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

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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.

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

    Manuel Del Pino

  2. Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

    Michał Kowalczyk

  3. Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong

    Juncheng Wei

Authors
  1. Manuel Del Pino
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  2. Michał Kowalczyk
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  3. Juncheng Wei
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Correspondence to Michał Kowalczyk.

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Communicated by D. Kinderlehrer

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Del Pino, M., Kowalczyk, M. & Wei, J. The Toda System and Clustering Interfaces in the Allen–Cahn equation. Arch Rational Mech Anal 190, 141–187 (2008). https://doi.org/10.1007/s00205-008-0143-3

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  • DOI: https://doi.org/10.1007/s00205-008-0143-3

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