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We study the well-posedness of the vector-field Peierls-Nabarro model forcurved dislocations with a double well potential and a bi-states limit at farfield. Using the Dirichlet to Neumann map, the 3D Peierls-Nabarro model isreduced to a nonlocal scalar Ginzburg-Landau equation. We derive an integralformulation of the nonlocal operator, whose kernel is anisotropic and positivewhen Poisson's ratio $\nu\in(-\frac12, \frac13)$. We then prove that anybounded stable solutions to this nonlocal scalar Ginzburg-Landau equation has a1D profile, which corresponds to the PDE version of flatness result for minimalsurfaces with anisotropic nonlocal perimeter. Based on this, we finally obtainthat steady states to the nonlocal scalar equation, as well as the originalPeierls-Nabarro model, can be characterized as a one-parameter family ofstraight dislocation solutions to a rescaled 1D Ginzburg-Landau equation withthe half Laplacian.

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