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When $\\rho=0$, the Zakharov-Ito equation reduces to the KdV equation, hence has solitary waves with speeds $c\\in(0,+\\infty)$. We prove the orbital stability of these solitary waves in $H^1\\times L^2$ by combining a variational approach and the framework of Grillakis, Shatah and Strauss \\cite{GSS1987}."},"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":"2506.07053","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-08T09:27:55Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"9eaf23aa3b8ecbb0f9caf249d101e4a61655b0b95fdfa972dcc464cd18852d8a","abstract_canon_sha256":"300e573c2931b72f9cb8c633bc24cc76c9a865fb5f78ddb0c26a9ed7df6427cd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:21:48.489962Z","signature_b64":"trmOB9MFp3DuDVerHzlSN+Deh8buucUzg02XJSBN856xkSJN+rTva6x6w30rD+NdPzlD5EAiBJx6LEd0H/5oCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e7c6121eab66a674bf7f58c63cf94f04d03f12afb08ff027c20f31d5362164ef","last_reissued_at":"2026-07-05T11:21:48.489483Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:21:48.489483Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global well-posedness and orbital stability of solitary waves for Zakharov-Ito equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Fan Wu, Feng Shao","submitted_at":"2025-06-08T09:27:55Z","abstract_excerpt":"In this paper, we consider the Zakharov-Ito equation \\begin{equation*} \\begin{cases} u_t+u_{xxx}+3uu_x+\\rho\\rho_x=0,\\\\ \\rho_t+{(u\\rho)}_x=0. \\end{cases} \\end{equation*} We prove the local well-posedness in $H^s\\times H^s$ for $s>3/2$ and global well-posedness in $H^s\\times H^s$ for $s\\geq2$. When $\\rho=0$, the Zakharov-Ito equation reduces to the KdV equation, hence has solitary waves with speeds $c\\in(0,+\\infty)$. We prove the orbital stability of these solitary waves in $H^1\\times L^2$ by combining a variational approach and the framework of Grillakis, Shatah and Strauss \\cite{GSS1987}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.07053","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.07053/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2506.07053","created_at":"2026-07-05T11:21:48.489539+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.07053v4","created_at":"2026-07-05T11:21:48.489539+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.07053","created_at":"2026-07-05T11:21:48.489539+00:00"},{"alias_kind":"pith_short_12","alias_value":"47DBEHVLM2TH","created_at":"2026-07-05T11:21:48.489539+00:00"},{"alias_kind":"pith_short_16","alias_value":"47DBEHVLM2THJP37","created_at":"2026-07-05T11:21:48.489539+00:00"},{"alias_kind":"pith_short_8","alias_value":"47DBEHVL","created_at":"2026-07-05T11:21:48.489539+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/47DBEHVLM2THJP37LDDDZ6KPAT","json":"https://pith.science/pith/47DBEHVLM2THJP37LDDDZ6KPAT.json","graph_json":"https://pith.science/api/pith-number/47DBEHVLM2THJP37LDDDZ6KPAT/graph.json","events_json":"https://pith.science/api/pith-number/47DBEHVLM2THJP37LDDDZ6KPAT/events.json","paper":"https://pith.science/paper/47DBEHVL"},"agent_actions":{"view_html":"https://pith.science/pith/47DBEHVLM2THJP37LDDDZ6KPAT","download_json":"https://pith.science/pith/47DBEHVLM2THJP37LDDDZ6KPAT.json","view_paper":"https://pith.science/paper/47DBEHVL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.07053&json=true","fetch_graph":"https://pith.science/api/pith-number/47DBEHVLM2THJP37LDDDZ6KPAT/graph.json","fetch_events":"https://pith.science/api/pith-number/47DBEHVLM2THJP37LDDDZ6KPAT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/47DBEHVLM2THJP37LDDDZ6KPAT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/47DBEHVLM2THJP37LDDDZ6KPAT/action/storage_attestation","attest_author":"https://pith.science/pith/47DBEHVLM2THJP37LDDDZ6KPAT/action/author_attestation","sign_citation":"https://pith.science/pith/47DBEHVLM2THJP37LDDDZ6KPAT/action/citation_signature","submit_replication":"https://pith.science/pith/47DBEHVLM2THJP37LDDDZ6KPAT/action/replication_record"}},"created_at":"2026-07-05T11:21:48.489539+00:00","updated_at":"2026-07-05T11:21:48.489539+00:00"}