Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one

María del Mar González; Regis Monneau

doi:10.3934/dcds.2012.32.1255

Article Contents

2012, Volume 32, Issue 4: 1255-1286. Doi: 10.3934/dcds.2012.32.1255

This issue Previous Article On isotopy and unimodal inverse limit spaces Next Article Box dimension and bifurcations of one-dimensional discrete dynamical systems

Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one

María del Mar González^1, and
Regis Monneau^2,

1.
Universitat Politècnica de Catalunya, ETSEIB - Departament de MA1, Av. Diagonal, 647, 08028 Barcelona
2.
Université Paris-Est, Cermics, Ecole des Ponts ParisTech, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2

Received: June 30, 2010

Revised: July 31, 2011

Published: March 2012

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Abstract

We consider a reaction-diffusion equation with a half-Laplacian. In the case where the solution is independent on time, the model reduces to the Peierls-Nabarro model describing dislocations as transition layers in a phase field setting. We introduce a suitable rescaling of the evolution equation, using a small parameter $\varepsilon$. As $\varepsilon$ goes to zero, we show that the limit dynamics is characterized by a system of ODEs describing the motion of particles with two-body interactions. The interaction forces are in $1/x$ and correspond to the well-known interaction between dislocations.

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Mathematics Subject Classification: 35Q99, 35B40, 35J25, 35D30, 35G25, 70F99.

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