Travelling heteroclinic waves in a Frenkel–Kontorova chain with anharmonic on-site potential

摘要

The Frenkel-Kontorova model for dislocation dynamics from 1938 is given by achain of atoms, where neighbouring atoms interact through a linear spring andare exposed to a smooth periodic on-site potential. A dislocation moving withconstant speed corresponds to a heteroclinic travelling wave, making atransition from one well of the on-site potential to another. The ensuingsystem is nonlocal, nonlinear and nonconvex. We present an existence result fora class of smooth nonconvex on-site potentials. Previous results in mathematicsand mechanics have been limited to on-site potentials with harmonic wells. Toovercome this restriction, we first develop a global centre manifold theory foranharmonic wave trains, then parametrise the centre manifold to obtainasymptotically correct approximations to the solution sought, and finallyobtain the heteroclinic wave via a fixed point argument.