{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:FLDZ37WS2ATKGBJAOTGISGD5JL","short_pith_number":"pith:FLDZ37WS","schema_version":"1.0","canonical_sha256":"2ac79dfed2d026a3052074cc89187d4ae44ae3d04a62624a2152f7126ed479f0","source":{"kind":"arxiv","id":"1411.2143","version":4},"attestation_state":"computed","paper":{"title":"Time-averaging for weakly nonlinear CGL equations with arbitrary potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alberto Maiocchi, Guan Huang, Sergei Kuksin","submitted_at":"2014-11-08T17:48:38Z","abstract_excerpt":"Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the form: $$ u_t+i(-\\Delta u+V(x)u)=\\epsilon\\mu\\Delta u+\\epsilon \\mathcal{P}( u),\\quad x\\in {R^d}\\,, \\quad(*)\n  $$ under the periodic boundary conditions, where $\\mu\\geqslant0$ and $\\mathcal{P}$ is a smooth function. Let $\\{\\zeta_1(x),\\zeta_2(x),\\dots\\}$ be the $L_2$-basis formed by eigenfunctions of the operator $-\\Delta +V(x)$. For a complex function $u(x)$, write it as $u(x)=\\sum_{k\\geqslant1}v_k\\zeta_k(x)$ and set $I_k(u)=\\frac{1}{2}|v_k|^2$. Then for any solution $u(t,x)$ of the linear equation $(*)_{\\epsilon=0}$ we have"},"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":"1411.2143","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-11-08T17:48:38Z","cross_cats_sorted":[],"title_canon_sha256":"1512dba0003d16670eaf2d8ee5f58af2cac2e1782adabec1bf115e0879508582","abstract_canon_sha256":"ef3863a17746ab1578826023f44b0acf1349995691931c752ab4f2c90c848bef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:33.262792Z","signature_b64":"dazkp9lXiFj9ivhx22QbqWGRTyyPDat3bUX6z87v0fNMq8U+UF2S0OjWLuqBcHeaMRCyMW28echDWfJ74aBdBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ac79dfed2d026a3052074cc89187d4ae44ae3d04a62624a2152f7126ed479f0","last_reissued_at":"2026-05-18T01:24:33.262114Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:33.262114Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Time-averaging for weakly nonlinear CGL equations with arbitrary potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alberto Maiocchi, Guan Huang, Sergei Kuksin","submitted_at":"2014-11-08T17:48:38Z","abstract_excerpt":"Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the form: $$ u_t+i(-\\Delta u+V(x)u)=\\epsilon\\mu\\Delta u+\\epsilon \\mathcal{P}( u),\\quad x\\in {R^d}\\,, \\quad(*)\n  $$ under the periodic boundary conditions, where $\\mu\\geqslant0$ and $\\mathcal{P}$ is a smooth function. Let $\\{\\zeta_1(x),\\zeta_2(x),\\dots\\}$ be the $L_2$-basis formed by eigenfunctions of the operator $-\\Delta +V(x)$. For a complex function $u(x)$, write it as $u(x)=\\sum_{k\\geqslant1}v_k\\zeta_k(x)$ and set $I_k(u)=\\frac{1}{2}|v_k|^2$. Then for any solution $u(t,x)$ of the linear equation $(*)_{\\epsilon=0}$ we have"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.2143","kind":"arxiv","version":4},"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":"1411.2143","created_at":"2026-05-18T01:24:33.262211+00:00"},{"alias_kind":"arxiv_version","alias_value":"1411.2143v4","created_at":"2026-05-18T01:24:33.262211+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.2143","created_at":"2026-05-18T01:24:33.262211+00:00"},{"alias_kind":"pith_short_12","alias_value":"FLDZ37WS2ATK","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_16","alias_value":"FLDZ37WS2ATKGBJA","created_at":"2026-05-18T12:28:28.263976+00:00"},{"alias_kind":"pith_short_8","alias_value":"FLDZ37WS","created_at":"2026-05-18T12:28:28.263976+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/FLDZ37WS2ATKGBJAOTGISGD5JL","json":"https://pith.science/pith/FLDZ37WS2ATKGBJAOTGISGD5JL.json","graph_json":"https://pith.science/api/pith-number/FLDZ37WS2ATKGBJAOTGISGD5JL/graph.json","events_json":"https://pith.science/api/pith-number/FLDZ37WS2ATKGBJAOTGISGD5JL/events.json","paper":"https://pith.science/paper/FLDZ37WS"},"agent_actions":{"view_html":"https://pith.science/pith/FLDZ37WS2ATKGBJAOTGISGD5JL","download_json":"https://pith.science/pith/FLDZ37WS2ATKGBJAOTGISGD5JL.json","view_paper":"https://pith.science/paper/FLDZ37WS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1411.2143&json=true","fetch_graph":"https://pith.science/api/pith-number/FLDZ37WS2ATKGBJAOTGISGD5JL/graph.json","fetch_events":"https://pith.science/api/pith-number/FLDZ37WS2ATKGBJAOTGISGD5JL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FLDZ37WS2ATKGBJAOTGISGD5JL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FLDZ37WS2ATKGBJAOTGISGD5JL/action/storage_attestation","attest_author":"https://pith.science/pith/FLDZ37WS2ATKGBJAOTGISGD5JL/action/author_attestation","sign_citation":"https://pith.science/pith/FLDZ37WS2ATKGBJAOTGISGD5JL/action/citation_signature","submit_replication":"https://pith.science/pith/FLDZ37WS2ATKGBJAOTGISGD5JL/action/replication_record"}},"created_at":"2026-05-18T01:24:33.262211+00:00","updated_at":"2026-05-18T01:24:33.262211+00:00"}