{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:ANTOWAYZUBPJ4ETJ7N25PDS3VG","short_pith_number":"pith:ANTOWAYZ","schema_version":"1.0","canonical_sha256":"0366eb0319a05e9e1269fb75d78e5ba9bb707c15f3868b5827026b34d7c844c3","source":{"kind":"arxiv","id":"1007.0962","version":3},"attestation_state":"computed","paper":{"title":"Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Manwai Yuen","submitted_at":"2010-07-06T17:19:04Z","abstract_excerpt":"In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \\begin{equation} \\left\\{ \\begin{array} [c]{c}% \\rho_{t}+u\\rho_{x}+\\rho u_{x}=0\n  m_{t}+2u_{x}m+um_{x}+\\sigma\\rho\\rho_{x}=0 \\end{array} \\right. \\end{equation} with \\begin{equation} m=u-\\alpha^{2}u_{xx}. \\end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\\sigma=1$ or $-1$. In particular, for the integrable system with $\\sigma=1$, we have the global solutions:% \\begin{equation} \\left\\{ \\begin{array} [c]{c}% \\rho(t,x)=\\left\\{ \\begin{array} [c]{c}% \\frac{f\\l"},"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":"1007.0962","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2010-07-06T17:19:04Z","cross_cats_sorted":["math.AP","math.MP"],"title_canon_sha256":"fb1540c3d3eecd9852bedba6e085ea813aba74530af1d13e7b7a84e1570c54cf","abstract_canon_sha256":"f259fe22608df6a35b25c6cc0411e94b49d0a4730896af91618837f82e11a35d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:00.529665Z","signature_b64":"ANVPDkVvGulA1aKg+6ipBX68czOZ1JF97Nx7COOumA6fmB1Mtq1/+dvhcBWOcPrVsBP5DReWQHoLt1pAiUJSAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0366eb0319a05e9e1269fb75d78e5ba9bb707c15f3868b5827026b34d7c844c3","last_reissued_at":"2026-05-18T04:40:00.529265Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:00.529265Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MP"],"primary_cat":"math-ph","authors_text":"Manwai Yuen","submitted_at":"2010-07-06T17:19:04Z","abstract_excerpt":"In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \\begin{equation} \\left\\{ \\begin{array} [c]{c}% \\rho_{t}+u\\rho_{x}+\\rho u_{x}=0\n  m_{t}+2u_{x}m+um_{x}+\\sigma\\rho\\rho_{x}=0 \\end{array} \\right. \\end{equation} with \\begin{equation} m=u-\\alpha^{2}u_{xx}. \\end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\\sigma=1$ or $-1$. In particular, for the integrable system with $\\sigma=1$, we have the global solutions:% \\begin{equation} \\left\\{ \\begin{array} [c]{c}% \\rho(t,x)=\\left\\{ \\begin{array} [c]{c}% \\frac{f\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1007.0962","kind":"arxiv","version":3},"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":"1007.0962","created_at":"2026-05-18T04:40:00.529323+00:00"},{"alias_kind":"arxiv_version","alias_value":"1007.0962v3","created_at":"2026-05-18T04:40:00.529323+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1007.0962","created_at":"2026-05-18T04:40:00.529323+00:00"},{"alias_kind":"pith_short_12","alias_value":"ANTOWAYZUBPJ","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_16","alias_value":"ANTOWAYZUBPJ4ETJ","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_8","alias_value":"ANTOWAYZ","created_at":"2026-05-18T12:26:05.355336+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/ANTOWAYZUBPJ4ETJ7N25PDS3VG","json":"https://pith.science/pith/ANTOWAYZUBPJ4ETJ7N25PDS3VG.json","graph_json":"https://pith.science/api/pith-number/ANTOWAYZUBPJ4ETJ7N25PDS3VG/graph.json","events_json":"https://pith.science/api/pith-number/ANTOWAYZUBPJ4ETJ7N25PDS3VG/events.json","paper":"https://pith.science/paper/ANTOWAYZ"},"agent_actions":{"view_html":"https://pith.science/pith/ANTOWAYZUBPJ4ETJ7N25PDS3VG","download_json":"https://pith.science/pith/ANTOWAYZUBPJ4ETJ7N25PDS3VG.json","view_paper":"https://pith.science/paper/ANTOWAYZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1007.0962&json=true","fetch_graph":"https://pith.science/api/pith-number/ANTOWAYZUBPJ4ETJ7N25PDS3VG/graph.json","fetch_events":"https://pith.science/api/pith-number/ANTOWAYZUBPJ4ETJ7N25PDS3VG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ANTOWAYZUBPJ4ETJ7N25PDS3VG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ANTOWAYZUBPJ4ETJ7N25PDS3VG/action/storage_attestation","attest_author":"https://pith.science/pith/ANTOWAYZUBPJ4ETJ7N25PDS3VG/action/author_attestation","sign_citation":"https://pith.science/pith/ANTOWAYZUBPJ4ETJ7N25PDS3VG/action/citation_signature","submit_replication":"https://pith.science/pith/ANTOWAYZUBPJ4ETJ7N25PDS3VG/action/replication_record"}},"created_at":"2026-05-18T04:40:00.529323+00:00","updated_at":"2026-05-18T04:40:00.529323+00:00"}