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Namely, we show that, for generic $m$, most of the small amplitude invariant finite dimensional tori of the linear equation $(*)_{g=0}$, written as the system $$ u_t=-v,\\quad v_t=\\Delta^2 u+mu, $$, persist as invariant tori of the nonlinear equation $(*)$, re-written similarly. If $d\\ge2$, then not all the persisted tori are linearly stable, and we construct explic"},"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":"1412.2803","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-08T22:29:08Z","cross_cats_sorted":[],"title_canon_sha256":"4d9082646522995633c9e74d4949a0781e2368b4b543824acdf7e236982e607a","abstract_canon_sha256":"31e20594f7341490373bf7a729730c45800ce8ace4f3af9a8ffe231743b83d0b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:33.227256Z","signature_b64":"dtOJVGxXpueEmfGjCRH9EaSdeTHHFneUVJMn10kz9+zAz9MvLMKUwzLNHoz95XhEaeXoiH+splU93k0syqsrBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3812e4cd8b87a98521c8ca5e56ff9fe5aa2013b55c73c1c4cc53a98a02ee8ee8","last_reissued_at":"2026-05-18T01:24:33.226627Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:33.226627Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"KAM for the nonlinear beam equation 1: small-amplitude solutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Benoit Grebert, Hakan L. Eliasson, Sergei B. Kuksin","submitted_at":"2014-12-08T22:29:08Z","abstract_excerpt":"In this paper we prove a KAM result for the non linear beam equation on the d-dimensional torus $$u_{tt}+\\Delta^2 u+m u + g(x,u)=0\\ ,\\quad t\\in { \\mathbb{R}} , \\; x\\in {\\mathbb T}^d, \\qquad \\qquad (*) $$ where $g(x,u)=4u^3+ O(u^4)$. Namely, we show that, for generic $m$, most of the small amplitude invariant finite dimensional tori of the linear equation $(*)_{g=0}$, written as the system $$ u_t=-v,\\quad v_t=\\Delta^2 u+mu, $$, persist as invariant tori of the nonlinear equation $(*)$, re-written similarly. If $d\\ge2$, then not all the persisted tori are linearly stable, and we construct explic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.2803","kind":"arxiv","version":3},"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":"1412.2803","created_at":"2026-05-18T01:24:33.226721+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.2803v3","created_at":"2026-05-18T01:24:33.226721+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.2803","created_at":"2026-05-18T01:24:33.226721+00:00"},{"alias_kind":"pith_short_12","alias_value":"HAJOJTMLQ6UY","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_16","alias_value":"HAJOJTMLQ6UYKIOI","created_at":"2026-05-18T12:28:30.664211+00:00"},{"alias_kind":"pith_short_8","alias_value":"HAJOJTML","created_at":"2026-05-18T12:28:30.664211+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/HAJOJTMLQ6UYKIOIZJPFN7474W","json":"https://pith.science/pith/HAJOJTMLQ6UYKIOIZJPFN7474W.json","graph_json":"https://pith.science/api/pith-number/HAJOJTMLQ6UYKIOIZJPFN7474W/graph.json","events_json":"https://pith.science/api/pith-number/HAJOJTMLQ6UYKIOIZJPFN7474W/events.json","paper":"https://pith.science/paper/HAJOJTML"},"agent_actions":{"view_html":"https://pith.science/pith/HAJOJTMLQ6UYKIOIZJPFN7474W","download_json":"https://pith.science/pith/HAJOJTMLQ6UYKIOIZJPFN7474W.json","view_paper":"https://pith.science/paper/HAJOJTML","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.2803&json=true","fetch_graph":"https://pith.science/api/pith-number/HAJOJTMLQ6UYKIOIZJPFN7474W/graph.json","fetch_events":"https://pith.science/api/pith-number/HAJOJTMLQ6UYKIOIZJPFN7474W/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HAJOJTMLQ6UYKIOIZJPFN7474W/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HAJOJTMLQ6UYKIOIZJPFN7474W/action/storage_attestation","attest_author":"https://pith.science/pith/HAJOJTMLQ6UYKIOIZJPFN7474W/action/author_attestation","sign_citation":"https://pith.science/pith/HAJOJTMLQ6UYKIOIZJPFN7474W/action/citation_signature","submit_replication":"https://pith.science/pith/HAJOJTMLQ6UYKIOIZJPFN7474W/action/replication_record"}},"created_at":"2026-05-18T01:24:33.226721+00:00","updated_at":"2026-05-18T01:24:33.226721+00:00"}