Title: From atomistic model to the Peierls-Nabarro model with $γ$-surface for dislocations,
Abstract: The Peierls-Nabarro (PN) model for dislocations is a hybrid model that
incorporates the atomistic information of the dislocation core structure into
the continuum theory. In this paper, we study the convergence from a full
atomistic model to the PN model with $\gamma$-surface for the dislocation in a
bilayer system (e.g. bilayer graphene). We prove that the displacement field of
and the total energy of the dislocation solution of the PN model are
asymptotically close to those of the full atomistic model. Our work can be
considered as a generalization of the analysis of the convergence from
atomistic model to Cauchy-Born rule for crystals without defects in the
literature.