{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:E6TAETF3PKJ6KUOTDF5RJMAVAB","short_pith_number":"pith:E6TAETF3","canonical_record":{"source":{"id":"1507.04098","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-15T06:31:22Z","cross_cats_sorted":[],"title_canon_sha256":"a058913fcab9ff822c621fc7b3fc7a7f9d15ca88d18844ec27bde7ebd5f556f5","abstract_canon_sha256":"284cacd3d9ea082361380540221f3c5a451be53e675aaecb639f397ab1ee0def"},"schema_version":"1.0"},"canonical_sha256":"27a6024cbb7a93e551d3197b14b0150066e52d758f275e2d854019e5f53c6c20","source":{"kind":"arxiv","id":"1507.04098","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.04098","created_at":"2026-05-18T01:36:49Z"},{"alias_kind":"arxiv_version","alias_value":"1507.04098v1","created_at":"2026-05-18T01:36:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.04098","created_at":"2026-05-18T01:36:49Z"},{"alias_kind":"pith_short_12","alias_value":"E6TAETF3PKJ6","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"E6TAETF3PKJ6KUOT","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"E6TAETF3","created_at":"2026-05-18T12:29:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:E6TAETF3PKJ6KUOTDF5RJMAVAB","target":"record","payload":{"canonical_record":{"source":{"id":"1507.04098","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-15T06:31:22Z","cross_cats_sorted":[],"title_canon_sha256":"a058913fcab9ff822c621fc7b3fc7a7f9d15ca88d18844ec27bde7ebd5f556f5","abstract_canon_sha256":"284cacd3d9ea082361380540221f3c5a451be53e675aaecb639f397ab1ee0def"},"schema_version":"1.0"},"canonical_sha256":"27a6024cbb7a93e551d3197b14b0150066e52d758f275e2d854019e5f53c6c20","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:49.142902Z","signature_b64":"c6HTDDnHtUQaXLKjy4vQPTriYVEAU0q0oHYepZSQJ3roG5FXOvcFDNUgPwY3I5aqyVXRNefL54FR+9lhc91jDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27a6024cbb7a93e551d3197b14b0150066e52d758f275e2d854019e5f53c6c20","last_reissued_at":"2026-05-18T01:36:49.142228Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:49.142228Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.04098","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Nf+vd0kZYdeRCj9R34r4/TzSDOcIGy3nrLx2UVe33IgxHQSEuusViCcOGz6RYk7H9nuNSd+7ug2ytThAHFPyCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T01:38:20.008703Z"},"content_sha256":"2456f1f4c6d1f518dcb483fc284ea8c963b59673c03661bc776f3c63aed33225","schema_version":"1.0","event_id":"sha256:2456f1f4c6d1f518dcb483fc284ea8c963b59673c03661bc776f3c63aed33225"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:E6TAETF3PKJ6KUOTDF5RJMAVAB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Degenerate Edge Bifurcation in the 1D Linearized Nonlinear Schrodinger Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Matt Coles, Stephen Gustafson","submitted_at":"2015-07-15T06:31:22Z","abstract_excerpt":"This work deals with the focusing Nonlinear Schrodinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish the degenerate bifurcation of these resonances to eigenvalues as the nonlinearity deviates from cubic. The leading-order expression for these eigenvalues is consistent with previous numerical computations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04098","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"t5qG2sni++5tgy8LEtK6MEEmlXqjHfmQxd7qh25IKcAzCY34+luRwP0lItqgmpBURhCfYjdKhiics2L7XoBxBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T01:38:20.009300Z"},"content_sha256":"4657323b420328631a83df6b25de0eb935582a77d699adc411a4e0522c951e77","schema_version":"1.0","event_id":"sha256:4657323b420328631a83df6b25de0eb935582a77d699adc411a4e0522c951e77"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E6TAETF3PKJ6KUOTDF5RJMAVAB/bundle.json","state_url":"https://pith.science/pith/E6TAETF3PKJ6KUOTDF5RJMAVAB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E6TAETF3PKJ6KUOTDF5RJMAVAB/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-08T01:38:20Z","links":{"resolver":"https://pith.science/pith/E6TAETF3PKJ6KUOTDF5RJMAVAB","bundle":"https://pith.science/pith/E6TAETF3PKJ6KUOTDF5RJMAVAB/bundle.json","state":"https://pith.science/pith/E6TAETF3PKJ6KUOTDF5RJMAVAB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E6TAETF3PKJ6KUOTDF5RJMAVAB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:E6TAETF3PKJ6KUOTDF5RJMAVAB","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":"284cacd3d9ea082361380540221f3c5a451be53e675aaecb639f397ab1ee0def","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-15T06:31:22Z","title_canon_sha256":"a058913fcab9ff822c621fc7b3fc7a7f9d15ca88d18844ec27bde7ebd5f556f5"},"schema_version":"1.0","source":{"id":"1507.04098","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.04098","created_at":"2026-05-18T01:36:49Z"},{"alias_kind":"arxiv_version","alias_value":"1507.04098v1","created_at":"2026-05-18T01:36:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.04098","created_at":"2026-05-18T01:36:49Z"},{"alias_kind":"pith_short_12","alias_value":"E6TAETF3PKJ6","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"E6TAETF3PKJ6KUOT","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"E6TAETF3","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:4657323b420328631a83df6b25de0eb935582a77d699adc411a4e0522c951e77","target":"graph","created_at":"2026-05-18T01:36:49Z","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":"This work deals with the focusing Nonlinear Schrodinger Equation in one dimension with pure-power nonlinearity near cubic. We consider the spectrum of the linearized operator about the soliton solution. When the nonlinearity is exactly cubic, the linearized operator has resonances at the edges of the essential spectrum. We establish the degenerate bifurcation of these resonances to eigenvalues as the nonlinearity deviates from cubic. The leading-order expression for these eigenvalues is consistent with previous numerical computations.","authors_text":"Matt Coles, Stephen Gustafson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-15T06:31:22Z","title":"A Degenerate Edge Bifurcation in the 1D Linearized Nonlinear Schrodinger Equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04098","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":"sha256:2456f1f4c6d1f518dcb483fc284ea8c963b59673c03661bc776f3c63aed33225","target":"record","created_at":"2026-05-18T01:36:49Z","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":"284cacd3d9ea082361380540221f3c5a451be53e675aaecb639f397ab1ee0def","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-15T06:31:22Z","title_canon_sha256":"a058913fcab9ff822c621fc7b3fc7a7f9d15ca88d18844ec27bde7ebd5f556f5"},"schema_version":"1.0","source":{"id":"1507.04098","kind":"arxiv","version":1}},"canonical_sha256":"27a6024cbb7a93e551d3197b14b0150066e52d758f275e2d854019e5f53c6c20","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"27a6024cbb7a93e551d3197b14b0150066e52d758f275e2d854019e5f53c6c20","first_computed_at":"2026-05-18T01:36:49.142228Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:49.142228Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"c6HTDDnHtUQaXLKjy4vQPTriYVEAU0q0oHYepZSQJ3roG5FXOvcFDNUgPwY3I5aqyVXRNefL54FR+9lhc91jDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:49.142902Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.04098","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2456f1f4c6d1f518dcb483fc284ea8c963b59673c03661bc776f3c63aed33225","sha256:4657323b420328631a83df6b25de0eb935582a77d699adc411a4e0522c951e77"],"state_sha256":"a86c57f8e329924349b0958d1dd2d118755262ee7453067a447676000e731b46"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"15HwQw3uDE5i8L3593ULT0GV53+lPeU0+bIynojZ2sGlJfgDS1BN8w5QYv4JV0G+BaU2aTAVMpkj10fG0ZhTAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T01:38:20.012345Z","bundle_sha256":"03c7530dc11ffcf121ddb7ad7927587a3ff0d314169326d14b9a8872aa893e01"}}