Asymptotic analysis of a system of algebraic equations arising in dislocation theory - Hall, C; Chapman, S; Ockendon, J (2010)
The system of algebraic equations given by $\sum_{j=0, j ≠ i}^n sgn(x_i - x_j) / |x_i - x_j|^a = 1, i = 1, 2, … n, x_0 = 0,$ 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 -> ∞, 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 only valid for i not 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.