{"paper":{"title":"Ancient multiple-layer solutions to the Allen-Cahn equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Konstantinos T. Gkikas, Manuel del Pino","submitted_at":"2017-03-26T09:42:17Z","abstract_excerpt":"We consider the parabolic one-dimensional Allen-Cahn equation $$u_t= u_{xx}+ u(1-u^2)\\quad (x,t)\\in \\mathbb{R}\\times (-\\infty, 0].$$ The steady state $w(x) =\\tanh (x/\\sqrt{2})$, 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 $t\\to -\\infty$. More precisely, we find $$ u(x,t) \\approx \\sum_{j=1}^k (-1)^{j-1}w(x-\\xi_j(t)) + \\frac 12 ((-1)^{k-1}- 1)\\quad \\hbox{as} t\\to -\\inf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.08796","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}