For an untilted Frenkel-Kontorova chain and any rational $p/q$, Aubry and Mather proved there are minimising equilibrium states that are left- and right-asymptotic to neighbouring pairs of spatially periodic minimisers of type $(p,q)$.  They are known as {\em discommensurations} (or kinks or fronts), {\em advancing }if the right-asymptotic equilibrium is to the right of the left-asymptotic one, {\em retreating} otherwise.  Following work of Middleton, Floria \& Mazo and Baesens \& MacKay, there is a threshold tilt $F_d(p/q)\ge 0$ up to which there continue to be periodic equilibria of type $(p,q)$ and above which there are none.

We prove that there are threshold tilts $F_d(p/q\pm) \le F_d(p/q)$, generically positive and less than $F_d(p/q)$, up to which there continue to be equilibrium advancing or retreating discommensurations, respectively, and beyond which there are periodically sliding discommensurations, apart perhaps from exceptional cases with both a degenerate type $(p,q)$ equilibrium and a degenerate advancing equilibrium discommensuration. Moreover, we prove that $F_d(\omega) \to F_d(p/q\pm)$ as $\omega \searrow p/q$ or $\nearrow p/q$ respectively, and $F_d(p/q\pm)=0$ is equivalent to the existence of a rotational invariant circle for the corresponding twist map on the cylinder, consisting of periodic orbits of type $(p,q)$ and right-going (respectively left-going) separatrices.

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