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In this paper, we prove the instability of the"},"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":"1804.02738","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-04-08T18:48:54Z","cross_cats_sorted":[],"title_canon_sha256":"84b8d33007103846867038d640477ac620f72e39c517b1b73d4d942fc9a10af7","abstract_canon_sha256":"fe425409f5f334f9965975579877972a0b1e90faa63182664028830b0527a629"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:18:58.532524Z","signature_b64":"ytYW49/AJebhAAFigCfiqvZSlT3jg8TZU0AMmAIdODcjMWZhSeIJs2QlejDWV+Y1xAnRLbPTB//KB/SSQ1B3Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f310e4550c036ef5dec2c07776291cb98d505b1dbeeb65fff75e0116bfece622","last_reissued_at":"2026-05-18T00:18:58.530364Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:18:58.530364Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Instability of the solitary wave solutions for the generalized derivative nonlinear Schr\\\"odinger equation in the endpoint case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bing Li, Cui Ning","submitted_at":"2018-04-08T18:48:54Z","abstract_excerpt":"We consider the stability theory of solitary wave solutions for the generalized derivative nonlinear Schr\\\"odinger equation\n  $$\n  i\\partial_{t}u+\\partial_{x}^{2}u+i|u|^{2\\sigma}\\partial_x u=0,\n  $$\n  where $1<\\sigma<2$.\n  The equation has a two-parameter family of solitary wave solutions of the form\n  $$ u_{\\omega,c}(t,x)=e^{i\\omega t+i\\frac c2(x-ct)-\\frac{i}{2\\sigma+2}\\int_{-\\infty}^{x-ct}\\varphi^{2\\sigma}_{\\omega,c}(y)dy}\\varphi_{\\omega,c}(x-ct).\n  $$\n  The stability theory in the frequency region of $|c|<2\\sqrt{\\omega}$ was studied previously. 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