Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T18:26:43.019Z Has data issue: false hasContentIssue false
  • English
  • Français

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 ([email protected])
Konstantinos T. Gkikas
Affiliation:
Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile ([email protected])
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)