The dynamic characters of excitations in aone-dimensional Frenkel--Kontorova model

A one-dimensional (1D) Frenkel--Kontorova (FK) model is studied numerically in this paper, and two new analytical solutions (a supersonic kink and a nonlinear periodic wave) of the Sine--Gordon (SG) equation (continuum limit approximation of the FK model) are obtained by using the Jacobi elliptic function expansion method. Taking these new solutions as initial conditions for the FK model, we numerically find there exist the supersonic kink and the nonlinear periodic wave in these systems and obtain a lot of interesting and significant results. Moreover, we also investigate the subsonic kink and the breather in these systems and obtain some new feature.