{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:YSJH5JMXBVNDNOTIVNYYCP67VU","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"78a6a3c619e5884d6d73c8314d173ac9ad35a7a49d370181c8dae8eb1b5956ff","cross_cats_sorted":["math.AP","math.MP","nlin.PS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-06-21T14:27:29Z","title_canon_sha256":"c4b54e02c441166fab9817e7a92e28b235ef5f483e93ae6b6a31c7b34812c27e"},"schema_version":"1.0","source":{"id":"1306.5150","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.5150","created_at":"2026-05-18T02:38:59Z"},{"alias_kind":"arxiv_version","alias_value":"1306.5150v4","created_at":"2026-05-18T02:38:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.5150","created_at":"2026-05-18T02:38:59Z"},{"alias_kind":"pith_short_12","alias_value":"YSJH5JMXBVND","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_16","alias_value":"YSJH5JMXBVNDNOTI","created_at":"2026-05-18T12:28:09Z"},{"alias_kind":"pith_short_8","alias_value":"YSJH5JMX","created_at":"2026-05-18T12:28:09Z"}],"graph_snapshots":[{"event_id":"sha256:9675720f49bf034ec0f2ba4fbb2e8288ba556d9f55b8e35f2ea924f5a38398b2","target":"graph","created_at":"2026-05-18T02:38:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study the linear stability of localized modes in self-interacting spinor fields, analyzing the spectrum of the operator corresponding to linearization at solitary waves. Following the generalization of the Vakhitov--Kolokolov approach, we show that the bifurcation of real eigenvalues from the origin is completely characterized by the Vakhitov--Kolokolov condition $dQ/d\\omega=0$ and by the vanishing of the energy functional. We give the numerical data on the linear stability in the generalized Gross--Neveu model and the generalized massive Thirring model in the charge-subcritical, critical, ","authors_text":"Alim Sukhtayev, Andrew Comech, Gregory Berkolaiko","cross_cats":["math.AP","math.MP","nlin.PS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-06-21T14:27:29Z","title":"Vakhitov-Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classical self-interacting spinor fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.5150","kind":"arxiv","version":4},"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":"sha256:0744d482079a5d3f313d8b78c845ffead92f96e878947c3fd62c1ec429b8d632","target":"record","created_at":"2026-05-18T02:38:59Z","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":"78a6a3c619e5884d6d73c8314d173ac9ad35a7a49d370181c8dae8eb1b5956ff","cross_cats_sorted":["math.AP","math.MP","nlin.PS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2013-06-21T14:27:29Z","title_canon_sha256":"c4b54e02c441166fab9817e7a92e28b235ef5f483e93ae6b6a31c7b34812c27e"},"schema_version":"1.0","source":{"id":"1306.5150","kind":"arxiv","version":4}},"canonical_sha256":"c4927ea5970d5a36ba68ab71813fdfad2a0f41a348d7abb0a19628793c4e937f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c4927ea5970d5a36ba68ab71813fdfad2a0f41a348d7abb0a19628793c4e937f","first_computed_at":"2026-05-18T02:38:59.557423Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:38:59.557423Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Npj0dJjh0ieGh2hzaJdDIv39RsrOYB+zK/waX2sMZE6oz90Kvrg5uUpELeTjFT2tOv7NvV+COG3WvvJ+pJL+Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T02:38:59.557818Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.5150","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0744d482079a5d3f313d8b78c845ffead92f96e878947c3fd62c1ec429b8d632","sha256:9675720f49bf034ec0f2ba4fbb2e8288ba556d9f55b8e35f2ea924f5a38398b2"],"state_sha256":"d4885e72c875c8da320244c6a24e14c8255d272ac80e12e0decf5d449e9beb11"}