We give an existence proof for heteroclinic wave solutions to the Frenkel-Kontorova model of dislocation dynamics. The on-site potential is assumed to be piecewise quadratic. The framework employed here allows us to determine explicitly a regime of subsonic velocities for which travelling waves exist.
Kreiner, C. F., & Zimmer, J. (2011). Existence of subsonic heteroclinic waves for the Frenkel-Kontorova model with piecewise quadratic on-site potential. Nonlinearity, 24(4), 1137-1163. https://doi.org/10.1088/0951-7715/24/4/007