{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:TCEGGT3KPNK4ICMID7ZP635QWL","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":"ffeefbf895ffd31d02ac9d0005a2382f92e7994f63df200e3a3ad1ab9f1ea5a5","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-22T17:31:54Z","title_canon_sha256":"b63627b7fac1e780ec8290a0035c11670e2da9b000558d2723d47910787921c1"},"schema_version":"1.0","source":{"id":"1202.4983","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.4983","created_at":"2026-05-18T03:40:25Z"},{"alias_kind":"arxiv_version","alias_value":"1202.4983v2","created_at":"2026-05-18T03:40:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.4983","created_at":"2026-05-18T03:40:25Z"},{"alias_kind":"pith_short_12","alias_value":"TCEGGT3KPNK4","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"TCEGGT3KPNK4ICMI","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"TCEGGT3K","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:401e76cb2a9c313e45885d65d38702bd27971d682763732bd1744106d5e07cca","target":"graph","created_at":"2026-05-18T03:40:25Z","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":"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","authors_text":"Francesco Gargano, Giuseppe Maria Coclite, Vincenzo Sciacca","cross_cats":["math.NA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-22T17:31:54Z","title":"Analytic solutions and Singularity formation for the Peakon b--Family equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.4983","kind":"arxiv","version":2},"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":"sha256:2bed62b04cbcef554fbeb046c3289fc54c819e48c9ba5634365a1fd07f3317ff","target":"record","created_at":"2026-05-18T03:40:25Z","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":"ffeefbf895ffd31d02ac9d0005a2382f92e7994f63df200e3a3ad1ab9f1ea5a5","cross_cats_sorted":["math.NA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-22T17:31:54Z","title_canon_sha256":"b63627b7fac1e780ec8290a0035c11670e2da9b000558d2723d47910787921c1"},"schema_version":"1.0","source":{"id":"1202.4983","kind":"arxiv","version":2}},"canonical_sha256":"9888634f6a7b55c409881ff2ff6fb0b2d5f2e3bd48d2e32c702e841fca2a55e7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9888634f6a7b55c409881ff2ff6fb0b2d5f2e3bd48d2e32c702e841fca2a55e7","first_computed_at":"2026-05-18T03:40:25.690036Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:40:25.690036Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XglyX7JR1JT/IRAovN8wHYQ8kPJO+TKhg30vSLLNl7eQBWdQ0TmjdXk82kSdeqxG/wPa06weUjAhFHHqeq2+Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:40:25.690709Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.4983","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2bed62b04cbcef554fbeb046c3289fc54c819e48c9ba5634365a1fd07f3317ff","sha256:401e76cb2a9c313e45885d65d38702bd27971d682763732bd1744106d5e07cca"],"state_sha256":"4967610bb6bb3fd82addc3122352c5faf8fed5561becd3a6a3a08c97d3e93b6b"}