{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:TCEGGT3KPNK4ICMID7ZP635QWL","short_pith_number":"pith:TCEGGT3K","schema_version":"1.0","canonical_sha256":"9888634f6a7b55c409881ff2ff6fb0b2d5f2e3bd48d2e32c702e841fca2a55e7","source":{"kind":"arxiv","id":"1202.4983","version":2},"attestation_state":"computed","paper":{"title":"Analytic solutions and Singularity formation for the Peakon b--Family equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.AP","authors_text":"Francesco Gargano, Giuseppe Maria Coclite, Vincenzo Sciacca","submitted_at":"2012-02-22T17:31:54Z","abstract_excerpt":"Using the Abstract Cauchy-Kowalewski Theorem we prove that the $b$-family equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic and it belongs to $H^s$ with $s > 3/2$, and the momentum density $u_0 - u_{0,{xx}}$ does not change sign, we prove that the solution stays analytic globally in time, for $b\\geq 1$. Using pseudospectral numerical methods, we study, also, the singularity formation for the $b$-family equations with the singularity tracking method. This method allows us to follow the process of the singularity formation in the complex"},"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":"1202.4983","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-22T17:31:54Z","cross_cats_sorted":["math.NA"],"title_canon_sha256":"b63627b7fac1e780ec8290a0035c11670e2da9b000558d2723d47910787921c1","abstract_canon_sha256":"ffeefbf895ffd31d02ac9d0005a2382f92e7994f63df200e3a3ad1ab9f1ea5a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:25.690709Z","signature_b64":"XglyX7JR1JT/IRAovN8wHYQ8kPJO+TKhg30vSLLNl7eQBWdQ0TmjdXk82kSdeqxG/wPa06weUjAhFHHqeq2+Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9888634f6a7b55c409881ff2ff6fb0b2d5f2e3bd48d2e32c702e841fca2a55e7","last_reissued_at":"2026-05-18T03:40:25.690036Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:25.690036Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Analytic solutions and Singularity formation for the Peakon b--Family equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA"],"primary_cat":"math.AP","authors_text":"Francesco Gargano, Giuseppe Maria Coclite, Vincenzo Sciacca","submitted_at":"2012-02-22T17:31:54Z","abstract_excerpt":"Using the Abstract Cauchy-Kowalewski Theorem we prove that the $b$-family equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic and it belongs to $H^s$ with $s > 3/2$, and the momentum density $u_0 - u_{0,{xx}}$ does not change sign, we prove that the solution stays analytic globally in time, for $b\\geq 1$. Using pseudospectral numerical methods, we study, also, the singularity formation for the $b$-family equations with the singularity tracking method. This method allows us to follow the process of the singularity formation in the complex"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.4983","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":"1202.4983","created_at":"2026-05-18T03:40:25.690175+00:00"},{"alias_kind":"arxiv_version","alias_value":"1202.4983v2","created_at":"2026-05-18T03:40:25.690175+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.4983","created_at":"2026-05-18T03:40:25.690175+00:00"},{"alias_kind":"pith_short_12","alias_value":"TCEGGT3KPNK4","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_16","alias_value":"TCEGGT3KPNK4ICMI","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_8","alias_value":"TCEGGT3K","created_at":"2026-05-18T12:27:23.164592+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/TCEGGT3KPNK4ICMID7ZP635QWL","json":"https://pith.science/pith/TCEGGT3KPNK4ICMID7ZP635QWL.json","graph_json":"https://pith.science/api/pith-number/TCEGGT3KPNK4ICMID7ZP635QWL/graph.json","events_json":"https://pith.science/api/pith-number/TCEGGT3KPNK4ICMID7ZP635QWL/events.json","paper":"https://pith.science/paper/TCEGGT3K"},"agent_actions":{"view_html":"https://pith.science/pith/TCEGGT3KPNK4ICMID7ZP635QWL","download_json":"https://pith.science/pith/TCEGGT3KPNK4ICMID7ZP635QWL.json","view_paper":"https://pith.science/paper/TCEGGT3K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1202.4983&json=true","fetch_graph":"https://pith.science/api/pith-number/TCEGGT3KPNK4ICMID7ZP635QWL/graph.json","fetch_events":"https://pith.science/api/pith-number/TCEGGT3KPNK4ICMID7ZP635QWL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TCEGGT3KPNK4ICMID7ZP635QWL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TCEGGT3KPNK4ICMID7ZP635QWL/action/storage_attestation","attest_author":"https://pith.science/pith/TCEGGT3KPNK4ICMID7ZP635QWL/action/author_attestation","sign_citation":"https://pith.science/pith/TCEGGT3KPNK4ICMID7ZP635QWL/action/citation_signature","submit_replication":"https://pith.science/pith/TCEGGT3KPNK4ICMID7ZP635QWL/action/replication_record"}},"created_at":"2026-05-18T03:40:25.690175+00:00","updated_at":"2026-05-18T03:40:25.690175+00:00"}