Exceptional discretisations of the sine-Gordon equation

TL;DR

This paper introduces novel discretisations of the sine-Gordon equation that support kinks at arbitrary positions and velocities, enhancing the understanding of discrete models with continuous-like properties.

Contribution

It extends the method of one-dimensional maps to create exceptional discretisations of the sine-Gordon equation, enabling kinks to be centered at any lattice position and to move without radiation.

Findings

Supported kinks at arbitrary positions relative to the lattice

Constructed a discrete sine-Gordon model with radiationless moving kinks

Demonstrated enhanced flexibility of discrete sine-Gordon models

Abstract

Recently, the method of one-dimensional maps was introduced as a means of generating exceptional discretisations of the $\phi^4$-theories, i.e., discrete $\phi^4$-models which support kinks centred at a continuous range of positions relative to the lattice. In this paper, we employ this method to obtain exceptional discretisations of the sine-Gordon equation (i.e. exceptional Frenkel-Kontorova chains). We also use one-dimensional maps to construct a discrete sine-Gordon equation supporting kinks moving with arbitrary velocities without emitting radiation.