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We show, via variational methods, that if the set of solutions to the one dimensional system $-\\ddot q(x)+\\nabla W(q(x))=0,\\ x\\in\\R$, which connect the two minima of $W$ as $x\\to\\pm\\infty$ has a discrete structure, then the given system has infinitely many layered solutions."},"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":"1211.5751","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-25T10:07:25Z","cross_cats_sorted":[],"title_canon_sha256":"bd51c04051e328a5a4cea1506615d1e296243d4b6f46e492c3c2444f94d00344","abstract_canon_sha256":"373231b35e9ccd0ddc1e14c826d0e71b0f775f8805d191d143585b44cda31aca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:54.033129Z","signature_b64":"Ia2HbxG7Uq1IFU35az5qmaMNbDkFQcq/sxuhvYZlipcuvySAHHDosJRLTeBiVnc1C4T15DkCYBfsT1U4MMl0DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95877fecf2b209184348d94338912e97e038afc8a695d7999f11fed02b07f5f2","last_reissued_at":"2026-05-18T02:53:54.032344Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:54.032344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stationary layered solutions for a system of Allen-Cahn type equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Francesca Alessio","submitted_at":"2012-11-25T10:07:25Z","abstract_excerpt":"We consider a class of semilinear elliptic system of the form $-\\Delta u(x,y)+\\nabla W(u(x,y))=0,\\quad (x,y)\\in\\R^{2}$ where $W:\\R^{2}\\to\\R$ is a double well non negative symmetric potential. We show, via variational methods, that if the set of solutions to the one dimensional system $-\\ddot q(x)+\\nabla W(q(x))=0,\\ x\\in\\R$, which connect the two minima of $W$ as $x\\to\\pm\\infty$ has a discrete structure, then the given system has infinitely many layered solutions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5751","kind":"arxiv","version":2},"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":"1211.5751","created_at":"2026-05-18T02:53:54.032479+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.5751v2","created_at":"2026-05-18T02:53:54.032479+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.5751","created_at":"2026-05-18T02:53:54.032479+00:00"},{"alias_kind":"pith_short_12","alias_value":"SWDX73HSWIER","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_16","alias_value":"SWDX73HSWIERQQ2I","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_8","alias_value":"SWDX73HS","created_at":"2026-05-18T12:27:23.164592+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/SWDX73HSWIERQQ2I3FBTREJOS7","json":"https://pith.science/pith/SWDX73HSWIERQQ2I3FBTREJOS7.json","graph_json":"https://pith.science/api/pith-number/SWDX73HSWIERQQ2I3FBTREJOS7/graph.json","events_json":"https://pith.science/api/pith-number/SWDX73HSWIERQQ2I3FBTREJOS7/events.json","paper":"https://pith.science/paper/SWDX73HS"},"agent_actions":{"view_html":"https://pith.science/pith/SWDX73HSWIERQQ2I3FBTREJOS7","download_json":"https://pith.science/pith/SWDX73HSWIERQQ2I3FBTREJOS7.json","view_paper":"https://pith.science/paper/SWDX73HS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.5751&json=true","fetch_graph":"https://pith.science/api/pith-number/SWDX73HSWIERQQ2I3FBTREJOS7/graph.json","fetch_events":"https://pith.science/api/pith-number/SWDX73HSWIERQQ2I3FBTREJOS7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SWDX73HSWIERQQ2I3FBTREJOS7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SWDX73HSWIERQQ2I3FBTREJOS7/action/storage_attestation","attest_author":"https://pith.science/pith/SWDX73HSWIERQQ2I3FBTREJOS7/action/author_attestation","sign_citation":"https://pith.science/pith/SWDX73HSWIERQQ2I3FBTREJOS7/action/citation_signature","submit_replication":"https://pith.science/pith/SWDX73HSWIERQQ2I3FBTREJOS7/action/replication_record"}},"created_at":"2026-05-18T02:53:54.032479+00:00","updated_at":"2026-05-18T02:53:54.032479+00:00"}