Asymptotic Analysis of a System of Algebraic Equations Arising in Dislocation Theory
Cameron L. Hall, Stephen Jonathan Chapman, John Ockendon · SIAM Journal on Applied Mathematics · 2010
The system of algebraic equations given by $\sum_{j=0,\,j eq i}^n\frac{sgn}(x_i-x_j)}{|x_i-x_j|^a}=1$, $i=1,2,\dots,n$, $x_0=0$, 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 $02$ 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.