{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:XB5HRG6YYDXPTO4QAOYMNLXHFL","short_pith_number":"pith:XB5HRG6Y","canonical_record":{"source":{"id":"1106.1158","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-06-06T19:31:20Z","cross_cats_sorted":["math.AP","math.MP"],"title_canon_sha256":"bdb5c62b52ff2236301b351ab21eff1671d360df0e0e9553d370c41fe1cfb0bf","abstract_canon_sha256":"d990c2086a1c5f971d037ebb49f7a1b31a6e8aecfe84e31bdd9c3c67994a8d28"},"schema_version":"1.0"},"canonical_sha256":"b87a789bd8c0eef9bb9003b0c6aee72af2d845b1997d15610dcd27603a06a63e","source":{"kind":"arxiv","id":"1106.1158","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.1158","created_at":"2026-05-18T03:13:00Z"},{"alias_kind":"arxiv_version","alias_value":"1106.1158v1","created_at":"2026-05-18T03:13:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.1158","created_at":"2026-05-18T03:13:00Z"},{"alias_kind":"pith_short_12","alias_value":"XB5HRG6YYDXP","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"XB5HRG6YYDXPTO4Q","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"XB5HRG6Y","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:XB5HRG6YYDXPTO4QAOYMNLXHFL","target":"record","payload":{"canonical_record":{"source":{"id":"1106.1158","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-06-06T19:31:20Z","cross_cats_sorted":["math.AP","math.MP"],"title_canon_sha256":"bdb5c62b52ff2236301b351ab21eff1671d360df0e0e9553d370c41fe1cfb0bf","abstract_canon_sha256":"d990c2086a1c5f971d037ebb49f7a1b31a6e8aecfe84e31bdd9c3c67994a8d28"},"schema_version":"1.0"},"canonical_sha256":"b87a789bd8c0eef9bb9003b0c6aee72af2d845b1997d15610dcd27603a06a63e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:00.628099Z","signature_b64":"6qF/jA0HYpXF//tIrXmMM7+gi7XRAU5/Qtlo+huY6if0wDKK44Wn+mCnMDOUrq6tKtJGUbuh+R/oWEmrrTyLAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b87a789bd8c0eef9bb9003b0c6aee72af2d845b1997d15610dcd27603a06a63e","last_reissued_at":"2026-05-18T03:13:00.627351Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:00.627351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1106.1158","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:13:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tImwL1QHMbHUqq6hMsaV2gRczfHUrI7wJG+hzQUh3eBEqzv+cWXix+1AGHQ14VMwl6IGPD8m/b8unHVgxGnIAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T17:52:04.785150Z"},"content_sha256":"533d4a98b01554c15c35b77c9d022a92f7f441612a4ed0e1e245606c5fb70018","schema_version":"1.0","event_id":"sha256:533d4a98b01554c15c35b77c9d022a92f7f441612a4ed0e1e245606c5fb70018"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:XB5HRG6YYDXPTO4QAOYMNLXHFL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weakly nonlinear stochastic CGL equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Sergei B. Kuksin","submitted_at":"2011-06-06T19:31:20Z","abstract_excerpt":"We consider the linear Schr\\\"odinger equation under periodic boundary condition, driven by a random force and damped by a quasilinear damping: $$ \\frac{d}{dt}u+i\\big(-\\Delta+V(x)\\big) u=\\nu \\Big(\\Delta u-\\gr |u|^{2p}u-i\\gi |u|^{2q}u \\Big) +\\sqrt\\nu\\, \\eta(t,x).\\qquad (*) $$ The force $\\eta$ is white in time and smooth in $x$. We are concerned with the limiting, as $\\nu\\to0$, behaviour of its solutions on long time-intervals $0\\le t\\le\\nu^{-1}T$, and with behaviour of these solutions under the double limit $t\\to\\infty$ and $\\nu\\to0$. We show that these two limiting behaviours may be described i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.1158","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:13:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YZJQ+uwfeORneh51dG25czSPsiUchw6D/w5HVF9SRywIOJWL3Gpd9V8UuMl5fgU0m7qv4EjgEEjIf0xWIIGzAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T17:52:04.785520Z"},"content_sha256":"6fb4dd3c4ef02b1711357bf1ffef0f8f378501286bcb7174a2c9bebdd10d5940","schema_version":"1.0","event_id":"sha256:6fb4dd3c4ef02b1711357bf1ffef0f8f378501286bcb7174a2c9bebdd10d5940"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XB5HRG6YYDXPTO4QAOYMNLXHFL/bundle.json","state_url":"https://pith.science/pith/XB5HRG6YYDXPTO4QAOYMNLXHFL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XB5HRG6YYDXPTO4QAOYMNLXHFL/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T17:52:04Z","links":{"resolver":"https://pith.science/pith/XB5HRG6YYDXPTO4QAOYMNLXHFL","bundle":"https://pith.science/pith/XB5HRG6YYDXPTO4QAOYMNLXHFL/bundle.json","state":"https://pith.science/pith/XB5HRG6YYDXPTO4QAOYMNLXHFL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XB5HRG6YYDXPTO4QAOYMNLXHFL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:XB5HRG6YYDXPTO4QAOYMNLXHFL","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d990c2086a1c5f971d037ebb49f7a1b31a6e8aecfe84e31bdd9c3c67994a8d28","cross_cats_sorted":["math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-06-06T19:31:20Z","title_canon_sha256":"bdb5c62b52ff2236301b351ab21eff1671d360df0e0e9553d370c41fe1cfb0bf"},"schema_version":"1.0","source":{"id":"1106.1158","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1106.1158","created_at":"2026-05-18T03:13:00Z"},{"alias_kind":"arxiv_version","alias_value":"1106.1158v1","created_at":"2026-05-18T03:13:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.1158","created_at":"2026-05-18T03:13:00Z"},{"alias_kind":"pith_short_12","alias_value":"XB5HRG6YYDXP","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"XB5HRG6YYDXPTO4Q","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"XB5HRG6Y","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:6fb4dd3c4ef02b1711357bf1ffef0f8f378501286bcb7174a2c9bebdd10d5940","target":"graph","created_at":"2026-05-18T03:13:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We consider the linear Schr\\\"odinger equation under periodic boundary condition, driven by a random force and damped by a quasilinear damping: $$ \\frac{d}{dt}u+i\\big(-\\Delta+V(x)\\big) u=\\nu \\Big(\\Delta u-\\gr |u|^{2p}u-i\\gi |u|^{2q}u \\Big) +\\sqrt\\nu\\, \\eta(t,x).\\qquad (*) $$ The force $\\eta$ is white in time and smooth in $x$. We are concerned with the limiting, as $\\nu\\to0$, behaviour of its solutions on long time-intervals $0\\le t\\le\\nu^{-1}T$, and with behaviour of these solutions under the double limit $t\\to\\infty$ and $\\nu\\to0$. We show that these two limiting behaviours may be described i","authors_text":"Sergei B. Kuksin","cross_cats":["math.AP","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-06-06T19:31:20Z","title":"Weakly nonlinear stochastic CGL equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.1158","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:533d4a98b01554c15c35b77c9d022a92f7f441612a4ed0e1e245606c5fb70018","target":"record","created_at":"2026-05-18T03:13:00Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d990c2086a1c5f971d037ebb49f7a1b31a6e8aecfe84e31bdd9c3c67994a8d28","cross_cats_sorted":["math.AP","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2011-06-06T19:31:20Z","title_canon_sha256":"bdb5c62b52ff2236301b351ab21eff1671d360df0e0e9553d370c41fe1cfb0bf"},"schema_version":"1.0","source":{"id":"1106.1158","kind":"arxiv","version":1}},"canonical_sha256":"b87a789bd8c0eef9bb9003b0c6aee72af2d845b1997d15610dcd27603a06a63e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b87a789bd8c0eef9bb9003b0c6aee72af2d845b1997d15610dcd27603a06a63e","first_computed_at":"2026-05-18T03:13:00.627351Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:00.627351Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6qF/jA0HYpXF//tIrXmMM7+gi7XRAU5/Qtlo+huY6if0wDKK44Wn+mCnMDOUrq6tKtJGUbuh+R/oWEmrrTyLAA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:00.628099Z","signed_message":"canonical_sha256_bytes"},"source_id":"1106.1158","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:533d4a98b01554c15c35b77c9d022a92f7f441612a4ed0e1e245606c5fb70018","sha256:6fb4dd3c4ef02b1711357bf1ffef0f8f378501286bcb7174a2c9bebdd10d5940"],"state_sha256":"2a8ee9737a110aca0b1b5cecfc97c00d1c4eafe2e89736323008f5118f473867"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xHiRW0NZr2+qHRPD02yxSC+j1HVSa98IQvw/XiMoK5sB2ki8E49giZTDMeSat0W2vLaL6HSfsBz6MgEOj0AbAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T17:52:04.787570Z","bundle_sha256":"e3e18b24b2bb700ce1a56da8229a9ffbc6934fb4c18669bc76dd01eee682df0a"}}