{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:U5LGFYKKM3KMYNDH5SCQBU5WDE","short_pith_number":"pith:U5LGFYKK","canonical_record":{"source":{"id":"2501.07732","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-01-13T22:47:26Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"71d6c6d764223a0bc902ce602eebd310cd12799da25e356461375a5a3cad5c9b","abstract_canon_sha256":"0cdec84a86c7ff66315b7dc3f3145c24c8a8d0b9ae29925a674a6624a484ed32"},"schema_version":"1.0"},"canonical_sha256":"a75662e14a66d4cc3467ec8500d3b6191d9655d6eacb278ad7fd3b86d9722def","source":{"kind":"arxiv","id":"2501.07732","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2501.07732","created_at":"2026-07-05T10:00:49Z"},{"alias_kind":"arxiv_version","alias_value":"2501.07732v1","created_at":"2026-07-05T10:00:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.07732","created_at":"2026-07-05T10:00:49Z"},{"alias_kind":"pith_short_12","alias_value":"U5LGFYKKM3KM","created_at":"2026-07-05T10:00:49Z"},{"alias_kind":"pith_short_16","alias_value":"U5LGFYKKM3KMYNDH","created_at":"2026-07-05T10:00:49Z"},{"alias_kind":"pith_short_8","alias_value":"U5LGFYKK","created_at":"2026-07-05T10:00:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:U5LGFYKKM3KMYNDH5SCQBU5WDE","target":"record","payload":{"canonical_record":{"source":{"id":"2501.07732","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-01-13T22:47:26Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"71d6c6d764223a0bc902ce602eebd310cd12799da25e356461375a5a3cad5c9b","abstract_canon_sha256":"0cdec84a86c7ff66315b7dc3f3145c24c8a8d0b9ae29925a674a6624a484ed32"},"schema_version":"1.0"},"canonical_sha256":"a75662e14a66d4cc3467ec8500d3b6191d9655d6eacb278ad7fd3b86d9722def","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:00:49.178572Z","signature_b64":"WrLKRfKW87CfQPEdIlDTJImIznyfHZ5DY0ADL5U8avP4xt6vu0hFpIi4ytK4tIpm4f0R/ByRF/4hP2URQXCLAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a75662e14a66d4cc3467ec8500d3b6191d9655d6eacb278ad7fd3b86d9722def","last_reissued_at":"2026-07-05T10:00:49.178098Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:00:49.178098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2501.07732","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":"2026-07-05T10:00:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1BRbb6Avv89zwvSaD/mTh/vdVLmLMYLsiJxRSkqKpJ31Sio2CHfD3B+agdWaQhT+hnEBpmmNNjkjW8Tlxn38Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T10:02:42.164939Z"},"content_sha256":"21deda1c38de84201d2349df88980d70b0d778fc7d52c6f1316119e225adfede","schema_version":"1.0","event_id":"sha256:21deda1c38de84201d2349df88980d70b0d778fc7d52c6f1316119e225adfede"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:U5LGFYKKM3KMYNDH5SCQBU5WDE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The large time asymptotics of nonlinear multichannel Schroedinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Avy Soffer, Baoping Liu","submitted_at":"2025-01-13T22:47:26Z","abstract_excerpt":"We consider the Schroedinger equation with a general interaction term, which is localized in space. The interaction may be x, t dependent and non-linear. Purely non-linear parts of the interaction are localized via the radial Sobolev embedding. Under the assumption of radial symmetry and boundedness in H1(R3) of the solution, uniformly in time. we prove it is asymptotic in L2 (and H1) in the strong sense, to a free wave and a weakly localized solution. The general properties of the localized solutions are derived. The proof is based on the introduction of phase-space analysis of the nonlinear "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.07732","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.07732/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T10:00:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hT2mEX7hC2BZRFxkw/XNIkS3ZEnLexj638eUFYZmbAEkDnTWz5YYsFPa+FobLVRieykOZbOGbeOKJGRIkXKKDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-11T10:02:42.165868Z"},"content_sha256":"44ebe642ce5885622077537b46d3766ca4b600854056877ca0b93ca24f798dc1","schema_version":"1.0","event_id":"sha256:44ebe642ce5885622077537b46d3766ca4b600854056877ca0b93ca24f798dc1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U5LGFYKKM3KMYNDH5SCQBU5WDE/bundle.json","state_url":"https://pith.science/pith/U5LGFYKKM3KMYNDH5SCQBU5WDE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U5LGFYKKM3KMYNDH5SCQBU5WDE/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":"2026-08-11T10:02:42Z","links":{"resolver":"https://pith.science/pith/U5LGFYKKM3KMYNDH5SCQBU5WDE","bundle":"https://pith.science/pith/U5LGFYKKM3KMYNDH5SCQBU5WDE/bundle.json","state":"https://pith.science/pith/U5LGFYKKM3KMYNDH5SCQBU5WDE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U5LGFYKKM3KMYNDH5SCQBU5WDE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:U5LGFYKKM3KMYNDH5SCQBU5WDE","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":"0cdec84a86c7ff66315b7dc3f3145c24c8a8d0b9ae29925a674a6624a484ed32","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-01-13T22:47:26Z","title_canon_sha256":"71d6c6d764223a0bc902ce602eebd310cd12799da25e356461375a5a3cad5c9b"},"schema_version":"1.0","source":{"id":"2501.07732","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2501.07732","created_at":"2026-07-05T10:00:49Z"},{"alias_kind":"arxiv_version","alias_value":"2501.07732v1","created_at":"2026-07-05T10:00:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.07732","created_at":"2026-07-05T10:00:49Z"},{"alias_kind":"pith_short_12","alias_value":"U5LGFYKKM3KM","created_at":"2026-07-05T10:00:49Z"},{"alias_kind":"pith_short_16","alias_value":"U5LGFYKKM3KMYNDH","created_at":"2026-07-05T10:00:49Z"},{"alias_kind":"pith_short_8","alias_value":"U5LGFYKK","created_at":"2026-07-05T10:00:49Z"}],"graph_snapshots":[{"event_id":"sha256:44ebe642ce5885622077537b46d3766ca4b600854056877ca0b93ca24f798dc1","target":"graph","created_at":"2026-07-05T10:00:49Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2501.07732/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider the Schroedinger equation with a general interaction term, which is localized in space. The interaction may be x, t dependent and non-linear. Purely non-linear parts of the interaction are localized via the radial Sobolev embedding. Under the assumption of radial symmetry and boundedness in H1(R3) of the solution, uniformly in time. we prove it is asymptotic in L2 (and H1) in the strong sense, to a free wave and a weakly localized solution. The general properties of the localized solutions are derived. The proof is based on the introduction of phase-space analysis of the nonlinear ","authors_text":"Avy Soffer, Baoping Liu","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-01-13T22:47:26Z","title":"The large time asymptotics of nonlinear multichannel Schroedinger equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.07732","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":"sha256:21deda1c38de84201d2349df88980d70b0d778fc7d52c6f1316119e225adfede","target":"record","created_at":"2026-07-05T10:00:49Z","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":"0cdec84a86c7ff66315b7dc3f3145c24c8a8d0b9ae29925a674a6624a484ed32","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2025-01-13T22:47:26Z","title_canon_sha256":"71d6c6d764223a0bc902ce602eebd310cd12799da25e356461375a5a3cad5c9b"},"schema_version":"1.0","source":{"id":"2501.07732","kind":"arxiv","version":1}},"canonical_sha256":"a75662e14a66d4cc3467ec8500d3b6191d9655d6eacb278ad7fd3b86d9722def","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a75662e14a66d4cc3467ec8500d3b6191d9655d6eacb278ad7fd3b86d9722def","first_computed_at":"2026-07-05T10:00:49.178098Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:00:49.178098Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WrLKRfKW87CfQPEdIlDTJImIznyfHZ5DY0ADL5U8avP4xt6vu0hFpIi4ytK4tIpm4f0R/ByRF/4hP2URQXCLAA==","signature_status":"signed_v1","signed_at":"2026-07-05T10:00:49.178572Z","signed_message":"canonical_sha256_bytes"},"source_id":"2501.07732","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:21deda1c38de84201d2349df88980d70b0d778fc7d52c6f1316119e225adfede","sha256:44ebe642ce5885622077537b46d3766ca4b600854056877ca0b93ca24f798dc1"],"state_sha256":"0f55e6d17f46ec260731b9114bc0f304fb0aedaf3f8353c0cefa99c237c08a05"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jfsLHy7tpyF/xYsdPEekB46nScggmVvR18zbW9ifeN8TXAiU1WcAONxbNOx+iQxPE0MD3Gq8TIxaXbJHjUXqDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-11T10:02:42.189004Z","bundle_sha256":"2fee97fa5b25ac925710f834fde8a102f9b8891bbb4b88cb6ff1f4c81cd185bb"}}