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The equation has a two-parameter family of solitary waves $$u_{\\omega,c}(t,x)=\\Phi_{\\omega,c}(x)e^{i\\omega t+\\frac{ic}2x-\\frac i{2\\sigma+2}\\int_0^x\\Phi_{\\omega,c}(y)^{2\\sigma}dy},$$ with $(\\omega,c)$ satisfying $\\omega>c^2/4$, or $\\omega=c^2/4$ and $c>0$. The stability theory in the frequency region $\\omega>c^2/4$ was studied previously. In this paper, we prove the sta"},"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":"1705.04458","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-05-12T07:56:34Z","cross_cats_sorted":[],"title_canon_sha256":"e83fdb1bcb0b34cf96d4707c6a8e979577a25c44fea0807e43975ca60c799ab0","abstract_canon_sha256":"28a3daf92ec008980c4c74940cc615eb3186c744a7bb3f16d4d795e262246158"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:07:01.187575Z","signature_b64":"3JgK/mtcG+MJ/TAWvJVptXYR/hSg+fAy4SrEIGZ7Uvz1IKC4LulyPIVGrnUKVVEIticGoYQHiYTjpxKpZBIsCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1ec6e0e8cd8a9f600b6e0ef8ce8898887adf7ab29b5b89fab805ba8050f53921","last_reissued_at":"2026-05-18T00:07:01.186925Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:07:01.186925Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Orbital stability of solitary waves for generalized derivative nonlinear Schr\\\"odinger equations in the endpoint case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Qing Guo","submitted_at":"2017-05-12T07:56:34Z","abstract_excerpt":"We consider the following generalized derivative nonlinear Schr\\\"odinger equation \\begin{equation*} i\\partial_tu+\\partial^2_xu+i|u|^{2\\sigma}\\partial_xu=0,\\ (t,x)\\in\\mathbb R\\times\\mathbb R \\end{equation*} when $\\sigma\\in(0,1)$. The equation has a two-parameter family of solitary waves $$u_{\\omega,c}(t,x)=\\Phi_{\\omega,c}(x)e^{i\\omega t+\\frac{ic}2x-\\frac i{2\\sigma+2}\\int_0^x\\Phi_{\\omega,c}(y)^{2\\sigma}dy},$$ with $(\\omega,c)$ satisfying $\\omega>c^2/4$, or $\\omega=c^2/4$ and $c>0$. The stability theory in the frequency region $\\omega>c^2/4$ was studied previously. In this paper, we prove the sta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.04458","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":"1705.04458","created_at":"2026-05-18T00:07:01.187036+00:00"},{"alias_kind":"arxiv_version","alias_value":"1705.04458v2","created_at":"2026-05-18T00:07:01.187036+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1705.04458","created_at":"2026-05-18T00:07:01.187036+00:00"},{"alias_kind":"pith_short_12","alias_value":"D3DOB2GNRKPW","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_16","alias_value":"D3DOB2GNRKPWAC3O","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_8","alias_value":"D3DOB2GN","created_at":"2026-05-18T12:31:10.602751+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/D3DOB2GNRKPWAC3OB34M5CEYRB","json":"https://pith.science/pith/D3DOB2GNRKPWAC3OB34M5CEYRB.json","graph_json":"https://pith.science/api/pith-number/D3DOB2GNRKPWAC3OB34M5CEYRB/graph.json","events_json":"https://pith.science/api/pith-number/D3DOB2GNRKPWAC3OB34M5CEYRB/events.json","paper":"https://pith.science/paper/D3DOB2GN"},"agent_actions":{"view_html":"https://pith.science/pith/D3DOB2GNRKPWAC3OB34M5CEYRB","download_json":"https://pith.science/pith/D3DOB2GNRKPWAC3OB34M5CEYRB.json","view_paper":"https://pith.science/paper/D3DOB2GN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1705.04458&json=true","fetch_graph":"https://pith.science/api/pith-number/D3DOB2GNRKPWAC3OB34M5CEYRB/graph.json","fetch_events":"https://pith.science/api/pith-number/D3DOB2GNRKPWAC3OB34M5CEYRB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/D3DOB2GNRKPWAC3OB34M5CEYRB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/D3DOB2GNRKPWAC3OB34M5CEYRB/action/storage_attestation","attest_author":"https://pith.science/pith/D3DOB2GNRKPWAC3OB34M5CEYRB/action/author_attestation","sign_citation":"https://pith.science/pith/D3DOB2GNRKPWAC3OB34M5CEYRB/action/citation_signature","submit_replication":"https://pith.science/pith/D3DOB2GNRKPWAC3OB34M5CEYRB/action/replication_record"}},"created_at":"2026-05-18T00:07:01.187036+00:00","updated_at":"2026-05-18T00:07:01.187036+00:00"}