{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:7T63RGO5HFID2G3YQRDX2FP3A3","short_pith_number":"pith:7T63RGO5","canonical_record":{"source":{"id":"1312.1132","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-12-04T12:17:21Z","cross_cats_sorted":[],"title_canon_sha256":"3d688d140f9ced4298324cf2ab6f16001fb2b2fef41e5499f6dc5c26a240cac3","abstract_canon_sha256":"b5557bd2b0ba7cf32ed84897be43bae95cc70d7e7cd8e27d133c17c7ba1bfed6"},"schema_version":"1.0"},"canonical_sha256":"fcfdb899dd39503d1b7884477d15fb06edcace6ff614cc7ed62d505dfde91581","source":{"kind":"arxiv","id":"1312.1132","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.1132","created_at":"2026-05-18T00:43:18Z"},{"alias_kind":"arxiv_version","alias_value":"1312.1132v3","created_at":"2026-05-18T00:43:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.1132","created_at":"2026-05-18T00:43:18Z"},{"alias_kind":"pith_short_12","alias_value":"7T63RGO5HFID","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7T63RGO5HFID2G3Y","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7T63RGO5","created_at":"2026-05-18T12:27:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:7T63RGO5HFID2G3YQRDX2FP3A3","target":"record","payload":{"canonical_record":{"source":{"id":"1312.1132","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-12-04T12:17:21Z","cross_cats_sorted":[],"title_canon_sha256":"3d688d140f9ced4298324cf2ab6f16001fb2b2fef41e5499f6dc5c26a240cac3","abstract_canon_sha256":"b5557bd2b0ba7cf32ed84897be43bae95cc70d7e7cd8e27d133c17c7ba1bfed6"},"schema_version":"1.0"},"canonical_sha256":"fcfdb899dd39503d1b7884477d15fb06edcace6ff614cc7ed62d505dfde91581","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:18.942085Z","signature_b64":"tp+xzyHQayKzd7Qh3fae398dRfjtfkTfeJs3paXJ2HQt3X/R/FFBZHeZFVvYZUGlAxeHOc4wdhLIZ32WBLulBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fcfdb899dd39503d1b7884477d15fb06edcace6ff614cc7ed62d505dfde91581","last_reissued_at":"2026-05-18T00:43:18.941398Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:18.941398Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1312.1132","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:43:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ljJzMc2XSHcR/PqUseJuTpWMny75JJFzz9wnOQ78vSSLkl61fg3/HlJsvYtJl9vq1bKjleoaMl04xni54M8hBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T19:11:46.754586Z"},"content_sha256":"812e90bb360dded2ba3598455116cabc6ebb6fb3a964fa9ae0a9c161a3a80c93","schema_version":"1.0","event_id":"sha256:812e90bb360dded2ba3598455116cabc6ebb6fb3a964fa9ae0a9c161a3a80c93"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:7T63RGO5HFID2G3YQRDX2FP3A3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral and Modulational Stability of Periodic Wavetrains for the Nonlinear Klein-Gordon Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Christopher K. R. T. Jones, Peter D. Miller, Ramon G. Plaza, Robert Marangell","submitted_at":"2013-12-04T12:17:21Z","abstract_excerpt":"This paper is a detailed and self-contained study of the stability properties of periodic traveling wave solutions of the nonlinear Klein-Gordon equation $u_{tt}-u_{xx}+V'(u)=0$, where $u$ is a scalar-valued function of $x$ and $t$, and the potential $V(u)$ is of class $C^2$ and periodic. Stability is considered both from the point of view of spectral analysis of the linearized problem (spectral stability analysis) and from the point of view of wave modulation theory (the strongly nonlinear theory due to Whitham as well as the weakly nonlinear theory of wave packets). The aim is to develop and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.1132","kind":"arxiv","version":3},"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":"2026-05-18T00:43:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HzGlDeKZm1GkltGdM16YHkWMZ2s6W97G1gu2O9br/3LKRk0AcndF0wIt5d9o/WW/vKIjZgRToB9M/OoZc0m8CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T19:11:46.755299Z"},"content_sha256":"2a9711a3114593d726de8e911c1d5458247002c3760abf0c2c5d0adbee5d8376","schema_version":"1.0","event_id":"sha256:2a9711a3114593d726de8e911c1d5458247002c3760abf0c2c5d0adbee5d8376"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7T63RGO5HFID2G3YQRDX2FP3A3/bundle.json","state_url":"https://pith.science/pith/7T63RGO5HFID2G3YQRDX2FP3A3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7T63RGO5HFID2G3YQRDX2FP3A3/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":"2026-05-31T19:11:46Z","links":{"resolver":"https://pith.science/pith/7T63RGO5HFID2G3YQRDX2FP3A3","bundle":"https://pith.science/pith/7T63RGO5HFID2G3YQRDX2FP3A3/bundle.json","state":"https://pith.science/pith/7T63RGO5HFID2G3YQRDX2FP3A3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7T63RGO5HFID2G3YQRDX2FP3A3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:7T63RGO5HFID2G3YQRDX2FP3A3","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":"b5557bd2b0ba7cf32ed84897be43bae95cc70d7e7cd8e27d133c17c7ba1bfed6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-12-04T12:17:21Z","title_canon_sha256":"3d688d140f9ced4298324cf2ab6f16001fb2b2fef41e5499f6dc5c26a240cac3"},"schema_version":"1.0","source":{"id":"1312.1132","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1312.1132","created_at":"2026-05-18T00:43:18Z"},{"alias_kind":"arxiv_version","alias_value":"1312.1132v3","created_at":"2026-05-18T00:43:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.1132","created_at":"2026-05-18T00:43:18Z"},{"alias_kind":"pith_short_12","alias_value":"7T63RGO5HFID","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7T63RGO5HFID2G3Y","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7T63RGO5","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:2a9711a3114593d726de8e911c1d5458247002c3760abf0c2c5d0adbee5d8376","target":"graph","created_at":"2026-05-18T00:43:18Z","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 paper is a detailed and self-contained study of the stability properties of periodic traveling wave solutions of the nonlinear Klein-Gordon equation $u_{tt}-u_{xx}+V'(u)=0$, where $u$ is a scalar-valued function of $x$ and $t$, and the potential $V(u)$ is of class $C^2$ and periodic. Stability is considered both from the point of view of spectral analysis of the linearized problem (spectral stability analysis) and from the point of view of wave modulation theory (the strongly nonlinear theory due to Whitham as well as the weakly nonlinear theory of wave packets). The aim is to develop and","authors_text":"Christopher K. R. T. Jones, Peter D. Miller, Ramon G. Plaza, Robert Marangell","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-12-04T12:17:21Z","title":"Spectral and Modulational Stability of Periodic Wavetrains for the Nonlinear Klein-Gordon Equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.1132","kind":"arxiv","version":3},"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":"sha256:812e90bb360dded2ba3598455116cabc6ebb6fb3a964fa9ae0a9c161a3a80c93","target":"record","created_at":"2026-05-18T00:43:18Z","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":"b5557bd2b0ba7cf32ed84897be43bae95cc70d7e7cd8e27d133c17c7ba1bfed6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-12-04T12:17:21Z","title_canon_sha256":"3d688d140f9ced4298324cf2ab6f16001fb2b2fef41e5499f6dc5c26a240cac3"},"schema_version":"1.0","source":{"id":"1312.1132","kind":"arxiv","version":3}},"canonical_sha256":"fcfdb899dd39503d1b7884477d15fb06edcace6ff614cc7ed62d505dfde91581","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fcfdb899dd39503d1b7884477d15fb06edcace6ff614cc7ed62d505dfde91581","first_computed_at":"2026-05-18T00:43:18.941398Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:43:18.941398Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tp+xzyHQayKzd7Qh3fae398dRfjtfkTfeJs3paXJ2HQt3X/R/FFBZHeZFVvYZUGlAxeHOc4wdhLIZ32WBLulBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:43:18.942085Z","signed_message":"canonical_sha256_bytes"},"source_id":"1312.1132","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:812e90bb360dded2ba3598455116cabc6ebb6fb3a964fa9ae0a9c161a3a80c93","sha256:2a9711a3114593d726de8e911c1d5458247002c3760abf0c2c5d0adbee5d8376"],"state_sha256":"8255845acaf931410edd020e24768fb68dcc4387f234b45ef39045404458a351"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AzvJESYOl2D2FYwSJcvS4ZgNO8PKYI/7dz+b/1K6idwBhHExHZOo7yqxWI/Mt68yVda9JminjRGOoPgZ4+PSAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T19:11:46.759051Z","bundle_sha256":"4f5755cc666995b2439eb03efcd13df23ea7484b464ff3a690025c1f4be33380"}}