A new barrier to the existence of moving kinks in Frenkel-Kontorova lattices

An explanation is offered for an observed lower bound on the wave speed of travelling kinks in Frenkel-Kontorova lattices. Kinks exist at discrete wave speeds within a parameter regime where there is a resonance with linear waves (phonons). However, they fail to exist even in this codimension-one sense whenever there is more than one phonon branch in the dispersion relation. The results are presented for a discrete sine-Gordon lattice with an on-site potential that has a tunable amount of anharmonicity. Novel numerical methods are used to trace kinks with topological charge Q=1 and 2 in three parameters representing the propagation speed, lattice discreteness and anharmonicity. It is found that the lower bound is sharp, leading to a simple analytical criterion for predicting the possible wave speed of the moving kinks. For wave speeds that do not satisfy this criterion, only quasi-kinks with non-decaying oscillatory tails are possible. Recently, Savin et al [26] showed that the Peierls-Nabarro barrier between stationary kinks vanishes at discrete values of anharmonicity. The new barrier shows why such points do not lead to a bifurcation of kinks with small wave speed.