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It was proved in \\cite{LiSiSu1} that the solitary wave solutions are stable if $-2\\sqrt{\\omega }<c <2z_"},"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":"1803.07700","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-03-21T00:19:38Z","cross_cats_sorted":[],"title_canon_sha256":"c9ab77ea72c71a2db667f17bca3459422ec78fbde02b6ca1b8659f3325ee3439","abstract_canon_sha256":"2ea673d01dea4295535e2a798e17946e4fac6e7f8143e7bf73f98b1d45b0e3d4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:20:29.241611Z","signature_b64":"90I5uNFRlFne3CHgXIMA6I+egWt1iSX3BnVqQQxtPfvAycp5jhMkff5xqQiS34/CD1o8kxoMVwubr7OQIb0SAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ad2b91f239392f80eb23079b50b88ee0297d813a7630799334d6ea59cf863a4d","last_reissued_at":"2026-05-18T00:20:29.241154Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:20:29.241154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Instability of the solitary wave solutions for the genenalized derivative Nonlinear Schr\\\"odinger equation in the critical frequency case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Cui Ning, Yifei Wu, Zihua Guo","submitted_at":"2018-03-21T00:19:38Z","abstract_excerpt":"We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schr\\\"odinger equation $$ i\\partial_{t}u+\\partial_{x}^{2}u+i|u|^{2\\sigma}\\partial_x u=0. $$ The equation has a two-parameter family of solitary wave solutions of the form \\begin{align*} \\phi_{\\omega,c}(x)=\\varphi_{\\omega,c}(x)\\exp{\\big\\{ i\\frac c2 x-\\frac{i}{2\\sigma+2}\\int_{-\\infty}^{x}\\varphi^{2\\sigma}_{\\omega,c}(y)dy\\big\\}}. \\end{align*} Here $ \\varphi_{\\omega,c}$ is some real-valued function. 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