Existence and uniqueness of bounded stable solutions to the Peierls–Nabarro model for curved dislocations

We study the well-posedness of vector-field Peierls-Nabarro model for curved dislocations with a double well potential and bi-states limit at far field. Using Dirichlet to Neumann map, 3D is reduced nonlocal scalar Ginzburg-Landau equation. derive an integral formulation operator, whose kernel anisotropic positive when Poisson's ratio $\nu\in(-\frac12, \frac13)$. then prove that any bounded stable solutions this equation has 1D profile, which corresponds PDE version flatness result minimal surfaces perimeter. Based on this, we finally obtain steady states equation, as original model, can be characterized one-parameter family straight dislocation rescaled half Laplacian.