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More precisely, given a small $W(\\xi)$, we construct a solution $u$ such that \\begin{equation*}\n  u\\rightarrow (2\\pi)^{-1/2}(\\ii t)^{-1/2}e^{\\ii x^2/(2t)}\\, W\\!\\Big(\\frac{x}{t}\\Big)\\exp(-\\ii\\la|W(x/t)|^2\\log t). \\end{equation*} Crucially, we design a contraction map, so that we can run the analysis in the spirit of Kato--Pusateri \\cite{KP} for $w$ with a forcing term depe"},"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":"2605.23063","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2026-05-21T21:54:40Z","cross_cats_sorted":[],"title_canon_sha256":"f46cecb124f617af5503977282998d42ca566410aa0e41f23d7c1e150fd97cf3","abstract_canon_sha256":"1756ea9ce1e4ecb739fccf179f998400668693455a7383fd034d2b368e39cea3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-25T02:01:36.725343Z","signature_b64":"VaAQJ4dT4yM49VKWJ705cC968q4rmI/h1O4ykbusz7uZvi/p8y6DZVyP7PiG3mqaWNcs9nSOr45BegdUJQHhDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4f881155467e986f2620e8aa8aec6da86c78fab821af1a3785c16f2d81a5ea84","last_reissued_at":"2026-05-25T02:01:36.724785Z","signature_status":"signed_v1","first_computed_at":"2026-05-25T02:01:36.724785Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the final-state problem for the 1D cubic NLS","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gong Chen, Yongyu Qiang","submitted_at":"2026-05-21T21:54:40Z","abstract_excerpt":"We consider the one-dimensional cubic nonlinear Schr\\\"odinger equation $$ \\ii\\partial_tu+\\frac12\\partial_{xx}u=\\la|u|^2u,\\,\\lambda=\\pm 1 $$ and solve the final-state (modified wave operator) problem for small asymptotic data. More precisely, given a small $W(\\xi)$, we construct a solution $u$ such that \\begin{equation*}\n  u\\rightarrow (2\\pi)^{-1/2}(\\ii t)^{-1/2}e^{\\ii x^2/(2t)}\\, W\\!\\Big(\\frac{x}{t}\\Big)\\exp(-\\ii\\la|W(x/t)|^2\\log t). \\end{equation*} Crucially, we design a contraction map, so that we can run the analysis in the spirit of Kato--Pusateri \\cite{KP} for $w$ with a forcing term depe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.23063","kind":"arxiv","version":1},"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/2605.23063/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2605.23063","created_at":"2026-05-25T02:01:36.724872+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.23063v1","created_at":"2026-05-25T02:01:36.724872+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.23063","created_at":"2026-05-25T02:01:36.724872+00:00"},{"alias_kind":"pith_short_12","alias_value":"J6EBCVKGP2MG","created_at":"2026-05-25T02:01:36.724872+00:00"},{"alias_kind":"pith_short_16","alias_value":"J6EBCVKGP2MG6JRA","created_at":"2026-05-25T02:01:36.724872+00:00"},{"alias_kind":"pith_short_8","alias_value":"J6EBCVKG","created_at":"2026-05-25T02:01:36.724872+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/J6EBCVKGP2MG6JRA5CVIV3DNVB","json":"https://pith.science/pith/J6EBCVKGP2MG6JRA5CVIV3DNVB.json","graph_json":"https://pith.science/api/pith-number/J6EBCVKGP2MG6JRA5CVIV3DNVB/graph.json","events_json":"https://pith.science/api/pith-number/J6EBCVKGP2MG6JRA5CVIV3DNVB/events.json","paper":"https://pith.science/paper/J6EBCVKG"},"agent_actions":{"view_html":"https://pith.science/pith/J6EBCVKGP2MG6JRA5CVIV3DNVB","download_json":"https://pith.science/pith/J6EBCVKGP2MG6JRA5CVIV3DNVB.json","view_paper":"https://pith.science/paper/J6EBCVKG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.23063&json=true","fetch_graph":"https://pith.science/api/pith-number/J6EBCVKGP2MG6JRA5CVIV3DNVB/graph.json","fetch_events":"https://pith.science/api/pith-number/J6EBCVKGP2MG6JRA5CVIV3DNVB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/J6EBCVKGP2MG6JRA5CVIV3DNVB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/J6EBCVKGP2MG6JRA5CVIV3DNVB/action/storage_attestation","attest_author":"https://pith.science/pith/J6EBCVKGP2MG6JRA5CVIV3DNVB/action/author_attestation","sign_citation":"https://pith.science/pith/J6EBCVKGP2MG6JRA5CVIV3DNVB/action/citation_signature","submit_replication":"https://pith.science/pith/J6EBCVKGP2MG6JRA5CVIV3DNVB/action/replication_record"}},"created_at":"2026-05-25T02:01:36.724872+00:00","updated_at":"2026-05-25T02:01:36.724872+00:00"}