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It admits an eight-dimensional manifold of special solutions called ground state solitons.\n  We exhibit a codimension-one critical real-analytic manifold N of asymptotically stable solutions in a neighborhood of the soliton manifold. We then show that N is centre-stable, in the dynamical systems sense of Bates-Jones, and globally-in-time invariant.\n  Solutions in N are asymptotically stable and separate into two asymptotically free parts that decouple in the limit --- a so"},"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":"0909.1180","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2009-09-07T09:13:33Z","cross_cats_sorted":[],"title_canon_sha256":"d4ff44be3918bd8ce05cabdcee3101c7941b13de7dc046e345ef20d7de175561","abstract_canon_sha256":"1c5f13b779cda87677858f0cd8a5749d2cba1a117db2f0aefb03ed4fd3cfe969"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:22:11.594768Z","signature_b64":"fOQgcTpuD64TQaRK8lWPUhDZIjEJABWnMAkujDww1QKxez1gmY2V7a65XS/4OnfIPzBMyG3bZ2t3U3wyhvvWDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"59b400ec6db9328e798173abaeaca5dd6dba809a6304bc62226f4b04a61a2e8d","last_reissued_at":"2026-05-18T04:22:11.594353Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:22:11.594353Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Critical Centre-Stable Manifold for Schroedinger's Equation in R^3","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Marius Beceanu","submitted_at":"2009-09-07T09:13:33Z","abstract_excerpt":"Consider the focusing cubic semilinear Schroedinger equation in R^3 i \\partial_t \\psi + \\Delta \\psi + | \\psi |^2 \\psi = 0. It admits an eight-dimensional manifold of special solutions called ground state solitons.\n  We exhibit a codimension-one critical real-analytic manifold N of asymptotically stable solutions in a neighborhood of the soliton manifold. We then show that N is centre-stable, in the dynamical systems sense of Bates-Jones, and globally-in-time invariant.\n  Solutions in N are asymptotically stable and separate into two asymptotically free parts that decouple in the limit --- a so"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.1180","kind":"arxiv","version":2},"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":"0909.1180","created_at":"2026-05-18T04:22:11.594429+00:00"},{"alias_kind":"arxiv_version","alias_value":"0909.1180v2","created_at":"2026-05-18T04:22:11.594429+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.1180","created_at":"2026-05-18T04:22:11.594429+00:00"},{"alias_kind":"pith_short_12","alias_value":"LG2AB3DNXEZI","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_16","alias_value":"LG2AB3DNXEZI46MB","created_at":"2026-05-18T12:26:00.592388+00:00"},{"alias_kind":"pith_short_8","alias_value":"LG2AB3DN","created_at":"2026-05-18T12:26:00.592388+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/LG2AB3DNXEZI46MBOOV25LFF3V","json":"https://pith.science/pith/LG2AB3DNXEZI46MBOOV25LFF3V.json","graph_json":"https://pith.science/api/pith-number/LG2AB3DNXEZI46MBOOV25LFF3V/graph.json","events_json":"https://pith.science/api/pith-number/LG2AB3DNXEZI46MBOOV25LFF3V/events.json","paper":"https://pith.science/paper/LG2AB3DN"},"agent_actions":{"view_html":"https://pith.science/pith/LG2AB3DNXEZI46MBOOV25LFF3V","download_json":"https://pith.science/pith/LG2AB3DNXEZI46MBOOV25LFF3V.json","view_paper":"https://pith.science/paper/LG2AB3DN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0909.1180&json=true","fetch_graph":"https://pith.science/api/pith-number/LG2AB3DNXEZI46MBOOV25LFF3V/graph.json","fetch_events":"https://pith.science/api/pith-number/LG2AB3DNXEZI46MBOOV25LFF3V/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LG2AB3DNXEZI46MBOOV25LFF3V/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LG2AB3DNXEZI46MBOOV25LFF3V/action/storage_attestation","attest_author":"https://pith.science/pith/LG2AB3DNXEZI46MBOOV25LFF3V/action/author_attestation","sign_citation":"https://pith.science/pith/LG2AB3DNXEZI46MBOOV25LFF3V/action/citation_signature","submit_replication":"https://pith.science/pith/LG2AB3DNXEZI46MBOOV25LFF3V/action/replication_record"}},"created_at":"2026-05-18T04:22:11.594429+00:00","updated_at":"2026-05-18T04:22:11.594429+00:00"}