Ancient multiple-layer solutions to the Allen-Cahn equation

We consider the parabolic one-dimensional Allen-Cahn equation u_t = u_xx + 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-ξ_j(t)) + 1/2((-1)^k-1-1) as t →-where the functions ξ_j(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.