Ancient multiple-layer solutions to the Allen-Cahn equation

Manuel Del Pino, Konstantinos T. Gkikas

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7 Citations (SciVal)


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.

Original languageEnglish
Pages (from-to)1165-1199
Number of pages35
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Issue number6
Early online date18 Dec 2017
Publication statusPublished - 1 Dec 2018


  • Allen-Cahn equation
  • ancient solutions
  • nonlinear parabolic equation

ASJC Scopus subject areas

  • General Mathematics


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