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 \(\nu \in (-{\frac{1}{2}}, {\frac{1}{3}})\). We then prove that any bounded stable solution 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.

我们研究了在远场具有双阱势和双态极限的弯曲位错的矢量场Peierls-Nabarro模型的适定性。使用Dirichlet到Neumann的地图，将3D Peierls–Nabarro模型简化为非局部标量Ginzburg–Landau方程。我们推导出非局部算子的积分形式，当泊松比\（\ nu \ in（-{\ frac {1} {2}}，{\ frac {1} {3}}）\时，其非各向异性算子的正整数。 ）。然后，我们证明该非局部标量Ginzburg-Landau方程的任何有界稳定解都具有1D轮廓，这与具有各向异性非局部周长的最小曲面的平面度结果的PDE版本相对应。基于此，我们最终获得非局部标量方程的稳态以及原始的Peierls-Nabarro模型的特征，可以看作是一维准位错解的一参数族，它是一维缩放后的一维Ginzburg-Landau方程的一半。拉普拉斯人。