Contrast Structures with a Multizone Internal Layer

Abstract

A boundary value problem for a singularly perturbed second-order differential equation is considered in two cases, in each of which one of the roots of the degenerate equation is double. In the first case, a narrow internal layer is proven to be formed where there is a rapid transition of the solution from the double root of the degenerate equation to a simple root; in the second case, the solution is proven to have a “spike” in the internal layer. Such solutions are called step-like contrast structures and spike-like contrast structures, respectively. In each case, the asymptotic expansion of the contrast structure is constructed. It significantly differs from the known expansion when all roots of the degenerate equation are simple; in particular, the internal layer turns out to be multizone.