{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2006:CJXXQX5FKJYNIPGONPT7SW6KHA","short_pith_number":"pith:CJXXQX5F","schema_version":"1.0","canonical_sha256":"126f785fa55270d43cce6be7f95bca381b38b5e7ab1dff7abae61ea9861c273b","source":{"kind":"arxiv","id":"math/0605031","version":2},"attestation_state":"computed","paper":{"title":"Asymptotic stability of small solitons to 1D NLS with potential","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tetsu Mizumachi","submitted_at":"2006-05-01T05:59:38Z","abstract_excerpt":"We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinear Schr\\\"{o}dinger equations $$ iu_t+u_{xx}=Vu\\pm |u|^{p-1}u \\quad\\text{for $(x,t)\\in\\mathbb{R}\\times\\mathbb{R}$,}$$ in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai \\cite{GNT} in the 3-dimensional case using the endpoint Strichartz estimate.\n  To prove asymptotic stability of solitary waves, we need to show that a dispersive part $v(t,x)$ of a solution belongs to $L^2_t(0,\\infty;X)$ for some space $X$. In the 1-dimensional case, this property does not follow from the Strich"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0605031","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2006-05-01T05:59:38Z","cross_cats_sorted":[],"title_canon_sha256":"51401e5d51c3154a41b57795747530cf1e8e3643a6eb3a5337ed3c1d27911c7e","abstract_canon_sha256":"b29dd94cd31434948257a6bec12f03142e904971b36b8a8c39ebf904b1de836e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:35.494399Z","signature_b64":"llcOr6umnwRyrhnGuUV+Nnr1nQs8i+GxVVL9ncHZGHGDmMF5kKk3asd/zSs1K3vUPL5s9d/flVuyEbPc3nzyDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"126f785fa55270d43cce6be7f95bca381b38b5e7ab1dff7abae61ea9861c273b","last_reissued_at":"2026-05-18T04:42:35.493731Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:35.493731Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic stability of small solitons to 1D NLS with potential","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tetsu Mizumachi","submitted_at":"2006-05-01T05:59:38Z","abstract_excerpt":"We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinear Schr\\\"{o}dinger equations $$ iu_t+u_{xx}=Vu\\pm |u|^{p-1}u \\quad\\text{for $(x,t)\\in\\mathbb{R}\\times\\mathbb{R}$,}$$ in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai \\cite{GNT} in the 3-dimensional case using the endpoint Strichartz estimate.\n  To prove asymptotic stability of solitary waves, we need to show that a dispersive part $v(t,x)$ of a solution belongs to $L^2_t(0,\\infty;X)$ for some space $X$. In the 1-dimensional case, this property does not follow from the Strich"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0605031","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0605031","created_at":"2026-05-18T04:42:35.493833+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0605031v2","created_at":"2026-05-18T04:42:35.493833+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0605031","created_at":"2026-05-18T04:42:35.493833+00:00"},{"alias_kind":"pith_short_12","alias_value":"CJXXQX5FKJYN","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_16","alias_value":"CJXXQX5FKJYNIPGO","created_at":"2026-05-18T12:25:53.939244+00:00"},{"alias_kind":"pith_short_8","alias_value":"CJXXQX5F","created_at":"2026-05-18T12:25:53.939244+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CJXXQX5FKJYNIPGONPT7SW6KHA","json":"https://pith.science/pith/CJXXQX5FKJYNIPGONPT7SW6KHA.json","graph_json":"https://pith.science/api/pith-number/CJXXQX5FKJYNIPGONPT7SW6KHA/graph.json","events_json":"https://pith.science/api/pith-number/CJXXQX5FKJYNIPGONPT7SW6KHA/events.json","paper":"https://pith.science/paper/CJXXQX5F"},"agent_actions":{"view_html":"https://pith.science/pith/CJXXQX5FKJYNIPGONPT7SW6KHA","download_json":"https://pith.science/pith/CJXXQX5FKJYNIPGONPT7SW6KHA.json","view_paper":"https://pith.science/paper/CJXXQX5F","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0605031&json=true","fetch_graph":"https://pith.science/api/pith-number/CJXXQX5FKJYNIPGONPT7SW6KHA/graph.json","fetch_events":"https://pith.science/api/pith-number/CJXXQX5FKJYNIPGONPT7SW6KHA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CJXXQX5FKJYNIPGONPT7SW6KHA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CJXXQX5FKJYNIPGONPT7SW6KHA/action/storage_attestation","attest_author":"https://pith.science/pith/CJXXQX5FKJYNIPGONPT7SW6KHA/action/author_attestation","sign_citation":"https://pith.science/pith/CJXXQX5FKJYNIPGONPT7SW6KHA/action/citation_signature","submit_replication":"https://pith.science/pith/CJXXQX5FKJYNIPGONPT7SW6KHA/action/replication_record"}},"created_at":"2026-05-18T04:42:35.493833+00:00","updated_at":"2026-05-18T04:42:35.493833+00:00"}