{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:4G5W5HK55NBGGGKPH3CNR7B7IO","short_pith_number":"pith:4G5W5HK5","schema_version":"1.0","canonical_sha256":"e1bb6e9d5deb4263194f3ec4d8fc3f43ab28da895ff9507c0e8c424a63a82bd8","source":{"kind":"arxiv","id":"1903.05340","version":1},"attestation_state":"computed","paper":{"title":"Ground states of Nonlinear Schr\\\"{o}dinger System with Mixed Couplings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Juncheng Wei, Yuanze Wu","submitted_at":"2019-03-13T07:06:50Z","abstract_excerpt":"We consider the following $k$-coupled nonlinear Schr\\\"odinger system: \\begin{align*} \\begin{cases} &-\\Delta u_j + \\lambda_j u_j = \\mu_j u_j^3 + \\sum_{i=1, i\\not=j}^k \\beta_{i,j} u_i^2 u_j \\quad {\\rm in}\\ \\mathbb{R}^N,\\\\ &u_j>0 \\quad {\\rm in}\\ \\mathbb{R}^N, \\quad u_j(x) \\to 0 \\quad \\text{as }|x|\\to +\\infty, \\quad j=1,2,\\cdots,k, \\end{cases} \\end{align*} where $N\\leq 3$, $k\\geq3$, $\\lambda_j,\\mu_j>0$ are constants and $\\beta_{i,j}=\\beta_{j,i}\\not=0$ are parameters. There have been intensive studies for the above system when $k=2$ or the system is purely attractive ($ \\beta_{i,j}>0, \\forall i \\no"},"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":"1903.05340","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-03-13T07:06:50Z","cross_cats_sorted":[],"title_canon_sha256":"9b949d0d462730fe15ed6d964ace2e319f6bffe5e4d1ae3010d30efb294e3c96","abstract_canon_sha256":"692b80a184a4781f19c0d99d0d270e84d288b065a3dd8796013ce9522db9f558"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:24.591996Z","signature_b64":"EtcdhHgUTGLDNtZGfre+gq9Waktnlqswh+Af6KRSOYy++q5zrUFTjAFnSg4QXxAt3hXD4+CCDh1WGpdRoMchAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e1bb6e9d5deb4263194f3ec4d8fc3f43ab28da895ff9507c0e8c424a63a82bd8","last_reissued_at":"2026-05-17T23:51:24.591330Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:24.591330Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ground states of Nonlinear Schr\\\"{o}dinger System with Mixed Couplings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Juncheng Wei, Yuanze Wu","submitted_at":"2019-03-13T07:06:50Z","abstract_excerpt":"We consider the following $k$-coupled nonlinear Schr\\\"odinger system: \\begin{align*} \\begin{cases} &-\\Delta u_j + \\lambda_j u_j = \\mu_j u_j^3 + \\sum_{i=1, i\\not=j}^k \\beta_{i,j} u_i^2 u_j \\quad {\\rm in}\\ \\mathbb{R}^N,\\\\ &u_j>0 \\quad {\\rm in}\\ \\mathbb{R}^N, \\quad u_j(x) \\to 0 \\quad \\text{as }|x|\\to +\\infty, \\quad j=1,2,\\cdots,k, \\end{cases} \\end{align*} where $N\\leq 3$, $k\\geq3$, $\\lambda_j,\\mu_j>0$ are constants and $\\beta_{i,j}=\\beta_{j,i}\\not=0$ are parameters. 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