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

We consider the Allen-Cahn equation ε∆u+(1−u)u = 0 in a bounded, smooth domain Ω in R, under zero Neumann boundary conditions, where ε > 0 is a small parameter. Let Γ0 be a segment contained in Ω, connecting orthogonally the boundary. Under certain non-degeneracy and non-minimality assumptions for Γ0, 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(ε| log ε|) and which collapse onto Γ0 as ε→ 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.