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arXiv · 1010.0865

The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model

Ahmad Fino·Hassan Ibrahim·Régis Monneau
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Abstract

We study a generalization of the fully overdamped Frenkel-Kontorova model in dimension $n\geq 1.$ This model describes the evolution of the position of each atom in a crystal, and is mathematically given by an infinite system of coupled first order ODEs. We prove that for a suitable rescaling of this model, the solution converges to the solution of a Peierls-Nabarro model, which is a coupled system of two PDEs (typically an elliptic PDE in a domain with an evolution PDE on the boundary of the domain). This passage from the discrete model to a continuous model is done in the framework of viscosity solutions.

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Ahmad Fino, Hassan Ibrahim, Régis Monneau. 2010-10-05. The Peierls-Nabarro model as a limit of a Frenkel-Kontorova model. https://arxiv.org/abs/1010.0865

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