Analysis of the heteroclinic connection in a singularly perturbed system arising from the study of crystalline grain boundaries

Abstract

Mathematically, the problem considered here is that of heteroclinic connections for a system of two second order differential equations of Hamiltonian type, in which a small parameter \( \e \) conveys a singular perturbation. The motivation comes from a multi-order-parameter phase field model developed by Braun et al \cite{BCMcFW} and \cite{T} for the description of crystalline interphase boundaries. In this, the smallness of \( \e \) is related to large anisotropy. The existence of such a heteroclinic, and its dependence on \( \e \), is proved. In addition, its robustness is investigated by establishing its spectral stability.