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In \\cite{G1} and \\cite{yuan}, the mass-energy quantity $M(Q)^{\\frac{1-s_{c}}{s_{c}}}E(Q)$ has been shown to be a threshold for the dynamical behavior of solutions of the equation. 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.","authors_text":"Qing Guo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-24T02:26:59Z","title":"Threshold solutions for the focusing $L^{2}$ -supercritical NLS Equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5669","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f23f251e5948c449c6258b41ddbba182b0cd6ed71c6b4a22e9cd10e3c2d6eb22","target":"record","created_at":"2026-05-18T04:07:42Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"5efe517c0ac25774076add3f44fb19cb95704563acfee0c7b437cddb006c4e88","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-11-24T02:26:59Z","title_canon_sha256":"3699b4527766edc711e29c0ea88dd6864daa1d1a9f5e06123c8a599090c74b36"},"schema_version":"1.0","source":{"id":"1111.5669","kind":"arxiv","version":1}},"canonical_sha256":"67b5b8ad2eb753ffe8b2971ab4e51b655bf04eaad52eaf5862aeb14c5f774d14","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"67b5b8ad2eb753ffe8b2971ab4e51b655bf04eaad52eaf5862aeb14c5f774d14","first_computed_at":"2026-05-18T04:07:42.621585Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:07:42.621585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"f1D/lNXyeL3nCLJnF3GYHAjg8OTY+o1T/D0ra7FX2quLpUqpjREGu2R0OxRIQcOjerJiGJ1tbZZC7ZzqPouGBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:07:42.622063Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.5669","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f23f251e5948c449c6258b41ddbba182b0cd6ed71c6b4a22e9cd10e3c2d6eb22","sha256:8a71c1f226cd155d17a5dbd2dec625003ad1ed3bc5516ddf5b4b73435217e1bd"],"state_sha256":"890d5c28d368eb30d56e1eae7eb8163edbd62892d666fa3b639d8512403ed9bc"}