{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:72C4CV4L7Y6MBGVNDXJRTUWWOL","short_pith_number":"pith:72C4CV4L","schema_version":"1.0","canonical_sha256":"fe85c1578bfe3cc09aad1dd319d2d672dae63fa7c5c8a20f6212e159b0e576d8","source":{"kind":"arxiv","id":"1810.08974","version":1},"attestation_state":"computed","paper":{"title":"Scattering in the weighted $L^2$-space for a 2D nonlinear Schr\\\"odinger equation with inhomogeneous exponential nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Abdelwahab Bensouilah, Mohamed Majdoub, Van Duong Dinh","submitted_at":"2018-10-21T15:24:18Z","abstract_excerpt":"We investigate the defocusing inhomogeneous nonlinear Schr\\\"odinger equation\n  $$ i \\partial_tu + \\Delta u = |x|^{-b} \\left({\\rm e}^{\\alpha|u|^2} - 1- \\alpha |u|^2 \\right) u, \\quad u(0)=u_0, \\quad x \\in \\mathbb{R}^2,\n  $$ with $0<b<1$ and $\\alpha=2\\pi(2-b)$. First we show the decay of global solutions by assuming that the initial data $u_0$ belongs to the weighted space $\\Sigma(\\mathbb{R}^2)=\\{\\,u\\in H^1(\\mathbb{R}^2) \\ : \\ |x|u\\in L^2(\\mathbb{R}^2)\\,\\}$. Then we combine the local theory with the decay estimate to obtain scattering in $\\Sigma$ when the Hamiltonian is below the value $\\frac{2}{"},"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":"1810.08974","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-10-21T15:24:18Z","cross_cats_sorted":[],"title_canon_sha256":"e010a402f16fe52cb0b3d309e0f26bd814e98cf675bf606aa8c501132f8e6699","abstract_canon_sha256":"94cc363a44f4ce411b0d91cd8551d8acc47b3b7d389bc765e853ae21b55c91e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:41.316440Z","signature_b64":"kE/QDMhuIyHrR7ipBY63zWK4khKC6LVVS13uwKB0UoJuSNbLeOtQQ5TBMc0nJYXgoGhCnvZ2MS/KtpnCxqQyDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fe85c1578bfe3cc09aad1dd319d2d672dae63fa7c5c8a20f6212e159b0e576d8","last_reissued_at":"2026-05-18T00:02:41.316004Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:41.316004Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Scattering in the weighted $L^2$-space for a 2D nonlinear Schr\\\"odinger equation with inhomogeneous exponential nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Abdelwahab Bensouilah, Mohamed Majdoub, Van Duong Dinh","submitted_at":"2018-10-21T15:24:18Z","abstract_excerpt":"We investigate the defocusing inhomogeneous nonlinear Schr\\\"odinger equation\n  $$ i \\partial_tu + \\Delta u = |x|^{-b} \\left({\\rm e}^{\\alpha|u|^2} - 1- \\alpha |u|^2 \\right) u, \\quad u(0)=u_0, \\quad x \\in \\mathbb{R}^2,\n  $$ with $0<b<1$ and $\\alpha=2\\pi(2-b)$. First we show the decay of global solutions by assuming that the initial data $u_0$ belongs to the weighted space $\\Sigma(\\mathbb{R}^2)=\\{\\,u\\in H^1(\\mathbb{R}^2) \\ : \\ |x|u\\in L^2(\\mathbb{R}^2)\\,\\}$. Then we combine the local theory with the decay estimate to obtain scattering in $\\Sigma$ when the Hamiltonian is below the value $\\frac{2}{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.08974","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":""},"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":"1810.08974","created_at":"2026-05-18T00:02:41.316067+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.08974v1","created_at":"2026-05-18T00:02:41.316067+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.08974","created_at":"2026-05-18T00:02:41.316067+00:00"},{"alias_kind":"pith_short_12","alias_value":"72C4CV4L7Y6M","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_16","alias_value":"72C4CV4L7Y6MBGVN","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_8","alias_value":"72C4CV4L","created_at":"2026-05-18T12:32:11.075285+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/72C4CV4L7Y6MBGVNDXJRTUWWOL","json":"https://pith.science/pith/72C4CV4L7Y6MBGVNDXJRTUWWOL.json","graph_json":"https://pith.science/api/pith-number/72C4CV4L7Y6MBGVNDXJRTUWWOL/graph.json","events_json":"https://pith.science/api/pith-number/72C4CV4L7Y6MBGVNDXJRTUWWOL/events.json","paper":"https://pith.science/paper/72C4CV4L"},"agent_actions":{"view_html":"https://pith.science/pith/72C4CV4L7Y6MBGVNDXJRTUWWOL","download_json":"https://pith.science/pith/72C4CV4L7Y6MBGVNDXJRTUWWOL.json","view_paper":"https://pith.science/paper/72C4CV4L","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.08974&json=true","fetch_graph":"https://pith.science/api/pith-number/72C4CV4L7Y6MBGVNDXJRTUWWOL/graph.json","fetch_events":"https://pith.science/api/pith-number/72C4CV4L7Y6MBGVNDXJRTUWWOL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/72C4CV4L7Y6MBGVNDXJRTUWWOL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/72C4CV4L7Y6MBGVNDXJRTUWWOL/action/storage_attestation","attest_author":"https://pith.science/pith/72C4CV4L7Y6MBGVNDXJRTUWWOL/action/author_attestation","sign_citation":"https://pith.science/pith/72C4CV4L7Y6MBGVNDXJRTUWWOL/action/citation_signature","submit_replication":"https://pith.science/pith/72C4CV4L7Y6MBGVNDXJRTUWWOL/action/replication_record"}},"created_at":"2026-05-18T00:02:41.316067+00:00","updated_at":"2026-05-18T00:02:41.316067+00:00"}