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In the present paper, we study the dynamics at the critical level $M(u)^{\\frac{1-s_{c}}{s_{c}}}E(u)=M(Q)^{\\frac{1-s_{c}}{s_{c}}}E(Q)$ and classify the corresponding solutions using modulation theory, non-trivially generalize the results obtained in \\cite{holmer3} for the 3D cubic Schr\\\"{o}dinger equation."},"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":"1111.5669","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-24T02:26:59Z","cross_cats_sorted":[],"title_canon_sha256":"3699b4527766edc711e29c0ea88dd6864daa1d1a9f5e06123c8a599090c74b36","abstract_canon_sha256":"5efe517c0ac25774076add3f44fb19cb95704563acfee0c7b437cddb006c4e88"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:42.622063Z","signature_b64":"f1D/lNXyeL3nCLJnF3GYHAjg8OTY+o1T/D0ra7FX2quLpUqpjREGu2R0OxRIQcOjerJiGJ1tbZZC7ZzqPouGBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67b5b8ad2eb753ffe8b2971ab4e51b655bf04eaad52eaf5862aeb14c5f774d14","last_reissued_at":"2026-05-18T04:07:42.621585Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:42.621585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Threshold solutions for the focusing $L^{2}$ -supercritical NLS Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Qing Guo","submitted_at":"2011-11-24T02:26:59Z","abstract_excerpt":"We investigate the $L^2$-supercritical and $\\dot{H}^1$-subcritical nonlinear Schr\\\"{o}dinger equation in $H^1$. 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