{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:KNLCLVYZ4KL6RRGL66RQWBG44E","short_pith_number":"pith:KNLCLVYZ","canonical_record":{"source":{"id":"1004.0909","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-04-06T17:46:04Z","cross_cats_sorted":[],"title_canon_sha256":"4f0fd1571e0d20237490ac2f90fdbcbb5eb945bcb76e19f00ba91977303f2658","abstract_canon_sha256":"5e178ef690f33156979b611d4a642d9550a70ee59c0a6845366948a9892dd393"},"schema_version":"1.0"},"canonical_sha256":"535625d719e297e8c4cbf7a30b04dce10cdbd730cb1c74af05f60ee0d5c7c682","source":{"kind":"arxiv","id":"1004.0909","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.0909","created_at":"2026-05-18T02:08:14Z"},{"alias_kind":"arxiv_version","alias_value":"1004.0909v2","created_at":"2026-05-18T02:08:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.0909","created_at":"2026-05-18T02:08:14Z"},{"alias_kind":"pith_short_12","alias_value":"KNLCLVYZ4KL6","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"KNLCLVYZ4KL6RRGL","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"KNLCLVYZ","created_at":"2026-05-18T12:26:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:KNLCLVYZ4KL6RRGL66RQWBG44E","target":"record","payload":{"canonical_record":{"source":{"id":"1004.0909","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-04-06T17:46:04Z","cross_cats_sorted":[],"title_canon_sha256":"4f0fd1571e0d20237490ac2f90fdbcbb5eb945bcb76e19f00ba91977303f2658","abstract_canon_sha256":"5e178ef690f33156979b611d4a642d9550a70ee59c0a6845366948a9892dd393"},"schema_version":"1.0"},"canonical_sha256":"535625d719e297e8c4cbf7a30b04dce10cdbd730cb1c74af05f60ee0d5c7c682","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:08:14.075258Z","signature_b64":"XKfkLhLZtYeWSG9B7xeeyxJpt4hF9+ZWO/7AC3L4d5SH4IQiks07jQsAtr00ih93gcpFak5g+PCIInepMzz+Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"535625d719e297e8c4cbf7a30b04dce10cdbd730cb1c74af05f60ee0d5c7c682","last_reissued_at":"2026-05-18T02:08:14.074772Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:08:14.074772Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.0909","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":"2026-05-18T02:08:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OeT9EnV1cN7ny//3c87Bb1WLVP0NAWNLaH+ys4hiWSsH0dzr08ugR6e08YaM0uhEQA5kcq5Nh+veIT47aactCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T22:29:32.949496Z"},"content_sha256":"4dafabebaac57e9138c1891390cc144113509c8fb604c5afb1aea0834e376134","schema_version":"1.0","event_id":"sha256:4dafabebaac57e9138c1891390cc144113509c8fb604c5afb1aea0834e376134"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:KNLCLVYZ4KL6RRGL66RQWBG44E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kevin Zumbrun, Mathew Johnson","submitted_at":"2010-04-06T17:46:04Z","abstract_excerpt":"Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian rate of spatially periodic traveling-waves of systems of reaction diffusion equations. In the case that wave-speed is identically zero for all periodic solutions, we recover and slightly sharpen a well-known result of Schneider obtained by renormalization/Bloch transform techniques; by the same arguments, we are able to treat the open case of nonzero wave-s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.0909","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":"2026-05-18T02:08:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hv8IMTWYTNcHClC5Ap8HyQiLhrUpOWu+mE6VusDGSsIoNnubgIyNZdNZwtds7xIVnL+ebBhOq2fC9P339ApfBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T22:29:32.949882Z"},"content_sha256":"f88a1073d0b83dec0b59d58050749391f05dbfe4da5508268fcb3b1d7d236cda","schema_version":"1.0","event_id":"sha256:f88a1073d0b83dec0b59d58050749391f05dbfe4da5508268fcb3b1d7d236cda"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KNLCLVYZ4KL6RRGL66RQWBG44E/bundle.json","state_url":"https://pith.science/pith/KNLCLVYZ4KL6RRGL66RQWBG44E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KNLCLVYZ4KL6RRGL66RQWBG44E/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":"2026-06-22T22:29:32Z","links":{"resolver":"https://pith.science/pith/KNLCLVYZ4KL6RRGL66RQWBG44E","bundle":"https://pith.science/pith/KNLCLVYZ4KL6RRGL66RQWBG44E/bundle.json","state":"https://pith.science/pith/KNLCLVYZ4KL6RRGL66RQWBG44E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KNLCLVYZ4KL6RRGL66RQWBG44E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:KNLCLVYZ4KL6RRGL66RQWBG44E","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":"5e178ef690f33156979b611d4a642d9550a70ee59c0a6845366948a9892dd393","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-04-06T17:46:04Z","title_canon_sha256":"4f0fd1571e0d20237490ac2f90fdbcbb5eb945bcb76e19f00ba91977303f2658"},"schema_version":"1.0","source":{"id":"1004.0909","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.0909","created_at":"2026-05-18T02:08:14Z"},{"alias_kind":"arxiv_version","alias_value":"1004.0909v2","created_at":"2026-05-18T02:08:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.0909","created_at":"2026-05-18T02:08:14Z"},{"alias_kind":"pith_short_12","alias_value":"KNLCLVYZ4KL6","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"KNLCLVYZ4KL6RRGL","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"KNLCLVYZ","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:f88a1073d0b83dec0b59d58050749391f05dbfe4da5508268fcb3b1d7d236cda","target":"graph","created_at":"2026-05-18T02:08:14Z","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":"Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian rate of spatially periodic traveling-waves of systems of reaction diffusion equations. In the case that wave-speed is identically zero for all periodic solutions, we recover and slightly sharpen a well-known result of Schneider obtained by renormalization/Bloch transform techniques; by the same arguments, we are able to treat the open case of nonzero wave-s","authors_text":"Kevin Zumbrun, Mathew Johnson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-04-06T17:46:04Z","title":"Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.0909","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":"sha256:4dafabebaac57e9138c1891390cc144113509c8fb604c5afb1aea0834e376134","target":"record","created_at":"2026-05-18T02:08:14Z","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":"5e178ef690f33156979b611d4a642d9550a70ee59c0a6845366948a9892dd393","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-04-06T17:46:04Z","title_canon_sha256":"4f0fd1571e0d20237490ac2f90fdbcbb5eb945bcb76e19f00ba91977303f2658"},"schema_version":"1.0","source":{"id":"1004.0909","kind":"arxiv","version":2}},"canonical_sha256":"535625d719e297e8c4cbf7a30b04dce10cdbd730cb1c74af05f60ee0d5c7c682","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"535625d719e297e8c4cbf7a30b04dce10cdbd730cb1c74af05f60ee0d5c7c682","first_computed_at":"2026-05-18T02:08:14.074772Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:08:14.074772Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XKfkLhLZtYeWSG9B7xeeyxJpt4hF9+ZWO/7AC3L4d5SH4IQiks07jQsAtr00ih93gcpFak5g+PCIInepMzz+Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:08:14.075258Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.0909","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4dafabebaac57e9138c1891390cc144113509c8fb604c5afb1aea0834e376134","sha256:f88a1073d0b83dec0b59d58050749391f05dbfe4da5508268fcb3b1d7d236cda"],"state_sha256":"db89b64a5da23913cb326c6ee2eabbadc24eca323944cb3f4004dbf21dc9f39a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VS8gMKEo3G9Uh//DotG53+WjLCs8tLAg/OSWEOLXaz4H/1Uq+tA4gZ5qAIILnejVDdM/y6kbEXbnq6h6Qys/DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T22:29:32.952092Z","bundle_sha256":"4e7db6b4f21ad443fe140421324d183d05b92d30614e886970bca32d60e3bbd9"}}