Abstract We consider a family of evolution equations that generalize the Peierls-Nabarro model for crystal dislocations. They can be seen as semilinear parabolic reaction-diffusion equations in which the diffusion is regulated by a fractional Laplace operator of order 2 s ∈ ( 0 , 2 ) acting in one space dimension and the reaction is determined by a 1-periodic multi-well potential. We construct solutions of these equations that represent the typical propagation of N ⩾ 2 equally oriented dislocations of size 1. For large times, the dislocations occur around points that evolve according to a repulsive dynamical system. When s ∈ ( 1 / 2 , 1 ) , these solutions are shown to be asymptotically stable with respect to odd perturbations.

中文翻译：

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演化晶体位错模型的长时间渐近

摘要 我们考虑了一系列演化方程，它们概括了晶体位错的 Peierls-Nabarro 模型。它们可以看作是半线性抛物线反应扩散方程，其中扩散由作用于一个空间维度的 2 s ∈ ( 0 , 2 ) 阶分数拉普拉斯算子调节，反应由 1 周期多孔井确定潜在的。我们构建了这些方程的解，这些方程代表了大小为 1 的 N ⩾ 2 个相同取向的位错的典型传播。在很多时候，位错发生在根据排斥动力学系统演化的点周围。当 s ∈ ( 1 / 2 , 1 ) 时，这些解对于奇数扰动是渐近稳定的。