{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:VDFUS7DIWYZBIPKY35BXEWYYLP","short_pith_number":"pith:VDFUS7DI","schema_version":"1.0","canonical_sha256":"a8cb497c68b632143d58df43725b185be59ee4937a062fc3ccaebc8cd90afcb7","source":{"kind":"arxiv","id":"1509.08044","version":1},"attestation_state":"computed","paper":{"title":"Stability of the train of $N$ solitary waves for the two-component Camassa-Holm shallow water system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xingxing Liu","submitted_at":"2015-09-27T02:41:24Z","abstract_excerpt":"Considered herein is the integrable two-component Camassa-Holm shallow water system derived in the context of shallow water theory, which admits blow-up solutions and the solitary waves interacting like solitons. Using modulation theory, and combining the almost monotonicity of a local version of energy with the argument on the stability of a single solitary wave, we prove that the train of $N$ solitary waves, which are sufficiently decoupled, is orbitally stable in the energy space $H^1(\\R)\\times L^2(\\R)$."},"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":"1509.08044","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-09-27T02:41:24Z","cross_cats_sorted":[],"title_canon_sha256":"6cfdf56d3734a30f237dfe4815c1098d0de13829b8d6259db97438472fe1d8b6","abstract_canon_sha256":"319dd442861542d4217d361d0165980cc1b50fa4ba961ee31ce5df7271c2b9a1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:54.860532Z","signature_b64":"0CEzlEdHtMylCSlIijgH5q6O8VlR7XmF5p1dXQd/NItWTRl9jo/0QCaUUdpsye8GMKLaX8dn3BdDBfUuG2z+BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a8cb497c68b632143d58df43725b185be59ee4937a062fc3ccaebc8cd90afcb7","last_reissued_at":"2026-05-18T01:31:54.860163Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:54.860163Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Stability of the train of $N$ solitary waves for the two-component Camassa-Holm shallow water system","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xingxing Liu","submitted_at":"2015-09-27T02:41:24Z","abstract_excerpt":"Considered herein is the integrable two-component Camassa-Holm shallow water system derived in the context of shallow water theory, which admits blow-up solutions and the solitary waves interacting like solitons. Using modulation theory, and combining the almost monotonicity of a local version of energy with the argument on the stability of a single solitary wave, we prove that the train of $N$ solitary waves, which are sufficiently decoupled, is orbitally stable in the energy space $H^1(\\R)\\times L^2(\\R)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.08044","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":""},"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":"1509.08044","created_at":"2026-05-18T01:31:54.860219+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.08044v1","created_at":"2026-05-18T01:31:54.860219+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.08044","created_at":"2026-05-18T01:31:54.860219+00:00"},{"alias_kind":"pith_short_12","alias_value":"VDFUS7DIWYZB","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"VDFUS7DIWYZBIPKY","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"VDFUS7DI","created_at":"2026-05-18T12:29:44.643036+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/VDFUS7DIWYZBIPKY35BXEWYYLP","json":"https://pith.science/pith/VDFUS7DIWYZBIPKY35BXEWYYLP.json","graph_json":"https://pith.science/api/pith-number/VDFUS7DIWYZBIPKY35BXEWYYLP/graph.json","events_json":"https://pith.science/api/pith-number/VDFUS7DIWYZBIPKY35BXEWYYLP/events.json","paper":"https://pith.science/paper/VDFUS7DI"},"agent_actions":{"view_html":"https://pith.science/pith/VDFUS7DIWYZBIPKY35BXEWYYLP","download_json":"https://pith.science/pith/VDFUS7DIWYZBIPKY35BXEWYYLP.json","view_paper":"https://pith.science/paper/VDFUS7DI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.08044&json=true","fetch_graph":"https://pith.science/api/pith-number/VDFUS7DIWYZBIPKY35BXEWYYLP/graph.json","fetch_events":"https://pith.science/api/pith-number/VDFUS7DIWYZBIPKY35BXEWYYLP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VDFUS7DIWYZBIPKY35BXEWYYLP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VDFUS7DIWYZBIPKY35BXEWYYLP/action/storage_attestation","attest_author":"https://pith.science/pith/VDFUS7DIWYZBIPKY35BXEWYYLP/action/author_attestation","sign_citation":"https://pith.science/pith/VDFUS7DIWYZBIPKY35BXEWYYLP/action/citation_signature","submit_replication":"https://pith.science/pith/VDFUS7DIWYZBIPKY35BXEWYYLP/action/replication_record"}},"created_at":"2026-05-18T01:31:54.860219+00:00","updated_at":"2026-05-18T01:31:54.860219+00:00"}