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In the focusing case, we show that the scattering holds\nin $\\R^d (d\\ge 1)$ when the energy $E(u_0,u_1)<E(R,0)$ and $\\|\\Delta\nu_0\\|_{L^2}^2+m\\|u_0\\|_{L^2}^2<\\|\\Delta R\\|_{L^2}^2+m\\|R\\|_{L^2}^2$\nfor ground state $R$. The difficulties lie the absence of the\nscaling invariance and a Galilean transformation for the equation to\nco"},"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":"1108.0825","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-08-03T11:42:15Z","cross_cats_sorted":[],"title_canon_sha256":"5dbb74eef5d6451631e3aeb6219c8b68103d6e9e2f99c2d01409bb21c330084e","abstract_canon_sha256":"0dc1ce7075444c7f9c848d4450e2dcb8bae33afcc01c5089f518faf3df782cd7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:25.834881Z","signature_b64":"QeT590ZPgDxuYui9Sw7ue53Pa6vLxSn2SIn2/YtXlSmtWo+xlGVANq1lh/QtNnmqW/m1uXpzroacvKg3Xxz+Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b684015b529b77e3051ca8300b6791ce046f6c1fa32e6d4ade63293a0dce6c99","last_reissued_at":"2026-05-18T03:40:25.834024Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:25.834024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Scattering for the beam equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changxing Miao, Yifei Wu","submitted_at":"2011-08-03T11:42:15Z","abstract_excerpt":"In this paper, we study the scattering for the nonlinear beam\nequation $u_{tt}+\\Delta^2u+mu+\\mu |u|^{p-1}u=0$. 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