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The nonlinearity power $p$ and the dimension $d$ are such that the scaling index $s=\\frac{d}2-\\frac2{p-1}$ is between 0 and 1, thus, the NLS is mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$\n  For solutions with $\\ME[u_0]<1$ ($\\ME[u_0]$ stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is"},"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":"1203.6146","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-03-28T03:43:09Z","cross_cats_sorted":[],"title_canon_sha256":"e873debcc66e3f56d389edb19f5b07450970028aa10fbd984490b139d419825e","abstract_canon_sha256":"d37f30ea883f65a9ce6bacfa3952aafad6046a6f59f7ad02507e3262cd3896dd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:03:38.641900Z","signature_b64":"yGMOjbnW4v977pQTSF+dvSmRg+4rnM+erIsiJWRPi8pG3RvLOM3sFHzoHS2JQ1MqElq+CVhyD+LJwDD9c9AoCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7b250ead13549bea3d6059dfe575ea2665a19ff67e95eeaa30d07a967ce7a823","last_reissued_at":"2026-05-18T02:03:38.641070Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:03:38.641070Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global Behavior Of Finite Energy Solutions To The $d$-Dimensional Focusing Nonlinear Schr\\\"odinger Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Cristi Guevara","submitted_at":"2012-03-28T03:43:09Z","abstract_excerpt":"We study the global behavior of finite energy solutions to the $d$-dimensional focusing nonlinear Schr\\\"odinger equation (NLS), $i \\partial_t u+\\Delta u+ |u|^{p-1}u=0, $ with initial data $u_0\\in H^1,\\; x \\in R^n$. The nonlinearity power $p$ and the dimension $d$ are such that the scaling index $s=\\frac{d}2-\\frac2{p-1}$ is between 0 and 1, thus, the NLS is mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$\n  For solutions with $\\ME[u_0]<1$ ($\\ME[u_0]$ stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6146","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":"1203.6146","created_at":"2026-05-18T02:03:38.641201+00:00"},{"alias_kind":"arxiv_version","alias_value":"1203.6146v2","created_at":"2026-05-18T02:03:38.641201+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1203.6146","created_at":"2026-05-18T02:03:38.641201+00:00"},{"alias_kind":"pith_short_12","alias_value":"PMSQ5LITKSN6","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"PMSQ5LITKSN6UPLA","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"PMSQ5LIT","created_at":"2026-05-18T12:27:18.751474+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/PMSQ5LITKSN6UPLALHP6K5PKEZ","json":"https://pith.science/pith/PMSQ5LITKSN6UPLALHP6K5PKEZ.json","graph_json":"https://pith.science/api/pith-number/PMSQ5LITKSN6UPLALHP6K5PKEZ/graph.json","events_json":"https://pith.science/api/pith-number/PMSQ5LITKSN6UPLALHP6K5PKEZ/events.json","paper":"https://pith.science/paper/PMSQ5LIT"},"agent_actions":{"view_html":"https://pith.science/pith/PMSQ5LITKSN6UPLALHP6K5PKEZ","download_json":"https://pith.science/pith/PMSQ5LITKSN6UPLALHP6K5PKEZ.json","view_paper":"https://pith.science/paper/PMSQ5LIT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1203.6146&json=true","fetch_graph":"https://pith.science/api/pith-number/PMSQ5LITKSN6UPLALHP6K5PKEZ/graph.json","fetch_events":"https://pith.science/api/pith-number/PMSQ5LITKSN6UPLALHP6K5PKEZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PMSQ5LITKSN6UPLALHP6K5PKEZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PMSQ5LITKSN6UPLALHP6K5PKEZ/action/storage_attestation","attest_author":"https://pith.science/pith/PMSQ5LITKSN6UPLALHP6K5PKEZ/action/author_attestation","sign_citation":"https://pith.science/pith/PMSQ5LITKSN6UPLALHP6K5PKEZ/action/citation_signature","submit_replication":"https://pith.science/pith/PMSQ5LITKSN6UPLALHP6K5PKEZ/action/replication_record"}},"created_at":"2026-05-18T02:03:38.641201+00:00","updated_at":"2026-05-18T02:03:38.641201+00:00"}