{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:276X2U6JX7SWIV3MMZPJBGPDN5","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":"882fc3f9348dc6a5cfba3127e0ea28c3d44e90cd05d39a611358f45996aa564f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-10-04T07:59:43Z","title_canon_sha256":"40934f7065b4191d85fc0e1b603b3509109f11ddf462708b9636f63df6357637"},"schema_version":"1.0","source":{"id":"1210.1322","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.1322","created_at":"2026-05-18T01:22:26Z"},{"alias_kind":"arxiv_version","alias_value":"1210.1322v1","created_at":"2026-05-18T01:22:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.1322","created_at":"2026-05-18T01:22:26Z"},{"alias_kind":"pith_short_12","alias_value":"276X2U6JX7SW","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_16","alias_value":"276X2U6JX7SWIV3M","created_at":"2026-05-18T12:26:50Z"},{"alias_kind":"pith_short_8","alias_value":"276X2U6J","created_at":"2026-05-18T12:26:50Z"}],"graph_snapshots":[{"event_id":"sha256:0062d2f7c60134da7ae4541d906401d157ac6c8c1c8f4ac5e5c7acc6fd0f9485","target":"graph","created_at":"2026-05-18T01:22:26Z","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 study the stability/instability of the subsonic travelling waves of the Nonlinear Schr\\\"odinger Equation in dimension one. Our aim is to propose several methods for showing instability (use of the Grillakis-Shatah-Strauss theory, proof of existence of an unstable eigenvalue via an Evans function) or stability. For the later, we show how to construct in a systematic way a Liapounov functional for which the travelling wave is a local minimizer. These approaches allow to give a complete stability/instability analysis in the energy space including the critical case of the kink solution. We also","authors_text":"David Chiron","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-10-04T07:59:43Z","title":"Stability and instability for subsonic travelling waves of the Nonlinear Schr\\\"odinger Equation in dimension one"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.1322","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":"sha256:955520fd20b69fafa10759ab0dc6db98ae0132157c55dd2862e64d7897cc53a9","target":"record","created_at":"2026-05-18T01:22:26Z","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":"882fc3f9348dc6a5cfba3127e0ea28c3d44e90cd05d39a611358f45996aa564f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-10-04T07:59:43Z","title_canon_sha256":"40934f7065b4191d85fc0e1b603b3509109f11ddf462708b9636f63df6357637"},"schema_version":"1.0","source":{"id":"1210.1322","kind":"arxiv","version":1}},"canonical_sha256":"d7fd7d53c9bfe564576c665e9099e36f473b9c2cd126ab61fd768ea740c1d570","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d7fd7d53c9bfe564576c665e9099e36f473b9c2cd126ab61fd768ea740c1d570","first_computed_at":"2026-05-18T01:22:26.203455Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:26.203455Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"plkbXiG6CTErpW6SMsqXCk+BFPSRxtzq3Fs85KEtyrgrFZH4aAXYUxILNV6IEwyzTNgnsSEgozIz6fJbZBlpDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:26.204012Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.1322","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:955520fd20b69fafa10759ab0dc6db98ae0132157c55dd2862e64d7897cc53a9","sha256:0062d2f7c60134da7ae4541d906401d157ac6c8c1c8f4ac5e5c7acc6fd0f9485"],"state_sha256":"46b2bf0f9e28256ef3f3ef3a365adba30fe446872e93560f189868741c90eced"}