Depinning of discommensurations for tilted Frenkel–Kontorova chains

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 discommensurations (or kinks or fronts), advancing if the right-asymptotic equilibrium is to the right of the left-asymptotic one, retreating otherwise. Following work of Middleton, Floria & Mazo and Baesens & MacKay, there is a threshold tilt F d ( p / q ) ⩾ 0 up to which there continue to be periodic equilibria of type ( p , q ) and above which there is a globally attracting periodically sliding solution in the space of sequences of type ( p , q ). In this paper, we prove that there are values F d ( p / q ± ) of tilt with 0 ⩽ F d ( p / q ± ) ⩽ 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 such that for F d ( p / q ± ) < F < F d ( p / q ) there are periodically sliding discommensurations, apart perhaps from exceptional cases with both a degenerate type ( p , q ) equilibrium and a degenerate advancing equilibrium discommensuration. We give examples, however, to show that equilibrium and periodically sliding discommensurations may co-exist, both above and below F d ( p / q ± ) , so the case of discommensurations is not as clean as that of periodic configurations. On the way, we prove that F d ( ω ) → F d ( p / q ± ) as ω ↘ p / q or ↗ p / q respectively. Finally, we prove that F d ( p / q ± ) = 0 is equivalent to the existence of a rotational invariant circle consisting of periodic orbits of type ( p , q ) and right-going (respectively left-going) separatrices, for the corresponding twist map on the cylinder.