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

By Manuel del Pino, M. Kowalczyk, Jun-cheng Wei. 2008

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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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Materials Chemistry

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In Applied Mathematics

In Computational Theory and Mathematics

In Materials Chemistry

Citations in the corpus

Some new entire solutions of semilinear elliptic equations on Rn (2009, result): “Some related results are given in [ 19 ], [17] (also for the Allen-Cahn equation), but there the proflle of solutions is homogeneous, or nearly homogeneous, along the transitions, in strong contrast with our case.”
The toda system and multiple-end solutions of autonomous planar elliptic problems (2007, background): “) +O(e−ϑ|t|),as t→+∞, j= 1,...,k−1, cjt+dj +τ− j (c) +O(e−ϑ |t ),as t→−∞, j= 1,...,k−1, (9.18) where τ± j (c) are smooth functions of the vector c= (c1,...,ck−1). Proof. This Lemma has been proved in [12]. We include a proof here for completeness. Let qj, j= 1,...,kbe a solution of the system (9.6) ”