allow IST-like analysis, linking to the lattice SG models of Izergin and Korepin. The two-component formulation provides an elegant discretization while maintaining connection to the continuous SG. Overall, the work extends integrable discretizations to non-characteristic coordinates and clarifies relationships with known lattice SG frameworks.

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.