The Peierls-Nabarro model was first introduced by Peierls Nabarro to describe the continuum model of dislocation in materials. It incorporates the atomic effect into the continuum framework and give us an understanding of dislocation core structure. Our main goal is to determine the displacement in the whole space using Perierls-Nabarro (PN) model, which incorporates the atomic effect into the continuum framework. We focus on the existence of analytic solution to Peierls-Nabarro model including stationary model and dynamic model. Since we incorporate the misfit surface energy into total energy of the system, which leads to a nonlinear boundary condition, the strategy is to first decouple the displacement field and then to uniquely solve a nonlocal equation on boundary.