{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:WOVBKUBVE6FQR6VMOLUWIHFWHZ","short_pith_number":"pith:WOVBKUBV","schema_version":"1.0","canonical_sha256":"b3aa155035278b08faac72e9641cb63e4a1054a96892e4605ed8f0f98eca2f1c","source":{"kind":"arxiv","id":"1407.1156","version":1},"attestation_state":"computed","paper":{"title":"Long-time dynamics of resonant weakly nonlinear CGL equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Guan Huang","submitted_at":"2014-07-04T09:07:51Z","abstract_excerpt":"Consider a weakly nonlinear CGL equation on the torus~$\\mathbb{T}^d$: \\[u_t+i\\Delta u=\\epsilon [\\mu(-1)^{m-1}\\Delta^{m} u+b|u|^{2p}u+ ic|u|^{2q}u].\\eqno{(*)}\\] Here $u=u(t,x)$, $x\\in\\mathbb{T}^d$, $0<\\epsilon<<1$, $\\mu\\geqslant0$, $b,c\\in\\mathbb{R}$ and $m,p,q\\in\\mathbb{N}$. Define \\mbox{$I(u)=(I_{\\dk},\\dk\\in\\mathbb{Z}^d)$}, where $I_{\\dk}=v_{\\dk}\\bar{v}_{\\dk}/2$ and $v_{\\dk}$, $\\dk\\in\\mathbb{Z}^d$, are the Fourier coefficients of the function~$u$ we give. Assume that the equation $(*)$ is well posed on time intervals of order $\\epsilon^{-1}$ and its solutions have there a-priori bounds, indep"},"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":"1407.1156","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-07-04T09:07:51Z","cross_cats_sorted":[],"title_canon_sha256":"57f2171cd072ed04cc0ec81e5a843ed9a40cfa731c565e22ded18de49abbf4a1","abstract_canon_sha256":"2a114d6fdaf024a2b96e90e0d7ad4232caa4a76f3d0cc8789b2c8a82f6e721a2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:48:18.612830Z","signature_b64":"g6giujqqOqmyHTnZEsmqruGvBIWV7Z/LhEL+bGjpqZIybUxkoTDt9czIf+mTo1vOCDn38Kfg1/x4wU+BXlkHBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b3aa155035278b08faac72e9641cb63e4a1054a96892e4605ed8f0f98eca2f1c","last_reissued_at":"2026-05-18T02:48:18.612212Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:48:18.612212Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Long-time dynamics of resonant weakly nonlinear CGL equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Guan Huang","submitted_at":"2014-07-04T09:07:51Z","abstract_excerpt":"Consider a weakly nonlinear CGL equation on the torus~$\\mathbb{T}^d$: \\[u_t+i\\Delta u=\\epsilon [\\mu(-1)^{m-1}\\Delta^{m} u+b|u|^{2p}u+ ic|u|^{2q}u].\\eqno{(*)}\\] Here $u=u(t,x)$, $x\\in\\mathbb{T}^d$, $0<\\epsilon<<1$, $\\mu\\geqslant0$, $b,c\\in\\mathbb{R}$ and $m,p,q\\in\\mathbb{N}$. Define \\mbox{$I(u)=(I_{\\dk},\\dk\\in\\mathbb{Z}^d)$}, where $I_{\\dk}=v_{\\dk}\\bar{v}_{\\dk}/2$ and $v_{\\dk}$, $\\dk\\in\\mathbb{Z}^d$, are the Fourier coefficients of the function~$u$ we give. Assume that the equation $(*)$ is well posed on time intervals of order $\\epsilon^{-1}$ and its solutions have there a-priori bounds, indep"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.1156","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":"1407.1156","created_at":"2026-05-18T02:48:18.612294+00:00"},{"alias_kind":"arxiv_version","alias_value":"1407.1156v1","created_at":"2026-05-18T02:48:18.612294+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1407.1156","created_at":"2026-05-18T02:48:18.612294+00:00"},{"alias_kind":"pith_short_12","alias_value":"WOVBKUBVE6FQ","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_16","alias_value":"WOVBKUBVE6FQR6VM","created_at":"2026-05-18T12:28:54.890064+00:00"},{"alias_kind":"pith_short_8","alias_value":"WOVBKUBV","created_at":"2026-05-18T12:28:54.890064+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/WOVBKUBVE6FQR6VMOLUWIHFWHZ","json":"https://pith.science/pith/WOVBKUBVE6FQR6VMOLUWIHFWHZ.json","graph_json":"https://pith.science/api/pith-number/WOVBKUBVE6FQR6VMOLUWIHFWHZ/graph.json","events_json":"https://pith.science/api/pith-number/WOVBKUBVE6FQR6VMOLUWIHFWHZ/events.json","paper":"https://pith.science/paper/WOVBKUBV"},"agent_actions":{"view_html":"https://pith.science/pith/WOVBKUBVE6FQR6VMOLUWIHFWHZ","download_json":"https://pith.science/pith/WOVBKUBVE6FQR6VMOLUWIHFWHZ.json","view_paper":"https://pith.science/paper/WOVBKUBV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1407.1156&json=true","fetch_graph":"https://pith.science/api/pith-number/WOVBKUBVE6FQR6VMOLUWIHFWHZ/graph.json","fetch_events":"https://pith.science/api/pith-number/WOVBKUBVE6FQR6VMOLUWIHFWHZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WOVBKUBVE6FQR6VMOLUWIHFWHZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WOVBKUBVE6FQR6VMOLUWIHFWHZ/action/storage_attestation","attest_author":"https://pith.science/pith/WOVBKUBVE6FQR6VMOLUWIHFWHZ/action/author_attestation","sign_citation":"https://pith.science/pith/WOVBKUBVE6FQR6VMOLUWIHFWHZ/action/citation_signature","submit_replication":"https://pith.science/pith/WOVBKUBVE6FQR6VMOLUWIHFWHZ/action/replication_record"}},"created_at":"2026-05-18T02:48:18.612294+00:00","updated_at":"2026-05-18T02:48:18.612294+00:00"}