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Ancient multiple-layer solutions to the Allen–Cahn equation

Author:
del Pino, Manuel   Gkikas, Konstantinos T.   


Journal:
Proceedings of the Royal Society of Edinburgh: Section A Mathematics


Issue Date:
2017


Abstract(summary):

We consider the parabolic one-dimensional Allen-Cahn equation u(t) =3D u(xx) + u(1 - u(2)), (x, t) is an element of R x (-infinity, 0]. The steady state w(x) =3D tanh(x/root 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 -> -infinity. More precisely, we find u(x, t) approximate to Sigma(k )(j=3D1)(-1)(j-1) w(x-xi(j)(t)) + 1/2 ((-1)(k-1) - 1) as t -> -infinity, where the functions xi(j)(t) satisfy a first-order Toda-type system. They are given by xi(j)(t) =3D 1/root 2(j-k+1/2) log(-t) + gamma(jk), j=3D1,...,k, for certain explicit constants gamma(jk).


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