The Toda system and clustering interfaces in the Allen-Cahn equation

Abstract

We consider the Allen-Cahn equation "2 u+(1−u2)u = 0 in a bounded, smooth domain

in R2, under zero Neumann boundary conditions, where " > 0 is a small parameter. Let ð€€€0 be a segment contained in , connecting orthogonally the boundary. Under certain non-degeneracy and non-minimality assumptions for ð€€€0, satisfied for instance by the short axis in an ellipse, we construct, for any given N 1, a solution exhibiting N transition layers whose mutual distances are O("| log "|) and which collapse onto ð€€€0 as " ! 0. Asymptotic location of these interfaces is governed by a Toda type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary.