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SIAM J. Appl. Math. 70, pp. 2729-2749 (21 pages)
Asymptotic Analysis of a System of Algebraic Equations Arising in Dislocation Theory
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Add to FacebookThe system of algebraic equations given by $\sum_{j=0,\,j\neq i}^n\frac{sgn}(x_i-x_j)}{|x_i-x_j|^a}=1$, $i=1,2,\dots,n$, $x_0=0$, appears in dislocation theory in models of dislocation pile-ups. Specifically, the case $a=1$ corresponds to the simple situation where $n$ dislocations are piled up against a locked dislocation, while the case $a=3$ corresponds to $n$ dislocation dipoles piled up against a locked dipole. We present a general analysis of systems of this type for $a>0$ and $n$ large. In the asymptotic limit $n\rightarrow\infty$, it becomes possible to replace the system of discrete equations with a continuum equation for the particle density. For $0<a<2$, this takes the form of a singular integral equation, while for $a>2$ it is a first-order differential equation. The critical case $a=2$ requires special treatment, but, up to corrections of logarithmic order, it also leads to a differential equation. The continuum approximation is valid only for $i$ neither too small nor too close to $n$. The boundary layers at either end of the pile-up are also analyzed, which requires matching between discrete and continuum approximations to the main problem.
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