{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:NHUQFM6YQE2LAQYYR46PYZV2YA","short_pith_number":"pith:NHUQFM6Y","canonical_record":{"source":{"id":"2411.11703","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2024-11-18T16:30:19Z","cross_cats_sorted":[],"title_canon_sha256":"187dcb6ee36bae411a217ae969e5338233aa4ea03adcdbb0fc79e6ecf51f1ff3","abstract_canon_sha256":"ad2ed99f3452b3a96487136f42a4f58ec8fd7bfbf8fa4097062e5d6a4fc7cc68"},"schema_version":"1.0"},"canonical_sha256":"69e902b3d88134b043188f3cfc66bac01ff2f9bed68f57ee074d2180bef050a2","source":{"kind":"arxiv","id":"2411.11703","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.11703","created_at":"2026-06-29T01:14:01Z"},{"alias_kind":"arxiv_version","alias_value":"2411.11703v2","created_at":"2026-06-29T01:14:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.11703","created_at":"2026-06-29T01:14:01Z"},{"alias_kind":"pith_short_12","alias_value":"NHUQFM6YQE2L","created_at":"2026-06-29T01:14:01Z"},{"alias_kind":"pith_short_16","alias_value":"NHUQFM6YQE2LAQYY","created_at":"2026-06-29T01:14:01Z"},{"alias_kind":"pith_short_8","alias_value":"NHUQFM6Y","created_at":"2026-06-29T01:14:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:NHUQFM6YQE2LAQYYR46PYZV2YA","target":"record","payload":{"canonical_record":{"source":{"id":"2411.11703","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2024-11-18T16:30:19Z","cross_cats_sorted":[],"title_canon_sha256":"187dcb6ee36bae411a217ae969e5338233aa4ea03adcdbb0fc79e6ecf51f1ff3","abstract_canon_sha256":"ad2ed99f3452b3a96487136f42a4f58ec8fd7bfbf8fa4097062e5d6a4fc7cc68"},"schema_version":"1.0"},"canonical_sha256":"69e902b3d88134b043188f3cfc66bac01ff2f9bed68f57ee074d2180bef050a2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-29T01:14:01.010658Z","signature_b64":"XYtKrDZddY8YZdNwlbaCNRto3ElZ1QnTiwBCz2km6pCRpzfzoXSxYY8lIgSmX8OmclpzDD+a2wgcf93z4zV4BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"69e902b3d88134b043188f3cfc66bac01ff2f9bed68f57ee074d2180bef050a2","last_reissued_at":"2026-06-29T01:14:01.010132Z","signature_status":"signed_v1","first_computed_at":"2026-06-29T01:14:01.010132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.11703","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-29T01:14:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"bSZ1IsWmTN/ih2JgTZLqcXvAa7t68D5EuY7sMj+ZRH73qWyTcXT9/4w4pSniuY7ixfqy9TwduguxkPshXdzhBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T04:06:05.239704Z"},"content_sha256":"67f35cd074526a53603b7acc837dac4dab6d96a758985f63fc2555cb064ffc5b","schema_version":"1.0","event_id":"sha256:67f35cd074526a53603b7acc837dac4dab6d96a758985f63fc2555cb064ffc5b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:NHUQFM6YQE2LAQYYR46PYZV2YA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Construction of multi solitary waves with symmetry for the damped nonlinear Klein-Gordon equation","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Haiming Du, Rapha\\\"el C\\^ote","submitted_at":"2024-11-18T16:30:19Z","abstract_excerpt":"We are interested in the nonlinear damped Klein-Gordon equation \\[ \\partial_t^2 u+2\\alpha \\partial_t u-\\Delta u+u-|u|^{p-1}u=0 \\] on $\\mathbb{R}^d$ for $2\\le d\\le 5$ and energy sub-critical exponents $2 < p < \\frac{d+2}{d-2}$.\n  We construct multi-solitons, that is, solutions which behave for large times as a sum of decoupled solitons, in various configurations with symmetry: this includes multi-solitons whose soliton centers lie at the vertices of an expanding regular polygon (with or without a center), of a regular polyhedron (with a center), or of a higher dimensional regular polytope. We g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.11703","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.11703/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-06-29T01:14:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jDLjrybeu01TGP5MngS1JPp7S8p9zZL16jJDHrKN5t90np2ka9/1OF6LF9ROpeMoMEmNTOgJq3/qdSNzZhDvDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T04:06:05.240119Z"},"content_sha256":"dddb6481f12aeee225fe45fb31898e5493cb0fe3ba29df146a89cfa33b763c5b","schema_version":"1.0","event_id":"sha256:dddb6481f12aeee225fe45fb31898e5493cb0fe3ba29df146a89cfa33b763c5b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NHUQFM6YQE2LAQYYR46PYZV2YA/bundle.json","state_url":"https://pith.science/pith/NHUQFM6YQE2LAQYYR46PYZV2YA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NHUQFM6YQE2LAQYYR46PYZV2YA/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":"2026-08-13T04:06:05Z","links":{"resolver":"https://pith.science/pith/NHUQFM6YQE2LAQYYR46PYZV2YA","bundle":"https://pith.science/pith/NHUQFM6YQE2LAQYYR46PYZV2YA/bundle.json","state":"https://pith.science/pith/NHUQFM6YQE2LAQYYR46PYZV2YA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NHUQFM6YQE2LAQYYR46PYZV2YA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:NHUQFM6YQE2LAQYYR46PYZV2YA","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":"ad2ed99f3452b3a96487136f42a4f58ec8fd7bfbf8fa4097062e5d6a4fc7cc68","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2024-11-18T16:30:19Z","title_canon_sha256":"187dcb6ee36bae411a217ae969e5338233aa4ea03adcdbb0fc79e6ecf51f1ff3"},"schema_version":"1.0","source":{"id":"2411.11703","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.11703","created_at":"2026-06-29T01:14:01Z"},{"alias_kind":"arxiv_version","alias_value":"2411.11703v2","created_at":"2026-06-29T01:14:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.11703","created_at":"2026-06-29T01:14:01Z"},{"alias_kind":"pith_short_12","alias_value":"NHUQFM6YQE2L","created_at":"2026-06-29T01:14:01Z"},{"alias_kind":"pith_short_16","alias_value":"NHUQFM6YQE2LAQYY","created_at":"2026-06-29T01:14:01Z"},{"alias_kind":"pith_short_8","alias_value":"NHUQFM6Y","created_at":"2026-06-29T01:14:01Z"}],"graph_snapshots":[{"event_id":"sha256:dddb6481f12aeee225fe45fb31898e5493cb0fe3ba29df146a89cfa33b763c5b","target":"graph","created_at":"2026-06-29T01:14:01Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2411.11703/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We are interested in the nonlinear damped Klein-Gordon equation \\[ \\partial_t^2 u+2\\alpha \\partial_t u-\\Delta u+u-|u|^{p-1}u=0 \\] on $\\mathbb{R}^d$ for $2\\le d\\le 5$ and energy sub-critical exponents $2 < p < \\frac{d+2}{d-2}$.\n  We construct multi-solitons, that is, solutions which behave for large times as a sum of decoupled solitons, in various configurations with symmetry: this includes multi-solitons whose soliton centers lie at the vertices of an expanding regular polygon (with or without a center), of a regular polyhedron (with a center), or of a higher dimensional regular polytope. We g","authors_text":"Haiming Du, Rapha\\\"el C\\^ote","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2024-11-18T16:30:19Z","title":"Construction of multi solitary waves with symmetry for the damped nonlinear Klein-Gordon equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.11703","kind":"arxiv","version":2},"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":"sha256:67f35cd074526a53603b7acc837dac4dab6d96a758985f63fc2555cb064ffc5b","target":"record","created_at":"2026-06-29T01:14:01Z","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":"ad2ed99f3452b3a96487136f42a4f58ec8fd7bfbf8fa4097062e5d6a4fc7cc68","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2024-11-18T16:30:19Z","title_canon_sha256":"187dcb6ee36bae411a217ae969e5338233aa4ea03adcdbb0fc79e6ecf51f1ff3"},"schema_version":"1.0","source":{"id":"2411.11703","kind":"arxiv","version":2}},"canonical_sha256":"69e902b3d88134b043188f3cfc66bac01ff2f9bed68f57ee074d2180bef050a2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"69e902b3d88134b043188f3cfc66bac01ff2f9bed68f57ee074d2180bef050a2","first_computed_at":"2026-06-29T01:14:01.010132Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-29T01:14:01.010132Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XYtKrDZddY8YZdNwlbaCNRto3ElZ1QnTiwBCz2km6pCRpzfzoXSxYY8lIgSmX8OmclpzDD+a2wgcf93z4zV4BA==","signature_status":"signed_v1","signed_at":"2026-06-29T01:14:01.010658Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.11703","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:67f35cd074526a53603b7acc837dac4dab6d96a758985f63fc2555cb064ffc5b","sha256:dddb6481f12aeee225fe45fb31898e5493cb0fe3ba29df146a89cfa33b763c5b"],"state_sha256":"02c6e9826984f36abf0c69dbc1ef622ce59ee08b6c91b161b98338abfe7133d1"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZRAULvTgVZN8n6s+sNWo4yTFIu/QfZV684wef+TzK7plbF0/GQv07ExC/XMdOOnsdRPkficLHHr7OrJ9k0aEAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T04:06:05.242891Z","bundle_sha256":"c7b2776052fabdb1546fcd485bfc99d9d2246c81b6c2728e3614a03329a4c383"}}