2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/156224Recently, the method of one-dimensional maps was introduced as a means of generating exceptional discretisations of the $ϕ^4$-theories, i.e., discrete $ϕ^4$-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.20 pagesPattern Formation and SolitonsExactly Solvable and Integrable Systemstext