Ancient multiple-layer solutions to the Allen–Cahn equation

Abstract

We consider the parabolic one-dimensional Allen–Cahn equationThe 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 findwhere the functions ξj (t) satisfy a first-order Toda-type system. They are given byfor certain explicit constants γjk.