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If $\\kappa =1/2$, we say the equation satisfies mass-resonance condition. We are interested in the scattering problem of this equation under the condition $M(u,v)<M(\\phi ,\\psi)$, where $M(u,v)$ denotes the mass and $(\\phi ,\\psi)$ is a ground state. In the mass-resonance"},"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":"1810.07904","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-10-18T05:31:14Z","cross_cats_sorted":[],"title_canon_sha256":"fa69669a196d2e414848069b471ad93d5fb4290e09fcfd3a9e49c122eb472f44","abstract_canon_sha256":"d46fd3adcb5e75271ba262a442a99550dbfe245e6627e9a377824965466c8dd6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:01:42.982062Z","signature_b64":"FqbuZVDR5SY2tYc+lPpD1IO8GR2aidXZtLNiwvJdN89WuVuo7/WECSdlFYlhCHLN4Lk4HVjhGqpZqXj7N5WyCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"94bced0d5c536cbe4824588c96cf41be799412844dfaafc8353060c9ae022509","last_reissued_at":"2026-05-18T00:01:42.981524Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:01:42.981524Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kuranosuke Nishimura, Nobu Kishimoto, Takahisa Inui","submitted_at":"2018-10-18T05:31:14Z","abstract_excerpt":"We consider a mass-critical system of nonlinear Sch\\\"{o}dinger equations \\begin{align*} \\begin{cases} i\\partial_t u +\\Delta u =\\bar{u}v,\\\\ i\\partial_t v +\\kappa \\Delta v =u^2, \\end{cases} (t,x)\\in \\mathbb{R}\\times \\mathbb{R}^4, \\end{align*} where $(u,v)$ is a $\\mathbb{C}^2$-valued unknown function and $\\kappa >0$ is a constant. If $\\kappa =1/2$, we say the equation satisfies mass-resonance condition. We are interested in the scattering problem of this equation under the condition $M(u,v)<M(\\phi ,\\psi)$, where $M(u,v)$ denotes the mass and $(\\phi ,\\psi)$ is a ground state. 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