{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:W7ZGUHLSC2NXJXHDKDFH4OUT63","short_pith_number":"pith:W7ZGUHLS","canonical_record":{"source":{"id":"1508.04664","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-19T14:46:31Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"bff0d45c07acce5d1ce76432178f13194ceec75bc311cdb24558a0b999ee61ad","abstract_canon_sha256":"ca49e21a369c953771a250b459b59590086ec63e79bc3374b57e5201c0768cf9"},"schema_version":"1.0"},"canonical_sha256":"b7f26a1d72169b74dce350ca7e3a93f6d2f7962188dee9b4bde8187f48e543b6","source":{"kind":"arxiv","id":"1508.04664","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.04664","created_at":"2026-05-17T23:40:50Z"},{"alias_kind":"arxiv_version","alias_value":"1508.04664v2","created_at":"2026-05-17T23:40:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.04664","created_at":"2026-05-17T23:40:50Z"},{"alias_kind":"pith_short_12","alias_value":"W7ZGUHLSC2NX","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"W7ZGUHLSC2NXJXHD","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"W7ZGUHLS","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:W7ZGUHLSC2NXJXHDKDFH4OUT63","target":"record","payload":{"canonical_record":{"source":{"id":"1508.04664","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-19T14:46:31Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"bff0d45c07acce5d1ce76432178f13194ceec75bc311cdb24558a0b999ee61ad","abstract_canon_sha256":"ca49e21a369c953771a250b459b59590086ec63e79bc3374b57e5201c0768cf9"},"schema_version":"1.0"},"canonical_sha256":"b7f26a1d72169b74dce350ca7e3a93f6d2f7962188dee9b4bde8187f48e543b6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:50.141797Z","signature_b64":"uH+gJqNBjRfkhsnT1Hd2vy1rjDo90ZgdTU7rL8PYEO99lGwSpxp2Ds8wJEGYzxLKRBIKnHaGZvDCepRFyG6TBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b7f26a1d72169b74dce350ca7e3a93f6d2f7962188dee9b4bde8187f48e543b6","last_reissued_at":"2026-05-17T23:40:50.140875Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:50.140875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.04664","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":"2026-05-17T23:40:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RGuKylxTXwPfD4QLRjzOQWOSbwyzaujmv4bRZZIbcCajJw9uooV4XCfF9SWbEzzeP7j/nFIfsv1PhlUBf3VnDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T19:23:55.201772Z"},"content_sha256":"5eb7ea36820010cbf076d816b01ee9870919851dbba0ed717d254ace747f2dc0","schema_version":"1.0","event_id":"sha256:5eb7ea36820010cbf076d816b01ee9870919851dbba0ed717d254ace747f2dc0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:W7ZGUHLSC2NXJXHDKDFH4OUT63","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Traveling gravity water waves with critical layers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.AP","authors_text":"Ailo Aasen, Kristoffer Varholm","submitted_at":"2015-08-19T14:46:31Z","abstract_excerpt":"We establish the existence of small-amplitude uni- and bimodal steady periodic gravity waves with an affine vorticity distribution, using a bifurcation argument that differs slightly from earlier theory. The solutions describe waves with critical layers and an arbitrary number of crests and troughs in each minimal period. An important part of the analysis is a fairly complete description of the local geometry of the so-called kernel equation, and of the small-amplitude solutions. Finally, we investigate the asymptotic behavior of the bifurcating solutions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.04664","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:40:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YOnJOYVpuIeD6oChxYLHNQ6S3K0VwD/T5pSgimHyStKajjk7BFzysauniBK2iRm+q42PP7gKu34hWoYt03DxCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T19:23:55.202113Z"},"content_sha256":"79efcce4006fe6cde64303490d798130da582362ba086ed1e4fbfc4eaf8d8dda","schema_version":"1.0","event_id":"sha256:79efcce4006fe6cde64303490d798130da582362ba086ed1e4fbfc4eaf8d8dda"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/W7ZGUHLSC2NXJXHDKDFH4OUT63/bundle.json","state_url":"https://pith.science/pith/W7ZGUHLSC2NXJXHDKDFH4OUT63/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/W7ZGUHLSC2NXJXHDKDFH4OUT63/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":"2026-06-26T19:23:55Z","links":{"resolver":"https://pith.science/pith/W7ZGUHLSC2NXJXHDKDFH4OUT63","bundle":"https://pith.science/pith/W7ZGUHLSC2NXJXHDKDFH4OUT63/bundle.json","state":"https://pith.science/pith/W7ZGUHLSC2NXJXHDKDFH4OUT63/state.json","well_known_bundle":"https://pith.science/.well-known/pith/W7ZGUHLSC2NXJXHDKDFH4OUT63/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:W7ZGUHLSC2NXJXHDKDFH4OUT63","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":"ca49e21a369c953771a250b459b59590086ec63e79bc3374b57e5201c0768cf9","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-19T14:46:31Z","title_canon_sha256":"bff0d45c07acce5d1ce76432178f13194ceec75bc311cdb24558a0b999ee61ad"},"schema_version":"1.0","source":{"id":"1508.04664","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.04664","created_at":"2026-05-17T23:40:50Z"},{"alias_kind":"arxiv_version","alias_value":"1508.04664v2","created_at":"2026-05-17T23:40:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.04664","created_at":"2026-05-17T23:40:50Z"},{"alias_kind":"pith_short_12","alias_value":"W7ZGUHLSC2NX","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"W7ZGUHLSC2NXJXHD","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"W7ZGUHLS","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:79efcce4006fe6cde64303490d798130da582362ba086ed1e4fbfc4eaf8d8dda","target":"graph","created_at":"2026-05-17T23:40:50Z","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 establish the existence of small-amplitude uni- and bimodal steady periodic gravity waves with an affine vorticity distribution, using a bifurcation argument that differs slightly from earlier theory. The solutions describe waves with critical layers and an arbitrary number of crests and troughs in each minimal period. An important part of the analysis is a fairly complete description of the local geometry of the so-called kernel equation, and of the small-amplitude solutions. Finally, we investigate the asymptotic behavior of the bifurcating solutions.","authors_text":"Ailo Aasen, Kristoffer Varholm","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-19T14:46:31Z","title":"Traveling gravity water waves with critical layers"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.04664","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":"sha256:5eb7ea36820010cbf076d816b01ee9870919851dbba0ed717d254ace747f2dc0","target":"record","created_at":"2026-05-17T23:40:50Z","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":"ca49e21a369c953771a250b459b59590086ec63e79bc3374b57e5201c0768cf9","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-08-19T14:46:31Z","title_canon_sha256":"bff0d45c07acce5d1ce76432178f13194ceec75bc311cdb24558a0b999ee61ad"},"schema_version":"1.0","source":{"id":"1508.04664","kind":"arxiv","version":2}},"canonical_sha256":"b7f26a1d72169b74dce350ca7e3a93f6d2f7962188dee9b4bde8187f48e543b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b7f26a1d72169b74dce350ca7e3a93f6d2f7962188dee9b4bde8187f48e543b6","first_computed_at":"2026-05-17T23:40:50.140875Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:40:50.140875Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uH+gJqNBjRfkhsnT1Hd2vy1rjDo90ZgdTU7rL8PYEO99lGwSpxp2Ds8wJEGYzxLKRBIKnHaGZvDCepRFyG6TBw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:40:50.141797Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.04664","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5eb7ea36820010cbf076d816b01ee9870919851dbba0ed717d254ace747f2dc0","sha256:79efcce4006fe6cde64303490d798130da582362ba086ed1e4fbfc4eaf8d8dda"],"state_sha256":"24f1803599b554620b8ddcf5d4e197854c23523cc9c17728a851b0afa26f71bc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OBEMPfw4M0cj98qAq2HIfqmbzBeK1hdM6SDP6cJom1beNOoOpuYEBrY54g9eH1JIKiJuE8/VK+lCVwA2VCvyAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T19:23:55.203995Z","bundle_sha256":"5e8da0b44f1946ed1b1768c0e0f2b668d5498faf94cd67adef220f97ab7de907"}}