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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 $$ u(x,t) \\approx \\sum_{j=1}^k (-1)^{j-1}w(x-\\xi_j(t)) + \\frac 12 ((-1)^{k-1}- 1)\\quad \\hbox{as} t\\to -\\inf"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1703.08796","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-26T09:42:17Z","cross_cats_sorted":[],"title_canon_sha256":"a5005979608e2db2fe0b3c15377bb97b65d7e3a2d6df4a55aa8d1f44741a4681","abstract_canon_sha256":"131b2894bf5cde182d735878e4bd875e320b089b175158271316fa7f1f9f0a95"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:55.530214Z","signature_b64":"2nqX34hQncPFvk5Dw4nLJNQ4O8gPxkT+ii+9gq+ajhc4uaRPead1RgPVpqQhuPPPdLDBV5JmLb4wqwjVRSMVBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"988f15f66b8fb7e8be1cabdd67385c0c3dc96bd35f1ba2a760ac715790f0dfd5","last_reissued_at":"2026-05-18T00:47:55.529432Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:55.529432Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ancient multiple-layer solutions to the Allen-Cahn equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Konstantinos T. Gkikas, Manuel del Pino","submitted_at":"2017-03-26T09:42:17Z","abstract_excerpt":"We consider the parabolic one-dimensional Allen-Cahn equation $$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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1703.08796","created_at":"2026-05-18T00:47:55.529539+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.08796v1","created_at":"2026-05-18T00:47:55.529539+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.08796","created_at":"2026-05-18T00:47:55.529539+00:00"},{"alias_kind":"pith_short_12","alias_value":"TCHRL5TLR636","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_16","alias_value":"TCHRL5TLR636RPQ4","created_at":"2026-05-18T12:31:46.661854+00:00"},{"alias_kind":"pith_short_8","alias_value":"TCHRL5TL","created_at":"2026-05-18T12:31:46.661854+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TCHRL5TLR636RPQ4VPOWOOC4BQ","json":"https://pith.science/pith/TCHRL5TLR636RPQ4VPOWOOC4BQ.json","graph_json":"https://pith.science/api/pith-number/TCHRL5TLR636RPQ4VPOWOOC4BQ/graph.json","events_json":"https://pith.science/api/pith-number/TCHRL5TLR636RPQ4VPOWOOC4BQ/events.json","paper":"https://pith.science/paper/TCHRL5TL"},"agent_actions":{"view_html":"https://pith.science/pith/TCHRL5TLR636RPQ4VPOWOOC4BQ","download_json":"https://pith.science/pith/TCHRL5TLR636RPQ4VPOWOOC4BQ.json","view_paper":"https://pith.science/paper/TCHRL5TL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.08796&json=true","fetch_graph":"https://pith.science/api/pith-number/TCHRL5TLR636RPQ4VPOWOOC4BQ/graph.json","fetch_events":"https://pith.science/api/pith-number/TCHRL5TLR636RPQ4VPOWOOC4BQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TCHRL5TLR636RPQ4VPOWOOC4BQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TCHRL5TLR636RPQ4VPOWOOC4BQ/action/storage_attestation","attest_author":"https://pith.science/pith/TCHRL5TLR636RPQ4VPOWOOC4BQ/action/author_attestation","sign_citation":"https://pith.science/pith/TCHRL5TLR636RPQ4VPOWOOC4BQ/action/citation_signature","submit_replication":"https://pith.science/pith/TCHRL5TLR636RPQ4VPOWOOC4BQ/action/replication_record"}},"created_at":"2026-05-18T00:47:55.529539+00:00","updated_at":"2026-05-18T00:47:55.529539+00:00"}