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