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

Abstract

An explanation is offered for an observed lower bound on the wave speed of travelling kinks in Frenkel–Kontorova lattices. Kinks exist at discrete wavespeeds 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; inside such bands only quasi-kinks with nondecaying oscillatory tails are possible. The results are presented for a discrete sine-Gordon lattice with an onsite 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. Although none of the analysis is presented as rigorous mathematics, numerical results suggest that the bound on allowable wavespeeds is sharp. The results also explain why the vanishing, at discrete values of anharmonicity, of the Peierls–Nabarro barrier between stationary kinks as discovered by Savin et al., does not lead to bifurcation of kinks with small wave speed.