Reduced ODE dynamics as formal relativistic asymptotics of a Peierls-Nabarro model
Résumé
In this paper, we consider a scalar Peierls-Nabarro model describing the motion of dislocations in the plane $(x_1,x_2)$, along the line $x_2=0$. Each dislocation can be seen as a phase transition and creates a scalar displacement field in the plane. This displacement field solves a simplified elasto-dynamics equation which is simply the linear wave equation. The total displacement field creates a stress which makes move the dislocation themselves. By symmetry, we can reduce the system to the wave equation in the half plane $x_2>0$ coupled with an equation for the dynamics of dislocations on the boundary of the half plane, i.e. on $x_2=0$. Our goal is to understand the dynamics of well-separated dislocations in the limit when the distance between dislocations is very large of order $1/\varepsilon$. After rescaling, this corresponds to introduce a small parameter $\varepsilon$ in the system. In the limit $\varepsilon \to 0$, we are formally able to identify a reduced ODE model describing the dynamics of relativistic dislocations.
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