### Abstract

We consider the parabolic one-dimensional Allen-Cahn equation u_{t} = u_{xx} + u(1-u^{2}), (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.

Original language | English |
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Pages (from-to) | 1165-1199 |

Number of pages | 35 |

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

Volume | 148 |

Issue number | 6 |

Early online date | 18 Dec 2017 |

DOIs | |

Publication status | Published - 1 Dec 2018 |

### Keywords

- Allen-Cahn equation
- ancient solutions
- nonlinear parabolic equation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Proceedings of the Royal Society of Edinburgh Section A: Mathematics*,

*148*(6), 1165-1199. https://doi.org/10.1017/S0308210517000245