Ancient multiple-layer solutions to the Allen-Cahn equation

2017/03/26 by Manuel del Pino, del Pino, Manuel, Konstantinos T. Gkikas +1
Computer Science · Materials Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Solidification and crystal growth phenomena #Theoretical and Computational Physics #math.AP

paper · pdf · doi:10.48550/arxiv.1703.08796

arxiv created 2017/03/26 · openalex publication_date 2017/03/26 · arxiv updated 2017/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the parabolic one-dimensional Allen-Cahn equation u_t= u_xx+ u(1-u²) (x,t)∈ ℝ× (-∞, 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. At main order they consist of k time-traveling copies of w with interfaces diverging one to each other as t→ -∞. More precisely, we find u(x,t) ≈ ∑_j=1^k (-1)^j-1w(x-ξ_j(t)) + \frac 12 ((-1)^k-1- 1) \hboxas t→ -∞, where the functions ξ_j(t) satisfy a first order Toda-type system. They are given by ξ_j(t)=(1)/(√(2))(j-(k+1)/(2))log(-t)+γ_jk, j=1,...,k, for certain explicit constants γ_jk.

Citations

• Type II ancient compact solutions to the Yamabe flow

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