The Toda System and Clustering Interfaces in the Allen–Cahn equation

Abstract

We consider the Allen–Cahn equation \({\varepsilon^{2}\Delta u + (1-u^2)u = 0}\) in a bounded, smooth domain Ω in \({\mathbb{R}^2}\) , under zero Neumann boundary conditions, where \({\varepsilon > 0}\) is a small parameter. Let Γ0 be a segment contained in Ω, connecting orthogonally the boundary. Under certain nondegeneracy and nonminimality 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 \({O(\varepsilon|\log\varepsilon|)}\) and which collapse onto Γ0 as \({\varepsilon\to 0}\) . 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.