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Ancient multiple-layer solutions to the Allen–Cahn equation

Published online by Cambridge University Press:  18 December 2017

Manuel del Pino  and
Konstantinos T. Gkikas
Manuel del Pino
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile (delpino@dim.uchile.cl)
Konstantinos T. Gkikas
Affiliation:
Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile (kgkikas@dim.uchile.cl)
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Extract

We consider the parabolic one-dimensional Allen–Cahn equation

The steady state 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

where the functions ξj (t) satisfy a first-order Toda-type system. They are given by

for certain explicit constants γjk.

Keywords

nonlinear parabolic equationAllen–Cahn equationancient solutions35K5535D05
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 148 , Issue 6 , December 2018 , pp. 1165 - 1199
Copyright
Copyright © Royal Society of Edinburgh 2018 

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