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.
Original languageEnglish
Publication statusUnpublished - 2003

Bibliographical note

Additional information: Later published by Elsevier Science, (2003) Physica D: Nonlinear Phenomena, 186 (3-4), pp.148-170. ISSN 0167-2789

Sponsorship: The authors would like to thank Michel Peyrard for giving us access to his numerical data for reproduction in Fig. 4, and also Yaroslav Zolotaryuk for sharing details of his numerical implementation. We also acknowledge useful
conversations with Jonathan Wattis, Sebius Doedel, Yuri Kivshar and Boris Malomed. The work was supported by EPSRC grant GR/R02719/01, and that of ARC was partially supported by an EPSRC Advanced Fellowship.


  • discrete sine-Gordon equation
  • dispersion relations
  • kinks
  • anharmonicity
  • nonlinear lattices
  • topological solitons
  • numerical continuation


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