Ancient multiple-layer solutions to the Allen-Cahn equation
Author
dc.contributor.author
Pino, Manuel del
Author
dc.contributor.author
Gkikas, Konstantinos
Admission date
dc.date.accessioned
2019-05-31T15:18:58Z
Available date
dc.date.available
2019-05-31T15:18:58Z
Publication date
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2018
Cita de ítem
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Proceedings of the Royal Society of Edinburgh Section A: Mathematics, Volumen 148, Issue 6, 2018, Pages 1165-1199
Identifier
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14737124
Identifier
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03082105
Identifier
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10.1017/S0308210517000245
Identifier
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https://repositorio.uchile.cl/handle/2250/169284
Abstract
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We consider the parabolic one-dimensional Allen-Cahn equation ut = uxx + u(1-u2), (x, t) R x (-0]. The steady state w(x) = tanh(x/2) connects, as a 'transition layer', the stable phases-1 and +1. We construct a solution u with any given number k of transition layers between-1 and +1. Mainly they consist of k time-travelling copies of w, with each interface diverging as t →. More precisely, we find u(x, t) ≈ ∑k j=1(-1)j-1 w(x-ξj(t)) + 1/2((-1)k-1-1) as t →-where the functions ξj(t) satisfy a first-order Toda-type system. They are given by ξ(t) = 1/2(j-k+1/2)log(-t)+ γjk, j =1,...,k, for certain explicit constants γjk.